Assessment of Optimization Methods Used to Determine Plasticity Parameters Based on DIC and back Calculation Methods
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Abstract
In this paper, we analyse various minimization algorithms applied to the problem of determining elastoplastic material parameters using an inverse analysis and digital image correlation (DIC) system. As the DIC system, ARAMIS is used, while for the finite element solution of boundary value problems, Abaqus software is applied. Different minimization algorithms, implemented in the SciPy Python library, were initially juxtaposed, compared and evaluated based on benchmark functions. Next the proper evaluation of the algorithms was performed to determine the material parameters for isotropic metal plasticity with the HuberMises yield criterion and isotropic or combined kinematicisotropic plastic hardening models. For all researchers utilizing back calculation methods based on a DIC measuring system, such analysis results may be interesting. It was concluded that among the local minimization methods, derivative free optimization algorithms, especially the Powell algorithm, perform the most efficiently.
Keywords
Plasticity DIC Inverse analysis Optimization algorithms Python AbaqusIntroduction
In recent decades, the application of measuring techniques that enable us to register whole displacement fields has spread dramatically, mainly owing to companies that produce such systems for commercial applications (see [1, 2, 3, 4]). Digital image correlation (DIC) systems can be configured to track selected measurement points or to track microsubregions, where points are defined unambiguously by randomly generated patterns (facets). These configurations have different working field ranges [4]: in the former case, the working/measurement field can have large dimensions (reaching a few metres); however, in the latter case, with a satisfactory measuring accuracy, the measuring field is only a few square centimetres. Generally, DIC systems are used for twodimensional (2D) measurements (utilizing one camera) or for threedimensional measurements (utilizing 2 or more cameras, depending on the visibility of the measured subregions).
When DIC is used with a randomly generated pattern on the analysed subregion, strain localization, plastic strain developments after yielding [5, 6, 7, 8, 9], and crack propagation can be monitored [10]. Every measurement can be done without risking damage to the measuring system, in contrast with observations carried out using conventional measurement methods. The application of such measurement methods leads to the design of experiments with homogenous/uniform stress or strain fields to explicitly determine one of the constitutive model parameters on the basis of the specific experiment (with the other parameters having no influence on the particular experimental results or being already determined at the given stage of the experiment).
Unfortunately, in the case of complex constitutive models, such as isotropic and anisotropic models of elastoplasticity or hyperelasticity, such an approach is unfeasible [11, 12, 13]. Due to this limitation, it is worthwhile to use the ability to measure displacements in the whole subregion, as is presented in many applications [9, 13, 14, 15, 16, 17, 18]. Arguably, the measurement of homogeneous deformation fields using DIC systems does not make much sense, as the full potential of this method has not yet been fully utilized. Such measurements can possibly be treated as checking the homogeneity of the measured field. Therefore, to determine the constitutive model parameters, introducing the heterogeneity of the deformation (strain) field and the strain concentration regions in the case of elastoplasticity is reasonable. The observed deformation field may then be assumed to be influenced by all of the constitutive parameters – if the strains are large enough [19]. This influence is another problem for interpretation; to determine the material parameters, we must obtain the solution of the boundary value problem that describes the analysed study with the constitutive relationships for which we need to find the parameters. Having such solutions and the results of the experimental tests, we can then define the goal/cost function, with the material parameters being variables whose norm should be minimized. Obtaining a boundary value problem solution for any sample shape and complex load condition requires a universal solution method (e.g., finite element method, FEM) [20, 21]. In this study, the ARAMIS system [1] was used to measure the displacement field and the ABAQUS/Standard FEM system [20, 21] was used to solve the boundary value problems with elastoplasticity constitutive relationships [22].
This article is a continuation of previous work [14] in which inverse analysis and DIC tools were used to determine material parameters of isotropic metal plasticity with the HuberMises plasticity condition and isotropic strain hardening [12, 22]. The aforementioned article [14] presents results of nonlinear optimization that was performed using the same optimization method for two differently shaped subregions (a rectangular sample with a hole or with two semicircular notches). In addition, various forms of the goal/cost function were considered, which included not only displacements but also a global response of the sample (reaction force) to the considered displacement boundary conditions. Using this approach, we concluded that using a sample with asymmetric notches and a cost function that takes into account the global reaction of the sample is preferable. The importance of including a global reaction function in the cost function has also been discussed by other researchers [13]. Apparently, no analysis of the influence of the selected optimization method on the convergence and efficiency of the material parameter determination process in the case of isotropic elastoplasticity with isotropic or combined strain hardening has been reported, even in very similar papers [23, 24, 25]. This article attempts to fill this gap, pointing out the most efficient and fastest convergence method. By contrast, the literature contains several papers that propose an analytical approach to the problem of determining constitutive material parameters [26]. There are also papers in which different optimization strategies [19] and multiobjective approaches [27] are analysed.
Experimental Approach
The Concept of Determining Material Parameters Using DIC and Inverse Analysis
The concept presented in this article for determining material parameters is based on a reversal of the classical boundary value problem. In this problem for a given geometry, the displacement field is determined on the basis of boundary conditions (stresstype and displacementtype boundary conditions) and the constitutive relationship with the given material parameters. In the case of the inverse analysis, the displacement state of the sample is given (based on the experiment) but the material parameters in the constitutive relationship are unknown. To determine them, the FEM tasks that correspond to the experiments are solved and the resulting displacement map (and possibly the global reaction force function) is compared with the experimental results. The closer the results are to each other, the smaller the value of the objective function. Finding the optimal solution (objective function minimizer) requires the use of nonlinear optimization algorithms.
Constitutive Models and Determined Parameters
In equation (6), I is the secondorder identity tensor, and in equation (7), λ and μ are Lamé material parameters. In summary, the presented constitutive relations show that there are 7 material parameters: E and ν for the elasticity; σ_{0}, Q_{∞} and b for isotropic plasticity with hardening; Cand γ_{k} for kinematic hardening.
Description of the Samples and Laboratory Methods
The tests were performed on an Instron testing equipment, which applied displacement to the sample edge and measured the reaction force response to generate a displacement map of the sample. The sample was stretched between machine grips (steered with displacement) in the direction of the long edge to 2 mm, which caused plastification of the regions between notches; the sample was then unloaded. The applied displacement as a function of the real time is shown in Fig. 1. The displacements were recorded using an ARAMIS DIC system equipped with two cameras (for threedimensional (3D) measurement). Such systems enable noncontact measurement of displacement in the whole subregion of the sample being tested (usually lying flat) by taking a series of pictures during the test and then processing them using appropriate software (postprocessing). In the pictures, the characteristic points are recognized and their location and relative displacements during the experiment are determined.
Figure 2(a) shows ARAMIS measuring points and the measuring points located between the notches that correspond to the nodes of the FEM mesh shown in Fig. 2(b), which are selected as those that will be taken into account when comparing displacements in a goal function. The analysis of the selected measurement points included in the objective function is presented in [14]. The selected points should be located in the area with the largest heterogeneity of the displacement field, which, in this case, is in the region between notches (Figs. 2(a) and 2(c)).
Formulation of the Optimization Problem
The most important element in the definition of the optimization problem is the determination of the cost function that will evaluate the “distance” of the FEM solution for the given parameter values from the results of the laboratory experiment. The cost function accounts for all comparable quantities, i.e., the global reaction force of the sample to the assumed edge displacement and the displacement map of the sample surface. Due to the discrete character of the data obtained from the experiment and the FEM task, comparing displacement maps is simply done by comparing/subtracting the displacements of multiple pairs of correlated points. For each node in the chosen node set of the FEM model, the closest measurement point was determined from the DIC system and, on this basis, a reference displacement function was assigned to it. A tolerance was established that specified the maximum permissible distance between the FEM node and the corresponding DIC measurement point.
The problem of determining parameters p is reduced to the minimization of the goal function f(p) with norms (9) and (10).
Implementation
The main loop of the designed program is performed by the optimizer module, which utilizes one of many implemented optimization procedures. DIC software provides the optimizer module with exemplary data for later comparison with FEM results. The optimizer module manages the optimization process by evaluating the value of the cost function and choosing the next point/points to query for the FEM results. At each point, the FEM job is set up and solved, and the result data are collected. These data are then used to calculate the value of the cost function. The choice of ARAMIS and Abaqus software does not depend on the optimization procedures, and vice versa.
The displacement boundary conditions for the FEM jobs were established on the basis of experimental DIC data [9]. The nodes at the boundary of the measured region were given the displacements of their corresponding DIC points. Only the region inside the measured area was the subject of comparison between FEM and DIC (as shown in Fig. 2).
Results and Discussion
Analysis of the Optimization Methods for their Efficiency
The types of optimization algorithms used for benchmark tests and parameter determination problems
DFO  Powell 

NelderMead  
Gradient  BFGS 
LBFGSB  
Conjugate Gradient (CG)  
Truncated Newton (TNC)  
Sequential Least Squares (SLSQP)  
Trustregion  Dogleg 
Newton CG trustregion (trustncg)  
Stochastic  Differential Evolution 
Basinhopping (with LBFGSB for local minimization) 
The DFO, gradient and trustregion algorithms are used for local minimization. DFO algorithms do not calculate the gradient of the minimized functions, whereas gradient algorithms require gradient calculations. Trustregion methods approximate the subset of the search region with a model function (e.g., a quadratic). On the other hand, the differential evolution and basinhopping algorithms are designed for global minimization. Both are nondeterministic and are therefore based on an assignment to the stochastic group.
The implementation details of all minimization algorithms are presented elsewhere [30]. The algorithms were initially tested on the four 2D benchmark functions listed in subsection 5.1. Afterwards, the minimization of the cost function presented in section 3 was performed. The parameters sought were Q_{∞} and b, whereas the E, ν and σ_{0} parameters were assumed in advance on the basis of independent tests.
The optimization task is, in general, a constrained minimization problem. However, the use of unconstrained minimization algorithms, such as Powell or BFGS, does not cause any problems after a careful choice of the search region boundaries as well as the artificial addition of bounds in the form of infinite values of the cost function outside the search region.
The algorithms were compared based on the value of the cost function obtained during the optimization process after a fixed number of cost function evaluations as well as the final value of the cost function achieved with the total number of required cost function evaluations. In the material parameters minimization problem, the reliability of algorithms (whether or not they were error free) was also considered.
Benchmark Tests

Validation of the opensource implementation of optimization algorithms that are used [30];

Initial evaluation of the efficiency of each algorithm with a low timeconsuming numerical calculation; and

Choosing the test functions that have similar properties, such as the objective function in the plasticity parameters determination problem presented in point 5.2.

Ackley: \( a\exp \left(b\sqrt{\frac{1}{2}\left({x}^2+{y}^2\right)}\right)\exp \left(\frac{1}{2}\left(\cos (cx)+\cos (cy)\right)\right)+a+\exp (1) \),

Levy n.13: sin^{2}(3πx) + (x − 1)^{2}(1 + sin^{2}(3πy)) + (y − 1)^{2}(1 + sin^{2}(3πy)),

Schaffer n.2: \( 0.5+\frac{\sin^2\left({x}^2{y}^2\right)0.5}{{\left(1+0.001\left({x}^2+{y}^2\right)\right)}^2} \)
StyblinskiTang: 4 2 4 2 \( \frac{1}{2}+{x}^416{x}^2+5x+{y}^416{y}^2+5y \)
In the previously presented analysis, the derivativefree Powell algorithm stands out (excluding stochastic differential evolution algorithm) as the most costeffective approach, especially in the case of the Levi and Ackley functions for which the Powell algorithm is better than all gradient algorithms in avoiding local minima. The implementation of the differential evolution algorithm finds the minimum of a global function correctly in almost every case. However, it requires an orderofmagnitude larger number of function evaluations than simpler algorithms.
Target Tests: Determination of ElastoPlasticity Material Parameters
Because the evaluation of the cost function is a timeconsuming process (mainly because of the need for the calculation of an FEM job at every point), the convergence speed, as well as the final result, is considered to be the most important factor in the evaluation of the algorithms.
Furthermore, the stochastic algorithms (basinhopping and differential evolution) have been excluded from the comparison presented in this section. Both algorithms can be characterized by the best effectiveness regarding the minimization results among all; however, they require the largest number of function evaluations. Basinhopping repeatedly performs a local minimization using the LBFGSB algorithm, whereas differential evolution is a bruteforce algorithm. Therefore, only the deterministic local minimization algorithms are analysed below.
The optimization problem has only two parameters; as such, a graph of the cost function in the search region can be plotted (Fig. 6). A large subregion with little variation in the value of the cost function exists where the global minimum is located. Additionally, due to the noise present in the cost function [34], which is caused by, among other things, the noise in the laboratory data, the cost function is barely differentiable; thus, the algorithms that required calculation of the Hessian of the cost function did not perform well. In addition, the gradient algorithms, such as CG, displayed a higher susceptibility to errors than the DFO algorithms. In the following summary, only the algorithms that succeeded without errors in most cases are included.
Arguably, the best behaviour was demonstrated by the derivativefree Powell algorithm. This algorithm appears to be the most reliable, converging successfully in every case and achieving good results. Among the other algorithms, BFGS stands out as the best gradient method, although it often does not converge successfully; this shortcoming is highlighted further in problems of minimization with a greater number of variables. The exceptional performance of SLSQP is a coincidence, as the algorithm performs extremely ineffectively on problems with larger dimensions. The BFGS and LBFGSB can perform better than the Powell algorithm from certain starting points; however, for the presented set of tests, the BFGS shows no significant disadvantage over the Powell algorithm. However, in some cases, gradient algorithms fail to achieve any satisfactory result (when they terminate due to an error), whereas the Powell algorithm finds a local minimum in every case.
Overall, the best behaviour, mostly due to reliability, was demonstrated by the DFO Powell algorithm.
Conclusions
 a)
Derivativefree algorithms, especially the Powell algorithm, performed better than gradient algorithms in the presented optimization task.
 b)
Algorithms that require a Hessian of the function to be calculated appears to be susceptible to noise present in the minimization task.
 c)
BFGS appears to be the best gradient algorithm, whereas Powell behaves arguably better than any other local minimization algorithm. BFGS converges faster, but Powell is more reliable.
 d)
Both the Powell and the BFGS algorithms are methods for unconstrained optimization. However, practical tests show that they can perform well in a problem with strict bounds.
 e)
More complex algorithms—specifically, basinhopping and differential evolution—perform better when used to find the global minima in a single run of an algorithm. However, the basinhopping algorithm simply repeats the minimization task from (mostly) randomly chosen starting points, whereas differential evolution does not perform any local minimization, thus requiring a large number of function evaluations to converge to any solution.
 f)
Among the benchmark functions, the minimizations of the Levi and Ackley functions show results similar to the target minimization task, as the Powell algorithm has a substantial advantage. However, gradient algorithms (mainly BFGS) perform better on the target task than on benchmark functions.
 g)
Among all the local minimization methods, the Powell algorithm was found to be the most efficient.
Notes
Acknowledgements
The authors would like to express their gratitude to Prof. Grzegorz Dzierżanowski, whose remarks led to an improved final version of the paper.
Compliance with Ethical Standards
Conflict of Interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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