On the peculiarities of using sparse matrices in problems of mathematical modeling
Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science and eng...
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PROBLEMS IN PROGRAMMING
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pp_isofts_kiev_ua-article-5252023-06-25T06:38:49Z On the peculiarities of using sparse matrices in problems of mathematical modeling Про особливості використання розріджених матриць в задачах математичного моделювання Khimich, O.M. Popov, A.V. Chystiakov, О.V. mathematical modeling; structural strength problems; sparse matrices; linear algebra algorithms; methods of structuring sparse matrices UDC 004.415.2.045 (076.5) математичне моделювання; проблеми з міцністю конструкції; розріджені матриці; алгоритми лінійної алгебри; методи структурування розріджених матриць УДК 004.415.2.045 (076.5) Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were effectively applied.Prombles in programming 2022; 3-4: 240-248 Математичне моделювання і пов'язаний з ним комп'ютерний експеримент в даний час є одним з основних засобів дослідження об'єктів, процесів і явищ різної природи: в науці, техніці, економіці, суспільстві і т.д. Значне підвищення якості математичного моделювання в багатьох областях науки і інжиніринг можливий лише за умови використання принципово нових тривимірних моделей, переходу від комп'ютерного моделювання окремих вузлів і вузлів до розрахунку й оптимізації виробу в цілому. Очевидно, що розгляд задач у такій постановці призводить до дискретних математичних моделей надвеликих розмірів. Існуючі суперкомп’ютери різних паралельних архітектур дозволяють ефективно вирішувати такі задачі. Але час розв’язування задач на паралельних комп’ютерах складається з часу фактичного розв’язання та часу виконання додаткових операцій, необхідних для обміну інформацією між обчислювальними пристроями, тобто накладних витрат. Особливо це стосується задач лінійної алгебри з різними структурами розріджених матриць великих об’ємів, які виникають при математичному моделюванні процесів. Схеми ущільнення розріджених матриць, декомпозиція масивів даних між процесорами є одними з основних факторів ефективного вирішення цих проблем на паралельних комп'ютерах. У статті розглянуто ефективні методи обробки розріджених матриць довільної структури з метою ефективного математичного моделювання задач міцності конструкцій на паралельних комп’ютерах. Запропоновано різні методи регуляризації та декомпозиції розріджених матриць довільної структури, ефективні схеми зберігання даних, технологію дослідження обумовленості матриці наближеними даними на ЕОМ. Такий спосіб використання розріджених матриць у математичному моделюванні забезпечує більш ефективне використання обчислювальних ресурсів і достовірність результатів комп’ютера. Представлено проблеми математичного моделювання, де ефективно застосовано розглянуті методи обробки розріджених матриць.Prombles in programming 2022; 3-4: 240-248 PROBLEMS IN PROGRAMMING ПРОБЛЕМЫ ПРОГРАММИРОВАНИЯ ПРОБЛЕМИ ПРОГРАМУВАННЯ 2023-01-23 Article Article application/pdf https://pp.isofts.kiev.ua/index.php/ojs1/article/view/525 10.15407/pp2022.03-04.240 PROBLEMS IN PROGRAMMING; No 3-4 (2022); 240-248 ПРОБЛЕМЫ ПРОГРАММИРОВАНИЯ; No 3-4 (2022); 240-248 ПРОБЛЕМИ ПРОГРАМУВАННЯ; No 3-4 (2022); 240-248 1727-4907 10.15407/pp2022.03-04 en https://pp.isofts.kiev.ua/index.php/ojs1/article/view/525/577 Copyright (c) 2023 PROBLEMS IN PROGRAMMING |
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Problems in programming |
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2023-06-25T06:38:49Z |
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| topic |
mathematical modeling structural strength problems sparse matrices linear algebra algorithms methods of structuring sparse matrices UDC 004.415.2.045 (076.5) |
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mathematical modeling structural strength problems sparse matrices linear algebra algorithms methods of structuring sparse matrices UDC 004.415.2.045 (076.5) Khimich, O.M. Popov, A.V. Chystiakov, О.V. On the peculiarities of using sparse matrices in problems of mathematical modeling |
| topic_facet |
mathematical modeling structural strength problems sparse matrices linear algebra algorithms methods of structuring sparse matrices UDC 004.415.2.045 (076.5) математичне моделювання проблеми з міцністю конструкції розріджені матриці алгоритми лінійної алгебри методи структурування розріджених матриць УДК 004.415.2.045 (076.5) |
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Article |
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Khimich, O.M. Popov, A.V. Chystiakov, О.V. |
| author_facet |
Khimich, O.M. Popov, A.V. Chystiakov, О.V. |
| author_sort |
Khimich, O.M. |
| title |
On the peculiarities of using sparse matrices in problems of mathematical modeling |
| title_short |
On the peculiarities of using sparse matrices in problems of mathematical modeling |
| title_full |
On the peculiarities of using sparse matrices in problems of mathematical modeling |
| title_fullStr |
On the peculiarities of using sparse matrices in problems of mathematical modeling |
| title_full_unstemmed |
On the peculiarities of using sparse matrices in problems of mathematical modeling |
| title_sort |
on the peculiarities of using sparse matrices in problems of mathematical modeling |
| title_alt |
Про особливості використання розріджених матриць в задачах математичного моделювання |
| description |
Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were effectively applied.Prombles in programming 2022; 3-4: 240-248 |
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PROBLEMS IN PROGRAMMING |
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2023 |
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https://pp.isofts.kiev.ua/index.php/ojs1/article/view/525 |
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240
Програмні засоби моделювання процесів і об′єктів
UDC 004.415.2.045 (076.5) http://doi.org/10.15407/pp2022.03-04.240
ABOUT THE PECULIARITIES
OF USING SPARSE MATRICES IN PROBLEMS
OF MATHEMATICAL MODELING
Olexand Khimich, Alexandr Popov, Oleksii Chystiakov
Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena
of various nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in
many areas of science and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from
computer simulation of individual components and assemblies to the calculation and optimization of the product as a whole. It is obvious
that the consideration of problems in such a formulation leads to discrete mathematical models of super-large sizes. Existing supercomput-
ers of different parallel architectures make it possible to efficiently solve such problems. However, the time for solving problems on parallel
computers consists of the time of the actual solution and the time of performing additional operations, that are necessary for the exchange of
information between computing devices, that is overhead costs. This is especially true for problems of linear algebra with different structures
of sparse matrices of large volumes, that arise in the mathematical modeling of processes. Sparse matrix compaction schemes, decomposition
of data arrays between processors are one of the main factors for the effective solution of these problems on parallel computers. The paper
considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of effective mathematical modeling of struc-
tural strength problems on parallel computers. Various methods of regularization and decomposition of sparse matrices of arbitrary structure,
efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on a computer are proposed. This
way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and reliability of computer results.
Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were effectively applied.
Key words: mathematical modeling, structural strength problems, sparse matrices, linear algebra algorithms, methods of structuring
sparse matrices
Introduction
Nowadays, one of the main directions of scientific and technical progress is the mathematical modeling of com-
plex objects, processes, and phenomena, which is the basis of modern applied developments in various fields of science
and technology. With the advent and constant development of supercomputers of various architectures, the problems
of their effective use, the expansion of the range of tasks that need to be solved, and the assurance of the reliability of
computer results become very relevant. As a practice and literary sources show, mathematical modeling of 80-85% of all
scientific and technical problems is reduced to solving problems of linear algebra - systems of linear algebraic equations
(SLAE) and algebraic eigenvalue problem (AEVP) for sparse matrices of various structures. The efficiency of solving the
entire problem of mathematical modeling largely depends on the efficiency of solving these problems.
Despite the fact that there is a huge amount of software for solving linear algebra problems on various comput-
er architectures on the international market, new applied problems are appearing that require new efficient algorithms.
This is primarily due to the difference in the types of matrices (dense and sparse), structural properties (band, arbitrary
structure, triangular and others), mathematical properties (singular and non-singular, positive definite, sign-definite
etc.), and ways of processing them on different computer architectures.
Mathematical modeling of structural strength problems and stability of new composite materials
When designing objects for various purposes, it often becomes necessary to carry out calculations of the
strength of structures of the whole objects and their individual elements. Such problems arise, in particular, in various
branches of mechanical engineering (ship building, aircraft building, rocket building, engine building, etc.), in indus-
trial and civil engineering (calculation of individual structural elements or structures as a whole), etc.
Increasing requirements for the quality of design solutions, the use of new constructive materials lead to the
development of refined three-dimensional mathematical models with huge amounts of processed information. Despite
the availability of powerful supercomputers of different architectures, there is a need to use efficient methods of com-
puter data processing, the volume of which is constantly growing.
Mathematical formulations of structural strength problems. Mathematical problems of calculating the strength
of structures using the principle of possible displacements are reduced to the following variational problems [1]:
It is necessary to find a vector function that
Програмні засоби моделювання процесів і об'єктів
© O.M. Khimich, A.V. Popov, O.V. Chystiakov, 2022
ISSN 1727-4907. Проблеми програмування. 2022. №3-4. Спеціальний випуск
UDC 004.415.2.045 (076.5) http://doi.org/10.15407/pp2022.03-04.....
ABOUT THE PECULIARITIES OF USING SPARSE MATRICES IN
PROBLEMS OF MATHEMATICAL MODELING
Olexand Khimich, Alexandr Popov, Oleksii Chystiakov
Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various
nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science
and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual
components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a
formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to
efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time
of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is
especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of
processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution
of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of
effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of
sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on
a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and
reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were
effectively applied.
Key words: mathematical modeling, structural strength problems, sparse matrices, linear algebra algorithms, methods of structuring sparse matrices
Introduction
Nowadays, one of the main directions of scientific and technical progress is the mathematical modeling of
complex objects, processes, and phenomena, which is the basis of modern applied developments in various fields of
science and technology. With the advent and constant development of supercomputers of various architectures, the
problems of their effective use, the expansion of the range of tasks that need to be solved, and the assurance of the
reliability of computer results become very relevant. As a practice and literary sources show, mathematical modeling of
80-85% of all scientific and technical problems is reduced to solving problems of linear algebra - systems of linear
algebraic equations (SLAE) and algebraic eigenvalue problem (AEVP) for sparse matrices of various structures. The
efficiency of solving the entire problem of mathematical modeling largely depends on the efficiency of solving these
problems.
Despite the fact that there is a huge amount of software for solving linear algebra problems on various computer
architectures on the international market, new applied problems are appearing that require new efficient algorithms.
This is primarily due to the difference in the types of matrices (dense and sparse), structural properties (band, arbitrary
structure, triangular and others), mathematical properties (singular and non-singular, positive definite, sign-definite
etc.), and ways of processing them on different computer architectures.
Mathematical modeling of structural strength problems and stability of new composite
materials
When designing objects for various purposes, it often becomes necessary to carry out calculations of the strength
of structures of the whole objects and their individual elements. Such problems arise, in particular, in various branches
of mechanical engineering (ship building, aircraft building, rocket building, engine building, etc.), in industrial and civil
engineering (calculation of individual structural elements or structures as a whole), etc.
Increasing requirements for the quality of design solutions, the use of new constructive materials lead to the
development of refined three-dimensional mathematical models with huge amounts of processed information. Despite
the availability of powerful supercomputers of different architectures, there is a need to use efficient methods of
computer data processing, the volume of which is constantly growing.
Mathematical formulations of structural strength problems. Mathematical problems of calculating the strength
of structures using the principle of possible displacements are reduced to the following variational problems [1]:
It is necessary to find a vector function that 0Uu , for any possible displacement 0Uv satisfies the integral
identity
• for a static problem
),(),( vflvua == ; (1)
• for dynamic problem
),(),(),(),( vflvu'cvu"bvua ==++++ ; (2)
, for any possible displacement
Програмні засоби моделювання процесів і об'єктів
© O.M. Khimich, A.V. Popov, O.V. Chystiakov, 2022
ISSN 1727-4907. Проблеми програмування. 2022. №3-4. Спеціальний випуск
UDC 004.415.2.045 (076.5) http://doi.org/10.15407/pp2022.03-04.....
ABOUT THE PECULIARITIES OF USING SPARSE MATRICES IN
PROBLEMS OF MATHEMATICAL MODELING
Olexand Khimich, Alexandr Popov, Oleksii Chystiakov
Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various
nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science
and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual
components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a
formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to
efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time
of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is
especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of
processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution
of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of
effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of
sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on
a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and
reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were
effectively applied.
Key words: mathematical modeling, structural strength problems, sparse matrices, linear algebra algorithms, methods of structuring sparse matrices
Introduction
Nowadays, one of the main directions of scientific and technical progress is the mathematical modeling of
complex objects, processes, and phenomena, which is the basis of modern applied developments in various fields of
science and technology. With the advent and constant development of supercomputers of various architectures, the
problems of their effective use, the expansion of the range of tasks that need to be solved, and the assurance of the
reliability of computer results become very relevant. As a practice and literary sources show, mathematical modeling of
80-85% of all scientific and technical problems is reduced to solving problems of linear algebra - systems of linear
algebraic equations (SLAE) and algebraic eigenvalue problem (AEVP) for sparse matrices of various structures. The
efficiency of solving the entire problem of mathematical modeling largely depends on the efficiency of solving these
problems.
Despite the fact that there is a huge amount of software for solving linear algebra problems on various computer
architectures on the international market, new applied problems are appearing that require new efficient algorithms.
This is primarily due to the difference in the types of matrices (dense and sparse), structural properties (band, arbitrary
structure, triangular and others), mathematical properties (singular and non-singular, positive definite, sign-definite
etc.), and ways of processing them on different computer architectures.
Mathematical modeling of structural strength problems and stability of new composite
materials
When designing objects for various purposes, it often becomes necessary to carry out calculations of the strength
of structures of the whole objects and their individual elements. Such problems arise, in particular, in various branches
of mechanical engineering (ship building, aircraft building, rocket building, engine building, etc.), in industrial and civil
engineering (calculation of individual structural elements or structures as a whole), etc.
Increasing requirements for the quality of design solutions, the use of new constructive materials lead to the
development of refined three-dimensional mathematical models with huge amounts of processed information. Despite
the availability of powerful supercomputers of different architectures, there is a need to use efficient methods of
computer data processing, the volume of which is constantly growing.
Mathematical formulations of structural strength problems. Mathematical problems of calculating the strength
of structures using the principle of possible displacements are reduced to the following variational problems [1]:
It is necessary to find a vector function that 0Uu , for any possible displacement 0Uv satisfies the integral
identity
• for a static problem
),(),( vflvua == ; (1)
• for dynamic problem
),(),(),(),( vflvu'cvu"bvua ==++++ ; (2)
satisfies the integral
identity
• for a static problem
Програмні засоби моделювання процесів і об'єктів
© O.M. Khimich, A.V. Popov, O.V. Chystiakov, 2022
ISSN 1727-4907. Проблеми програмування. 2022. №3-4. Спеціальний випуск
UDC 004.415.2.045 (076.5) http://doi.org/10.15407/pp2022.03-04.....
ABOUT THE PECULIARITIES OF USING SPARSE MATRICES IN
PROBLEMS OF MATHEMATICAL MODELING
Olexand Khimich, Alexandr Popov, Oleksii Chystiakov
Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various
nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science
and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual
components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a
formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to
efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time
of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is
especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of
processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution
of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of
effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of
sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on
a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and
reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were
effectively applied.
Key words: mathematical modeling, structural strength problems, sparse matrices, linear algebra algorithms, methods of structuring sparse matrices
Introduction
Nowadays, one of the main directions of scientific and technical progress is the mathematical modeling of
complex objects, processes, and phenomena, which is the basis of modern applied developments in various fields of
science and technology. With the advent and constant development of supercomputers of various architectures, the
problems of their effective use, the expansion of the range of tasks that need to be solved, and the assurance of the
reliability of computer results become very relevant. As a practice and literary sources show, mathematical modeling of
80-85% of all scientific and technical problems is reduced to solving problems of linear algebra - systems of linear
algebraic equations (SLAE) and algebraic eigenvalue problem (AEVP) for sparse matrices of various structures. The
efficiency of solving the entire problem of mathematical modeling largely depends on the efficiency of solving these
problems.
Despite the fact that there is a huge amount of software for solving linear algebra problems on various computer
architectures on the international market, new applied problems are appearing that require new efficient algorithms.
This is primarily due to the difference in the types of matrices (dense and sparse), structural properties (band, arbitrary
structure, triangular and others), mathematical properties (singular and non-singular, positive definite, sign-definite
etc.), and ways of processing them on different computer architectures.
Mathematical modeling of structural strength problems and stability of new composite
materials
When designing objects for various purposes, it often becomes necessary to carry out calculations of the strength
of structures of the whole objects and their individual elements. Such problems arise, in particular, in various branches
of mechanical engineering (ship building, aircraft building, rocket building, engine building, etc.), in industrial and civil
engineering (calculation of individual structural elements or structures as a whole), etc.
Increasing requirements for the quality of design solutions, the use of new constructive materials lead to the
development of refined three-dimensional mathematical models with huge amounts of processed information. Despite
the availability of powerful supercomputers of different architectures, there is a need to use efficient methods of
computer data processing, the volume of which is constantly growing.
Mathematical formulations of structural strength problems. Mathematical problems of calculating the strength
of structures using the principle of possible displacements are reduced to the following variational problems [1]:
It is necessary to find a vector function that 0Uu , for any possible displacement 0Uv satisfies the integral
identity
• for a static problem
),(),( vflvua == ; (1)
• for dynamic problem
),(),(),(),( vflvu'cvu"bvua ==++++ ; (2)
; (1)
• for dynamic problem
Програмні засоби моделювання процесів і об'єктів
© O.M. Khimich, A.V. Popov, O.V. Chystiakov, 2022
ISSN 1727-4907. Проблеми програмування. 2022. №3-4. Спеціальний випуск
UDC 004.415.2.045 (076.5) http://doi.org/10.15407/pp2022.03-04.....
ABOUT THE PECULIARITIES OF USING SPARSE MATRICES IN
PROBLEMS OF MATHEMATICAL MODELING
Olexand Khimich, Alexandr Popov, Oleksii Chystiakov
Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various
nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science
and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual
components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a
formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to
efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time
of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is
especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of
processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution
of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of
effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of
sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on
a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and
reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were
effectively applied.
Key words: mathematical modeling, structural strength problems, sparse matrices, linear algebra algorithms, methods of structuring sparse matrices
Introduction
Nowadays, one of the main directions of scientific and technical progress is the mathematical modeling of
complex objects, processes, and phenomena, which is the basis of modern applied developments in various fields of
science and technology. With the advent and constant development of supercomputers of various architectures, the
problems of their effective use, the expansion of the range of tasks that need to be solved, and the assurance of the
reliability of computer results become very relevant. As a practice and literary sources show, mathematical modeling of
80-85% of all scientific and technical problems is reduced to solving problems of linear algebra - systems of linear
algebraic equations (SLAE) and algebraic eigenvalue problem (AEVP) for sparse matrices of various structures. The
efficiency of solving the entire problem of mathematical modeling largely depends on the efficiency of solving these
problems.
Despite the fact that there is a huge amount of software for solving linear algebra problems on various computer
architectures on the international market, new applied problems are appearing that require new efficient algorithms.
This is primarily due to the difference in the types of matrices (dense and sparse), structural properties (band, arbitrary
structure, triangular and others), mathematical properties (singular and non-singular, positive definite, sign-definite
etc.), and ways of processing them on different computer architectures.
Mathematical modeling of structural strength problems and stability of new composite
materials
When designing objects for various purposes, it often becomes necessary to carry out calculations of the strength
of structures of the whole objects and their individual elements. Such problems arise, in particular, in various branches
of mechanical engineering (ship building, aircraft building, rocket building, engine building, etc.), in industrial and civil
engineering (calculation of individual structural elements or structures as a whole), etc.
Increasing requirements for the quality of design solutions, the use of new constructive materials lead to the
development of refined three-dimensional mathematical models with huge amounts of processed information. Despite
the availability of powerful supercomputers of different architectures, there is a need to use efficient methods of
computer data processing, the volume of which is constantly growing.
Mathematical formulations of structural strength problems. Mathematical problems of calculating the strength
of structures using the principle of possible displacements are reduced to the following variational problems [1]:
It is necessary to find a vector function that 0Uu , for any possible displacement 0Uv satisfies the integral
identity
• for a static problem
),(),( vflvua == ; (1)
• for dynamic problem
),(),(),(),( vflvu'cvu"bvua ==++++ ; (2) ; (2)
© О.М. Хіміч, О.В. Попов, О.В. Чистяков, 2022
ISSN 1727-4907. Проблеми програмування. 2022. № 3-4. Спеціальний випуск
241
Програмні засоби моделювання процесів і об′єктів Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
(3)
• for the problem of eigen oscillations
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
; (4)
where
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
– is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
,
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of de-
formation and braking energy, and the linear form
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
is denoted
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
, and the second
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
.
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of
the estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
, of nonzeros only on elements contain-
ing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees of free-
dom and basis functions
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
, (5)
where
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
– is the Kronecker symbol, and the result of the operation
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
is the value of the component component
of the vector-function
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
for the degrees of freedom
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
.
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
of space
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
. Vector-functions from subspace
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
are piecewise polynomial and can be represented as a linear com-
bination of basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis
vector-functions
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
(6)
where
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
– the piecewise polynomial basis mentioned above
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
.
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
(7)
• problems with initial conditions for dynamic problem (2), (3)
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
(8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
(9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustain-
ability of composite materials, the static method of the three-dimensional linearized theory of stability is used within
the framework of the second version of the theory of small subcritical deformations. The subcritical state is determined
from the solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various
boundary conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-
layer element of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of
the form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the correspond-
ing function determines
Програмні засоби моделювання процесів і об'єктів
)0(
0 )( utu == , )1(
0 )( utu" == ; (3)
• for the problem of eigen oscillations
),(),( vubvua == ; (4)
where 0U – is the infinite-dimensional functional space of possible displacements; symmetrical bilinear forms ),( vua ,
),( vu"b , ),( vu'c symmetrical bilinear forms are proportional in accordance with the potential, kinetic energy of
deformation and braking energy, and the linear form ),( vfl is proportional to the work of the applied (external) forces
under load. The first time derivative of the vector function ),( tu is denoted u' , and the second u" .
Only linear problems are considered here, since it is assumed that the solution of a nonlinear problem can be
reduced to solving a sequence of linear problems.
Discretization of problems of mathematical modeling of structural strength. The theoretical basis of most
software tools for calculating the strength of structures, for example, the SP LIRA software package designed to solve
problems of calculating the strength of building structures [2–4], is the finite element method (FEM) [5], implemented
in the form of displacements. The choice of this form is explained by the simplicity of its algorithmization and physical
interpretation, the availability of unified methods for constructing stiffness matrices and load vectors for various types
of finite elements, the possibility of taking into account arbitrary boundary conditions and the complex geometry of the
estimated structure.
To obtain a discrete problem with the help of the FEM, the area occupied by the structure is distributed into finite
elements, nodes and their degrees of freedom (displacement and rotation angles of nodes) are assigned. The degrees of
freedom correspond to the basic (coordinate, approximating) vector-functions i , of nonzeros only on elements
containing a node corresponding to a given degree of freedom. In addition, the following relations hold for the degrees
of freedom and basis functions
ijijL ==)( , (5)
where ij – is the Kronecker symbol, and the result of the operation )( ijL is the value of the component
component of the vector-function i for the degrees of freedom jL .
Approximate solutions of the corresponding problems are sought in the finite-dimensional subspace hU 0 of space
0U . Vector-functions from subspace hU 0 are piecewise polynomial and can be represented as a linear combination of
basis vector-functions, which satisfies the main (kinetic) conditions for a linear combination of basis vector-functions
)()(
1
i
n
i
ih xu
==
== , (6)
where i )...,,2,1( ni == – the piecewise polynomial basis mentioned above hU 0 .
Then the discrete problems, that should be solved have the form [6]:
• systems linear algebraic equations for a statistical problem (1)
bAx == ; (7)
;
• problems with initial conditions for dynamic problem (2), (3)
)()()()( tbtAxtCx'tBx'' ==++++ , )0(
0 )( xtx == , )1(
0 )( xtx' == ; (8)
• algebraic eigenvalue problem for the problem of eigen oscillations (4)
BxAx h== . (9)
Mathematical formulations of structural (composite materials) stability problems. To investigate the sustainability of
composite materials, the static method of the three-dimensional linearized theory of stability is used within the frame-
work of the second version of the theory of small subcritical deformations. The subcritical state is determined from the
solution of the plane problem of the linear theory of elasticity of piecewise homogeneous bodies for various boundary
conditions, which correspond to a three-layer element of periodicity in the case of a material and a three-layer element
of a structure with lateral sides free from stresses in the case of a composite sample [7].
When using the static Euler method, the stability problem is reduced to a generalized eigenvalue problem of the
form (9), in which the minimum (in general, modulo) eigenvalue determines the critical load, and the corresponding
function determines ),( 2,1, kkk uuu – the form of stability loss. The following are accepted as the main mathematical
models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
– the form of stability loss. The following are accepted as the main mathemati-
cal models: the equation of the linear theory of elasticity of piecewise homogeneous environment to determine the
subcritical stress-strain state; equation of the three-dimensional linearized theory of stability at small deformations to
determine the critical parameters. The model of a piecewise homogeneous medium is used as a mechanical model.
Composite components are modeled by linearly elastic isotropic bodies [7].
Thus, for a plane deformation, we obtain an eigenvalue problem for a system of partial differential equations
with variable coefficients depending on two variables. In this case (the investigation of the stability of a composite
material under a compressive load), the conditions of self-adjointness of the operators of the problem of determining
the initial stress-strain state and the generalized eigenvalue problem are satisfied, which corresponds to the problem
of three-dimensional stability theory. In addition, the differential operators of the problems under consideration are
positive-definite under the condition that there is no rigid displacement.
Discretization of problems of mathematical modeling of structural (composite materials) stability. For
the numerical solution of the problem of the three-dimensional linearized theory of stability and the problem of
the linear theory of elasticity, a discrete model is used, which is constructed by the variational-difference method
242
Програмні засоби моделювання процесів і об′єктів
(finite differences) using the concept of basic schemes. In the computational domain, with the help of straight lines,
a difference grid nonuniform in each direction is introduced so that the material is homogeneous within one grid
cell. Thus, the discrete problem of determining the subcritical stress-strain state (the problem of the linear theory of
elasticity) is written in the form of an SLAE of the form (7) with a sparse symmetric positive definite matrix. The
discrete problem for the stability problem is written in the form of a generalized AEVP of the form (9) of sparse
symmetric positive definite matrices [7].
Increasing the efficiency of mathematical modeling on a computer. In order to provide an effective computer
solution of problems (7)–(9), it is necessary to take into account the following features [4, 7, 8]:
• matrices of discrete problems – stiffness A, mass B, damping C are symmetrical, positive definite or posi-
tive semidefinite, and have a sparse structure (band, profile, block, etc.; a matrix is called sparse if the number of its
non-zero elements is much less than their total number n2, where is the order of the matrix) [1, 6].
• the order of the matrices of the discrete problems to be solved is
Програмні засоби моделювання процесів і об'єктів
Composite components are modeled by linearly elastic isotropic bodies [7].
Thus, for a plane deformation, we obtain an eigenvalue problem for a system of partial differential equations
with variable coefficients depending on two variables. In this case (the investigation of the stability of a composite ma-
terial under a compressive load), the conditions of self-adjointness of the operators of the problem of determining the
initial stress-strain state and the generalized eigenvalue problem are satisfied, which corresponds to the problem of
three-dimensional stability theory. In addition, the differential operators of the problems under consideration are posi-
tive-definite under the condition that there is no rigid displacement.
Discretization of problems of mathematical modeling of structural (composite materials) stability. For the
numerical solution of the problem of the three-dimensional linearized theory of stability and the problem of the linear
theory of elasticity, a discrete model is used, which is constructed by the variational-difference method (finite
differences) using the concept of basic schemes. In the computational domain, with the help of straight lines, a
difference grid nonuniform in each direction is introduced so that the material is homogeneous within one grid cell.
Thus, the discrete problem of determining the subcritical stress-strain state (the problem of the linear theory of
elasticity) is written in the form of an SLAE of the form (7) with a sparse symmetric positive definite matrix. The
discrete problem for the stability problem is written in the form of a generalized AEVP of the form (9) of sparse
symmetric positive definite matrices [7].
Increasing the efficiency of mathematical modeling on a computer. In order to provide an effective computer
solution of problems (7)–(9), it is necessary to take into account the following features [4, 7, 8]:
• matrices of discrete problems – stiffness A , mass B , damping С are symmetrical, positive definite or posi-
tive semidefinite, and have a sparse structure (band, profile, block, etc.; a matrix is called sparse if the number of its
non-zero elements is much less than their total number 2n , where n is the order of the matrix) [1, 6].
• the order of the matrices of the discrete problems to be solved is )10()10( 85 OO −− , in the near future the era of
exascale computing is coming [8, 9];
• elements of matrices and vectors that occur in the process of mathematical modeling are calculated with errors
caused by errors in the original data, discretization errors, and errors in calculating the values of these data on the com-
puter [8].
Therefore, for the effective use of computing resources of modern supercomputers in the mathematical modeling
of structural strength problems with approximate data and ensuring the reliability of the obtained computer results, it is
necessary:
− identify the original structure of the sparse matrix;
− determine the need for structural regularization and choose the most efficient (for a given computer
architecture) regular structure of matrix;
− carry out structural regularization of sparse matrices - reducing an arbitrary initial structure of matrix to a
regular block-sparse one (block-tape, block-profile, block-skyscraper, block-diagonal with bordering, etc.);
− choose or develop the most efficient (for a given computer architecture, taking into account the regularized
structure of matrices) parallel algorithm for solving the problem;
− choose, in accordance with the algorithm, a decomposition of matrices scheme (the distribution between
processor devices of a parallel computer);
− choose, in accordance with the algorithm, the scheme for storage of matrix elements in a computer;
− investigate of the correspondence the mathematical properties of the matrix, entered into the computer, with
approximate data and the functional purpose of the algorithm for solving the problem.
Such a scheme for researching and solving SLAE and AEVP for sparse matrices of various structures is
implemented in the created parallel algorithms for modern computers with parallel organization of computing, namely
for MIMD-architecture computers with multi-core processors and hybrid architecture [4, 7, 8].
Ways for structural regularization, decomposition and storage of sparse matrices for efficient
modeling of structural strength
The structure of non-zero elements (the structure of a sparse matrix) is determined by the numbering of unknown
problems and can be either regular (for example, band, profile, block-diagonal with framing, etc.), or irregular
(arbitrary). The structures of factorization matrices can be obtained by performing the so-called symbol factorization of
the initial matrix. In most software tools, before forming the matrices of calculation problems, the unknowns are
renumbered (i.e., the sparse structure is reorganized) in order to reduce the total number of floating-point arithmetic
operations when solving calculation problems. Figure 1 shows the structures of matrices (actually factorized matrices)
of calculation problems that arise when modeling the stress-strain state of structures.
, in the near future the
era of exascale computing is coming [8, 9];
• elements of matrices and vectors that occur in the process of mathematical modeling are calculated with
errors caused by errors in the original data, discretization errors, and errors in calculating the values of these data on
the computer [8].
Therefore, for the effective use of computing resources of modern supercomputers in the mathematical model-
ing of structural strength problems with approximate data and ensuring the reliability of the obtained computer results,
it is necessary:
− identify the original structure of the sparse matrix;
− determine the need for structural regularization and choose the most efficient (for a given computer archi-
tecture) regular structure of matrix;
− carry out structural regularization of sparse matrices – reducing an arbitrary initial structure of matrix to a
regular block-sparse one (block-tape, block-profile, block-skyscraper, block-diagonal with bordering, etc.);
− choose or develop the most efficient (for a given computer architecture, taking into account the regularized
structure of matrices) parallel algorithm for solving the problem;
− choose, in accordance with the algorithm, a decomposition of matrices scheme (the distribution between
processor devices of a parallel computer);
− choose, in accordance with the algorithm, the scheme for storage of matrix elements in a computer;
− investigate of the correspondence the mathematical properties of the matrix, entered into the computer, with
approximate data and the functional purpose of the algorithm for solving the problem.
Such a scheme for researching and solving SLAE and AEVP for sparse matrices of various structures is imple-
mented in the created parallel algorithms for modern computers with parallel organization of computing, namely for
MIMD-architecture computers with multi-core processors and hybrid architecture [4, 7, 8].
Ways for structural regularization, decomposition and storage of sparse matrices for
efficient modeling of structural strength
The structure of non-zero elements (the structure of a sparse matrix) is determined by the numbering of unknown
problems and can be either regular (for example, band, profile, block-diagonal with framing, etc.), or irregular (arbitrary).
The structures of factorization matrices can be obtained by performing the so-called symbol factorization of the initial
matrix. In most software tools, before forming the matrices of calculation problems, the unknowns are renumbered (i.e.,
the sparse structure is reorganized) in order to reduce the total number of floating-point arithmetic operations when solv-
ing calculation problems. Figure 1 shows the structures of matrices (actually factorized matrices) of calculation problems
that arise when modeling the stress-strain state of structures.
Програмні засоби моделювання процесів і об'єктів
Composite components are modeled by linearly elastic isotropic bodies [7].
Thus, for a plane deformation, we obtain an eigenvalue problem for a system of partial differential equations
with variable coefficients depending on two variables. In this case (the investigation of the stability of a composite ma-
terial under a compressive load), the conditions of self-adjointness of the operators of the problem of determining the
initial stress-strain state and the generalized eigenvalue problem are satisfied, which corresponds to the problem of
three-dimensional stability theory. In addition, the differential operators of the problems under consideration are posi-
tive-definite under the condition that there is no rigid displacement.
Discretization of problems of mathematical modeling of structural (composite materials) stability. For the
numerical solution of the problem of the three-dimensional linearized theory of stability and the problem of the linear
theory of elasticity, a discrete model is used, which is constructed by the variational-difference method (finite
differences) using the concept of basic schemes. In the computational domain, with the help of straight lines, a
difference grid nonuniform in each direction is introduced so that the material is homogeneous within one grid cell.
Thus, the discrete problem of determining the subcritical stress-strain state (the problem of the linear theory of
elasticity) is written in the form of an SLAE of the form (7) with a sparse symmetric positive definite matrix. The
discrete problem for the stability problem is written in the form of a generalized AEVP of the form (9) of sparse
symmetric positive definite matrices [7].
Increasing the efficiency of mathematical modeling on a computer. In order to provide an effective computer
solution of problems (7)–(9), it is necessary to take into account the following features [4, 7, 8]:
• matrices of discrete problems – stiffness A , mass B , damping С are symmetrical, positive definite or posi-
tive semidefinite, and have a sparse structure (band, profile, block, etc.; a matrix is called sparse if the number of its
non-zero elements is much less than their total number 2n , where n is the order of the matrix) [1, 6].
• the order of the matrices of the discrete problems to be solved is )10()10( 85 OO −− , in the near future the era of
exascale computing is coming [8, 9];
• elements of matrices and vectors that occur in the process of mathematical modeling are calculated with errors
caused by errors in the original data, discretization errors, and errors in calculating the values of these data on the com-
puter [8].
Therefore, for the effective use of computing resources of modern supercomputers in the mathematical modeling
of structural strength problems with approximate data and ensuring the reliability of the obtained computer results, it is
necessary:
− identify the original structure of the sparse matrix;
− determine the need for structural regularization and choose the most efficient (for a given computer
architecture) regular structure of matrix;
− carry out structural regularization of sparse matrices - reducing an arbitrary initial structure of matrix to a
regular block-sparse one (block-tape, block-profile, block-skyscraper, block-diagonal with bordering, etc.);
− choose or develop the most efficient (for a given computer architecture, taking into account the regularized
structure of matrices) parallel algorithm for solving the problem;
− choose, in accordance with the algorithm, a decomposition of matrices scheme (the distribution between
processor devices of a parallel computer);
− choose, in accordance with the algorithm, the scheme for storage of matrix elements in a computer;
− investigate of the correspondence the mathematical properties of the matrix, entered into the computer, with
approximate data and the functional purpose of the algorithm for solving the problem.
Such a scheme for researching and solving SLAE and AEVP for sparse matrices of various structures is
implemented in the created parallel algorithms for modern computers with parallel organization of computing, namely
for MIMD-architecture computers with multi-core processors and hybrid architecture [4, 7, 8].
Ways for structural regularization, decomposition and storage of sparse matrices for efficient
modeling of structural strength
The structure of non-zero elements (the structure of a sparse matrix) is determined by the numbering of unknown
problems and can be either regular (for example, band, profile, block-diagonal with framing, etc.), or irregular
(arbitrary). The structures of factorization matrices can be obtained by performing the so-called symbol factorization of
the initial matrix. In most software tools, before forming the matrices of calculation problems, the unknowns are
renumbered (i.e., the sparse structure is reorganized) in order to reduce the total number of floating-point arithmetic
operations when solving calculation problems. Figure 1 shows the structures of matrices (actually factorized matrices)
of calculation problems that arise when modeling the stress-strain state of structures.
Figure 1. Examples of Sparse Matrix Structures
Reduction of sparse matrices of arbitrary structure to a regular form. Problems of solving linear algebra
problems with sparse matrices and ways to solve them are considered in [10–13]. There are a number of methods
that allows to control the filling of the matrix when solving problems on computers. Both methods of general pur-
pose [10, 11] and those oriented to matrices of a specific type [12, 13] have been developed. Depending on the task and
features of the matrix, it is possible to optimize its portrait by changing the width of the tape, reducing the profile of the
matrix, adjusting the total amount of filling, and bringing the structure to a certain form [10, 11].
243
Програмні засоби моделювання процесів і об′єктів
Among these methods, the following should be mentioned:
• the Cuthill-McKee method and the factor tree method – the use of these methods ensures the concentration of
non-zero elements as close as possible to the main diagonal, which, as a rule, makes it possible to represent the matrix
in a band or in profile form;
• method of parallel sections – the use of this method makes it possible to represent the matrix as a block-
diagonal (with large diagonal blocks) with a border (the order of the diagonal frame block should be much less than the
orders of the main diagonal of blocks) [10, 11];
• minimum degree method – when using this method, as a rule, a reordered matrix is obtained, for the develop-
ment of which fewer arithmetic operations are required; the resulting structure is called «skyscraper» [2–4] because this
structure resembles the skyscrapers of Manhattan (figure 2).
Програмні засоби моделювання процесів і об'єктів
Figure 1. Examples of Sparse Matrix Structures
Reduction of sparse matrices of arbitrary structure to a regular form. Problems of solving linear algebra
problems with sparse matrices and ways to solve them are considered in [10–13]. There are a number of methods that
allows to control the filling of the matrix when solving problems on computers. Both methods of general pur-
pose [10, 11] and those oriented to matrices of a specific type [12, 13] have been developed. Depending on the task and
features of the matrix, it is possible to optimize its portrait by changing the width of the tape, reducing the profile of the
matrix, adjusting the total amount of filling, and bringing the structure to a certain form [10, 11].
Among these methods, the following should be mentioned:
• the Cuthill-McKee method and the factor tree method – the use of these methods ensures the concentration of
non-zero elements as close as possible to the main diagonal, which, as a rule, makes it possible to represent the matrix
in a band or in profile form;
• method of parallel sections – the use of this method makes it possible to represent the matrix as a block-
diagonal (with large diagonal blocks) with a border (the order of the diagonal frame block should be much less than the
orders of the main diagonal of blocks) [10, 11];
• minimum degree method – when using this method, as a rule, a reordered matrix is obtained, for the
development of which fewer arithmetic operations are required; the resulting structure is called "skyscraper" [2–4]
because this structure resembles the skyscrapers of Manhattan (figure 2).
Figure 2. Portrait of the upper triangle of the “skyscraper” matrix
Discrete problems with matrices of this type arise in the problems of calculating the stability of structures. For
effective implementation in algorithms, for example, matrix-matrix operations of sparse matrices, it is advisable to first
reorder (or immediately form) them so that the vast majority of nonzero elements are located in the most dense matrix
blocks: i.e, determine zero blocks (all elements are identically equal to 0 ) and non-zero (have at least one non-zero
element). It is also desirable that non-zero blocks are filled as much as possible. The block sizes are determined based
on the architecture of the computer.
If in the future it is supposed to use the decomposition of the sparse matrix into the product of triangular matrices
(e.g., LU , TLL , etc.), then it is necessary to determine the block-sparse structure of the decomposition matrices using
the symbolic decomposition of the original matrix. After that, it is necessary to optimize the block-sparse structure of
the original matrix or decomposition matrices, using one of the above algorithms and replacing the matrix elements in
the algorithms with blocks. In many cases, it is expedient to combine adjacent (in a row or column) non-zero off-
diagonal blocks of the optimized matrix structure into one rectangular block (tile), which will be further processed as a
whole. In some cases, it is advisable to optimize the structure of diagonal blocks to reduce the number of arithmetic op-
erations with individual blocks of the resulting block-sparse matrix.
Consider another type of sparse matrix – a bordered block-diagonal matrix (figure 3).
Figure 3. Block-Diagonal Bordered Matrix (p = 4)
Discrete problems with matrices of this type appear, for example, when solving boundary value problems by the
Figure 2. Portrait of the upper triangle of the “skyscraper” matrix
Discrete problems with matrices of this type arise in the problems of calculating the stability of structures. For
effective implementation in algorithms, for example, matrix-matrix operations of sparse matrices, it is advisable to first
reorder (or immediately form) them so that the vast majority of nonzero elements are located in the most dense matrix
blocks: i.e, determine zero blocks (all elements are identically equal to 0 ) and non-zero (have at least one non-zero
element). It is also desirable that non-zero blocks are filled as much as possible. The block sizes are determined based
on the architecture of the computer.
If in the future it is supposed to use the decomposition of the sparse matrix into the product of triangular
matrices
Програмні засоби моделювання процесів і об'єктів
Figure 1. Examples of Sparse Matrix Structures
Reduction of sparse matrices of arbitrary structure to a regular form. Problems of solving linear algebra
problems with sparse matrices and ways to solve them are considered in [10–13]. There are a number of methods that
allows to control the filling of the matrix when solving problems on computers. Both methods of general pur-
pose [10, 11] and those oriented to matrices of a specific type [12, 13] have been developed. Depending on the task and
features of the matrix, it is possible to optimize its portrait by changing the width of the tape, reducing the profile of the
matrix, adjusting the total amount of filling, and bringing the structure to a certain form [10, 11].
Among these methods, the following should be mentioned:
• the Cuthill-McKee method and the factor tree method – the use of these methods ensures the concentration of
non-zero elements as close as possible to the main diagonal, which, as a rule, makes it possible to represent the matrix
in a band or in profile form;
• method of parallel sections – the use of this method makes it possible to represent the matrix as a block-
diagonal (with large diagonal blocks) with a border (the order of the diagonal frame block should be much less than the
orders of the main diagonal of blocks) [10, 11];
• minimum degree method – when using this method, as a rule, a reordered matrix is obtained, for the
development of which fewer arithmetic operations are required; the resulting structure is called "skyscraper" [2–4]
because this structure resembles the skyscrapers of Manhattan (figure 2).
Figure 2. Portrait of the upper triangle of the “skyscraper” matrix
Discrete problems with matrices of this type arise in the problems of calculating the stability of structures. For
effective implementation in algorithms, for example, matrix-matrix operations of sparse matrices, it is advisable to first
reorder (or immediately form) them so that the vast majority of nonzero elements are located in the most dense matrix
blocks: i.e, determine zero blocks (all elements are identically equal to 0 ) and non-zero (have at least one non-zero
element). It is also desirable that non-zero blocks are filled as much as possible. The block sizes are determined based
on the architecture of the computer.
If in the future it is supposed to use the decomposition of the sparse matrix into the product of triangular matrices
(e.g., LU , TLL , etc.), then it is necessary to determine the block-sparse structure of the decomposition matrices using
the symbolic decomposition of the original matrix. After that, it is necessary to optimize the block-sparse structure of
the original matrix or decomposition matrices, using one of the above algorithms and replacing the matrix elements in
the algorithms with blocks. In many cases, it is expedient to combine adjacent (in a row or column) non-zero off-
diagonal blocks of the optimized matrix structure into one rectangular block (tile), which will be further processed as a
whole. In some cases, it is advisable to optimize the structure of diagonal blocks to reduce the number of arithmetic op-
erations with individual blocks of the resulting block-sparse matrix.
Consider another type of sparse matrix – a bordered block-diagonal matrix (figure 3).
Figure 3. Block-Diagonal Bordered Matrix (p = 4)
Discrete problems with matrices of this type appear, for example, when solving boundary value problems by the
, then it is necessary to determine the block-sparse structure of the decomposition
matrices using the symbolic decomposition of the original matrix. After that, it is necessary to optimize the block-
sparse structure of the original matrix or decomposition matrices, using one of the above algorithms and replacing
the matrix elements in the algorithms with blocks. In many cases, it is expedient to combine adjacent (in a row or
column) non-zero off-diagonal blocks of the optimized matrix structure into one rectangular block (tile), which will
be further processed as a whole. In some cases, it is advisable to optimize the structure of diagonal blocks to reduce
the number of arithmetic operations with individual blocks of the resulting block-sparse matrix.
Consider another type of sparse matrix – a bordered block-diagonal matrix (figure 3).
Програмні засоби моделювання процесів і об'єктів
Figure 1. Examples of Sparse Matrix Structures
Reduction of sparse matrices of arbitrary structure to a regular form. Problems of solving linear algebra
problems with sparse matrices and ways to solve them are considered in [10–13]. There are a number of methods that
allows to control the filling of the matrix when solving problems on computers. Both methods of general pur-
pose [10, 11] and those oriented to matrices of a specific type [12, 13] have been developed. Depending on the task and
features of the matrix, it is possible to optimize its portrait by changing the width of the tape, reducing the profile of the
matrix, adjusting the total amount of filling, and bringing the structure to a certain form [10, 11].
Among these methods, the following should be mentioned:
• the Cuthill-McKee method and the factor tree method – the use of these methods ensures the concentration of
non-zero elements as close as possible to the main diagonal, which, as a rule, makes it possible to represent the matrix
in a band or in profile form;
• method of parallel sections – the use of this method makes it possible to represent the matrix as a block-
diagonal (with large diagonal blocks) with a border (the order of the diagonal frame block should be much less than the
orders of the main diagonal of blocks) [10, 11];
• minimum degree method – when using this method, as a rule, a reordered matrix is obtained, for the
development of which fewer arithmetic operations are required; the resulting structure is called "skyscraper" [2–4]
because this structure resembles the skyscrapers of Manhattan (figure 2).
Figure 2. Portrait of the upper triangle of the “skyscraper” matrix
Discrete problems with matrices of this type arise in the problems of calculating the stability of structures. For
effective implementation in algorithms, for example, matrix-matrix operations of sparse matrices, it is advisable to first
reorder (or immediately form) them so that the vast majority of nonzero elements are located in the most dense matrix
blocks: i.e, determine zero blocks (all elements are identically equal to 0 ) and non-zero (have at least one non-zero
element). It is also desirable that non-zero blocks are filled as much as possible. The block sizes are determined based
on the architecture of the computer.
If in the future it is supposed to use the decomposition of the sparse matrix into the product of triangular matrices
(e.g., LU , TLL , etc.), then it is necessary to determine the block-sparse structure of the decomposition matrices using
the symbolic decomposition of the original matrix. After that, it is necessary to optimize the block-sparse structure of
the original matrix or decomposition matrices, using one of the above algorithms and replacing the matrix elements in
the algorithms with blocks. In many cases, it is expedient to combine adjacent (in a row or column) non-zero off-
diagonal blocks of the optimized matrix structure into one rectangular block (tile), which will be further processed as a
whole. In some cases, it is advisable to optimize the structure of diagonal blocks to reduce the number of arithmetic op-
erations with individual blocks of the resulting block-sparse matrix.
Consider another type of sparse matrix – a bordered block-diagonal matrix (figure 3).
Figure 3. Block-Diagonal Bordered Matrix (p = 4)
Discrete problems with matrices of this type appear, for example, when solving boundary value problems by the
Figure 3. Block-Diagonal Bordered Matrix (p = 4)
Discrete problems with matrices of this type appear, for example, when solving boundary value problems by
the finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging
244
Програмні засоби моделювання процесів і об′єктів
to two subregions, then three, and so on – these unknowns form a border. For efficient parallelization, it is necessary
that the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of
the border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by
a gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of pro-
cesses (flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the ma-
trix are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th
process is distributed elements of matrix rows with numbers
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
, then
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
-th process will have rows
with numbers i-th process will have rows with numbers
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
, where s is the number of rows in
the block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges
performed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
, where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p, one can practically balance the load of
parallel processes at each moment of time if
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
, where
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of pro-
cesses, and it is advisable to use block distribution schemes between computational elements. That means that each
diagonal block, together with the corresponding off-diagonal border block, is distributed to a separate process (or
thread). A diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in
the way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the
storage scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage
format, of course, affects memory requests, and therefore significantly affects the efficiency of software imple-
mentations of algorithms for processing sparse matrices (for example, decomposition of matrices into a product of
triangular matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matri-
ces are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models
describing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathemati-
cal models with approximate data is that their mathematical properties are a priori unknown. Within a given level of
error, problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well
conditioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degener-
ate matrix can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer
model of the problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set,
is called a hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the
obtained mathematical solution may not have a physical meaning, that is, such a solution will not contain the
solution of a physical problem. Therefore, it is necessary to determine the scope of the mathematical model, be-
cause the hereditary error cannot be corrected by mathematical methods of solving the problem. To reduce the
hereditary error, you can either increase the accuracy of the input data or reformulate the problem with respect
to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two
relations:
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
(10)
where
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
– is the condition number of the system matrix,
Програмні засоби моделювання процесів і об'єктів
finite element method or the finite difference method, if the domain is divided into subregions and discretization is
carried out, and the unknowns of the discrete problem are numbered in the following order. First, the unknowns that
belong to only one subregions, sequentially through the subregions. Then, in the same order, the unknowns belonging to
two subregions, then three, and so on - these unknowns form a border. For efficient parallelization, it is necessary that
the dimensions of the diagonal blocks corresponding to the unknowns of one subregions exceed the dimensions of the
border of the diagonal block by an order of magnitude or more.
Decomposition (distribution schemes) of sparse matrices. Many methods for solving linear algebra problems,
that is used in modeling structural strength problems, are based on algorithms for decomposing a matrix into a product
of matrices of standard types, for example, lower and upper triangular matrices. Such algorithms are characterized by a
gradual decrease from step to step in the size of the processed part of the matrix. Therefore, it is important to ensure
approximately the same amount of calculations, exchanges and synchronizations performed by each process or thread
on a parallel computing model, that is, to exclude the influence of the Haydn effect [8, 14, 15].
As research has shown, parallel versions of these algorithms, in which the so-called cyclic distribution schemes
and processing matrices are created (see, for example, [8, 14, 15]), provide a sufficiently good load balance of processes
(flows).
Quite often, one-dimensional block-cyclic schemes are used: rows or columns of non-zero blocks of the matrix
are distributed cyclically between processes or threads of the upper level of parallelism, for example, if the q-th process
is distributed elements of matrix rows with numbers rsrsr ++++ ...,,1 , then )1( ++q -th process will have rows with
numbers i-th process will have rows with numbers srsrs ++++++++ )1(...,,1)1( , where s is the number of rows in the
block (we can talk about the r-th and (r + 1)-th rows of square matrix blocks of order s ).
In the case of a regular sparse matrix structure, parallel algorithms that use one-dimensional block-cyclic
schemes for allocating matrix elements can provide an approximately equal amount of computations and exchanges per-
formed by each parallel process or thread at each instant of time (see, for example, [8, 14, 15]).
Such a regular structure is primarily the band structure of the matrix under the obvious condition that the half-
width of the matrix band m exceeds the product ps , where p is the number of processes or threads. In the case of a
regular profile structure of the system matrix, by varying the value of s and p , one can practically balance the load of
parallel processes at each moment of time if spnce / , where ec is the total number of subdiagonal elements in the
matrix profile.
For block-diagonal bordered matrices, cyclic distribution schemes do not allow balancing the load of processes,
and it is advisable to use block distribution schemes between computational elements. That means that each diagonal
block, together with the corresponding off-diagonal border block, is distributed to a separate process (or thread). A
diagonal border block can also be allocated in a separate process (or thread).
Storage schemes for sparse matrices. There are various storage schemes for sparse matrices, that differ in the
way zero elements usage. In some cases, it is allowed to store a portion of zeros in exchange for simplifying the storage
scheme; in others, all zeros of the matrix are used; thirdly, zeros are not used at all. The choice of storage format, of
course, affects memory requests, and therefore significantly affects the efficiency of software implementations of
algorithms for processing sparse matrices (for example, decomposition of matrices into a product of triangular
matrices).
In the algorithms created for solving SLAE and AEVP, different formats for storing elements of sparse matrices
are used: coordinate format, sparse row or column format, ELLPACK format, hybrid format, etc. [4, 8].
Investigation of the mathematical properties of matrices entered into the computer. Mathematical models de-
scribing applied problems always contain errors in the initial data [8]. That is, a characteristic feature of mathematical
models with approximate data is that their mathematical properties are a priori unknown. Within a given level of error,
problems can be both compatible and incompatible, both correctly and incorrectly stated, both poorly and well condi-
tioned. Due to the rounding error, a nonsingular matrix can become degenerate in a computer, and a degenerate matrix
can become nonsingular. There is a problem of choosing the necessary algorithm for solving the computer model of the
problem with the reliability of the results obtained.
An error in the solution of a mathematical problem, which is caused by an error in the initial data set, is called a
hereditary error. If the hereditary error of the solution of a mathematical problem is large, then the obtained
mathematical solution may not have a physical meaning, that is, such a solution will not contain the solution of a
physical problem. Therefore, it is necessary to determine the scope of the mathematical model, because the hereditary
error cannot be corrected by mathematical methods of solving the problem. To reduce the hereditary error, you can
either increase the accuracy of the input data or reformulate the problem with respect to other parameters [8].
The investigation of the correctness of the problem statement in a computer is reduced to checking two relations:
0.10.1 ++ , (( )) 1)( AhAA ; (10)
where )(Ah – is the condition number of the system matrix, (( ))Ah 1−−== . The first condition, fulfilled in floating-point
arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular), and the
second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is uni-
fied and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data in-
. The first condition, fulfilled in floating-
point arithmetic, means that the matrix is non-singular within the limits of machine accuracy (machine non-singular),
and the second one means that it is non-singular within the limits of the accuracy of the input data [8, 16].
Thus, when conditions (10) are met, the solution of the computer problem with approximate data exists, is
unified and stable. Such a computer task should be considered as correctly posed within the accuracy of the initial data
input. Otherwise, the matrix of the system may turn out to be a matrix of incomplete rank, and the computer problem
should be considered as incorrectly posed. The fulfillment of condition (10) at an increased computer bit rate in ac-
cordance with the value of the condition number of the matrix indicates that the original problem is poorly conditioned
relative to the previous bit rate and it is possible to obtain a single solution at an increased bit rate.
245
Програмні засоби моделювання процесів і об′єктів
So, the computer problem, which should to be solved in the end, always has approximate data relative to the
original problem (due to the hereditary error in the initial data, due to the discretization errors, due to the errors in
obtaining numerical data in the computer).
In the created parallel algorithms for solving SLAE and AEVP for sparse matrices of various structures on
modern high-performance computing systems with parallel organization of calculations, a research is provided in a
computer environment of the mathematical properties of matrices using various computer bit rate ( the technology is
described in the work [16]).
Using effective ways of structuring sparse matrices in some practical problems of math-
ematical modeling of structural strength
Let’s consider some practical problems of mathematical modeling of structural strength, which are reduced to
solving linear algebra problems with sparse matrices.
Solving the problem of calculating building structures. To analyze the strength of building structures on
single-processor computers, Software package Lira-Sapr is widely used – a multi-threaded version of PC Lira [17]. The
developed parallel algorithms for solving SLAE with sparse matrices of various structures were used as a processor
component of the parallel version of PC Lira – the software package Lira-cluster [2–4, 18].
Consider the problem of analyzing the strength of a 27-storey building (figure 4). The sparse symmetric
matrix obtained as a result of discretization by the finite element method (FEM-mesh) has an order of 661590 and
a band half-width of 34242, density (the matrix tape is filled with non-zero elements by 5%). After optimizing
the matrix structure by the minimum degree method, a skyscraper structure matrix was obtained (order – 661590,
ribbon width – 541257, density <1%). The time for solving the problem using Lira-Sapr on a personal computer
is 3254 sec, using Lira-cluster on the hybrid architecture of the SKIT-4 supercomputer is 57 sec. Thus, an ac-
celeration of 23 times was achieved.
Програмні засоби моделювання процесів і об'єктів
put. Otherwise, the matrix of the system may turn out to be a matrix of incomplete rank, and the computer problem
should be considered as incorrectly posed. The fulfillment of condition (10) at an increased computer bit rate in accord-
ance with the value of the condition number of the matrix indicates that the original problem is poorly conditioned rela-
tive to the previous bit rate and it is possible to obtain a single solution at an increased bit rate.
So, the computer problem, which should to be solved in the end, always has approximate data relative to the
original problem (due to the hereditary error in the initial data, due to the discretization errors, due to the errors in
obtaining numerical data in the computer).
In the created parallel algorithms for solving SLAE and AEVP for sparse matrices of various structures on
modern high-performance computing systems with parallel organization of calculations, a research is provided in a
computer environment of the mathematical properties of matrices using various computer bit rate ( the technology is
described in the work [16]).
Using effective ways of structuring sparse matrices in some practical problems of
mathematical modeling of structural strength
Let’s consider some practical problems of mathematical modeling of structural strength, which are reduced to
solving linear algebra problems with sparse matrices.
Solving the problem of calculating building structures. To analyze the strength of building structures on single-
processor computers, Software package Lira-Sapr is widely used – a multi-threaded version of PC Lira [17]. The
developed parallel algorithms for solving SLAE with sparse matrices of various structures were used as a processor
component of the parallel version of PC Lira – the software package Lira-cluster [2–4, 18].
Consider the problem of analyzing the strength of a 27-storey building (figure 4). The sparse symmetric matrix
obtained as a result of discretization by the finite element method (FEM-mesh) has an order of 661590 and a band half-
width of 34242, density (the matrix tape is filled with non-zero elements by 5%). After optimizing the matrix structure
by the minimum degree method, a skyscraper structure matrix was obtained (order – 661590, ribbon width – 541257,
density <1%). The time for solving the problem using Lira-Sapr on a personal computer is 3254 sec, using Lira-cluster
on the hybrid architecture of the SKIT-4 supercomputer is 57 sec. Thus, an acceleration of 23 times was achieved.
Figure 4. FEM-model of a 27-storey building
Supercomputer SKIT-4 of the V.M. Glushkov Institute of Cybernetics National Academy of Sciences of
Ukraine has the following specifications: CPU – Intel(R) Xeon(R) CPU E5-26700, clock speed 2.60 GHz, speed 8
GT/s, cache memory – 20 MB, per node: 2 CPUs with 8 cores + Hyperthrefding =32 cores, Max Memory Size –
384 GB; GPU – Nvidia Tesla M2050, 3 GB memory, peak performance – 515 Gflops [19].
Numerical investigation of the stability of composite materials. The created parallel algorithms for solving
SLAE and AEVP for sparse matrices of various structures were used in mathematical modeling of the problem of
structural stability - the problem of stability of a layered two-component composite material with a regular structure
(Fig. 5, left) under a uniform single-axis load on computers of various architectures. Since the stability problem is
solved for the computational domain of finite dimensions (Fig. 5, right) and the influence of the boundary conditions on
the lateral sides of the computational domain on the critical stability parameters is studied, we distinguish between the
loss of stability in a composite material and a composite sample corresponding to this material [7, 20, 21].
Figure 4. FEM-model of a 27-storey building
Supercomputer SKIT-4 of the V.M. Glushkov Institute of Cybernetics National Academy of Sciences of
Ukraine has the following specifications: CPU – Intel(R) Xeon(R) CPU E5-26700, clock speed 2.60 GHz, speed 8
GT/s, cache memory – 20 MB, per node: 2 CPUs with 8 cores + Hyperthrefding =32 cores, Max Memory Size –
384 GB; GPU – Nvidia Tesla M2050, 3 GB memory, peak performance – 515 Gflops [19].
Numerical investigation of the stability of composite materials. The created parallel algorithms for
solving SLAE and AEVP for sparse matrices of various structures were used in mathematical modeling of the
problem of structural stability - the problem of stability of a layered two-component composite material with a
regular structure (Fig. 5, left) under a uniform single-axis load on computers of various architectures. Since the
246
Програмні засоби моделювання процесів і об′єктів
stability problem is solved for the computational domain of finite dimensions (Fig. 5, right) and the influence of
the boundary conditions on the lateral sides of the computational domain on the critical stability parameters is
studied, we distinguish between the loss of stability in a composite material and a composite sample correspond-
ing to this material [7, 20, 21].
Програмні засоби моделювання процесів і об'єктів
Figure 5. Schematic representation of the stability problem
The problem of mathematical modeling of the stability of composite materials was solved on a parallel computer
of hybrid architecture SKIT-4.
Comparative results of solving the obtained partial generalized eigenvalue problem BxAx == in the problem of
mathematical modeling of the stability of new composite materials are given for band symmetric positive definite ma-
trices (the order of matrices A and B is 12282, the half-width band of matrix A is 6212, the half-width tape of the
matrix B is 71).
Compared to the sequential version of the algorithm, the case of solving the AEVP of band symmetric matrices, a
7-fold acceleration was obtained by the parallel algorithm on the MIMD architecture; on a hybrid architecture - 18 and
33 times, using one and two GPUs respectively compared. In the case of the AEVP solution for sparse matrices
reduced to the block-diagonal form with border, an acceleration of 43 times was obtained on the hybrid architecture
using two GPUs compared to the sequential version of the algorithm [21].
Conclusions
Methods for regularization, decomposition and storage of sparse matrices are proposed, which allow to
significantly increase the efficiency of solving problems of mathematical modeling of structural strength. New methods
and computer block-cyclic and block algorithms for high-performance computing created on their basis for solving
linear algebra problems (systems of linear algebraic equations, algebraic eigenvalue problem) with approximate data
can be used for mathematical modeling of different processes on modern supercomputers.
References
1. Timoshenko S.P., Goodier J. (1975) Theory of elasticity. M.: Nauka. 575 p. (in Russian)
2. Gorodetsky A.S., Evzerov I.D. (2007) Computer models of structures. K: FACT. 394 p. (in Russian)
3. Slobodyan Ya.E. (2009) Supercomputer technology for modeling the life cycle of especially complex technical objects. Cybernetics and Sys-
tems Analysis, Vol. 45, N. 6. P. 959–965.
4. Baranov A.Yu., Popov A.B., Slobodyan Y.E., Khimich A.N. (2017) Mathematical Modeling of Building Constructions Using Hybrid
Computing Systems. Journal of Automation and Information Sciences. Vol. 50 (7). P 18–32. https://doi.org/10.1615/jautomatinfscien.v49.i7.20.
5. G. Strang and J. Fix An (1973) Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ, 306 p.
6. Molchanov I.N., Nikolenko L.D. (1984) Fundamentals of the finite element method. Kiev: Naukova dumka. 267 p. (in Russian)
7. Khimich A.N., Dekret V.A., Popov A.V., Chistyakov A.V. (2018) Numerical Study of the Stability of Composite Materials on Computers of
Hybrid Architecture. Journal of Automation and Information Sciences. Vol. 50. N. 7. P. 7–24.
https://doi.org/10.1615/jautomatinfscien.v50.i7.20.
8. A.N. Khimich, I.N. Molchanov, A.V. Popov, Т.V. Chistyakova, M.F. Yakovlev. (2007) Parallel algorithms for solving problems in computa-
tional mathematics. Kiev: Naukova dumka. 248 p. (in Russian)
9. Sergienko I.V., Khimich O.M. (2019) Mathematical modeling: From small to exaflops. Bulletin of the National Academy of Sciences of
Ukraine. N 8. P. 37–50. (in Russian)
10. Pissanetzky S. (1984) Sparse Matrix Technology. London: Academic Press. 321 p.
11. Gene H. Golub, Charlee F. Van Lone. (2013) Matrix calculations. JHUPress. 756 p.
12. George, A., and Liu, J. W.-H. (1981) Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs, NJ, USA.
324 р.
13. Saad Y . (2000) Iterative Methods for Sparse Linear Systems. PWS Publishing Company. 448 p.
14. Molchanov I.N., Khimich A.N., Popov A.V. (2005) Efficient Implementation of Computational Algorithms on MIMD Computers. Artificial In-
telligence. N. 3. P. 175–184. (in Russian)
15. Popov O.V. (2013) About parallel algorithms for factorization of sparsed ed matrices. Computer Mathematics: Collection of scientific papers.
N. 2. P. 115–124. (in Ukrainian)
16. Khimich О.М., Chistjakova T.V., Sidoruk V.A., Ershov P.S. (2021) Adaptive computer technologies for solving problems of computational
and applied mathematics. Cybernetics and Systems Analysis. Vol. 57. N 6, P. 990–997. https://doi.org/10.1007/s10559-021-00424-z.
17. LIRA-SAPR [Online] Available from: http://www.liraland.ua. [Accessed: 1 June 2022]
18. Programs for calculation and design of structures [Online] Available from: http://www.lira.com.ua. [Accessed: 1 June 2022]
Програмні засоби моделювання процесів і об'єктів
Figure 5. Schematic representation of the stability problem
The problem of mathematical modeling of the stability of composite materials was solved on a parallel computer
of hybrid architecture SKIT-4.
Comparative results of solving the obtained partial generalized eigenvalue problem BxAx == in the problem of
mathematical modeling of the stability of new composite materials are given for band symmetric positive definite ma-
trices (the order of matrices A and B is 12282, the half-width band of matrix A is 6212, the half-width tape of the
matrix B is 71).
Compared to the sequential version of the algorithm, the case of solving the AEVP of band symmetric matrices, a
7-fold acceleration was obtained by the parallel algorithm on the MIMD architecture; on a hybrid architecture - 18 and
33 times, using one and two GPUs respectively compared. In the case of the AEVP solution for sparse matrices
reduced to the block-diagonal form with border, an acceleration of 43 times was obtained on the hybrid architecture
using two GPUs compared to the sequential version of the algorithm [21].
Conclusions
Methods for regularization, decomposition and storage of sparse matrices are proposed, which allow to
significantly increase the efficiency of solving problems of mathematical modeling of structural strength. New methods
and computer block-cyclic and block algorithms for high-performance computing created on their basis for solving
linear algebra problems (systems of linear algebraic equations, algebraic eigenvalue problem) with approximate data
can be used for mathematical modeling of different processes on modern supercomputers.
References
1. Timoshenko S.P., Goodier J. (1975) Theory of elasticity. M.: Nauka. 575 p. (in Russian)
2. Gorodetsky A.S., Evzerov I.D. (2007) Computer models of structures. K: FACT. 394 p. (in Russian)
3. Slobodyan Ya.E. (2009) Supercomputer technology for modeling the life cycle of especially complex technical objects. Cybernetics and Sys-
tems Analysis, Vol. 45, N. 6. P. 959–965.
4. Baranov A.Yu., Popov A.B., Slobodyan Y.E., Khimich A.N. (2017) Mathematical Modeling of Building Constructions Using Hybrid
Computing Systems. Journal of Automation and Information Sciences. Vol. 50 (7). P 18–32. https://doi.org/10.1615/jautomatinfscien.v49.i7.20.
5. G. Strang and J. Fix An (1973) Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ, 306 p.
6. Molchanov I.N., Nikolenko L.D. (1984) Fundamentals of the finite element method. Kiev: Naukova dumka. 267 p. (in Russian)
7. Khimich A.N., Dekret V.A., Popov A.V., Chistyakov A.V. (2018) Numerical Study of the Stability of Composite Materials on Computers of
Hybrid Architecture. Journal of Automation and Information Sciences. Vol. 50. N. 7. P. 7–24.
https://doi.org/10.1615/jautomatinfscien.v50.i7.20.
8. A.N. Khimich, I.N. Molchanov, A.V. Popov, Т.V. Chistyakova, M.F. Yakovlev. (2007) Parallel algorithms for solving problems in computa-
tional mathematics. Kiev: Naukova dumka. 248 p. (in Russian)
9. Sergienko I.V., Khimich O.M. (2019) Mathematical modeling: From small to exaflops. Bulletin of the National Academy of Sciences of
Ukraine. N 8. P. 37–50. (in Russian)
10. Pissanetzky S. (1984) Sparse Matrix Technology. London: Academic Press. 321 p.
11. Gene H. Golub, Charlee F. Van Lone. (2013) Matrix calculations. JHUPress. 756 p.
12. George, A., and Liu, J. W.-H. (1981) Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs, NJ, USA.
324 р.
13. Saad Y . (2000) Iterative Methods for Sparse Linear Systems. PWS Publishing Company. 448 p.
14. Molchanov I.N., Khimich A.N., Popov A.V. (2005) Efficient Implementation of Computational Algorithms on MIMD Computers. Artificial In-
telligence. N. 3. P. 175–184. (in Russian)
15. Popov O.V. (2013) About parallel algorithms for factorization of sparsed ed matrices. Computer Mathematics: Collection of scientific papers.
N. 2. P. 115–124. (in Ukrainian)
16. Khimich О.М., Chistjakova T.V., Sidoruk V.A., Ershov P.S. (2021) Adaptive computer technologies for solving problems of computational
and applied mathematics. Cybernetics and Systems Analysis. Vol. 57. N 6, P. 990–997. https://doi.org/10.1007/s10559-021-00424-z.
17. LIRA-SAPR [Online] Available from: http://www.liraland.ua. [Accessed: 1 June 2022]
18. Programs for calculation and design of structures [Online] Available from: http://www.lira.com.ua. [Accessed: 1 June 2022]
Figure 5. Schematic representation of the stability problem
The problem of mathematical modeling of the stability of composite materials was solved on a parallel
computer of hybrid architecture SKIT-4.
Comparative results of solving the obtained partial generalized eigenvalue problem
Програмні засоби моделювання процесів і об'єктів
Figure 5. Schematic representation of the stability problem
The problem of mathematical modeling of the stability of composite materials was solved on a parallel computer
of hybrid architecture SKIT-4.
Comparative results of solving the obtained partial generalized eigenvalue problem BxAx == in the problem of
mathematical modeling of the stability of new composite materials are given for band symmetric positive definite ma-
trices (the order of matrices A and B is 12282, the half-width band of matrix A is 6212, the half-width tape of the
matrix B is 71).
Compared to the sequential version of the algorithm, the case of solving the AEVP of band symmetric matrices, a
7-fold acceleration was obtained by the parallel algorithm on the MIMD architecture; on a hybrid architecture - 18 and
33 times, using one and two GPUs respectively compared. In the case of the AEVP solution for sparse matrices
reduced to the block-diagonal form with border, an acceleration of 43 times was obtained on the hybrid architecture
using two GPUs compared to the sequential version of the algorithm [21].
Conclusions
Methods for regularization, decomposition and storage of sparse matrices are proposed, which allow to
significantly increase the efficiency of solving problems of mathematical modeling of structural strength. New methods
and computer block-cyclic and block algorithms for high-performance computing created on their basis for solving
linear algebra problems (systems of linear algebraic equations, algebraic eigenvalue problem) with approximate data
can be used for mathematical modeling of different processes on modern supercomputers.
References
1. Timoshenko S.P., Goodier J. (1975) Theory of elasticity. M.: Nauka. 575 p. (in Russian)
2. Gorodetsky A.S., Evzerov I.D. (2007) Computer models of structures. K: FACT. 394 p. (in Russian)
3. Slobodyan Ya.E. (2009) Supercomputer technology for modeling the life cycle of especially complex technical objects. Cybernetics and Sys-
tems Analysis, Vol. 45, N. 6. P. 959–965.
4. Baranov A.Yu., Popov A.B., Slobodyan Y.E., Khimich A.N. (2017) Mathematical Modeling of Building Constructions Using Hybrid
Computing Systems. Journal of Automation and Information Sciences. Vol. 50 (7). P 18–32. https://doi.org/10.1615/jautomatinfscien.v49.i7.20.
5. G. Strang and J. Fix An (1973) Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ, 306 p.
6. Molchanov I.N., Nikolenko L.D. (1984) Fundamentals of the finite element method. Kiev: Naukova dumka. 267 p. (in Russian)
7. Khimich A.N., Dekret V.A., Popov A.V., Chistyakov A.V. (2018) Numerical Study of the Stability of Composite Materials on Computers of
Hybrid Architecture. Journal of Automation and Information Sciences. Vol. 50. N. 7. P. 7–24.
https://doi.org/10.1615/jautomatinfscien.v50.i7.20.
8. A.N. Khimich, I.N. Molchanov, A.V. Popov, Т.V. Chistyakova, M.F. Yakovlev. (2007) Parallel algorithms for solving problems in computa-
tional mathematics. Kiev: Naukova dumka. 248 p. (in Russian)
9. Sergienko I.V., Khimich O.M. (2019) Mathematical modeling: From small to exaflops. Bulletin of the National Academy of Sciences of
Ukraine. N 8. P. 37–50. (in Russian)
10. Pissanetzky S. (1984) Sparse Matrix Technology. London: Academic Press. 321 p.
11. Gene H. Golub, Charlee F. Van Lone. (2013) Matrix calculations. JHUPress. 756 p.
12. George, A., and Liu, J. W.-H. (1981) Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs, NJ, USA.
324 р.
13. Saad Y . (2000) Iterative Methods for Sparse Linear Systems. PWS Publishing Company. 448 p.
14. Molchanov I.N., Khimich A.N., Popov A.V. (2005) Efficient Implementation of Computational Algorithms on MIMD Computers. Artificial In-
telligence. N. 3. P. 175–184. (in Russian)
15. Popov O.V. (2013) About parallel algorithms for factorization of sparsed ed matrices. Computer Mathematics: Collection of scientific papers.
N. 2. P. 115–124. (in Ukrainian)
16. Khimich О.М., Chistjakova T.V., Sidoruk V.A., Ershov P.S. (2021) Adaptive computer technologies for solving problems of computational
and applied mathematics. Cybernetics and Systems Analysis. Vol. 57. N 6, P. 990–997. https://doi.org/10.1007/s10559-021-00424-z.
17. LIRA-SAPR [Online] Available from: http://www.liraland.ua. [Accessed: 1 June 2022]
18. Programs for calculation and design of structures [Online] Available from: http://www.lira.com.ua. [Accessed: 1 June 2022]
in the problem of
mathematical modeling of the stability of new composite materials are given for band symmetric positive definite matri-
ces (the order of matrices A and B is 12282, the half-width band of matrix A is 6212, the half-width tape of the matrix
B is 71).
Compared to the sequential version of the algorithm, the case of solving the AEVP of band symmetric matrices, a
7-fold acceleration was obtained by the parallel algorithm on the MIMD architecture; on a hybrid architecture – 18 and 33
times, using one and two GPUs respectively compared. In the case of the AEVP solution for sparse matrices reduced to
the block-diagonal form with border, an acceleration of 43 times was obtained on the hybrid architecture using two GPUs
compared to the sequential version of the algorithm [21].
Conclusions
Methods for regularization, decomposition and storage of sparse matrices are proposed, which allow to signifi-
cantly increase the efficiency of solving problems of mathematical modeling of structural strength. New methods and
computer block-cyclic and block algorithms for high-performance computing created on their basis for solving linear
algebra problems (systems of linear algebraic equations, algebraic eigenvalue problem) with approximate data can be
used for mathematical modeling of different processes on modern supercomputers.
References
1. Timoshenko S.P., Goodier J. (1975) Theory of elasticity. M.: Nauka. 575 p. (in Russian)
2. Gorodetsky A.S., Evzerov I.D. (2007) Computer models of structures. K: FACT. 394 p. (in Russian)
3. Slobodyan Ya.E. (2009) Supercomputer technology for modeling the life cycle of especially complex technical objects. Cybernetics and Systems
Analysis, Vol. 45, N. 6. P. 959–965.
4. Baranov A.Yu., Popov A.B., Slobodyan Y.E., Khimich A.N. (2017) Mathematical Modeling of Building Constructions Using Hybrid Computing
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Програмні засоби моделювання процесів і об′єктів
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Diagonal Bordered Matrices. Cybernetics and Systems Analisys. V. 53. N. 6. P. 913–923. https://doi.org/10.1007/s10559-020-00311-z.
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Analysis, Vol. 45, N. 6, P. 959–965.
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20. Khimich A.N., Popov A.V., Chistyakov A.V. (2017) Hybrid Algorithms for Solving the Algebraic Eigenvalue Problem with Sparse Matrices. Cy-
bernetics and Systems Analysis. N 6. P. 132–146. https://doi.org/10.1007/s10559-017-9996-5.
21. Khimich A.N., Popov A.V., Chistyakov O.V., Sidoruk V.A. (2020) A Parallel Algorithm for Solving a Partial Eigenvalue Problem for Block-
Diagonal Bordered Matrices. Cybernetics and Systems Analisys. V. 53. N. 6. P. 913–923.
Received 02.08.2022
About the authors:
Khimich Olexandr Mykolayovych
Doctor of Sciences, Academician of National Academy of Sciences of Ukraine,
Deputy director of Glushkov Institute of Cybernetics,
head of the computer science department of National Academy of Sciences of Ukraine,
head of department №150 of Glushkov Institute of Cybernetics.
number of publications in domestic publications – 220,
among them 11 monographs (four published abroad)
https://orcid.org/0000-0002-8103-4223
h-index – 7
Popov Aleksandr Volodymyrovych
Doctor of Sciences (Physics and mathematics),
senior scientist,Glushkov Institute of Cybernetics
of National Academy of Sciences of Ukraine,
Academic Gluskkov Avenue, 40, Kyiv, 03187, Ukraine,
number of publications in domestic publications – ≈95,
https://orcid.org/0000-0002-1217-2534
h-index – 4.
248
Програмні засоби моделювання процесів і об′єктів
Chystiakov Oleksii Valeriyovych,
PhD, Physics and mathematics,
senior scientist, Glushkov Institute of Cybernetics
of National Academy of Sciences of Ukraine,
Academic Gluskkov Avenue, 40, Kyiv, 03187, Ukraine,
number of publications in domestic publications – 33,
https://orcid.org/0000-0001-6456-2094
h-index – 3.
Place of work:
V.M. Glushkov Institute of Cybernetics of National Academy
of Sciences of Ukraine
Academic Gluskkov Avenue, 40, Kyiv, 03187, Ukraine,
Tel.: (38)(044) 526 11 96,
E-mail: khimich505@gmail.com, alex50popov@gmail.com,
alexej.chystyakov@gmail.com
Прізвища та ініціали авторів і назва доповіді українською мовою:
Хіміч О.М., Попов О.В., Чистяков О.В
Про особливості використання розріджених матриць
в задачах математичного моделювання.
Прізвища та ініціали авторів і назва доповіді англійською мовою:
Khimich O.M., Popov A.V., Chystiakov О.V.
On the peculiarities of using sparse matrices in problems
of mathematical modeling
Контакти для редактора: Чистяков Олексій Валерійович,
старший науковий співробітник Інституту кібернетики
ім. В.М. Глушкова НАН України,
alexej.chystyakov@gmail.com, тел.: (050) 995-24-32
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