# Hierarchical Tensor Decomposition of Module Partition for the Mechanical Simulation
## 1 Introduction  
The rapid advances in fabrication techniques and the exponential increase in computational power in recent years have enabled researchers to explore new horizons in the fields of material and structural design. Accordingly, module partition [1] can be considered as a powerful numerical tool for optimizing the material arrangement of products at various scales, ranging from nanometers [2] to skyscrapers [3], depending upon the target application. The module partition has different names in different fields; for example, image segmentation [4] and feature extraction [5] are commonly used in the field of image processing. When it comes to mechanical simulation, simulation experiments are usually designed in conjunction with topology optimization [6]. A mechanical simulation using ANSYS [7] is usually required when studying the properties of mechanical structural materials. For accurate calculations of stress and strain fields, meshing is necessary for mechanical simulation experiments by ANSYS [8]. Dense meshes will provide a better representation of the actual stress distribution, whereas coarse meshes will introduce large deviations when it comes to stress concentration [9]. By partitioning the hierarchy of stress or strain contours, we can achieve more accurate simulations of structural materials [1]. Several methods have been developed to incorporate macroscale boundary conditions into the module partition process. For example, the hierarchical design method proposed by [PERSON] et al. [10] is based on the assumption that macroelements represent single-scale microstructures that are optimized iteratively using inverse homogenization, with minimal connectivity effects from points to points at the macroscale due to microstructureconfiguration variations. A novel optimization approach developed by [PERSON] et al. [11] is based on the assumption that the design space consists of a single designable microstructure whose material layout is optimized based on macroscale boundary conditions. Several researchers have proposed concurrent multiscale module partition to perform module partition at macroscales and microscales simultaneously [12]. An approach called concurrent multiscale topology optimization has been proposed in an attempt to optimize topology at both macro- and microscales [13]. In this method, module partition is performed to obtain both the optimal material layout at the macroscale and the optimal local material distribution at the finer scale. Various methods are used at macroscales to design partitions for achieving optimal topology, whereas at the local scale, they strive for optimal microstructure, generally by considering uniform microstructures throughout each macrodomain. By considering subdomains within the spatial macroscopic design domain, a grid-based hierarchical approach is applied in this regard, thereby providing the optimal solution that considers uniform microstructures across each macro subdomain [14; 15; 16]. To optimize macrostructures and microstructures, density-based multipatching has been applied [17; 18; 19]. Based on proximity or other variables, these methods optimize design at the macroscale. A clustering method based on principal stress/strain directions was developed by [PERSON] and [PERSON] [20], and a density- and strain-based clustering method were presented by [PERSON] and [PERSON] [21]. The macroscale design domain is subdivided into several subdomains by using a dynamic K-clustering strategy based on directions and ratios of principal stresses [22]. It is noteworthy that the K-clustering method used in these works is significantly different from that used in the current thesis in terms of topology optimization. The objective of this paper is to develop a multilayer module partition of the strain/stress contour based on the mechanical simulation by ANSYS in order to guide the improvements of mechanical simulation experiments.