# Approximation Method for Stress-Strain Using Metamodel Parameter Updating
###### Abstract  
The properties of the material applied to the finite element (FE) simulation can be expressed by constitutive models, and simple constitutive and complex constitutive models can be used to show the actual phenomenon. The technology to improve the accuracy of the constitutive model applied to FE simulation is the inverse method. The inverse method is a method to curve fit the FE simulation result to the test data by utilizing finite element model updating (FEMU). Inverse methods are general approaches to update material properties. The inverse method can iteratively run many FE simulations for constitutive model optimization and consider metamodel-based simulation optimization (MBSO) to reduce this resource waste. With MBSO, one can obtain significant results with fewer resources. However, the MBSO algorithm has the problem in that the optimization performance deteriorates as the number of parameters increases. The typical process of the inverse method is to adjust these factor values individually. If there are many factors in the constitutive model, the optimization result may deteriorate owing to the performance limit of the MBSO when the structural method is used. This paper proposes a method of fitting a stress-strain constitutive model with a scaling factor to improve the efficiency of the inversion method using MBSO. For this purpose, a process was performed to determine the curve characteristics during the pretreatment stage. The results show that the proposed method significantly improved the prediction efficiency of the combination function. Thus, we conclude that initializing the combination function and setting the parameters of the inverse method by applying the proposed approach improves the efficiency of large deformation analyses. t 2  
this relationship considers the cross-sectional area of the specimen as a variable [9; 10]. When a necking phenomenon occurs in a material, the stress-strain characteristics change non-linearly. Then, different stress-strain curve characteristics can be observed for each material [11; 12; 13; 14]. This nonlinearity of materials makes it difficult and complex to define the properties required for simulation [15; 16; 17]. A constitutive model can be used to solve this nonlinear problem. A constitutive model is an approximate model that can replace or represent the actual model of a specific phenomenon. It can be used as a means of optimization or to represent the aspects of resource-consuming tests and FE simulations in a simple equation [18; 19; 20]. Various attempts have been made to define the properties of constitutive models and parameter values for obtaining the true stress-strain curve; however, in many cases, they have been limited to only specific materials [21; 22]. To address this problem, methods to predict the properties of various materials have been investigated, and these attempts can be classified into two groups: direct method and inverse method. The direct method is based on the Bridgman method [23]. To determine the true stress, the actual triaxial strain of the specimen must be determined [24; 25]. Therefore, the equivalent stress at the center is predicted by considering the necking phenomenon that results in the specimen's change in shape at the center [26]. The direct method utilizes professional digital image correlation (DIC) equipment and image post-processing software to accurately measure the non-linearly changing central shape [27; 28; 29]. Professional equipment and software for DIC applications show highly accurate measurement results. However, purchasing equipment and software requires a lot of investment. If given simple boundary conditions with uni-axial loads, inverse methods are available as an alternative to professional equipment and software. The inverse method is based on continuously updating the material parameters of the FE simulation, and it requires two curve-fitting processes [30; 31; 32]. The curve fitting in the inverse method reduces the difference between the simulation results and the tensile test data [4].