Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 6392901, 14 pages

http://dx.doi.org/10.1155/2016/6392901

## Efficient Local Level Set Method without Reinitialization and Its Appliance to Topology Optimization

School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China

Received 5 July 2015; Accepted 21 December 2015

Academic Editor: Manuel Pastor

Copyright © 2016 Wenhui Zhang and Yaoting Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The local level set method (LLSM) is higher than the LSMs with global models in computational efficiency, because of the use of narrow-band model. The computational efficiency of the LLSM can be further increased by avoiding the reinitialization procedure by introducing a distance regularized equation (DRE). The numerical stability of the DRE can be ensured by a proposed conditionally stable difference scheme under reverse diffusion constraints. Nevertheless, the proposed method possesses no mechanism to nucleate new holes in the material domain for two-dimensional structures, so that a bidirectional evolutionary algorithm based on discrete level set functions is combined with the LLSM to replace the numerical process of hole nucleation. Numerical examples are given to show high computational efficiency and numerical stability of this algorithm for topology optimization.

#### 1. Introduction

Topology optimization is a numerical iterative procedure for making an optimal layout of a structure or the best distribution of material in the conceptual design stage [1]. The level set method (LSM) is a recently developed approach to topology optimization that uses a flexible implicit description of the material domain [2]. The central idea of the LSM is to employ an implicit boundary describing model to parameterize the geometric model, and the boundary of a structure is embedded in a high-dimensional level set function that is called its zero level set [3]. The level set-based method is able to not only fundamentally avoid checkerboards and mesh-dependence, but also maintain smooth boundaries and distinct material interfaces during the topological design process [4]. Hence, many level set-based methods [5] have been developed for topology optimization since the LSM was first introduced into structure optimization.

With an implicit local level set model, the computational efficiency of the local level set method (LLSM) [6] is much higher than that of the global level set methods, especially for shape optimization. However, the main shortcoming of the conventional LSM is that it possesses no mechanism to nucleate new holes in the material domain for two-dimensional structures, resulting in the final design heavily dependent on the initial guess [4]. A mechanism named the bubble-method [7] was first proposed to create new holes inside the structures in topology and shape optimization. This idea has been further developed into the mathematical concept of topological derivatives [8]. In the shape-sensitivity-based level set approaches, topological derivatives are incorporated to indicate the best place for introducing a new hole in a separate step of the optimization process [9] or as an additional term in the Hamilton-Jacobi equation [10]. The globally supported radial basis function (RBF) [11] and compactly supported RBF (CSRBF) [12] are typically used to discretize the original time-dependent initial value problem into an interpolation problem. The CSRBF brings about the strictly positive definiteness and sparseness properties of matrices under certain conditions. Hence the CSRBF has generalized the practical applications of RBFs to a larger set of scattered data [12].

In the conventional LSM [3], a reinitialization procedure usually needs to reshape the level set function (LSF) to a signed distance function (SDF) periodically. However, the zero level set may drift away from its initial position by iteratively solving a classical reinitialization equation [13]. To suppress this drift, an interface preserving level set redistancing algorithm is proposed by Sussman and Fatemi [14]. Nevertheless, it has been proved that the SDF is not a feasible solution to the equation [15]. In practice, it not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way and thus should be avoided as much as possible [16]. The need for reinitialization was originally eliminated by introducing a penalty term [17] into a variational level set approach [18]. The undesirable boundary effect of the penalty term can be eliminated by taking a distance regularized equation (DRE) instead of this term. Hence, a so-called distance regularized level set evolution (DRLSE) [16] is realized based on the variational approach. As an unnecessary diffusion effect of the DRLSE was found in some locations where the surface is too flat, the DRE was recently modified with a new and balanced formulation to eliminate this effect [19]. Although parts of the diffusion rates in the DRE are negative, the numerical stability can still be maintained by incorporating reverse diffusion constraints in the difference schemes of the DRE, as can the reverse diffusion equations with all negative diffusion rates [20].

The aim of this work is to solve the aforementioned numerical issues that still exist in the LLSM for topology optimization of two-dimensional structures. A bidirectional evolutionary algorithm based on the discrete level set functions (DLSFs) is proposed to find a stable topological solution first and then combined with the LLSM to further evolve the local details of the topology and shape of the structure. Transforming the DLSFs into the local level set function of the LLSM is achieved by iteratively solving the DRE. After that, the DRE is incorporated into the LLSM to avoid the reinitialization procedure. A difference scheme under reverse diffusion constraints is formulated for the DRE to improve its numerical stability. Typical examples are given to show the effectiveness of the proposed algorithm in terms of convergence, computational efficiency, and numerical stability.

#### 2. Optimization Algorithm

##### 2.1. Local Level Set Method Using Narrow-Band Model

In the local level set method (LLSM) [6], the local level set equation is defined aswhere is defined as the local level set function (LLSF) and is the normal velocity in normal direction ; the truncation function is with being a narrow band with the half-band width . The narrow-band model and the corresponding LLSF are described as shown in Figure 1.