# Multi-Scale Concurrent Topology Optimization Based on BESO, Implemented in MATLAB
## 1 Introduction
The focus of the ESO-based TO problem was to slowly remove elements from the structural domain during the optimization procedure. The initial main criterion for the element removal was set as the stress field. [PERSON] et al. in [17] proposed a change in the criterion from stress to strain energy. The modified version of ESO called the BESO approach was proposed by [PERSON] et al. in [18; 19] in which new elements can be added by the algorithm. To avoid completely removing elements during the optimization procedure, a soft-BESO approach is proposed. [PERSON] et al. in [20] and [PERSON] and [PERSON] in [21] utilized material interpolation schemes similar to the Solid Isotropic Material with Penalization (SIMP) to include the soft elements during the optimization procedure. The soft-kill version of BESO combined with a modified version of SIMP is utilized in the presented implementation. Regarding multi-scale concurrent topology optimization (TO) and optimal material design, several notable studies have been conducted. [PERSON] et al. in [22; 23] introduced a formulation for optimal material design encompassing cellular and composite materials, employing the BESO approach. In a similar vein, [PERSON] et al. in [24] proposed a concurrent multi-scale TO formulation utilizing the BESO approach for both macro and micro-scales, employing composite microstructures of two-phase materials. Furthermore, [PERSON] et al. in [25] presented a formulation concerning multi-scale TO based on the BESO approach, utilizing dynamic compliance as the objective function for the formulation. Regarding the TO formulation, many different code implementations have been published. The first code implementation was published by [PERSON] in [26] and is referred to as top99, 99 lines of MATLAB code for the SIMP-based TO formulation in 2D. The top99 was later formatted to the top88, 88 lines of MATLAB code published by [PERSON] et al. in [27]. Later, the top99 code was expanded to the 3D space by [PERSON] and [PERSON] in [28] and a more optimized version for both 2D and 3D by [PERSON] et al. in [29]. Based on the SIMP MATLAB implementations, [PERSON] and [PERSON] in [30] published a BESO-based TO MATLAB implementation for the 2D space. Based on the homogenization method and their MATLAB implementations mentioned above, [PERSON] and [PERSON] in [31] as well as [PERSON] and [PERSON] [32; 33] presented MATLAB implementations for the optimal design of materials using the TO formulation. In these implementations, only the geometry of the periodic unit cell was optimized. Lastly, [PERSON] et al. [34] presented a MATLAB implementation for the multi-scale concurrent TO formulation based on the SIMP approach. In this optimization problem, the structural domain consists of periodically repeated cellular material in the form of a unit cell. Thus, the focus of the optimization procedure is to find the optimal geometry of both structure and material for minimizing the structural compliance for a given volume fraction. These types of Topology Optimization (TO) problem formulations can be solved using two different approaches, the mono-scale and multi-scale approach. In the mono-scale approach, only one scale is considered for the optimization procedure, whereas in the multi-scale approach, typically two different scales are considered for the optimization procedure. These scales consist of the macro-scale and micro-scale. The macro-scale refers to the scale of the structural domain, whereas the micro-scale refers to the scale of the periodically repeated unit cell representing the cellular material. The main objective of this work is to present a simple formulation for the multi-scale concurrent topology optimization problem based on the Bi-directional Evolutionary Structural Optimization (BESO) approach. In the presented formulation, the multi-scale approach will be implemented for the solution of the optimization procedure. This study is structured into seven sections, together with the Introduction and Conclusions sections.