Level-set method for design of multi-phase elastic and thermoelastic materials
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Springer 2005
Level-set method for design of multi-phase elastic and thermoelastic materials XIAOMING WANG1, YULIN MEI1 and MICHAEL YU WANG2,* 1 School of Mechanical Engineering Dalian, University of Technology, Dalian, 116024, China; 2Department of Automation & Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong *Author for correspondence (E-mail: [email protected])
Received 9 August 2004; accepted in revised form 7 January 2005 Abstract. The problem of designing composite materials with desired mechanical properties is to specify the materials microstructures in terms of the topology and distribution of their constituent material phases within a unit cell of periodic microstructures. In this paper we present an approach based on a multi-phase level-set model for the geometric and material representation and for numerical solution of a least squares optimization problem. The level-set model precisely specifies the material regions and their sharp boundaries in contrast to a raster discretization of the conventional homogenization-based approaches. Combined with the classical shape derivatives, the level-set method yields a computational system of partial differential equations. In using the Eulerian computation scheme with a fixed rectilinear grid and a fixed mesh in the unit cell, the gradient descent solution of the optimization captures the interfacial boundaries naturally and performs topological changes accurately. The proposed method is illustrated with several 2D examples for the synthesis of heterogeneous microstructures of elastic and/or thermoelastic composites composed of two and three material phases. Key words: material design, level-set method, composite material, microstructures, topology optimization
1. Introduction Material design refers to the synthesis of composite materials with prescribed properties. The materials are made of a single or multiple constituent material phases, and their desired properties may be highly special and unusual such as negative PoissonÕs behavior (Lakes, 1993). This might be accomplished with continuously varying material properties satisfying prescribed material conditions. The continuously varying material composition produces gradation in material properties, as they are often known as functionally gradient materials (FGM). Another method is to modify the microstructure of the composite materials to yield optimal components of the elasticity and/or thermoelasticity tensors. In this case, the material composition has a discontinuous change across the interface of material regions in the composite. This material heterogeneity is typically referred to as the structural topology (Bendsoe and Sigmund, 2003). The procedure to design a microstructure consists of finding an optimal distribution of material phases within a unit cell of periodic microstructures subject to volume fraction constraints of the constituent phases. There are three crucial ingredients in the mathematical modeling and computation of the design problem, which ar
