Topology Optimization Using Hyper Radial Basis Function Network

Abstract

*† In this paper we present the application of hyper radial basis function network (HRBFN) as a topology description function (TDF). HRBFN is used to parameterize the material distribution (density) for topology optimization. Thus, the topology optimization problem is to determine the parameters governing HRBFN in order to satisfy certain design criteria. Here we present two kinds of problems: minimum compliance and maximum fundamental natural frequency design. HRBFN can be used to parameterize material density during the finite element analysis stage or after analysis i.e. to post-process the results obtained by traditional topology optimization. Both the solutions are presented in this paper. An efficient optimization algorithm to obtain the HRBFN parameters was developed that makes use of optimality criteria. Examples are presented to demonstrate the proposed approach as also a comparison with the traditional topology optimization. The advantage of proposed approach is that it can yield checkerboard-free manufacturable topologies using coarse-mesh FE analysis models as opposed to the traditional approach that requires fine meshes. This results in reduction in solution-time for a response using FE analysis while conserving the ability to yield smooth topology. We feel the proposed approach has a great deal of application in topology optimization type of problems. I. Introduction OPOLOGY optimization 1 aims at obtaining the “best” possible arrangement of the given volume of structural material within a spatial domain to obtain an optimal mechanical performance of the concept design. The topology optimization is a material distribution problem in which the material content of each of the points of reference domain must be determined. The traditional approach to solve this problem is to mesh the reference domain and obtain the material content of each element to determine the optimal material distribution. If the material content of an element is above the cut-off level (corresponding to the percentage of total material), material density of 1 is assigned to that element, whereas if the material content of an element is below the cut-off level, void (or material density of 0) is assigned to that element. Also, various methods have been suggested by previous researchers to model the intermediate densities (eg. level set method 2 ) to improve the performance of this traditional approach. All these methods compute the optimal density of each element and hence there must be as many design variables as the number of finite elements. The number of elements must be sufficient to obtain a correct representation of geometrical features as well as obtain a correct physical response. The idea of using topology description function 3, 4 (TDF) stems from the fact that the number of elements required to represent the geometrical features far exceed that required to obtain a correct physical response. As such, describing the geometrical features using a TDF instead of the traditional approach would lead to a drastic reduction in number of parameters required to describe the geometrical features, assuming that a TDF can be represented by a small number of parameter as compared to the corresponding number of elements required. This would permit the use of a coarse mesh (sufficient to obtain correct physical response) in solving the FE analysis problem. In this work, the application of hyper radial basis function (HRBFN) as TDF is presented. The layout of the rest of the paper is as follows. HRBFN is briefly described in the second section. In the third section HRBFN is coupled with FE analysis to obtain compliance (strain energy) and the fundamental natural frequency of vibration. Semi-analytical sensitivity (to the HRBFN parameters) analysis for the aforementioned two types of problems is briefly outlined. The fourth section describes the algorithm to solve the topology optimization problem based on optimality criteria