# Structural Topology Design Optimization Using the Binary Bat Algorithm
## 1 Introduction  
Structural optimization is considered as one of the most important fields in engineering because it can propose innovative designs that satisfy different multidiscipline design requirements. There are three kinds of structural optimization: geometry, size, and topology. Geometry optimization involves finding the best arrangement of the joints. In size optimization, the main goal is to find the dimensions of the elements where the locations of the joints are fixed. Structural topology optimization is used to determine the best number, location, and size of voids as well as the best connectivity of the design domain. For any structural topology optimization problem, the applied loads, the support condition, the possible design requirements or any other restrictions, and the volume of structure to be constructed are the only known quantities. Then, in an iterative manner, the material is redistributed to meet the design requirements [1]. Structural topology optimization methods have received much attention in the relevant literature. They have been improved because they are very attractive for use in many industrial applications in civil and mechanical engineering [2]. These methods can help designers select suitable structures and reduce the effort required by trial and error. In structural topology optimization, similar to any other type of optimization, two methods are used: the deterministic (i.e., calculus-based or gradient-based) method and stochastic (i.e., metaheuristic or nature-inspired) algorithms. In the deterministic method, the Solid Isotopic Microstructure with Penalization (SIMP) or the power law method introduced by [PERSON] [1] is a generally accepted and efficient tool for structural topology optimization. In the SIMP method, the design variables are the density of each element, and the material property of each element is constant [3]. The gradient-based SIMP method is used to search for the optimum point in the neighborhood of the current estimate. It tends to converge to a local optimum with a blurry boundary. Moreover, the SIMP method cannot directly solve 0/1 problems, so it requires a filtering method to resolve the final design [4; 5; 6]. The Canonical Duality Theory (CDT) developed by [PERSON] [7] is used to analytically solve structural topology optimization problems with solid or void material distribution. The numerical results of solving benchmark topology optimization problems showed that the CDT method is powerful in comparison with SIMP and Basic Evolutionary Structural Optimization (BESO) methods in terms of computational time, the final compliance, not generating grayscale areas, and fewer checkerboard patterns [8; 9; 10]. For problems with mesh sizes of \(180\times 60\) and \(150\times 50\), the CDT method showed some checkerboard patterns. Rational Approximation of Material Properties (RAMP) developed by [PERSON] and [PERSON] [11] and the generalized form of it (called GRAMP) were used to solve two benchmark topology optimization problems for minimum compliance. Using a penalization factor of value equal to 3, [PERSON] showed that GRAMP was able to produce an almost \"0-1\" layout of material [12]. In comparison with the SIMP method, GRAMP and RAMP were able to obtain designs with fewer grayscale areas and better compliance. However, GRAMP and RAMP were not able to produce exactly void/solid material distribution. In addition to the compliance minimization topology design methods, the stress-constrained topology optimization methods have received much attention since the pioneering work of [PERSON] and [PERSON] [13]. It involves stress-related objectives or constraints to overcome three main issues in the minimizing compliance design: highly non-linear stress behavior, the local nature of stress constraints, and the stress singularity [14; 15; 16].