A level set method for structural topology optimization

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Abstract

This paper presents a new approach to 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 well-founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints. The result is a 3D topology optimization technique that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation. We have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods. The benefit and the advantages of the proposed method are illustrated with several 2D examples that are widely used in the recent literature of topology optimization, especially in the homogenization based methods.

Introduction

The goal of this paper is to advance methodology for structural topology optimization. We present a powerful method based on level set models for optimizing linearly elastic structures which satisfy a design objective and certain constraints. In the proposed method, the structure under optimization is implicitly represented by a moving boundary embedded in a scalar function (the level set function) of a higher dimensionality. While the shape and topology of the structure may undergo major changes, the level set function remains to be simple in its topology. Therefore, by a direct and efficient computation in the embedding space, the movement of the design boundaries under a relevant speed function can be tracked to capture changes in the shape and topology of the structure. The level set models may also be referred to as implicit moving boundary (IMB) models and they can easily represent complex boundaries that can form holes, split into multiple pieces, or merge with others to form a single one. Based on the concept of propagation of the level set surface, the design changes are carried out as a mathematical programming for the problem of optimization.

We have developed a numerical procedure for the structural optimization problem using the level set models. Necessary conditions for the optimum solution and for the convergence of the procedure are derived. We have implemented the proposed algorithm with the use of several robust and efficient numerical techniques of level set methods. The benefit and the advantages of the proposed method are illustrated with several 2D examples that are widely used in the recent literature of topology optimization, especially in the homogenization based methods.

Section snippets

Background

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. The following is a brief review of the key approaches.

One main approach to structural design for variable topologies is the method of homogenization [1], [2], [3], [4], [5], [6], [7], [8], in which a material model with micro-scale voids is introduced and the

Level set models of implicit moving boundaries

In practice, the method of IMBs for structural optimization must be computed using some specific boundary representation. Generally speaking, it is desirable to have the boundary representation as general as the underlying physical theory. More importantly, the representation should not rely on any kind of explicit parameterization, along with no direct specification of the topology of the structure. These capabilities would allow the boundary models to easily change the structural topology

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 asMinimize∂DJ(u)=∫DF(u)dΩ,subjectto:∫DEijklεij(u)εkl(v)dΩ=∫DpvdΩ+∫∂DtτvdS,u|∂Du=u0∀v∈U,∫Dd

Optimization algorithm

With the formulation of Eq. (8) we now describe an optimization procedure. The optimization process operates on the scalar function Φ which is defined over the fixed domain D. The process can be implemented as a mathematical programming problem. The principal guideline for the optimization process is to move the design boundary represented by the level set model according to its variation sensitivities with respect of the objective function. The process would terminate when the objective cannot

Conditions of the optimum solution

In this section we shall derive the necessary conditions for the optimum solution of Eq. (5) and show the convergence characteristics of the algorithm described above. The topology optimization algorithm is developed based on the following proposition:

Proposition 1

The series of embedding function{Φ(x,ti),i=0,1,2,…}generated by the process of the optimization algorithm are the descent series of the topology optimization problem(5).

We shall describe a proof of the proposition. First, we derive the Fréchet

Numerical implementation

There are a number of numerical issues that are important to the implementation of the proposed level set method. In the algorithm presented here, the geometric boundary of the structure under optimization is described as the zero level set of Φ(x,t)=0. In its numerical implementation, the embedding function Φ may be represented in any convenient form. It is often described as a rectangular sampling on a rectilinear grid of x over D[18]. Conventional interpolation functions may be used on a set

Numerical examples

In this section we present several examples of structural optimization obtained with the proposed algorithm and implementation. The optimization problem of choice is the mean compliance problem that has been widely studied in the relevant literature (e.g., [8], [22]). The objective function of the problem is the strain energy of the structure with a material volume constraint, i.e.,J(u)=∫DEijklεij(u)εkl(u)dΩ.

For all examples, the material used is steel with a modulus of elasticity of 210 GPa

Conclusions

We have presented a numerical method for structural shape and topology optimization. The method relies on a novel approach to the representation of the design boundaries with level set models. A structural optimization is formulated as a mathematical programming problem with a design objective and a set of constraints, utilizing the level set models for the incremental shape changes. The movements of the IMBs of the structure are driven by a transformation of the objective and the constraints

Acknowledgements

This research work is supported in part by the USA National Science Foundation (CMS9634717), the Chinese University of Hong Kong (Direct Research Grant 2050254), the Ministry of Education of China (a Visiting Scholar Grant at the Sate Key Laboratory of Manufacturing Systems in Xi’an Jiaotong University), and the Natural Science Foundation of China (NSFC) (grant no. 59775065 and Young Overseas Investigator Collaboration Award 50128503).

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