Finite Element Mesh Decomposition Using Evolving Ant Colony Optimization

Abstract

Combinatorial optimization problems arise in many areas of science and engineering. Unfortunately, due to the NP (non-polynomial) nature of these problems, the computations increase with the size of the problem (Bahreininejad & Topping, 1996; Topping & Bahreininejad, 1997). Finite Elements (FE) mesh decomposition (partitioning) is a well known NP-hard optimization problem and is used to split a computationally expensive FE mesh into smaller subdomains (parts) for parallel FE analysis. Partitioning must be performed to ensure: • Load balancing: for a mesh idealized using a single element type, then the number of elements in each partition must be the same, and • Inter-processor communication: the partitions must be performed so that the number of nodes or edges shared between the subdomains is minimized to ensure that the minimum inter-processor communication during the subsequent parallel FE analysis is achieved (Topping & Bahreininejad, 1997; Topping & Khan, 1996). Numerous methods have been used to decompose FE meshes (Farhat, 1988; Simon, 1991; Toping & Khan, 1996; Topping & Bahreininejad, 1997). For automatic partitioning of FE meshes, Farhat (1988) proposed a domain decomposition method which is based on a greedy algorithm. The method provides FE mesh partitions in relatively short duration of time. The division of a mesh with respect to assigning a certain number of mesh elements to a mesh partition may be accomplished with simple arithmetic. In this method, the partitions are created sequentially from an overall FE mesh until the number of partitions become equal to the desired number. Each FE element node is assigned a weight factor which is equal to the number of elements connecting to that particular node. The inner boundary of a partition is defined as the common boundary between two partitions. Two elements are considered to be adjacent if they share a vertex (node). The number of elements per partitions is determined by the total number of elements in the mesh, the number of different type of elements used in the mesh (triangular, quadrilateral, etc.), and the number of required partitions. In the case of a single type of elements, it is equal to the ratio between the total number of elements within the mesh and the number of required partitions (Farhat, 1988). Although Farhat’s method provides quick partitioning of FE meshes, the optimality of the mesh partitions with respect to the number of interfaces between adjacent partitions is not