Level Set Models for Structural Topology Optimization

This paper presents a new model for structural topology optimization. We represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension. Such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure. Furthermore, a gradient-based procedure leads to a numerical algorithm for the optimum solution satisfying specified constraints. The result is a 3D topology optimization technique with outstanding flexibility of handling topological changes, without resorting to homogenization-based relaxations that are widely used in the recent literature. INTRODUCTION Structural optimization, in particular the shape and topology optimization, has been identified as one of the most challenging tasks in structural design. Various techniques and approaches have been developed during the past decade. One main approach to structural design for variable topologies is the method of homogenization [1-8], in which a material model with micro-scale voids is introduced and the topology optimization problem is defined by seeking the optimal porosity of such a porous medium using one of the optimality criteria. A number of variations of the homogenization method have been investigated to deal with these issues by penalization of intermediate densities, especially the “solid isotropic material with penalization” (SIMP) approach for its conceptual simplicity [9-11]. Material properties are assumed constant within each element used to discretize the design domain and the design variables are the element densities. The material properties are modeled to be proportional to the relative material density raised to some power. The power-law based approach to topology optimization has been widely applied to problems with multiple constraints, multiple physics and multiple materials. However, numerical instability and computational complexity remain to be the major difficulties for realistic requirements. A simple method for shape and layout optimization, called “evolutionary structural optimization” (ESO), has been proposed by Xie and Steven [12] which is based on the concept of gradually removing material to achieve an optimal design. This approach is essentially based on an evolutionary strategy focusing on local consequences but not on the global optimum. It is typically computationally expensive. A similar approach called “reverse adaptivity” was proposed by Reynolds et al. [13] at which a fixed percentage of relatively under-stressed material is removed to find approximately fully stressed structures. Adopting the same principle of redesigning the structure based on the stress distribution in the current design, another approach was developed by Sethian and Wiegmann [14] with a focus on the resolution of the boundaries. An explicit jump immersed interface method is used for computing the solution of the elliptic problem in complex geometries without using meshes. The approach is also an evolutionary one. Our proposed method is to use implicit, moving boundaries for topology optimization. The structural boundaries are viewed as moving during the optimization process – interior boundaries (or holes) may merge with each other or with the exterior boundary and new holes may be created. Our idea is to combine level set methods [14,15] for the boundary representation and a mathematical programming method for optimization. LEVEL SET MODELS OF BOUNDARY REPRESENTATION We use the method of level set models [15-17] for an implicit representation of the structural boundaries. The fundamental concept of level set methods is described here for 1 Copyright © 2003by ASME a general three-dimensional structure with surface boundaries to provide necessary background for later sections. A level set model specifies a surface in an implicit form as an iso-surface of a scalar function, Φ , embedded in 3D, i.e., R R : 3 a ( ) { k x x S = Φ = : } (1) where is the iso-value and is arbitrary, and k x is a point in space on the iso-surface . In other words, Φ x is the set of points in 3 R that composes the k iso-surface of Φ . The embedding can be specified in any specific form, e.g., as a regular sampling on a rectilinear grid. A process of structural optimization can be described by letting the level set function dynamically change in time. Thus, the dynamic model is expressed as Φ ( ) ( ) ( ) ( ) { k t t x t x t S = Φ = , : } (2) By differentiating both sides of Eq. (2) with respect to time and applying the chain rule, we obtain the so-called “HamiltonJacobi-type” equation ( ) ( ) 0 , , = Φ ∇ + ∂ Φ ∂ dt dx t x t t x (3) This equation defines an initial value problem for the time dependent function . Φ In this dynamic level set model, the structural optimization process can be viewed as follows. Let dt dx be the movement of a point on a surface driven by the objective of the optimization, such that it can be expressed in terms of the position of x and the geometry of the surface at that point. Then, the optimal structural boundary is expressed as a solution of a partial differential equation on Φ : ( ) ( ) ( ) ( ) Φ Γ ⋅ Φ −∇ ≡ Φ −∇ = ∂ Φ ∂ , x x dt dx x t x (4) where denotes the “speed vector” of the level set surface, which depends on the objective of the optimization. ( Φ Γ , x ) This formulation with level set models has two major theoretical and practical advantages over conventional surface models, especially in the context of topology optimization. First, level set models are topologically flexible. The 3D scalar function is defined to always have a simple topology; its level sets can easily represent complicated surface shapes that can form holes, split to form multiple boundaries, or merge with other boundaries to form a single surface. There is no need to re-parameterize the model as it undergoes significant changes in shape, in contrast to any conventional boundary shape design [15]. Further, the models can incorporate a large number of degrees of freedom and a number of numerical techniques have been developed [15] to make the initial value problem of Eq. (4) computationally robust and efficient. In fact, the computational complexity can be made proportional to the level set’s surface area rather than the size of the volume in Φ which it is embedded. We shall describe the details of our proposed approach as follows. THE LEVEL SET FORMULATION In this section we present a formulation of the level set method for finding the optimum design of a linearly elastic structure. In this context the optimum design of the structure includes information on the topology, shape and sizing of the structure and the level set models allow for addressing all three problems simultaneously. In the general case, the problem of structural optimization can be specified as