# Multi-Scale Concurrent Topology Optimization Based on BESO, Implemented in MATLAB
## 1 Introduction  
A topology optimization problem formulation can be adopted to determine the optimal geometry for either the structural domain, the material, or both, with the objective of minimizing, e.g., structural compliance while considering a specific volume fraction. The structural domain involved in such a problem consists of a periodically repeated cellular material, which is represented by a unit cell. To solve these types of Topology Optimization (TO) problems, two approaches are commonly employed: the mono-scale approach and the multi-scale approach. In the multi-scale approach, establishing a connection between the different scales can be achieved through various methodologies. One popular method is the utilization of the Homogenization method. This technique enables the translation of the periodic unit cell geometry into a homogenized elasticity tensor, which can then be employed in the finite element analysis of the structural domain. The theoretical foundations of the homogenization methodology were initially presented by [PERSON] and [PERSON] in [1] and [PERSON] in [2]. The process of numerical homogenization involves implementing the homogenization theory using finite elements and the finite element analysis method to compute the homogenized elasticity tensor. Further insights into numerical homogenization can be found in the works of [PERSON] and [PERSON] in [3] as well as [PERSON] and [PERSON] in [4, 5, 6]. The homogenization theory has been effectively employed to solve the TO formulation in the context of homogenization-based TO problems, as demonstrated by [PERSON] and [PERSON] in [7]. Moreover, [PERSON] and [PERSON] in [8] utilized homogenizationto compute the mechanical properties of various periodic unit cells with different relative densities. Notably, numerical implementations of the homogenization method in the form of MATLAB codes have been provided by [PERSON] and [PERSON] in [9] for 2D spaces and by [PERSON] et al. in [10] for 3D spaces. These resources offer practical tools for applying the homogenization method in topology optimization. The optimization procedure of the multi-scale TO problem formulation can be implemented in either hierarchical or concurrent form. In the case of the hierarchical implementation, introduced by [PERSON] et al. in [11] the optimization procedure consists of two separate problems, the inner and outer problem. In the inner problem, the geometry of the unit cell is optimized for a fixed structural domain. In the outer problem, the geometry of the structural domain is optimized for a fixed periodic unit cell. Thus, at every optimization step, the inner and outer problems are solved hierarchically, providing each other with new geometries. In the concurrent optimization procedure, the geometries of the micro and macro-scales are updated simultaneously, like the procedure presented by [PERSON] et al. in [12]. This is achieved by augmenting the TO problem with the two sets of design variables representing the two different scales. The first set of design variables consists of the relative material densities of the finite elements discretizing the structural domain, whereas the second set of design variables consists of the relative material densities of the finite elements discretizing the periodic unit cell. Thus, in the presented methodology, the periodic unit cell geometry is optimized by directly changing all parts of the unit cell geometry through the assigned material densities. To gain additional insights into the multi-scale TO formulations, readers are encouraged to refer to the review published by [PERSON] et al. in [13]. The solution of the concurrent multi-scale TO formulation is based on the BESO approach. BESO in contrast to the popular SIMP approach is based on the discrete implementation of the TO problem and is categorized as a soft-kill approach. The BESO-based TO optimization approach was based on the Evolutionary Structural Optimization (ESO) approach, which was first introduced by [PERSON] and [PERSON] in [14; 15; 16].