Improvement of the efficiency of the inverse analysis (IA) for various material tests was the objective of the paper. Flow stress models and microstructure evolution models of various complexity of mathematical formulation were considered. Different types of experiments were performed and the results were used for the identification of models. Sensitivity analysis was performed for all the models and the importance of parameters in these models was evaluated. Metamodels based on artificial neural network were proposed to simulate experiments in the inverse solution. Performed analysis has shown that significant decrease of the computing times could be achieved when metamodels substitute finite element model in the inverse analysis, which is the case in the identification of flow stress models. Application of metamodels gave good results for flow stress models based on closed form equations accounting for an influence of temperature, strain, and strain rate (4 coefficients) and additionally for softening due to recrystallization (5 coefficients) and for softening and saturation (7 coefficients). Good accuracy and high efficiency of the IA were confirmed. On the contrary, identification of microstructure evolution models, including phase transformation models, did not give noticeable reduction of the computing time.
Continuous progress in numerical modelling of metals processing has been observed during more than half of the century. It became evident that the accuracy of simulations depends mainly on the correctness of the description of boundary conditions and properties of processed materials. The latter problem was the subject of the present work. A number of material models can be found in the scientific literature. Potential extensive predictive capabilities of these models are useful only when proper identification was performed on the basis of experiments. Interpretation of the results of various experiments is the main part of the identification, which usually uses inverse analysis (IA) with finite element (FE) simulation of the test [
Various material models used in simulations of thermomechanical processing and various experimental tests performed to identify these models were investigated in the present work. These models describe flow stress, microstructure evolution, and phase transformations. Plastometric tests [
The idea of numerical models substitution with metamodels in the inverse solution has been explored for some time now and solutions for structural mechanics [
Modelling of materials processing requires knowledge of material properties, which depend on many factors like grain size, grain boundaries, dislocation density, stacking fault energy, and so forth. Due to their complexity and scale, accounting for all these factors is difficult. To overcome this problem polycrystals are described by homogenized models, which represent statistically all mentioned microstructural phenomena.
A large number of flow stress models for metal forming were published in the second half of the XXth century. These models are characterized by various complexity of mathematical formulation and various predictive capabilities. There were several attempts to classify these models; see, for example, [
By conventional model, we understand closed form equations, which describe flow stress as a function of temperature, strain, and strain rate. Introduction of the internal variables instead of strain as independent variables allowed accounting for the inertia of microstructural phenomena. Dislocation density is the most commonly used internal variable and a variety of dislocation density based models were developed following fundamental works of Estrin, Kocks, and Mecking [
Conventional models give good results when conditions of deformation are reasonably monotonous and these models are commonly used in simulations of industrial metal forming processes. The first attempt to describe the flow stress as a function of process parameters is attributed to Hollomon, who proposed the power equation describing flow stress relation on strain. To account for the influence of temperature and strain rate, Hollomon equation was extended to the following form:
Equation (
At lower
For larger strains, flows stress calculated from (
Typical responses of metals subjected to deformation and equations used to describe these responses.
Large number of amendments of (
Mathematical form of this model allows the distinguishing of hardening and softening terms; the former is multiplied by
The conventional flow stress model is based on the assumption that stress
Coefficients
This method is capable of predicting the delay in response due to microstructural processes that take place during deformation, which has been proved experimentally. Details of the IVM solution with one internal variable are described in [
Changes in microstructure are connected mainly with transformations. JMAK equation was selected to describe kinetics of transformations:
St3S and DP steel strips were considered in the paper. End of rolling temperature for these steels is about 870°C; therefore, static recrystallization is the main mechanism which controls microstructure evolution. Equations based on works of Sellars [
Phase transformation model is based on (
The general idea of metamodelling relates to a postulation that metamodel approximates the model of a considered process. Metamodel must correctly correspond to the model and the metamodel output has to be evaluated with a radically lower computing time than using the original model. Thorough discussion of application of metamodelling to optimization metal forming processes can be found in [
One of the objectives of this work was exploring capabilities of various metamodels as direct problem models in inverse analyses of material tests. Three types of tests were investigated; see Section
The choice of the metamodelling technique was based on the comparison of the metamodelling results of the following two benchmark functions. The first was Rastrigin test function (Figure
Rastrigin (a) and Michalewicz (b) benchmark functions.
The Rastrigin and Michalewicz functions were used mainly to compare metamodelling techniques in terms of accuracy and required memory size. Two different metamodelling techniques were tested: ANN and Kriging. The aim of research was the comparison of the accuracy of these two metamodelling techniques in relation to the number of experimental data points. Since the used benchmark functions were only twodimensional functions, the metamodelling was performed using 50, 100, and 200 experimental points. The accuracy of metamodels was evaluated using error defined by the following equation:
The obtained results of the ANN and Kriging metamodels of both considered benchmark functions are graphically shown in Figures
Comparison of the errors of metamodels of benchmark functions (
Test function  Number of experimental points  ANN error  Kriging error 

Rastrigin  50  0.0446  4.32 
100  0.0023  9.26  
200  1.89 
9.66  


Michalewicz  50  0.0131  0.0082 
100  5.12 
0.0015  
200  2.36 
2.15 
Metamodels of the Rastrigin function (
Metamodels of the Michalewicz function (
Performed analysis confirms that metamodel error decreases, while a number of experimental points increase (but, on the other hand, the increase of number of experimental data points escalates the research costs). For small number of experimental points, Kriging technique is better in the case of both tested functions. However, when the number of points is higher, ANN technique gives better results for function (
Memory size of Kriging metamodel with respect to the number of experimental points.
Number of experimental points  Memory size of Kriging metamodel [B] 

100  87 064 
200  331 864 
500  2 026 264 
1000  8 050 264 
2000  32 098 264 
4000  128 194 264 
Since the number of available training points was relatively high (in some cases more than 20 000 points, obtained in former authors’ research), metamodels used in the present work were built using the ANN technique.
The analysis of the six examples presented in Table
Numerical tests showed that the error of the ANN increases with the increasing complexity of the model (increasing number of variables). This error is further magnified in optimization. Therefore, the possibility of improvement of the accuracy of the ANN by constraining the domain of the variables values was explored. The advantage was made from the fact that in the optimization in the inverse analysis the location of the output of the ANN has to be close to the value obtained from measurements. It inspired the authors to apply the clusterization of the ANN on the basis of the value of the output. The schematic illustration of this approach for only two optimization variables is shown in Figure
The idea of application of clusterization of the ANN during optimization.
St3S and DP600 steels with the chemical compositions in Table
Chemical composition of the investigated steels St3S and DP600, wt%.
Steel  C  Mn  Si  Cr  Cu  P  S  Ti  V  Al  N 


St3S  0.16  0.43  0.23  0.01  0.03  0.006  0.015  —  —  0.03  0.004 


DP600  0.071  1.45  0.25  0.55  0.02  0.01  0.006  0.002  0.005  0.022  0.0039 
Chemical composition of the investigated CuCr alloy, wt%.
Cr  Ni  Si  Fe  As  Bi  Cu 


0.81  <0.001  <0.001  0.026  <0.001  <5 ppm  Balance 
In general, the stress versus strain, strain rate, temperature, and so forth (depending on conditions) relation is determined on the basis of the results of experiments, which are called plastometric tests. The tests can have various forms (tension, compression, and torsion) depending on further use of the flow curve. Advantages and disadvantages of plane strain compression (PSC), cylinder (UC), and ring (RC) compression tests as well as torsion tests (TT) are discussed in the literature; see, for example, [
Schematic illustration of the compression tests investigated in the present work.
In hot compression, UC samples measuring
Samples measuring
Loads measured in the UC tests for the DP600 steel, strain rate 1 s^{−1} (a), and for the RC tests for the St3S steel, strain rate 1 s^{−1} (b).
Loads measured in the UC tests (a) and the PSC tests (b) for the CuCr alloy.
In the conventional interpretation of the tests, the flow stress is calculated as forcetocontact area ratio. The strain is calculated as
Stress relaxation tests were performed on Gleeble 3800 simulator. The idea of this test is described in [
Identification of models using inverse analysis is usually preceded by the sensitivity analysis. Both of these procedures are described briefly in this chapter.
Sensitivity analysis (SA) allows us to assess the accuracy of the model of the analysed system or process, determine the parameters which contribute the most to the output variability, indicate the parameters which are insignificant and may be eliminated from the model, evaluate these parameters which interact with each other, and determine the input parameters region for subsequent calibration space [
sensitivity measure: the measure expresses the model solution (model output) changes to the model parameter variation;
selection of the parameter domain: design of experiment technique was used to select the lower number of points, which guaranteed the searching of the whole domain;
sensitivity calculation: the sensitivities were estimated by Morris OAT (One At a Time) Design method [
The information obtained from sensitivity analysis was applied to the inverse method:
To verify whether the objective function is well defined, SA gives information if the sensitivity of the objective function to the parameter changes is large enough to allow estimation of this parameter.
A preliminary step is to select the starting point or the first population for optimization.
In optimization process, we construct the hybrid algorithms (the combination of two or more methods) or modified algorithms to increase the procedure efficiency.
Selected Morris OAT Design method [
The elementary effects
Since the investigated models are based on closed form equations (
Sensitivity of the flow stress with respect to coefficients in (
Due to complex mutual influence of phase transformations, sensitivity analysis for the phase transformation model is a difficult problem. Changes of kinetics of one transformation (e.g., ferritic) may result in the occurrence or nonoccurrence of another one (e.g., bainitic). Therefore, the problem of the sensitivity analysis of the phase transformation was a subject of the separate work [
Substitution of the FE model with the metamodel was the objective of the present work. To be efficient, once trained, the metamodel should be used for identification of various materials, without additional training. This method will be robust and accurate when the number of input parameters to the metamodel is reasonably low. Beyond coefficients in the flow stress model, there are additional parameters, which have to be considered as model inputs:
temperature and strain rate,
friction coefficient,
sample dimensions,
thermophysical properties of the material, which influence simulation of the temperature.
These problems were investigated in earlier publication [
In consequence, the following parameters were introduced as an input to the metamodel: coefficients in the flow stress equations (
The inverse algorithm proposed in [
When vectors
When the problem is linear, the inverse function can be usually found and the problem can be solved analytically. In the investigated problem of materials processing, this relation is nonlinear and the problem is transformed into the optimization task. Thus, the objective of the inverse analysis is the determination of the optimum components of vector
Thus, inverse analysis is composed of three steps: experiment, FE simulation of the experiment, and optimization. Flow chart of this algorithm is shown with the solid line in Figure
Flow chart of the inverse algorithm with the FE model (solid line) and the metamodel (broken line).
Metamodels were developed for all investigated experiments and for all material models. These metamodels were used in identification of material models and the results were compared with the classical inverse solution based on the FE direct problem model.
The objective function (
The input of the network included friction coefficient, temperature, and strain rate of the test and coefficients in the flow stress model. Since dimensions of samples in plastometric tests are standardized, separate models were developed for each sample dimension. Metamodels for the UC, RC, and PSC tests, which were described in the previous section, were combined with flow stress equations (
Equation (
Coefficients in (
Coeff.  IA + FE  IA + ANN 


193.2  197.6 

0.3  0.264 

0.02  0.016 

2000.4  1975.6 


Φ  0.0983  0.0827 
Selected plots of recorded loads during uniaxial hot compression of the DP600 steel are shown in Figure
Coefficients in (
Equation 







Φ 

( 
6038.8  0.376  0.105  0.00337  0.521  —  —  0.13 
( 
22.59  0.278  0.107  1589.4  0.441  6339.5  2.956  0.124 
Comparison between the result of the direct inverse analysis (dotted lines) and the plots of functions with coefficients in Table
Analysis of the plots in Figure
The wide range of parameters was used in the present work because the main objective of the research was to investigate inverse with metamodel approach in extreme conditions of identification of the model. It should be emphasized, however, that when the practical range of temperatures is smaller (e.g., 850–1050°C for finishing rolling), (
Capability of the selected material model to reproduce behaviour of the material in the whole range of parameters properly is the main factor, which influences the error of the inverse analysis. To avoid influence of the flexibility of the function on the evaluation of the metamodel, all the results of identification using metamodel will be referred to the results of the classical inverse analysis with FE model of the direct problem.
Performed optimization for (
Coefficients in (
Coeff.  St3S  CuCr 


2622.2  1178.4 

0.37  0.279 

0.122  0.098 

0.0025  0.00532 

0.727  0.624 


Φ  0.0709  0.0888 
Flow stress calculated from (
This model was also used for hot forming of DP600 steel and performed optimization yielded coefficients
Coefficients in (
Method 







Φ 

IA + FE  2.423  37555  0.208  0.122  0.00519  91088  0.534  0.0754 
IA + CANN  1.509  42439  0.217  0.112  3.339  7727.5  0.211  0.0623 
Flow stress calculated from (
Plane state of strains, which is not reachable in other plastometric tests, has inspired for years the scientists to various applications of the PSC tests. Identification of the flow stress model is one of such applications and investigation of the microstructure evolution is another example. Among several research laboratories involved in investigations based on the PSC tests a team led by Sellars at the University of Sheffield should be mentioned. This test was commonly used there for investigation of materials and fundamental works on microstructure evolution [
Conventional twostage inverse analysis [
Flow stress calculated in a tabular form from the first stage of the inverse analysis.
Inverse analysis with the metamodel was performed for both UC and PSC tests. Coefficients in (
Coefficients in (
Equation 






( 
201.0  0.275  0.024  1718.6  — 
( 
735.6  0.497  0.0201  0.0014  0.547 
Capabilities of the IA with metamodel were further explored for the compression of rings (RC). Various inhomogeneities make the interpretation of results of this test difficult. Due to the fact that ring dimensions after compression are sensitive to friction, this test is frequently used for identification of the friction coefficient [
ANNbased metamodel of the RC test was built. Coefficients in (
Coefficients in (
Method 





Φ 

IA + FE  3694.7  0.461  1.214  0.115  0.0276  0.1172 
IA + ANN  3036.8  0.381  0.984  0.108  0.0271  0.0669 
Flow stress calculated from (
Performed research has shown that the number of ANN training data points varied within 1–10 thousand, depending on the considered plastometric test. Training time of the ANN metamodel for various plastometric tests did not exceed 30 minutes (in the case of the largest training data set), while identification with metamodels required 2–10 minutes computing time, depending on the number of identified parameters. One calculation of the objective function using simple FEM model requires 20–30 min. Optimization using simplex method requires about 50–100 calculations of the objective function. Application of more advanced optimization methods inspired by the nature is even more demanding.
Equations describing microstructure evolution are simple and the metamodel did not accelerate the inverse analysis. As far as the phase transformation model is considered, two types of neural networks were used. The first was PNN (Probabilistic Neural Network) and the task of this network was to indicate the probability, whether the considered transformation occurs or not. The second type (MLP) was used to predict the starting temperature for the transformations, which were selected by the PNN for the considered conditions; see Figure
The general idea of using PNN and MLP as phase transformations metamodel.
An attempt to apply trained networks to identification coefficients in the phase transformation model was made. Experimental data in the form of dilatometric tests results for the DP600 steel were used. The primary results of optimization were not satisfactory, due to the following problems:
Even small errors in the predictions of the probabilistic network PPN involved large errors in the identification of coefficients. This aspect is the objective of further research and possibilities of improvement of classification will be searched for.
The networks predicted kinetics of transformation (transient process) for given boundary conditions, which were determined from the equilibrium diagram (ThermoCalc software was used). It means that trained networks could be used for steels with similar phase equilibrium diagram, which made inverse analysis less efficient.
Recapitulating this part of the research, it was concluded that application of the metamodel does not improve efficiency of the inverse analysis in the case of the phase transformation model. Therefore, conventional inverse solution based on the simplex optimization method was used in the present work.
Performed analysis has shown that, among models considered in this work, application of metamodels in the identification procedure was efficient for the flow stress models only. The decrease of the computing costs in the case of the microstructure evolution model and phase transformation model was negligible. Therefore, flow stress models were validated by comparing the results qualitatively and quantitatively with the experimental data. Comparison between forces measured in the tests and calculated by the FE model, with one of (
Loads predicted by the FE code with flow stress calculated from (
Loads predicted by the FE code with flow stress calculated from (
Observations made in Section
Comparison between loads measured in the tests (filled symbols) and calculated by the FE code with flow stress equations (
Analysis of the results presented in Section
Selected examples of comparison of the loads measured in the tests (solid lines) and calculated by the FE code (dotted lines) with flow stress model (
Loads predicted by the FE code with flow stress calculated from (
Loads measured in the tests (full symbols) and predicted by the FE code with flow stress calculated from (
This part of verification of the models was performed by comparison between forces measured in various tests and calculated by the FE code with the identified model introduced in the constitutive law. Identification of the models was performed on the basis of the UC and PSC tests. Figure
Comparison of forces measured in the uniaxial compression (a) and plane strain compression (b) and calculated by the FE code with (
Comparison of forces measured in the UC (a) and PSC (b) tests and calculated by the FE code with (
It is seen in Figures
Figure
Comparison of forces measured in the ring compression tests for the St3S steel and calculated by the FE code with (
Possibility of identification of the flow stress model on the basis of the inverse analysis with metamodel for the PSC and RC tests was confirmed, although it is more difficult and time consuming than the UC tests. Conventional inverse analysis performed for the uniaxial compression and plane strain compression gave similar flow stress model. The model determined on the basis of both tests gave very good prediction of forces.
The general conclusion from the performed research is that the accuracy of the inverse solution (final value of the objective function) depends on two factors:
capability of the selected material model to reproduce behaviour of the material in the whole range of parameters properly;
accuracy of the optimization methods and capability to avoid local minima.
Possibility of application of metamodels in the inverse analysis was confirmed. Metamodels for various tests and for various models were developed and applied in the inverse analysis. Significant decrease of the computing time was obtained when FE model was substituted by metamodel. Results of identification using metamodel are in good agreement with the classical inverse analysis with FE model of the direct problem. The general observations from this part of the work are as follows:
Metamodels are efficient in applications to identification of coefficients in flow stress models on the basis of various experimental tests. Inverse analysis with the metamodel is few orders of magnitude faster than the conventional approach with the FE model; see Section
Long computing times are needed for training the metamodel, but once trained the metamodel can be applied to any new material, assuming that the dimensions of the sample and the flow stress equation do not change.
Although the values of coefficients obtained from various tests may differ, the agreement between measured and calculated forces is good. It means that there is no unique solution of the problem, but the accuracy of the obtained solution is satisfactory.
The general observation in the paper was that good accuracy of the training of the metamodel (ANN) was obtained for all experiments and for all material models. However, when combined with the optimization in the inverse analysis the performance of the metamodel was decreasing with increasing number of coefficients in the model. Thus, good accuracy of the IA with the metamodel was obtained for (
Validation of the solution with the metamodel by comparison of measured and calculated compression forces confirmed its good accuracy for models with lower numbers of coefficients. Increase of number of coefficients resulted in a decrease of the accuracy.
Attempts to develop metamodel for the compression tests combined with equation of [
Generally all investigated flow stress models describe properly materials response for lower values of the ZenerHollomon parameter
An attempt to apply metamodelling to identification of microstructure evolution model was not successful. Identification based on the stress relaxation tests does not require long FE calculations; therefore, an advantage from using the metamodel was negligible.
Application of metamodelling to identification of phase transformation model showed that this task requires two metamodels, PPN and MPL networks. The first gives information whether a considered transformation occurs and the second gives information about transformation temperature. Numerical tests showed that this twonetwork approach gives reasonably good predictions of transformation temperatures, but it was not successful in combination with the optimization methods in the inverse analysis.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors acknowledge the financial assistance of the NCN, Project no. N508 629 740.