# Cellular Automata Approach to Topology Optimization of Graded Multi-Material Structures
###### Abstract
extended towards topology optimization of multi-material structures, including those of functionally graded material properties. The concept of gradation in material composition started in 1972 for polymeric materials. Since then, Japanese researchers began to develop that idea and established the concept of FGM in 1985. Since the late 1990s, the FGMs have evolved dynamically, induced by fundamental investigations on the physical and chemical properties of the newly invented materials. Their unique properties, but also the development of design methods and implementation of manufacturing techniques have led to the great success of the FGM concept. The detailed, impressive history of FGM, including manufacturing, modeling, and designing, together with a literature review, one can find, for example, in recently published papers [3]. The multi-material topology optimization can involve two approaches. The first one regards searching for the optimal distribution of multiple materials in the specified design domain in order to meet design criteria, e.g., [4; 5; 6; 7; 8; 9; 10; 11], designing interfaces made of functionally graded materials (FGM) between them [12; 13] or generation of optimal layout and gradation of the FGM in the specified design domain, e.g., [14; 15; 16; 17; 18]. The extended literature overview on these subjects one can find also in [6] or [19]. As to the related studies, the wide topic of topology optimization of functionally graded cellular materials can also be mentioned, e.g., [20; 21]. The second approach assumes that the optimal layout of the structure is sought for a predefined material properties distribution, e.g., a predefined FGM layout. This approach was investigated first in [22; 23; 24; 25] and has been continued for topology optimization of functionally graded piezoelectric structures, which can be illustrated by papers [26; 24]. While mentioning the above approach, the paper [27] can also be pointed out, where optimization of multi-material and FGM structures including cracks was investigated. Although the optimal distribution of two and three materials is the main part of the article, examples of predefined FGM are also investigated [27]. Based on the above-mentioned information, one can observe that the number of papers discussing topology optimization of graded multi-material structures is rather low. Additionally, according to the authors' knowledge, papers on topology optimization of graded multi-material structures using efficient heuristic methods are practically absent in the literature. Meanwhile, heuristic techniques are easy for numerical implementation, do not require gradient calculations, and can easily be combined with any finite element structural analysis code. These features caused these techniques to have gained popularity within the research community. It is therefore proposed to implement the efficient heuristic algorithm based on the concept of Cellular Automata into the optimization of FGM structures. The concept of Cellular Automata (CA) was proposed by [PERSON] and [PERSON] in the late 1940s. The idea of CA is to mimic the behavior of complex systems by applying simple rules built based on local information. This method, thanks to its efficiency and simplicity, has been found useful by researchers representing various branches of modern engineering, including topology optimization. The Cellular Automata approach was discussed within the topology optimization area for the first time by [PERSON] et al. [28], but the majority of papers appeared only during the last two decades, see [29; 30; 31; 32; 33; 34; 35; 36]. This paper presents the idea of Cellular Automata utilized in the topology optimization of graded multi-material structures, that is optimization of FGM structures as well as multi-material structures including the FGM interface. The optimal topologies of structures are sought under predefined material properties distribution. The outline of the paper is as follows.