# Inverse Approach of Parameter Optimization for Nonlinear Meta-Model Using Finite Element Simulation
## 1 Introduction  
The number of safety and environmental problems encountered in the construction and automotive industries is continuously increasing [1, 2]. Therefore, the utilization of lightweight materials and high-strength materials for user safety is increasing [3, 4]. However, part design and plastic working for such high-strength materials become difficult owing to the nonlinear properties resulting from hardening during plastic working. A typical method for identifying the nonlinear properties of materials is a tensile test, and tensile test results can be expressed using a force-displacement graph. This graph is then converted to an engineering stress-strain curve [5]. During this process, the curve can be transformed into a true stress-strain curve, which reflects the changes in the cross-sectional area, using an extrapolation, interpolation, or regression method [6, 7, 8, 9, 10, 11]. An extrapolation, interpolation, or regression model is used as a meta-model in the form of a constitutive equation to simulate the test data. A property meta-model (typically called a material model, hardening model, etc. ), which is an approximate model that accounts for practical physical phenomena, is widely used in engineering design and optimization processes. Property meta-models are used to represent elastic and plastic properties against the strength of materials [12]. In general, elastic properties are expressed by a linear function such as the Hook model, whereas plastic properties are expressed by power-law, exponential, and combination models [13, 14, 15, 16, 17, 18, 19]. When a curve-fit meta-model is applied to afinite element (FE) simulation, an engineering stress-strain curve is obtained, along with the differences between the FE simulation results and the actual test results. Thus, the inverse method is being investigated to reduce the differences between these results by applying a meta-model for the FE simulation in order to obtain the true stress-strain curve [20; 21; 22]. The inverse method is a meta-model-based optimization process that searches for the optimal property values by repeatedly performing FE simulations and updating the parameters of a meta-model to reduce errors. For the inverse method, nonlinear models such as power-law and exponential models can be considered as property meta-models. In addition, a modified meta-model such as a combination model can be considered to represent the nonlinear properties of high-strength, large-strain materials [23; 24; 25; 26]. These meta-models are being researched to increase the order or term for representing nonlinear properties [27; 28]. When the parameters of a meta-model increase, the resources and costs required by the inverse method increase. Therefore, a meta-model that exhibits a higher accuracy with a smaller number of parameters than that used for the typical meta-models is required. To compare the estimated performance of the properties for different meta-models, this study compared the RMSE values, the number of iterations, and the sum of FE simulations of the typical meta-models with those of the proposed advanced meta-model corresponding to the plastic property of high-strength materials. First, the engineering stress-strain curves were obtained by conducting ASTM E8/E8M [29] uniaxial tensile tests for 10705 MBU, a high-strength metal. Then, curve fitting was performed for the true stress-strain curves of three typical property meta-models (i.e., Gosh, Hockett-Sherby, and combination meta-models) and the advanced meta-models (Gaussian mixture, sum of sine, and polynomial). The MATLAB Curve Fitting Toolbox (CFT) was used for the parameter initializing process. In addition, HyperMesh was used for FE modeling and LS-DYNA for FE simulations. The property meta-models applied to the FE simulation were optimized using LS-OPT. The simulation and optimization results were compared in terms of the RMSE values and the iteration of convergence.