Modeling and Simulation of Stochastic Inverse Problems in Viscoplasticity
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Abstract
In this paper, we have proposed an approach for parameter identification of random field for rigid viscoplastic material. It is assumed that the random field is stationary and Gaussian with known autocovariance function. Karhunen–Loève decomposition has been used to quantify the effects of random inputs. A method is presented that takes into account only the first two statistical moments of the analyzed displacement field, and only two values of searched process are identified—mean value and coefficient of variation in autocovariation function. It has been shown that this approach is desirable when complicated systems are analyzed. The discretization of the governing equations has been described by the finite element method. The sparse grid stochastic collocation method has been used to solve the stochastic direct problem. It is shown that for the described nonlinear equations, the response function due to searched parameters with wide bounds and with reduced number of measurement points has many local extrema and global optimization technique is required. Genetic algorithm has been adopted to compute the functional cost. Numerical example shows the identification problem for compressed cylindrical sample. It is revealed that the key factor determining the convergence of the method is the degree of reduction in the height of the tested sample.
Keywords
Metal forming Collocation method Inverse problem Genetic algorithm Stochastic modeling1 Introduction
Deterministic metalforming problems have been considered in many publications [3, 18, 23, 34, 36]. In these problems, the parameters characterizing the material properties or boundary conditions are assumed to be known (achieved as the approximation of experimental studies). This is a good approach for a theoretical study of the problem, but it fails when the real object is analyzed. Experiments show that random fluctuations of geometrical and material parameters, or load values, significantly affect the behavior of the test object. For this reason, in metalforming processes, a stochastic approach has been used. Due to the considerable progress in computer technology and an associated significant increase in computing power, this approach, in recent years, has been widely used [1, 2, 17, 33].
The most common method of solving the aboveposed problem is the Monte Carlo method and the perturbation method [21, 22, 40]. Monte Carlo method is easy to implement and does not require changes in the existing code. Despite these advantages, it often requires a huge amount of sampling that the complex nonlinear problems can lead to many hours of calculations. Perturbation method is based on the extension of a classical deterministic finite element method or the finite difference method [16]. This method based on the Taylor series of expansion allows the calculation of secondorder statistics and is suitable for a small level of uncertainty, typically less than 10% for firstorder technique and less than 15% for secondorder [21].
In recent years, the stochastic spectral finite element method (SSFEM) and the stochastic collocation method (SCM) [5, 46] have gained a growing popularity. First one was proposed by Ghanem and Spanos [13]; subsequently, it was developed by other researchers [25, 26, 45] and applied to the problem of finite deformations [2]. Despite its high effectiveness and the possibility of its application to a number of problems, the use of the SSFEM may cause problems if governing equations are complicated and complex. In this case, it may be difficult to derive explicit forms of coefficients in the equations (large number of integration is needed). The equations in this method are coupled, but the domain decomposition method can be used to reduce the computation time [14, 41]. The idea behind the second method resembles the Monte Carlo method. The main difference is the definition of the sampling points defined as a tensor product of a set of onedimensional quadrature points associated with probability distribution function. This method was initially applied to the numerical integration problems with the large number of dimensions [12]. It gained its significance after the introduction of the sparse grid technique [6, 39], which contributed to the considerable reduction in the sampling points in the multidimensional random space. Due to the possibility of using the alreadyexisting source code of the deterministic program (used as the socalled black box), it has been used in the following paper. A similar approach to the analysis of the hyperelastic–viscoplastic large deformation problem has been proposed in paper [1].
Inverse problems, as ill posed, may be the cause of huge problems connected with the identification of the specified parameter in the deterministic approach, especially in cases of large sensitivity to small changes. The abovementioned reasons may cause significant errors in solving stochastic problems, by overlapping variety of errors: measurement errors, approach errors and a random perturbation of a searched parameter [11]. Classical methods of solving inverse problems, usually based on the sensitivity analysis, may be found in the numerous works [4, 8, 28, 30, 31, 42]. Solutions of stochastic inverse problems usually associated with heat conduction can be found in the work of Zabaras and coworkers [25, 47, 48] as well as associated with current flow [37]. In these works, to solve the inverse problem, the authors have used gradient methods. Problems of parameter identification in plastic forming of porous materials can be found in [38]. The abovementioned works assume the existence of a global minimum of response function. In general, except the global minimum, there can exist local minima that disqualify local search methods. Therefore, a nondeterministic method of the minimization of a functional achieved by the use of recently popular genetic algorithms [15, 19, 20, 24] has been proposed. This method is used to optimize problems for the deterministic plastic forming in works [9, 32].
2 Brief Description of the Direct Deterministic Problem
This chapter will first present basic equations governing the problem. These equations can be found in a number of publications on the plastic metalforming problem [18, 34].
2.1 Rigid Viscoplasticity
2.2 Finite Element Approximation^{1}
3 Stochastic Direct Problem
In the stochastic collocation method, the output stochastic process is formed on the basis of the solution of the deterministic problem. The input random process in a form of coefficients in governing equations or in boundary conditions is discretized. This method will be discussed in detail further in this article.
3.1 Method of Random Field Discretization
In this work, it is assumed that the considered random fields are Gaussian processes, or such processes, for which the random variable at each point \({\varvec{x}}\) has normal distribution, as for the material parameters is the most common. It is also assumed that the random fields are stationary in a wide sense [43] by which they satisfy the following assumptions: \(E\left( {\alpha ({\varvec{x}},\omega )^{2} } \right) < \infty\), \(E\left( {\alpha ({\varvec{x}},\omega )} \right) = {\text{const}}\), \({\text{Cov}}(\varvec{x}_{1} ,\varvec{x}_{2} ) = \left. {{\text{Cov}}({\varvec{\uptau}})} \right_{{{\varvec{\uptau}} = \varvec{x}_{1}  \varvec{x}_{2} }}\).
3.2 Sparse Grid Stochastic Collocation method (SGSCM)
In order to discuss in detail, the SGSCM method in the first place will present definitions necessary for the interpolation problems.
Let \({\varvec{\upxi}}\) represent a point in random space \(\Gamma \subset {\mathbb{R}}^{N}\), \(\Pi_{N}\) is a Ndimensional space of polynomials and \(\Pi_{N}^{p}\)—subspace of polynomials of degree p. Interpolation problem, as in [46], can be formulated as follows: For a set of nodes \(\Theta_{N} = \left\{ {{\varvec{\upxi}}_{i} } \right\}_{i = 1}^{N}\) in Ndimensional random space \(\Gamma\) and for a smooth function \(f:{\mathbb{R}}^{N} \to {\mathbb{R}}\), polynomial \(\mathcal{I}f\) is to be identified such that \(\mathcal{I}f\left( {{\varvec{\upxi}}_{i} } \right) = f\left( {{\varvec{\upxi}}_{i} } \right)\), \(\forall i = 1, \ldots ,M\).
The procedure of stochastic collocation for Eq. (29) is as follows:
The above formula requires \(M = m_{1} \ldots m_{N}\) points. In the case of using the same number of points in each space, the total number will be \(M = m^{N}\). This number rapidly increases with the number of the random spaces which leads to the need of rerunning the program solving the deterministic problem. To reduce the number of points, the socalled sparse grid method can be used.
4 The Inverse Problem
The accuracy of computer simulations of processes depends on the quality of mathematical models which describe those processes. The quality of models depends, in turn, on many different parameters, that need to be determined on an experimental way. If some values are determined through an experiment, and those values are then used to make a computer simulation of the same experiment, it may turn out that the result varies from the ones obtained during an experiment. Those differences result from the heterogeneity of stress fields, as well as deformations within a sample, which is a common element of the majority of metal deformation processes. To eliminate the described divergences, one needs to solve the inverse problem.
4.1 Optimization Algorithm
The genetic algorithms are a group of optimization methods based on random factor. They are grouped with artificial intelligence tools. The inventor of these algorithms, which are based on the rules that are also used in genetics, is John Holland [20].
The base of genetic algorithms is a wellknown fact of the evolution theory, which states that the organisms that are the most adjustable have the greatest chances of survival, and that their offspring greatly influences the subsequent generations of the species. This assumption is directly used in the genetic algorithms. Consecutively, the strategy used by the algorithm is similar to the natural selection; and that is the reason why the terminology is similar to the one used in genetics.
 A.Generation of initial population \({\mathbf{P}}_{i}\) for \(i = 1, \ldots ,n_{\text{pop}}\) of individuals with random values is initialized. These initial individual values are chosen within userdefined bound \(\left[ {P_{j}^{L} ,P_{j}^{U} } \right]\), where \(P_{j}^{L}\) are lower boundary constraints and \(P_{j}^{U}\) are upper boundary constraints for \(j = 1, \ldots ,n\), where \(n\) is the dimension of problem space. Hence,$$P_{i,j} = rnd_{i,j} \left( {P_{j}^{U}  P_{j}^{L} } \right) + P_{j}^{L} ;\;i = 1, \ldots ,n_{\text{pop}} ;\;j = 1, \ldots ,n$$
 B.
Evaluation of fitness function subject to boundary constrains (44) for each individual,
 C.
Selection The reproduction operation is carried out by choosing individuals according to their relative fitness. There are different methods to perform this operation. A method that lays out a line in which each parent corresponding to a section of the line of length proportional to its scaled value is used. The algorithm moves along the line in steps of equal size. At each step, the algorithm allocates a parent from the section it lands on. The first step is a uniform random number less than the step size,
 D.
Reproduction Genetic algorithm creating children for the next generation specifies the number of individuals that are guaranteed to survive to the next generation and specifies the fraction of the next generation that are produced by crossover,
 E.
Mutation Genetic algorithm makes small random changes in the individuals in the population to create mutation children. Mutation provides genetic diversity and enables the genetic algorithm to search a broader space,
 F.
Steps B to E are repeated until the specified convergence of fitness function or specified number of generations is satisfied.
5 Numerical Examples
Parameters used in the analysis
Parameter  Value 

Initial yield stress  \(k = 360.0\) MPa 
Material parameter 1 (Voce law)  \(R_{0} = 500.0\) MPa 
Simulated parameter 2 (Voce law)  \(R_{\inf ,0} = 52.0\) MPa 
Simulated parameter 3 (Voce law)  \(b_{0} = 17.2\) 
Fiction factor (upper, lower surface)  \(m = 0.12\) 
Internal diameter of specimen  \({\text{ID}} = 20.0\) mm 
External diameter of specimen  \({\text{ED}} = 40.0\) mm 
Height of specimen  \(H = 20.0\) mm 
Simulated parameter 1 (Perzyna model)  \(\delta_{0} = 0.357\) 
Simulated parameter 2 (Perzyna model)  \(\gamma_{0} = 0.55\) s^{−1} 
Simulated coefficient 1 (example 1)  \({\text{Var}}_{\delta } = 0.01\) 
Simulated coefficient 2 (example 1)  \({\text{Var}}_{\gamma } = 0.01\) s^{−1} 
Simulated coefficient 1 (example 2)  \({\text{Var}}_{{R_{\inf } }} = 16.0\) MPa 
Simulated coefficient 2 (example 2)  \({\text{Var}}_{b} = 2.0\) 
Crossover rate  0.6 
Generations  50 
Number of individuals  \(n_{\text{pop}} = 25\) 
Mutation rate  0.2 
Configurations of KL expansion order vs. SG interpolation level
Configuration  KL order  SG interpolation level  Number of collocation points  Dimension of stochastic space 

1  2  1  17  8 
2  2  2  177  8 
3  2  3  1409  8 
4  3  1  37  18 
5  3  2  757  18 
6  4  1  65  32 
7  4  2  2241  32 
Besides the choice of the appropriate measurement points, the degree of reduction in height of the sample is important. In the first step of analysis, 30% of reduction (Fig. 4) has been taken. On the graph of response function (Fig. 3), there are local extrema. Clearly visible is the global extreme (\(\delta_{0} = 0.357\), \(\gamma_{0} = 0.55\;{\text{s}}^{  1}\)).
Identified parameters (30% of reduction in height, first example)
Number of runs  \(\delta_{0}\)  \({\text{Var}}_{\delta }\)  \(\gamma_{0}\)  \({\text{Var}}_{\gamma }\)  Fitness function best 

1  \(0. 3 5 7 0 0 3 3\)  \(0. 0 1 0 0 1 2\)  \(0. { 54999845}\)  \(0. 0 0 9 9 9 7\)  \(3. 6 6 2\cdot 10^{  10}\) 
2  \(0. 3 5 6 9 4 7\)  \(0. 0 0 9 9 8\)  \(0. 5 5 0 0 1 1 8\)  \(0. 0 1 0 0 6 1\)  \(1. 4 7\cdot 10^{  8}\) 
3  \(0. { 3570305}\)  \(0. 0 1 0 0 2\)  \(0. 5 4 9 9 4 9 3\)  \(0. 0 0 9 9 8\)  \(1. 5 4 9\cdot 10^{  8}\) 
Identified parameters (50% of reduction in height, first example)
Number of runs  \(\delta_{0}\)  \({\text{Var}}_{\delta }\)  \(\gamma_{0}\)  \({\text{Var}}_{\gamma }\)  Fitness function best 

1  \(0. 3 6 0 1\)  \(0. 0 1 1 5 4\)  \(0. 5 7 9 2\)  \(0. 0 0 8 4 7\)  \(1.4797 \cdot 10^{  3}\) 
2  \(0. 3 5 8 6\)  \(0. 0 1 1 6 1\)  \(0. 5 7 0 2\)  \(0. 0 0 8 6 5\)  \(7.0806 \cdot 10^{  4}\) 
3  \(0. 3 5 7 1\)  \(0. 0 1 0 3 2\)  \(0. 5 5 1 8\)  \(0. 0 0 9 1 8\)  \(6.5732 \cdot 10^{  5}\) 
Identified parameters (30% of reduction in height, second example)
Number of runs  \(R_{\inf ,0}\)  \({\text{Var}}_{{R_{\inf } }}\)  \(b_{0}\)  \({\text{Var}}_{b}\)  Fitness function best 

1  \(5 1. 9 9 3 9 7\)  \(1 5. 9 9 3 5 9\)  \(1 7. 2 0 5 5 2\)  \(2. 0 0 1 2 4\)  \(3. 7 7 1\cdot 10^{  9}\) 
2  \(5 2. 0 3 4 5\)  \(1 5. 9 7 1 7 8\)  \(1 7. 1 6 9 1 8\)  \(1. 9 8 9 6 6\)  \(2. 1 1 3\cdot 10^{  8}\) 
3  \(5 1. 7 6 6 0 1\)  \(1 5. 9 5 9 3 8\)  \(1 7. 4 1 0 6 5\)  \(1. 9 7 3 4 9\)  \(1. 4 4 7\cdot 10^{  7}\) 
Identified parameters (50% of reduction in height, second example)
Number of runs  R _{inf,0}  \({\text{Var}}_{{R_{\inf } }}\)  b _{0}  Var_{b}  Fitness function best 

1  \(5 7. 5 8 6 0 7\)  \(1 4. 7 7 1 0 8\)  \(1 5. 8 4 9 4 5\)  \(1. 6 1 5 4\)  \(2. 2 2 1\cdot 10^{  5}\) 
2  \(5 1. 7 4 5 5 5\)  \(1 7. 4 6 5 6 3\)  \(1 7. 4 7 0 4 5\)  \(2. 2 3 8 1 7\)  \(2. 1 0 3 8\cdot 10^{  5}\) 
3  \(5 2. 7 8 0 5 9\)  \(1 6. 8 2 6 4 3\)  \(1 6. 4 2 9 5\)  \(2. 0 9 3 0 9\)  \(2. 0 5 4 7\cdot 10^{  6}\) 
Computations are performed on computer made of 32 physical cores, 2.53 GHz clock speed each. In the case of parameter’s identification in Perzyna model (example 1), computational time is 439.217 min for 30% reduction in height and 603.078 min for 50% reduction in height. In the second example, computational time in both cases is close to the first example.
6 Concluding Remarks
The paper proposes a method of parameter’s identification of twodimensional random field in rigid viscoplasticity. The investigated material is assumed to follow nonlinear isotropic hardening behavior (Voce hardening law), and the Perzyna flow stress model is adopted.

The degree of reduction in height significantly affects instability of solution, which is the cause of local extremes compaction in the surrounding of global extreme. This behavior can be explained by the significant plasticizing of the sample.

SGSCM method works well with viscoplastic forming problems, even with a small number of collocation points for multidimensional random space.

GA works well with stochastic inverse problems assuming the use of multithreading to which the method is well adapted.

The system response to the random disturbances modeled as a stationary Gaussian random field is also Gaussian random field.
The comparison shows that the proposed method works well with nonlinear problems and it is convergent to Monte Carlo method due to first and second statistical moments. The analysis of inverse problem has revealed that the proposed identification method is appropriate when the stochastic process with use of the first two statistical moments only and the knowledge of measurement only at external surfaces are considered.
Footnotes
 1.
Equations are presented for axially symmetric case.
Notes
Acknowledgements
The authors are grateful for granting access to the computing infrastructure built in the projects No. POIG.02.03.0000028/08 “PLATON  Science Services Platform” and No. POIG.02.03.0000110/13 “Deploying highavailability, critical services in Metropolitan Area Networks (MANHA).”
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