Mathematical Problems in Engineering

Volume 2018, Article ID 9804123, 12 pages

https://doi.org/10.1155/2018/9804123

## Parametric Level Set-Based Multimaterial Topology Optimization of Heat Conduction Structures

School of Sciences, Chang’an University, Xi’an 710064, China

Correspondence should be addressed to Yadong Shen; moc.621@770dys and Jianhu Feng; nc.ude.dhc@1002105102

Received 8 May 2018; Accepted 27 August 2018; Published 16 September 2018

Academic Editor: Filippo de Monte

Copyright © 2018 Yadong Shen and Jianhu Feng. 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

This paper presents a parametric level set-based method (PLSM) for multimaterial topology optimization of heat conduction structures with volume constraints. A parametric level set-based optimization model of heat conduction structures is built with multimaterial level set (MM-LS) model, which describes the boundaries of different materials by the combination of all level set functions. The heat dissipation efficiency which means the quadratic temperature gradient is conducted as the objective function. The adjoint method is utilized to calculate the sensitivities of the objective function with respect to expansion coefficients of the compactly supported radial basis functions (CSRBFs). The optimal configuration is achieved by updating the expansion coefficients gradually with the method of moving asymptotes (MMA). Several numerical examples are discussed to demonstrate effectiveness of the proposed method for multimaterial topology optimization of heat conduction structures.

#### 1. Introduction

In a high temperature environment, the performance, and efficiency of the structure tend to decrease. Therefore, the layout design of heat conduction structures is an important issue in the field of structural design. In conventional methods, size optimization and shape optimization are used in design of heat conduction structures. Unlike these passive methods, topology optimization is an active method to find the optimal layout of materials for achieving the minimized objective within a given design domain. Based on the advantages of convenience and high efficiency, topology optimization becomes an effective way for the design of heat conduction structures [1].

In recent years, different topology optimization methods have been applied to the layout design of heat conduction structures. Variable density method with material interpolation model of solid isotropic material with penalization (SIMP) was first introduced by Sigmund [1] to solve the heat conduction problem by using optimality criteria (OC) in two-dimensional space. And it is also extended into three-dimensional space by Liu et al. [2]. Gersborg-Hansen et al. [3] applied the variable density method into heat conduction problem, and the optimal structure was obtained by the finite volume method. Li et al. [4, 5] employed the evolutionary structural optimization (ESO) to solve the steady heat conduction problem. In this method, material elements which had less contribution to the objective function were allowed to be removed gradually. Bidirectional evolutionary structural optimization (BESO) also had been utilized to solve heat conduction problem by He et al. [6]. The optimal structure was achieved by gradually adding material elements as well as removing according to a certain criterion. However, these methods involve the problems of checkerboards pattern and mesh-dependent phenomena. Furthermore, the optimal structures which have zigzag boundaries are hard to manufacture.

Comparing to SIMP, ESO, and other material distribution methods, level set method (LSM) has the following advantages: the structural boundary is described by geometrical parameters explicitly which is easier to convert to CAD model for manufacturing; it is without checkerboards pattern and mesh-dependent phenomena; and the optimal configuration does not depend on the finite element meshing. Based on these advantages of LSM, it has been widely studied in different problems, such as elastic structure problem [7, 8], vibration problem [9], and compliant mechanisms problem [10]. For heat conduction problem [11], Ha and Cho [12] developed the weak form of equilibrium equation for heat conduction problem. The adjoint method and the Lagrange multiplier method were employed to derive the shape design sensitivity. Zhuang et al. [13] investigated the topology optimization of heat conduction problem under multiple load cases and developed an effective numerical procedure to implement the topological structure. Kim et al. [14] developed the topology optimization method which used the combination of topological derivatives to improve convergence of optimization process as well as avoid local minimum. However, there are also some shortcomings in classic level set method. Firstly, as the optimal structure is obtained by solving the Hamilton-Jacobi partial differential equation (HJ-PDE) with explicit time-marching algorithm, and the equations only can be implemented under some strict conditions [15, 16], it has a low computational efficiency. Secondly, the Courant-Friedrichs-Lewy (CFL) condition and reinitialization are also needed to keep the numerical stability. In addition, the optimal structure generally depends on the initial topology as the new holes cannot be created in the design domain.

Parametric level set-based method (PLSM) solves the topology optimization problems effectively without directly solving the HJ-PDE. In this method, the implicit level set function is interpolated by radial basis functions (RBFs); the time dependence of the implicit function is transformed into the time dependence of interpolation coefficients. Therefore, the HJ-PDE is converted into a series of ordinary differential equations which are easier to handle, and the propagation of boundaries is driven by updating interpolation coefficients using mathematical programming methods. The PLSM not only inherits the favorable features of the conventional LSM but also avoids the difficulties in solving the HJ-PDE. Wang et al. [17] applied globally supported radial basis functions (GSRBFs) into topology optimization of minimum compliance design. Their work illustrated the efficiency and accuracy of RBF-based LSM. Luo et al. [18] introduced the compactly supported radial basis functions (CSRBFs) [19, 20] into topology optimization of compliant mechanisms and established the optimization model of a nonconvex objective function with specified constraints. The evolution of the boundary was the process of updating the interpolation coefficients using method of moving asymptotes (MMA) [21]. Luo et al. [22, 23] also proposed effective methods for structural stiffness design which combined the CSRBFs with well-established optimization algorithms, such as MMA and OC. With the initial level set functions being interpolated by CSRBFs, the optimization algorithms were applied to iteratively updating the interpolation coefficients to achieve the optimal structure.

Up to date, multimaterial topology optimization design has been widely investigated [24–27]. LSM based method is an effective way that the boundaries of different materials can be described explicitly using the combination of all the level set functions. Wang et al. [28] developed the level set model of multimaterial with the minimization of the mean compliance under the volume constraints. They also showed the effectiveness of this approach in compliant mechanisms design of multimaterial [10]. Zhuang et al. [29] presented the multimaterial level set model of the heat conduction structures. The optimization problem was formulated as minimizing the heat compliance under the volume constraints for different materials. PLSM also had been applied into multimaterial topology optimization. Wang et al. [30] proposed a new multimaterial level set (MM-LS) model, in which materials and void material were represented by level set functions. The optimal configuration was achieved by iteratively updating the interpolation coefficients of different level set functions until convergence. Wang et al. [31] investigated the optimization design of mechanical metamaterials for multimaterial by PLSM. In their work, they utilized the homogenization method to evaluate the properties of microstructure, while PLSM was used to implement the evolution of microstructure. The combination of all level set functions represented the boundaries of multimaterial microstructure. Chu et al. [32] applied the stress interpolation scheme into design of compliant mechanisms for multimaterial. The final results with the proper distribution of multimaterial decreased the output displacement and the compliance of compliant mechanisms and satisfied the stress constraints of different materials simultaneously.

In this paper, the multimaterial topology optimization model of the heat conduction structures using PLSM is investigated. The optimization is conducted to minimize the quadratic temperature gradient in the whole design domain and subject to the volume constraints of different materials. The remainder of this paper is organized as follows. Section 2 reviews the MM-LS model for topology optimization of multimaterial. Section 3 discusses the multimaterial topology optimization model for heat conduction structures based on PLSM. Section 4 gives the optimization algorithm. Section 5 presents some numerical examples to demonstrate the validity of the proposed method. Conclusions are given in Section 6.

#### 2. MM-LS Model

In this section, the MM-LS model for multimaterial topology optimization is reviewed, in which level set functions are used to describe materials and the void phase.

##### 2.1. Level Set Representation for Multimaterial

In level set-based topology optimization, zero level sets embedded in a higher dimensional scalar function are used to describe the structural boundaries of different materials. The level set functions are used to denote the following:where is the th level set function, is the design domain, is the domain with the positive value of the th level set function, and denotes the zero level set of the th level set function.

A pseudo-time is introduced to manage . The HJ-PDE can be solved by differentiating both sides of with respect to :where is the normal velocity of the th level set function, as the normal component of velocity contributes to the shape evolution of the boundary. The finite difference method like up-wind scheme [33] is used to solve the HJ-PDE.

##### 2.2. MM-LS Model

To solve the multimaterial design problem, the MM-LS model is applied to represent the different materials. In this method, a combination of level set functions is used to describe the phases, including materials and the void area. For an -material structure, the characteristic function for the th material can be defined as follows:where is the Heaviside function of th level set function. The MM-LS model which involves two materials and the void area by the combination of two level set functions is illustrated in Figure 1.