Research paper
An efficient gradient projection method for structural topology optimization

https://doi.org/10.1016/j.advengsoft.2020.102863Get rights and content

Highlights

  • A novel and efficient method is proposed which greatly improves the efficiency of the gradient projection method with neglectable loss of accuracy for structural topology optimization.

  • An analytical approximation is provided to calculate projection onto the feasible region with negligible computation and memory costs.

  • The proposed method is implemented in MATLAB and open-sourced for educational usage.

Abstract

This paper presents an efficient gradient projection-based method for structural topological optimization problems characterized by a nonlinear objective function which is minimized over a feasible region defined by bilateral bounds and a single linear equality constraint. The specialty of the constraints type, as well as heuristic engineering experiences are exploited to improve the scaling scheme, projection, and searching step. In detail, gradient clipping and a modified projection of searching direction under certain condition are utilized to facilitate the efficiency of the proposed method. Besides, an analytical solution is proposed to approximate this projection with negligible computation and memory costs. Furthermore, the calculation of searching steps is largely simplified. Benchmark problems, including the MBB, the force inverter mechanism, and the 3D cantilever beam are used to validate the effectiveness of the method. The proposed method is implemented in MATLAB which is open-sourced for educational usage.

Introduction

Since the introduction of topology optimization to structural design, it has been successfully applied to many different types of structural design problems through various optimization schemes including density methods [1], boundary variation methods (like the level-set approach [2], [3], [4]), intelligent optimization methods (like the genetic evolutionary method [5]), etc. A comprehensive review can be found from Ref. [6], [7], [8], [9].

One typical large class of topology optimization problems can be characterized by a nonlinear objective function which is minimized over a feasible region defined by bilateral bounds and a single linear equality constraint. In this paper, we focus on solving this class of optimal design problems which have a wide range of applications [3, 9]. The optimization problem can be mathematically expressed as:minO(x)=ITUs.t.{V(x)/V0=fKU=Foxminxixmaxwherex=[x1,x2,,xn]in which x is the vector of design variables. Vector l in (1) takes different values for different problems. For example, l = Fo for the minimum compliance design problem [3]. n is the number of elements used to discretize the design domain. O(x) is the objective function, U and Fo are the global displacements and the force vector respectively. K is the global stiffness matrix. V(x) and V0 are the material volume and design domain volume, respectively. Finally, f is the prescribed volume fraction.

To solve the above list problem, different methods have been proposed and among which, the Solid Isotropic Material with Penalization for intermediate densities method (SIMP) [1] is considered the most effective material interpolation scheme thus has been widely implemented in industrial applications. The scheme is formulated as follows:Ee(x)=xrE0where Ee is the Youngs modulus, x is the vector of design variables which are constrained to [0, 1], E0 is the stiffness of the material, r is the penalization factor. Using this scheme, different approaches, including the Optimality Criteria (OC) method [10, 11], the Method of Moving Asymptote (MMA) [12], the Sequential Linear Programming (SLP) method [13, 14], and the Feasible Direction Method (FDM) [15], [16], [17], etc., have been proposed to solve the structural topology optimization problem and each approach has its desirable characteristics.

The OC method tends to convert the optimization to an equation-solving problem using the KT condition, which is attacked by iteratively approaching the fixed point [10]. The MMA method transforms the original problem into a series of localized strictly convex approximating subproblem which is solved by a dual method [12]. Similarly, the SLP method involves sequentially solving an approximate linear subproblem using the linearized objective and constraint functions [14]. The FDM, by utilizing the gradient at the current point, provides a feasible decent search direction and iteratively approach the optimal value. As a major feasible direction method, the gradient projection method (GPM) projects each step onto the feasible region [18].

To the best of our knowledge, comparing with the OC, MMA, and SLP, which have been extensively explored, relatively fewer investigations are performed on GPM for structural topology optimization. This may because that, people found using rudimentary methods like the raw GPM with no specialized scaling and projection scheme are less efficient [8]. While in a few current studies, people found that, given proper specialization, the GPM can be simple and effective for the current problem and has its advantages [15, 19]. Therefore, in the current study, we devote to improve this method and its variant in solving the problem defined in Eq. (1).

When applying the GPM to large scale problems like structural topology optimization, the main difficulties we may face are list as follows:

  • Effectively and efficiently obtain proper scaling for the gradient [18]

  • Effectively and efficiently calculate the projection [18, 19]

  • Effectively and efficiently obtain a proper searching step [18, 19]

  • Numerical problems and others [20]

Although lots of research are devoted to alleviating these problems and details are discussed in the following section, there is still space for further improving the GPM for the current problem. Particularly, many effective engineering experiences in improving structural topology optimization are not fully exploited such as various effective sensitivity filters and post processing techniques [8, 15, 21, 22]. In a board sense, these techniques implicitly improve the scaling scheme and searching step. Whether we could propose a more effective method using these techniques is discussed in this paper, and some effective and efficient techniques are proposed.

The scope of the current study is given as follows. The problem statement is presented in Section 2. The proposed method is presented in Section 3. Section 4 provides three benchmark problems to validate the effectiveness of the proposed method. Concluding remarks are provided in Section 5.

Section snippets

Problem statement

For the structural topology optimization problem, the differentiable cost function O is minimized over a closed convex polyhedron X ⊂ ℝn. In the feasible direction method, one seeks for a feasible sequence xX with an iteration of the formxk+1=xk+αkdkin which αk∈ℝ++++ and xk+αkdkX for small enough αk [23, 24]. (By stating small enough αk, we mean that the step size should be chosen properly, so that the new iterate belongs to the feasible region.) The subscript i denotes the iteration

Efficiency improvement

Gradient modification in structural topology optimization is not a new technique and often appears in the form of a sensitivity filter to avoid certain kinds of local minimum [22]. Other forms like magnitude modifications where the magnitude is modified by its squared root can also be seen for accelerating the process of solving certain problems [32]. We found that the effect of the latter technique is similar to a well-known technique called gradient clipping [31].

Gradient clipping, when used

Numerical examples

Three benchmark examples including an MBB beam (minimum compliance design problem), a compliant force inverter mechanism (compliant mechanism design problem), and a 3D cantilever beam are provided in this section to demonstrate the performance of the proposed method.

Conclusion and discussion

In this work, we proposed a modified gradient projection method with improved efficiency and neglectable loss of accuracy for structural topology optimization problems. Through gradient clipping, as well as the modified projection, the efficiency of the original gradient projection method has been greatly improved. It should be noted that since the projection is approximated by an analytical expression, the calculation involves negligible computation and memory costs. In addition, the

Replication of results

The method proposed in this paper is implemented in MATLAB and open-sourced on GitHub for educational usage. (https://github.com/zengzhi2015/EGP) For commercial usage, please contact the authors.

Compliance with ethical standards

The authors declare that they have no conflict of interest.

CRediT authorship contribution statement

Zhi Zeng: Investigation, Methodology, Software. Fulei Ma: Conceptualization, Writing - original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under Grant No. 51805397 and No. 61805185 and the Open Fund of State Key Laboratory of Robotics and System (HIT) under Grant No. SKLRS-2019-KF-07

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