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This paper analyses the definition of inverse and direct problems in engineering dimensional chains calculation based on discrete analogue data and the methodologies for solving these problems. It is shown that the direct dimensional chains calculation, which belongs to the class of inverse boundary...
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| author | Trofymova, Iryna Meniailov, Ievgen Chernysh, Serhii Yepifanov, Sergiy Khustochka, Olexandr Ugryumov, Mykhaylo Myenyaylov, Andriy Chumachenko, Dmytro |
| author_facet | Trofymova, Iryna Meniailov, Ievgen Chernysh, Serhii Yepifanov, Sergiy Khustochka, Olexandr Ugryumov, Mykhaylo Myenyaylov, Andriy Chumachenko, Dmytro |
| author_sort | Trofymova, Iryna |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2021-04-08T14:17:06Z |
| description | This paper analyses the definition of inverse and direct problems in engineering dimensional chains calculation based on discrete analogue data and the methodologies for solving these problems. It is shown that the direct dimensional chains calculation, which belongs to the class of inverse boundary value problems in a stochastic formulation, can be transformed into multi-criteria problems of stochastic optimization with mixed conditions. The new multi-step solutions search methodology for these problems is based on non-linear robust estimation methods. It can be achieved through hierarchical two-level decisions synthesis scheme development. At the first step, this scheme includes identification of surrogate models (in the form of regression equations). At the second step, the effective robust estimates are computed to determine unknown values; estimations of unknown quantities are carried out under a priori and parametric data uncertainties. Results of calculations of inverse and direct problems in engineering dimensional chains for two-stage axial compressors are presented. They were obtained using interactive computer systems for decision-making support “ROD&IDS”. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2020.4.06 |
| first_indexed | 2025-07-17T10:26:58Z |
| format | Article |
| fulltext |
I. Trofymova, I. Meniailov, S. Сhernysh, S. Yepifanov, O. Khustochka, M. Ugryumov, A. Myenyaylov,
D. Chumachenko, 2020
70 ISSN 1681–6048 System Research & Information Technologies, 2020, № 4
UDC 004.942:519.6:519.23
DOI: 10.20535/SRIT.2308-8893.2020.4.06
METHODOLOGY OF NON-LINEAR ROBUST ESTIMATION
FOR THE SOLUTIONS SYNTHESIS OF INVERSE AND DIRECT
MULTIDISCIPLINARY PROBLEMS IN ENGINEERING
DIMENSIONAL CHAINS CALCULATION BASED ON DISCRETE
ANALOG DATA
I. TROFYMOVA, I. MENIAILOV, S. СHERNYSH, S. YEPIFANOV,
O. KHUSTOCHKA, M. UGRYUMOV, A. MYENYAYLOV, D. CHUMACHENKO
Abstract. This paper analyses the definition of inverse and direct problems in engi-
neering dimensional chains calculation based on discrete analogue data and the
methodologies for solving these problems. It is shown that the direct dimensional
chains calculation, which belongs to the class of inverse boundary value problems in
a stochastic formulation, can be transformed into multi-criteria problems of stochas-
tic optimization with mixed conditions. The new multi-step solutions search meth-
odology for these problems is based on non-linear robust estimation methods. It can
be achieved through hierarchical two-level decisions synthesis scheme development.
At the first step, this scheme includes identification of surrogate models (in the form
of regression equations). At the second step, the effective robust estimates are com-
puted to determine unknown values; estimations of unknown quantities are carried
out under a priori and parametric data uncertainties. Results of calculations of in-
verse and direct problems in engineering dimensional chains for two-stage axial
compressors are presented. They were obtained using interactive computer systems
for decision-making support “ROD&IDS”.
Keywords: inverse boundary value problems in a stochastic formulation, a priori
and parametric uncertainties, methods and systems for estimating quantities and
processes, decision-making theory.
INTRODUCTION
Cost-cutting of the systems refinement is one of the most relevant issues in the
processes of project development and operation in state-of-the-art technology. It
should be noted that a good exact solution of the inverse problem in a determinis-
tic formulation (optimization problem) in practice during mass production, as a
rule, leads to a large scatter of the values of the integral characteristics of prod-
ucts. Then there is selective assembly or rejection of products for quality. It is
possible to resolve this general technical problem by putting the Robust Estima-
tion methods into practice.
The examples of partial problems that form this general problem are:
choice of equipment with a certain degree of accuracy for manufacturing
products (this will reduce the percentage of failures in serial production and avoid
selective assembly of products);
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 71
choice of measuring systems components with a certain degree of accu-
racy, which will ensure the specified measurement accuracy of controlled vari-
ables of systems and processes;
intelligent diagnostics of systems and processes based on monitoring of
controlled state variables;
prediction of the condition of patients in medical monitoring systems,
choice of an individual treatment program for each of them;
development of new drugs in the pharmaceutical industry;
robust optimal control of systems and processes;
machine-building including the product quality control (Design for Six
Sigma), and also the areas of industrial safety, ecology, the activities of banks,
insurance, audit, etc.
The following mathematical issues arise in the process of the above-
mentioned systems development: uncertainties evaluation, structuring of the regu-
larizing algorithms, and high computational complexity of methods for quasi-
solutions synthesis under uncertainties.
Such inverse problems are essentially ill-posed since it is not known in
which class of functions to search for the solutions and there are uncertainty deals
with the choice of the exact solution. However, these problems can be reduced to
conditionally well-posed.
Such an approach of reducing inverse problems to structural-parametric op-
timization problems at stochastics formulation, and for a given structure – to mul-
ti-parameter optimization problems in a stochastics formulation is widely used in
practice. The synthesis of solutions to such problems is performed using regulari-
zation methods based on machine learning algorithms.
The outcomes of this research are the development of new solutions synthe-
sis methods for the stochastic optimization problems with mixed conditions and
development of the software which implements them and can be used for practi-
cal problem-solving.
One problem is that during the development of state-of-the-art technic ob-
jects, it is necessary to take into account a optimal set of design parameters of sys-
tem elements as well as their resource issues. All these factors form the basis for
technological processes formation in manufacturing. For example, nowadays the
ratio of the technical defect in gas turbine engine blades manufacturing is around
5%, but it can reach 20% when the blades are checked under their workload fre-
quency. The quality of computation has a direct effect on the manufacturing and
operation quality of elements and whole systems.
Another problem is related to manufacturing tolerances that are a part of the
design technology. Initially, the tolerances are defined, and then the manufactur-
ing technology is designed to meet them. The technological manufacturing toler-
ances selection scheme is presented in Fig. 1. The technological tolerances of sys-
tem elements manufacturing are defined by the given values of confidence
intervals of design parameters, the junction type, the seating fit, the processing
equipment accuracy rating, and the assembly type. The confidence intervals of
design parameters are formed as the result of engineering dimension chains calcu-
lation.
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There are some methods for determination of confidence intervals of pa-
rameters, state variables, and decision selection criteria (objective functions) of
system elements based on engineering dimensional chains calculation. These
methods include methods based on limiting values, methods based on intervals
calculation, and probability methods.
The closing link tolerance is a separable function if the assembly of compo-
nents takes place. In general, the closing link tolerance is a nonlinear function of
its variables – chain links components tolerances.
A decrease of risks related to the high cost of complex technical systems re-
finement in serial production is possible due to the adoption of the Robust Estima-
tion methods based on the matching learning algorithms. The Robust Estimation
has the following issues that need to be addressed: the problem of improvement of
existing mathematical models, and the development of new mathematical models
and methods for solving stochastic optimization multi-criteria problems
(SOMCP).
It is known that random data triggers the uncertainties during the choice of
the metrics in objective functions and estimation of target values (parameters,
control variables, or state variables) during SOMCP structuring. Besides, the reg-
ularizing algorithms must be used for such type of ill-posed problem solving. This
provides the stable (robust) estimates of target values. At the same time, the
mathematical models will have robust quality, if they are synthesized with the use
of regularizing algorithms.
Up to now, many papers have been devoted to the development of the meth-
ods, including the methods for objective functions and estimation of target values
under conditions of data parametric uncertainties; multi-criteria identification of
mathematical models; optimization and decision-making in the robust designing,
improving and intellectual diagnostics of technical and medical-biological sys-
tems.
The model of optimization under conditions of uncertain multidisciplinary
design, which is described by L. Brevault et al. [1], is aimed at the organization of
interdisciplinary connections under uncertainties. The suggested method is based
on two levels of optimization and surrogate models to provide the convergence of
the optimization problem of the functional ratio of multidisciplinary system con-
nections. The solution search algorithm is based on the iterative construction of
the functional connections’ surrogate models. Coefficients are processed by a sys-
tem-level optimizer, while subsystem optimizers process local design variables
only.
Proceeding
equipment
accuracy rating
Junction
type
Seating
fit
Confidence
interval
Technological
tolerance selection
Assembly type
Fig. 1. Scheme of tolerance assignment
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 73
The model selection is a fundamental problem, which is widely discussed
during the process of data sets analysis. The hierarchical models with
uncertainties may not have a solution when classical approaches are used. Bayes
approaches could be applied with predictive distribution usage, but they have
disadvantages in certain cases.
Other approaches are also described in the literature: predicting the
replication of observed data [2]; the theory of estimation rules on common
probability spaces and cross-validation [3]; the probabilistic process described by
the Kriging surrogate model with Monte Carlo uncertainty modeling in
conjunction with the descriptive sampling method [4].
The solution of design and optimization problems is called stable (robust) if
it is resistant to the disturbance of input parameters of the model. The design
engineer may prefer to use a robust solution for engineering design optimization
problems rather than the optimal one to provide system robustness.
Various approaches to this problem exist, including multi-purpose
optimization based on a generalization of the class of functions, which allows
conducting orientation of the search area in the object space [5]; combined criteria
for stochastic optimization [6]; optimization under uncertainties using parallel
computing capabilities in various formulations of problems [7]; search for an
allowable range in the input parameters, given an acceptable range in the output
quantities [8]; a method for quantifying multidimensional interval uncertainties
[9]; interdisciplinary automated process optimization based on an optimization
strategy designed to overcome various limitations [10].
The following locally stochastic methods (including those, which are based
on self-organization) are used as computational methods for solutions synthesis in
stochastic optimization problems:
stochastic quasi-gradient algorithms;
evolutionary (genetic, immune) algorithms;
population methods (simulation of motion: migratory birds; ant or bee
colonies).
The overview of evolutionary and computational methods, which can be
successfully used for stochastic optimization problem solving, is presented in
A.P Karpenko monograph [11] and as publications by other authors [12–14].
I. Meniailov et al. proposed a computational method of solutions synthesis of
system modification multi-criteria problems in deterministic and stochastic (MV-
problems) formulations, which is based on a memetic algorithm [15]. This
method combines the method of the convergent neighborhood; the randomized
path relinking method; and the evolutionary method with parameters, which are
changing from epoch to epoch. These parameters are the real coding operators,
fitness, and relaxation functions. This approach provides an effective robust
estimation of target values when input data are a priori and parametric
uncertainties ones. Also, this approach reduces the informational complexity of
the method.
Analysis of existing literature shows that some mathematical issues appear
in the process of the development of solutions synthesis methods of multi-criteria
problems of mathematical model identification, optimization, and decision-
making, especially in the cases of the a priori uncertain data. These issues include:
I. Trofymova, I. Meniailov, S. Сhernysh, S. Yepifanov etc. ...
ISSN 1681–6048 System Research & Information Technologies, 2020, № 4 74
decision maker (DM) preferences system forming, i.e. forming of the
generalized (the scalar convolutions) objective functions, the limitations system,
and the correctness set;
structuring of the regularizing algorithms of quasi-solutions synthesis;
the high computational complexity of the defined methods.
Only a few interactive computer systems for decision-making support
(CSDMS), which provide robust optimal design feature, exist in the world. These
include “Dakota, A Multilevel Parallel Object-Oriented Framework for Design
Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity
Analysis” [16], “IOSO Technology, Robust design optimization” [17],
“ESTECO, modeFRONTIER” [18, 19], “Dassault Systems, Isight and Fiper”
[20], “DYNARDO, optiSLang” [21], “NUMECA International, FineDesign3D”
[22], “Concepts NREC’s, Agile Engineering Design System” [23], “AxSTREAM
Software” [24], “Propulsion Diagnostic Method Evaluation Strategy
(ProDiMES)” [25], and others.
These systems have the following disadvantages: high cost, inability to re-
solve stochastic optimization multi-criteria problems in MV-formulation (i.e. in-
verse and direct problems in engineering dimensional chains calculation).
This paper is dedicated to new methodology of non-linear robust estimation
for the solutions synthesis of inverse and direct engineering dimensional chains
calculation problems under the conditions of a priori and parametric uncertainties.
It is shown that this problem can be transformed into multi-criteria problems of
stochastic optimization with mixed conditions (to MV-problem). The suggested
methodology allows us to search for rational solutions of system modification
multi-criteria problems [26, 27] presented in deterministic and stochastic (MV-
problem) formulations. This is done through the development of a hierarchical
two-level decisions synthesis scheme, which includes:
inverse determination of the unknown equations governing the variation
of measured field quantities of given physical problem – shape identification of
robust meta-models or surrogate models (formal mathematical models in the form
of regression equations);
inverse determination of size(s) and shape(s) of the domain – effective
robust sought values estimations of unknown quantities are carried out under a
priori and parametric data uncertainties.
Results of an inverse and direct problem in engineering dimensional chains
calculation for Two-stage axial compressors are resolved by CSDMS
“ROD&IDS”. The result is presented below.
PROBLEM STATEMENT
Problem statement and a method for selection of functions and estimation
of unknown variables in multi-objective problems with the a priori uncer-
tainties data
Let us consider the problem statement, methodology and results of the solution of
Direct Multidisciplinary Problems in Engineering Dimensional Chains Calcula-
tion under Uncertainties in nonlinear statement (multi-criteria problems of sto-
chastic optimization with mixed conditions — MV-problems).
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 75
The following data are known, presented in a formalized form: structure,
functioning model (mathematical model, boundary conditions), properties and
general characteristics of the research object, basic requirements for its tactical,
technical and economic criteria; the class of admissible controls (methods and
devices implementing them).
Let it be known a set of alternatives – learning selection that contains values
and confidence intervals of subsystem (functional unit) parameters and control
variables. Additionally, the values of mathematical expectations and confidence
intervals of decision criteria (objective functions) values are known or phase vari-
ables of the system (subsystem) or process under consideration are given. These
values are set either by the decision-maker based on his experience or directives
or are known based on the results obtained using the measurement system.
So, it is necessary to determine the set of mathematical expectations and
confidence intervals of subsystem (functional unit) control variables (model pa-
rameters – in the case of an identification problem) and the corresponding values
of mathematical expectations and confidence intervals of decision criteria (objec-
tive functions) values or phase variables of the system (subsystem) or process un-
der consideration. At the same time, the last found values should be close to the
given values of objective functions or phase variables (mathematical expectations
and confidence intervals) for the selected metrics.
The Direct Multidisciplinary Problems in Engineering Dimensional Chains
Calculation under Uncertainties in nonlinear statement refer to inverse boundary
value problems in a stochastic formulation with restrictions on design and operat-
ing parameters, phase and control variables.
We will consider the solutions synthesis of the problem as a control process
that is based on the concept of invariant control by introducing into the computing
system a compensating connection with perturbations (input data and computing
errors).
Consider 0X as a vector of random variables of M dimensions (the model
parameters, control variables, state variables); and 0F as a vector of random
variables of I dimensions (measurement data, objective functions). The values
0F can be found using the initial mathematical model (IMM) of the investigation
subject reflected in the form of )( 00 XFF , where F is a vector function.
Let us define the projections of 0X and 0F as the random variables
following a normal distribution with given mathematical expectations, standard
deviations, and correlation matrices. This allows considering 0X and 0F as
systems of several random variables with the multidimensional normal
distribution.
Following Kolmogorov’s concept of power averages [28], we will use
Student’s statistics as a criterion for testing the equality of centers of distributions
hypothesis for representative samples of two multidimensional general
populations; and we will use the multidimensional analog of Romanovsky
criterion [29–31] for testing the equality hypothesis of covariance matrices Ro :
2
2
MD
n
t ,
I. Trofymova, I. Meniailov, S. Сhernysh, S. Yepifanov etc. ...
ISSN 1681–6048 System Research & Information Technologies, 2020, № 4 76
where n is the dimension of samples from the general populations; MD is Ma-
halanobis distance.
k
k
Ro
2
|| 2
, 3 nk ,
where 002 )( R
N
n T is the multidimensional analog of Pearson’s chi-
squared test; N is a dimension of 0X (or 0F );
*
0
n
n , Nn ..1 ;
*, nn are standard deviations of variables 0Xxn (* – given values); R is a
correlation matrix.
Let us define the log-likelihood function. The final form of the scalar convo-
lution for decision-making problems with (1–2) looks like:
LXXFFFF CRotRotRotXL )(
2
1
),/ˆ( 22 .
Scalar convolutions for the multi-objective problems with the a priori
uncertainties data
Further, let us assume that, ERR FX , where XR and FR are the correlation
matrices. In this case, the following scalar convolution of objective functions in
MV-problems is applied:
I
i
f
fitff
i
f
iifit
k
k
f
f
ff
I
E i
i
i
1
2
20
2
* 2
||
)1()(
2
1
M
m
xm
fitxxm
m
xm
mmfit
k
k
f
x
xf
M 1
2
20
2
* 2
||
)1()(
2
1
,
where
*][ iif ffM
i
,
2*
2
2
)(
]])[[(
i
i
f
ii
f
fMfM
n
;
*
i
i
i
f
f
f
;
nn
n
k
k
i
i
f
f 3
1)(
)3(22
2
2
;
*][ mmxm xxM ,
2*
2
2
)(
]])[[(
m
mm
xm
xMxM
n
;
*
xm
xm
xm
;
nn
n
k
k
xm
xm 3
1)(
)3(22
2
2
,
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 77
*
mx , *
m are the values of mathematical expectation and standard deviation of mx
variable for the prototype; xm is standard deviation value of 0Xxm variable;
*
if , *
if
are the values of mathematical expectation and standard deviation of if
decision selection criterion for the prototype;
if
is standard deviation value of
0Ffi decision selection criterion; fitf is a fitness function (FF);
)(exp1)( Cddf fit , 0C (it has been selected based on the condition that
initial value of )1(
avE was 1)1( avE ), where d is an argument of FF ( 0d );
)( ii f , )( mm x are membership functions; is a parameter of regularization
(if 0 this is еру identification, else if 1 this is optimization); xf , are
parameters of robustness.
Hence, the problem of )],[(ˆ 00
XXMX non-linear robust estimation can
be transformed to SOMCP with mixed conditions (to MV-problem in our case).
According to the principle of maximum likelihood [32] (M-estimate), the quasi-
solution of this problem is the following:
),/ˆ(infargˆ
ˆ
FF
DX
RotXEX
X
,
where XD is a correctness set, which is defined by the decision maker’s system
of preferences in the general case. It was assumed in this case that XD is a con-
vex set.
THE NEW METHODS NON-LINEAR ROBUST ESTIMATION FOR
THE SOLUTIONS SYNTHESIS OF INVERSE AND DIRECT
MULTIDISCIPLINARY PROBLEMS IN ENGINEERING DIMENSIONAL
CHAINS CALCULATION BASED ON DISCRETE ANALOG DATA
As a result of the decomposition of the methodology to solve the main problem
(3, 4), it decomposes into a sequence of interrelated methods, in particular:
Input data preparation: input data preliminary normalization methods. Da-
ta of alternatives (samplings) are used as input data: design values and operational
parameters, control and phase variables, decision selection criteria (objective
functions). The samplings are formed using either the solutions in the determinis-
tic formulation of the direct analysis problem or the results of discrete analog data
experimental research.
Methods of inverse determination of the unknown equations governing
the variation of measured field quantities of given physical problem – shape iden-
tification of robust meta-models or surrogate models (formal mathematical mod-
els in the form of regression equations). The methods of approximation of vector
functions of vector variables based on the application of a trainable artificial neu-
ral network (ANN), which are multilayer feedforward ones and radial-basis ANN.
The training of ANN is carried out by a stochastic approximation method based
on the conjugate gradients ravine method [33] together with the bee colonies
I. Trofymova, I. Meniailov, S. Сhernysh, S. Yepifanov etc. ...
ISSN 1681–6048 System Research & Information Technologies, 2020, № 4 78
simulation, in which the scalar convolution of objective functions (3) with 0
is used for rational decision selection. The proposed implementation allows us to
obtain effective stable (robust) estimates of the neural network model parameters
under the condition of input a priori and parametric uncertainties. It provides a
robust meta-models synthesis and the data approximation accuracy, which is suf-
ficient in system improvement problems. Application of the proposed methods
avoids the appearance of false ravines or valleys on response surfaces in case of
gross errors in the input data.
Graphical tools for the 3D-representation of meta-models.
Methods of meta-model variables informativeness (importance) estima-
tions (taking into account the pair correlation and the accuracy of variables meas-
urement) [34]. In particular, the solutions of controlled variables set synthesis for
systems designing or failure diagnostics under conditions of a priori and paramet-
ric uncertainties of input data may be obtained on the base of the received results.
Methods for solving the problem of systems and processes state classifi-
cation on the base of multilayer feedforward and radial-basis ANN application
with the usage of the monitoring data of controlled variables [35].
Solutions synthesis of system modification multi-criteria problems in de-
terministic and stochastic (MV-problem) formulations [15, 33].
The solution of the inverse problem of engineering dimensional chains cal-
culation/Monte Carlo analysis (MCA) is found for the prototype at the first stage:
the estimation of confidence intervals of decision selection criteria (objective
functions) mean values at the given confidence intervals of subsystems control
variables (functional elements) mean values.
The quasi-solutions synthesis of this problem is carried out by regularization
of the smoothing functional minimum search in the form of objective functions
scalar convolution (1) with. The comparative analysis of various systems produc-
tion technologies may be carried out on the base of the obtained results [15].
Solutions synthesis of the direct problem of engineering dimensional chains
calculation – system modification problem quasi-solutions in deterministic and
stochastic (MV-problem) formulations is carried out by the regularization method
at the second stage. The objective functions scalar convolution (1) was used as the
smoothing function [15]. The computational method based on meme’s algorithm
is applied. It includes the parameters changing from epoch to epoch such as the
operators of real coding, the fitness and relaxation functions, and also the method
of the convergent neighborhood and the randomized path relinking method.
This approach provides an effective robust estimation of target values when
input data are a priori and parametric uncertainties ones. Also, this approach re-
duces the informational complexity of the quasi-solutions synthesis method.
Thus, the direct problem solution of engineering dimensional chains calcula-
tion is solved by the probabilistic method (2). Mathematical expectations and con-
fidence intervals of subsystems control variables (functional elements) are found
according to the given values of mathematical expectations and the decision se-
lection criteria confidence intervals (of the objective functions). Also, they could
be found according to phase variables of the systems (subsystems).
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 79
NEW MEMETIC ALGORITHM OF THE STOCHASTIC OPTIMIZATION
PROBLEM WITH MIXED CONDITIONS
The quasi-solution of this problem (normal solution) may be find by the regulari-
zation method [32]. Quasi-solutions synthesis of system modification multi-
criteria problems in deterministic and stochastic (MV-problem) formulations is
carried out by the computational method based on memes algorithm.
Let us define an evolutionary method (EM) as a classical genetic algorithm
(GA) modification with the parameters, which are changing from epoch to epoch.
This paper presents features which distinguish the suggested EM from classi-
cal GA.
The real coded crossover operator that simulates the binary one is used [11].
Mikhalevich non-uniform mutation operator is used as the real coded mutation
operator, which relates to a class of nonstationary mutators.
After crossover operations and mutation, the most adapted individual is se-
lected and put to the set of individuals intended for the next epoch of the algo-
rithm. The most adapted individual is the individual, the selection criteria scalar
convolution values of which are the most acceptable.
It is known that clustering is one of the means for the EM convergence rate
to increase. Decremental Neighborhood Method (DNM), which realizes the idea
of clustering [15], has been developed to improve the convergence rate and accu-
racy of the extremum finding.
The suggested memetic algorithm is several times less complex in terms of
the information and time complexities compared to the classical GA, because of
combining parameters changing from epoch to epoch such as the operators of real
coding, the fitness and relaxation functions and also the method of the convergent
neighborhood and the randomized path relinking method.
NUMERICAL TESTS
The “ROD&IDS” interactive computer system for decision-making support was
developed. It implements the foregoing methodology.
As an example, let us consider the results of an inverse and direct problem in
engineering dimensional chains calculation for a two-stage axial compressor un-
der conditions when input data is stochastic.
The aerodynamic design of the compressor two axial stages was fulfilled for
gas-turbine drive power increasing from 8 to 10 MW. The “Axial” software
(Copyright © 1998-2017. Concepts NREC LLC) [23] was used during the stages
calculations basing on the average radius parameters. The problems of these
stages efficiency increasing (without gas-dynamic stability reducing) at the design
values of inlet mass flow rate and rotational speed was solved by the application
of “ROD&IDS”.
The compressor efficiency maximum search was carried out with following
elements changes of blades rows of the first and second compressor stages: the
gas path tip radius (6 variables), stagger angles (4 variables), entry and exit geo-
metrical angles (8 variables) and the cascade solidity (4 variables). For the inlet
guide vanes, only the gas path was changed. Experimental sampling was formed
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ISSN 1681–6048 System Research & Information Technologies, 2020, № 4 80
by the change of the gas path radius (within ±2 mm), geometrical angles of blades
airfoils (within ±2 degrees), and the blades row chord/pitch ratios (within ±5 %).
Approximation of the lines of constant referred speed, beginning from surge line
to choke measure is used to adequately describe the effect of varying parameters
on the stages map. The computation of inlet mass flow rate values between surge
and choke was performed by “Axial” macros. This way, the following quantity of
state variables under control was selected: 22 geometric parameters, 1 regime pa-
rameter (an inlet mass flow rate), and 450 points characterizing the experimental
sampling size.
The following parameters were selected as objective functions:
— out is flow outlet angle;
— * is compressor total pressure ratio at the design inlet mass flow rate
value;
— * is adiabatic efficiency calculated according to total parameters values
at the design inlet mass flow rate value;
— *
stall is total pressure ratio on surge line;
— chokeG is the choke inlet mass flow rate value;
— form-parameter is the parameter of response surface form corresponding
to the objective function selected at the design inlet mass flow rate value.
Further, the robust neural network models in the form of radial-basis ANNs
were used as meta-models of the systems.
Optimization was carried out based on the design inlet mass flow rate values
in deterministic and stochastic formulations.
Computational results in the deterministic formulation are presented in Fig. 2
and Table 1, where normalized values are: 0,
0 / pGGG , 0,
0 / р ,
0,
0 / р , 0,р is the efficiency change at the design inlet mass
flow rate value in comparison with the prototype, and уK is gas-dynamic stabil-
ity margin.
The first deterministic formulation of the optimization problem is selected in
such a way that the efficiency maximum at the design inlet mass flow rate value is
provided along with the required level of total pressure ratio. As a result, the effi-
ciency increment by 0,41 % in comparison with the prototype was achieved at the
design inlet mass flow rate value (MC_2_ (v.2.1) results). The efficiency maxi-
mum along the line of constant referred speed is close to the design point. At the
same time, the gas-dynamic stability margin increased from 12,6 % to 14,44 %.
The second deterministic formulation of the optimization problem (MC_2_
(v.4.4) results) is implemented to examine how much it is possible to vary the
limitations in the optimization problem. For example, let us consider the problem
of obtaining a greater efficiency maximum with a more steep behavior of line of
constant referred speed in comparison with the prototype. As shown in Fig. 2, the
line of constant referred speed with more steep behavior provides a higher level of
gas-dynamic stability margin (13,8% in comparison with prototype’s 12,16%) and
high maximal efficiency value is provided too. The difference of this optimization
from previous decisions lies in the fact that the compressor efficiency is 1,45%
less than the prototype’s one at the design inlet mass flow rate value.
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 81
Thus, according to the provided example, it was shown that obtaining such
blade rows geometry is possible. This solution can meet the requirements of the
compressor performances made by a design engineer.
As an example, the calculations intended to provide efficiency maximum
value at the designed inlet mass flow rate and efficiency maximum along the line
of constant referred speed were performed. These calculations meet the required
level of the total pressure ratio (the same as the prototype’s one). Implementation
of additional limitations to the objective functions gives the possibility to get
more or less steep behavior of the two-stage compressor line of constant referred
speed in comparison with the prototype.
Let us consider the manufacturing of a product series consisting of 100
copies. We chose a prototype, optimized it in a traditional deterministic
formulation. We enhanced of the efficiency for compressor optimal version at
deterministic formulation 406,0
d % in comparison with the prototype (see
in Table 1).
a
b
Fig. 2. Performances of the prototype and compressors optimal versions: а — ],[ 00 G
relation; b — ],[ 00 relation
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ISSN 1681–6048 System Research & Information Technologies, 2020, № 4 82
T a b l e 1 . Comparison of objective functions values for compressors versions
Values/Variants d
,% *
max ,% stallK ,%
Prototype 0 0,494 12,157
MC_2_(v.2.1)/ deterministic formulation 0,406 0,449 14,444
MC_2(v.4.4)/ deterministic formulation -1,448 0,756 13,80
Further, taking as a basis the calculated parameters of this optimal version,
we will make 100 copies of products. To conclude that the products comply with
the technical specifications, we will evaluate the results of numerical modeling of
work of 100 copies of products in accordance with the methodology presented in
the article.
Mathematical expectation, confidence intervals of regime parameters mean
values, and control variables, which are the prototype data, are known (the first
row in Tables 2 and 3). In Table 2 – 100
max,
0
m
m
mx x
x
is the accuracy class of
regime parameters and control variables, max,mx parameter is the maximum value
in the learning sample.
The stochastic formulation of the optimization problem is selected in such a
way that the maximal level of efficiency mathematical expectation at the design
inlet mass flow rate value and at the required level of objective functions mathe-
matical expectations and confidence intervals would be provided in the product
line.
Solving the inverse problem of engineering dimensional chains calcula-
tion/Monte Carlo analysis (MCA) was carried out for the prototype on the first
stage. In other words, estimation of confidence intervals of decision selection cri-
teria (objective functions) mean values were carried out under given confidence
intervals of subsystems control variables mean values (functional elements).
MCA results are presented in Tables 3 and 4. Table 3 shows that the confidence
intervals of objective functions mean values increased in comparison with as-
sumed measurement precision. At the same time, the values of mathematical ex-
pectations of objective functions were reduced for the considered compressor ver-
sions (see Table 4). These MCA results are typical for the results obtained in the
deterministic statement.
System modification quasi-solutions synthesis in the stochastic formulation
(MV-problem) is carried out on the second stage in order searching the mathe-
matical expectations and confidence intervals of subsystem (functional unit) con-
trol variables and the corresponding values of mathematical expectations and con-
fidence intervals of decision criteria (objective functions) values or phase
variables of the system (subsystem) or process under consideration at the design
inlet mass flow rate value that is close to the given values for the selected metrics.
Overall results of decisions synthesis of modification problems are presented
in the stochastic formulation, in other words, the confidence intervals of regime
parameters and control variables mean values are presented in Tables 2–4. A
comparison of mathematical expectations of objective function values is pre-
sented in Tables 2–4 too.
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 83
T a b l e 2 . Confidence intervals of regime parameters and control variables
mean values
Values/variants
0 G ,
%
0 of gas path
duct radii on the
periphery, %
0 of blades rows
geometrical angles,
%
0 of the
cascade
solidity, %
Prototype/given values 0,05 5,0 2,5 1,0
MC_2_(v.2.1)/deterministic
formulation/given values
0,05 5,0 2,5 1,0
MC_2_(v.2.1)/stochastic
formulation/calculation
results
0,04 4,0 2,0 0,8
T a b l e 3 . Confidence intervals of objective functions mean values
Values/
variants
Δ out ,
degrees
d d ,% Δ *
stall
Δ chokeG ,
kg/s
Δ Form-
parameter
Precision
of Measurements
0,2367 0,0121 0,47 0,0122 0,2680 0,0004
Prototype/
Monte Carlo analysis
0,0882 0,0180 0,58 0,0053 0,0575 0,0068
MC_2_(v.2.1)/
deterministic
formulation/
Monte Carlo analysis
0,1626 0,0660 2,49 0,0049 0,0547 0,0216
MC_2_(v.2.1)/
stochastic
formulation/desired
values
0,0900 0,0120 0,50 0,0050 0,0600 0,0070
MC_2_(v.2.1)/
stochastic
formulation/
calculation results
0,0830 0,0126 0,41 0,0055 0,0508 0,0053
Table 4. Comparison of mathematical expectations of objective functions values
for compressors versions
Values/versions out ,
degrees
d d , % stall chokeG ,
kg/s
Prototype/Monte Carlo analysis -0,03 -0,015 -0,58 -0,011 0,009
MC_2_(v.2.1)/deterministic
formulation/Monte Carlo analysis
-0,20 -0,083 -2,49 0,019 0,053
MC_2_(v.2.1)/stochastic
formulation/calculation results
0,02 0,016 0,24 -0,009 0,603
The result presented in Table 4 shows that the efficiency mathematical ex-
pectation value d for compressor optimal version at deterministic formulation
decreased by (2,49-0,58 )% = 1,91% in comparison with the prototype data ob-
tained by MCA method. The result presented in Table 3 shows that the efficiency
confidence interval value for compressor optimal version at deterministic formu-
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ISSN 1681–6048 System Research & Information Technologies, 2020, № 4 84
lation enhanced by (2,49-0,58)% = 1,91% in comparison with the prototype data
obtained by MCA method.
This is the main result of the research, confirming what said earlier that a
good exact solution of the inverse problem in a deterministic formulation (optimi-
zation problem) in practice in mass production, as a rule, leads to a large scatter of
the values of the integral characteristics of products.
The result presented in Table 4 shows that the efficiency mathematical ex-
pectation value d for compressor optimal version at stochastic formulation is
enhanced by (0,24+0,58)% = 0,82% in comparison with prototype. It should be
noted, that at the same time saved desired confidence intervals average values of
objective functions (see the last row in Table 3). But this could be achieved by
20% enhancement of compressor elements manufacturing accuracy (see the last
row in Table 2).
Also, there is a significant difference in the mathematical expectations’
values of the control variables, which were obtained by problem-solving in
deterministic and stochastic formulations.
Here is a justification for the need implementation of the concept and devel-
oped by us methodology of robust estimation based on the synthesis of robust sur-
rogate models and solutions to multidisciplinary MV-problems in mass manufac-
turing of products. The use of our developments in mass manufacturing of
products will reduce the percentage of rejects.
The sought quantities estimations in the result of solving of modification
problem in the stochastic formulation are the effective and stable (robust) ones.
CONCLUSIONS
The new methodology of non-linear robust estimation for the solutions synthesis
of inverse and direct multidisciplinary problems in engineering dimensional
chains calculation under conditions of a priori and parametric uncertainties was
developed. It is shown that this problem as a inverse boundary value problems in
a stochastic formulation can be transformed into multi-criteria problems of sto-
chastic optimization with mixed conditions (MV-problems). The methodology
allows us to obtain the rational solutions of system modification multi-criteria
problems in deterministic and stochastic (MV-problem) formulations. It can be
achieved through hierarchical two-level decisions synthesis scheme development,
which includes:
inverse determination of the unknown equations governing the variation
of measured field quantities of given physical problem – shape identification of
robust meta-models or surrogate models (formal mathematical models in the form
of regression equations);
inverse determination of size(s) and shape(s) of the domain – effective
robust sought values estimations of unknown quantities are carried out under a
priori and parametric data uncertainties.
The results of solving inverse and direct problems in engineering dimen-
sional chains calculation under the conditions where input data are stochastic for
two-stage axial compressors were obtained with the usage of “ROD&IDS” and
provided as an example.
Methodology of non-linear robust estimation for the solutions synthesis of …
Системні дослідження та інформаційні технології, 2020, № 4 85
Based on the analysis of numerical simulation results shown, that a good
exact solution of the inverse problem in a deterministic formulation (optimization
problem) in practice in mass production, as a rule, leads to a large scatter of the
values of the integral characteristics of products.
Application in practice of the concept and developed by us methodology of
robust estimations, based on the synthesis of robust surrogate models and solu-
tions to multidisciplinary MV-problems, in mass manufacturing of products will
reduce the percentage of rejects.
The developed interactive computer systems for decision-making support
“ROD&IDS” can be applied to various fields of science, medicine, and technology.
Particularly, this software may be used by the wide range of specialists, who work
on the issues of robust meta-models’ design (formal mathematical models in the
form of regression equations), as well as on the issues of optimization and deci-
sion-making in the process of robust design, improving and intellectual diagnos-
tics of technical and medical-biological systems, ecology, the activities of banks,
insurance, and audit companies etc.
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____________________________
From the Editorial Board: the article corresponds completely to submitted manuscript.
INFORMATION ON THE ARTICLE
Iryna O. Trofymova, ORCID: 0000-0002-1537-5601, National Aerospace University
“Kharkiv Aviation Institute”, Ukraine, e-mail: i.trofymova@khai.edu
Ievgen S. Meniailov, ORCID: 0000-0002-9440-8378, National Aerospace University
“Kharkiv Aviation Institute”, Ukraine, e-mail: j.menyailov@khai.edu
Serhii V. Chernysh, ORCID: 0000-0002-1750-5158, National Aerospace University
“Kharkiv Aviation Institute”, Ukraine, e-mail: mr.serhii.chernysh@gmail.com
Sergiy V. Yepifanov, ORCID: 0000-0003-0533-9524, National Aerospace University
“Kharkiv Aviation Institute”, Ukraine, e-mail: s.yepifanov@khai.edu
Olexandr M. Khustochka, Scientific Research and Design ComplexState Enterprise
Zaporizhzhia Machine-Building Design Bureau named after academician A.G. Ivchenko,
Ukraine, e-mail: 03527@ivchenko-progress.com
Mykhaylo L. Ugryumov, ORCID: 0000-0003-0902-2735, V.N. Karazin Kharkiv Na-
tional University, Ukraine, e-mail: m.ugryumov@karazin.ua
Andriy V. Myenyaylov, ORCID: 0000-0003-1407-1224, Commercial Aircraft Engine
Co., Ltd, China, e-mail: myenyaylov@ukr.net
Dmytro I. Chumachenko, ORCID: 0000-0003-2623-3294, National Aerospace Univer-
sity “Kharkiv Aviation Institute”, Ukraine, e-mail: d.chumachenko@khai.edu
I. Trofymova, I. Meniailov, S. Сhernysh, S. Yepifanov etc. ...
ISSN 1681–6048 System Research & Information Technologies, 2020, № 4 88
МЕТОДОЛОГІЯ НЕЛІНІЙНОГО РОБАСТНОГО ОЦІНЮВАННЯ ДЛЯ
СИНТЕЗУ РОЗВ’ЯЗКІВ ОБЕРНЕНИХ ТА ПРЯМИХ БАГАТОДИС-
ЦИПЛІНАРНИХ ЗАДАЧ У РОЗРАХУНКУ ІНЖЕНЕРНИХ РОЗМІРНИХ
ЛАНЦЮГІВ НА ОСНОВІ ДИСКРЕТНИХ ДАНИХ ПРО АНАЛОГИ / І.О. Тро-
фимова, Є.С. Меняйлов, С.В. Черниш, С.В. Єпіфанов, О.М. Хусточка, М.Л. Угрю-
мов, А.В. Мєняйлов, Д.І. Чумаченко
Анотація. Подано постановки обернених і прямих задач розрахунку інженер-
них розмірних ланцюгів на основі дискретних даних про аналоги і методології
розв’язання цих задач. Показано, що задачі розрахунку прямих розмірних лан-
цюгів належать до класу обернених крайових задач у стохастичній постановці,
які можна звести до багатокритеріальних задач стохастичної оптимізації зі
змішаними умовами. Нова багатоетапна методологія пошуку розв’язків таких
задач ґрунтується на методах нелінійного робастного оцінювання. Це може
бути досягнуто шляхом розроблення ієрархічної дворівневої схеми синтезу
розв’язків. На першому етапі схема включає ідентифікацію сурогатних моде-
лей (у вигляді рівнянь регресії). Другий етап — визначення ефективних робас-
тних оцінок шуканих величин; невідомі величини оцінюються за апріорної і
параметричної невизначеностей даних. Наведено результати розрахунків обер-
нених і прямих задач проектування розмірних ланцюгів для двоступеневих
осьових компресорів. Їх отримано за допомогою інтерактивної комп’ютерної
системи підтримання прийняття рішень «ROD & IDS».
Ключові слова: обернені крайові задачі в стохастичній постановці, апріорні і
параметричні невизначеності, методи і системи оцінювання величин і проце-
сів, теорія прийняття рішень.
МЕТОДОЛОГИЯ НЕЛИНЕЙНОГО РОБАСТНОГО ОЦЕНИВАНИЯ ДЛЯ
СИНТЕЗА РЕШЕНИЙ ОБРАТНЫХ И ПРЯМЫХ МНОГОДИСЦИП-
ЛИНАРНЫХ ЗАДАЧ РАСЧЕТА ИНЖЕНЕРНЫХ РАЗМЕРНЫХ ЦЕПЕЙ
НА ОСНОВЕ ДИСКРЕТНЫХ ДАННЫХ ОБ АНАЛОГАХ / И.А. Трофимова,
Е.С. Меняйлов, С.В. Черныш, С.В. Епифанов, А.Н. Хусточка, М.Л. Угрюмов,
А.В. Меняйлов, Д.И. Чумаченко
Аннотация. Представлены постановки обратной и прямой задач расчета ин-
женерных размерных цепей на основе дискретных данных об аналогах и мето-
дологии решения этих задач. Показано, что задачи расчета прямых размерных
цепей относятся к классу обратных краевых задач в стохастической постанов-
ке, которые можно свести к многокритериальным задачам стохастической оп-
тимизации со смешанными условиями. Новая многоэтапная методология по-
иска решений таких задач основана на методах нелинейного робастного
оценивания. Это может быть достигнуто путем разработки иерархической
двухуровневой схемы синтеза решений. На первом этапе схема включает
идентификацию суррогатных моделей (в виде уравнений регрессии). Второй
этап — определение эффективных робастных оценок искомых величин; неиз-
вестные величины оцениваються при априорной и параметрической неопреде-
ленностях данных. Приведены результаты расчетов обратной и прямой задач
проектирования размерных цепей для двухступенчатых осевых компрессоров.
Они были получены с помощью интерактивной компьютерной системы под-
держки принятия решений «ROD&IDS».
Ключевые слова: обратные краевые задачи в стохастической постановке, ап-
риорные и параметрические неопределенности, методы и системы оценивания
величин и процессов, теория принятия решений.
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| id | journaliasakpiua-article-218011 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:26:58Z |
| publishDate | 2020 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/ca/a93e5fcb4d66a5e0420be79d6f6f42ca.pdf |
| spelling | journaliasakpiua-article-2180112021-04-08T14:17:06Z Methodology of non-linear robust estimation for the solutions synthesis of inverse and direct multidisciplinary problems in engineering dimensional chains calculation based on discrete analog data Методология нелинейного робастного оценивания для синтеза решений обратных и прямых многодисциплинарных задач расчета инженерных размерных цепей на основе дискретных данных об аналогах Методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги Trofymova, Iryna Meniailov, Ievgen Chernysh, Serhii Yepifanov, Sergiy Khustochka, Olexandr Ugryumov, Mykhaylo Myenyaylov, Andriy Chumachenko, Dmytro обернені крайові задачі в стохастичній постановці апріорні і параметричні невизначеності методи і системи оцінювання величин і процесів теорія прийняття рішень inverse boundary value problems in a stochastic formulation a priori and parametric uncertainties methods and systems for estimating quantities and processes decision-making theory обратные краевые задачи в стохастической постановке априорные и параметрические неопределенности методы и системы оценивания величин и процессов теория принятия решений This paper analyses the definition of inverse and direct problems in engineering dimensional chains calculation based on discrete analogue data and the methodologies for solving these problems. It is shown that the direct dimensional chains calculation, which belongs to the class of inverse boundary value problems in a stochastic formulation, can be transformed into multi-criteria problems of stochastic optimization with mixed conditions. The new multi-step solutions search methodology for these problems is based on non-linear robust estimation methods. It can be achieved through hierarchical two-level decisions synthesis scheme development. At the first step, this scheme includes identification of surrogate models (in the form of regression equations). At the second step, the effective robust estimates are computed to determine unknown values; estimations of unknown quantities are carried out under a priori and parametric data uncertainties. Results of calculations of inverse and direct problems in engineering dimensional chains for two-stage axial compressors are presented. They were obtained using interactive computer systems for decision-making support “ROD&IDS”. Представлены постановки обратной и прямой задач расчета инженерных размерных цепей на основе дискретных данных об аналогах и методологии решения этих задач. Показано, что задачи расчета прямых размерных цепей относятся к классу обратных краевых задач в стохастической постановке, которые можно свести к многокритериальным задачам стохастической оптимизации со смешанными условиями. Новая многоэтапная методология поиска решений таких задач основана на методах нелинейного робастного оценивания. Это может быть достигнуто путем разработки иерархической двухуровневой схемы синтеза решений. На первом этапе схема включает идентификацию суррогатных моделей (в виде уравнений регрессии). Второй этап — определение эффективных робастных оценок искомых величин; неизвестные величины оцениваються при априорной и параметрической неопределенностях данных. Приведены результаты расчетов обратной и прямой задач проектирования размерных цепей для двухступенчатых осевых компрессоров. Они были получены с помощью интерактивной компьютерной системы поддержки принятия решений "ROD&IDS". Подано постановки обернених і прямих задач розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги і методології розв’язання цих задач. Показано, що задачі розрахунку прямих розмірних ланцюгів належать до класу обернених крайових задач у стохастичній постановці, які можна звести до багатокритеріальних задач стохастичної оптимізації зі змішаними умовами. Нова багатоетапна методологія пошуку розв’язків таких задач ґрунтується на методах нелінійного робастного оцінювання. Це може бути досягнуто шляхом розроблення ієрархічної дворівневої схеми синтезу розв’язків. На першому етапі схема включає ідентифікацію сурогатних моделей (у вигляді рівнянь регресії). Другий етап — визначення ефективних робастних оцінок шуканих величин; невідомі величини оцінюються за апріорної і параметричної невизначеностей даних. Наведено результати розрахунків обернених і прямих задач проектування розмірних ланцюгів для двоступеневих осьових компресорів. Їх отримано за допомогою інтерактивної комп’ютерної системи підтримання прийняття рішень "ROD & IDS". The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2020-12-29 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/218011 10.20535/SRIT.2308-8893.2020.4.06 System research and information technologies; No. 4 (2020); 70-88 Системные исследования и информационные технологии; № 4 (2020); 70-88 Системні дослідження та інформаційні технології; № 4 (2020); 70-88 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/218011/227495 |
| spellingShingle | обернені крайові задачі в стохастичній постановці апріорні і параметричні невизначеності методи і системи оцінювання величин і процесів теорія прийняття рішень Trofymova, Iryna Meniailov, Ievgen Chernysh, Serhii Yepifanov, Sergiy Khustochka, Olexandr Ugryumov, Mykhaylo Myenyaylov, Andriy Chumachenko, Dmytro Методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги |
| title | Методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги |
| title_alt | Methodology of non-linear robust estimation for the solutions synthesis of inverse and direct multidisciplinary problems in engineering dimensional chains calculation based on discrete analog data Методология нелинейного робастного оценивания для синтеза решений обратных и прямых многодисциплинарных задач расчета инженерных размерных цепей на основе дискретных данных об аналогах |
| title_full | Методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги |
| title_fullStr | Методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги |
| title_full_unstemmed | Методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги |
| title_short | Методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги |
| title_sort | методологія нелінійного робастного оцінювання для синтезу розв’язків обернених та прямих багатодисциплінарних задач у розрахунку інженерних розмірних ланцюгів на основі дискретних даних про аналоги |
| topic | обернені крайові задачі в стохастичній постановці апріорні і параметричні невизначеності методи і системи оцінювання величин і процесів теорія прийняття рішень |
| topic_facet | обернені крайові задачі в стохастичній постановці апріорні і параметричні невизначеності методи і системи оцінювання величин і процесів теорія прийняття рішень inverse boundary value problems in a stochastic formulation a priori and parametric uncertainties methods and systems for estimating quantities and processes decision-making theory обратные краевые задачи в стохастической постановке априорные и параметрические неопределенности методы и системы оценивания величин и процессов теория принятия решений |
| url | https://journal.iasa.kpi.ua/article/view/218011 |
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