Mathematical Problems in Engineering

Volume 2015, Article ID 521482, 11 pages

http://dx.doi.org/10.1155/2015/521482

## Improved Genetic Algorithm with Two-Level Approximation for Truss Optimization by Using Discrete Shape Variables

School of Astronautics, Beihang University, XueYuan Road No. 37, HaiDian District, Beijing 100191, China

Received 25 September 2014; Revised 10 April 2015; Accepted 17 April 2015

Academic Editor: P. Beckers

Copyright © 2015 Shen-yan Chen et al. 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 an Improved Genetic Algorithm with Two-Level Approximation (IGATA) to minimize truss weight by simultaneously optimizing size, shape, and topology variables. On the basis of a previously presented truss sizing/topology optimization method based on two-level approximation and genetic algorithm (GA), a new method for adding shape variables is presented, in which the nodal positions are corresponding to a set of coordinate lists. A uniform optimization model including size/shape/topology variables is established. First, a first-level approximate problem is constructed to transform the original implicit problem to an explicit problem. To solve this explicit problem which involves size/shape/topology variables, GA is used to optimize individuals which include discrete topology variables and shape variables. When calculating the fitness value of each member in the current generation, a second-level approximation method is used to optimize the continuous size variables. With the introduction of shape variables, the original optimization algorithm was improved in individual coding strategy as well as GA execution techniques. Meanwhile, the update strategy of the first-level approximation problem was also improved. The results of numerical examples show that the proposed method is effective in dealing with the three kinds of design variables simultaneously, and the required computational cost for structural analysis is quite small.

#### 1. Introduction

The optimal design of a truss structure has been an active research topic for many years. Important progress has been made in both optimality criteria and solution techniques. As is well known, the optimal shape design of a truss structure depends not only on its topology but also on the element cross-sectional areas. This inherent coupling of structural shape, topology, and element sections explicitly indicates that the truss shape or topology or sizing optimization should not be performed independently. To date, most researchers focus on the subject of truss shape and sizing optimization [1–3] or topology and sizing optimization [4, 5], while relatively little literature is available on truss shape, topology, and sizing simultaneous optimization [6, 7]. The main obstacle is that shape and topology and sizing variables are fundamentally different physical representations. Combining these three types of variables may entail considerable mathematical difficulties and, sometimes, lead to ill-conditioning problem because their changes are of widely different orders of magnitude.

GA has been widely applied in truss topology optimization, especially for mixed variable problems. Based on various given problems, many specific GA methods have been proposed [8, 9]. However, they are still not completely satisfactory owing to their high computational cost and unstable reliability, especially for large scale structures [10, 11]. It is, therefore, apparent that the efficiency and reliability of GA retain large space to be improved further.

To improve the efficiency of truss sizing/topology optimization, Dong and Huang [12] proposed a GA with a two-level approximation (GATA), which obtains an optimal solution by alternating topology optimization and size optimization. As the structural analyses are used for building a series of approximate problems and the GA is conducted based on the approximate functions, the computational efficiency is greatly improved and the number of structural analyses can be reduced to the order of tens. Later, Li et al. [5] improved the GATA to enhance its exploitation capabilities and convergence stability. However, the shape variables are not involved in their research.

In this paper, based on the truss topology and sizing optimization system using GATA [5], a series of techniques were proposed to implement the shape, topology, and sizing optimization simultaneously in a single procedure. Firstly, a new optimization model is established, in which the truss nodal coordinates are taken as shape variables. To avoid calculating sensitivity of shape variables, discrete variables are used by adding the length of the individual chromosome. Therefore, sizing variables are continuous and shape/topology variables are discrete. To solve this problem, a first-level approximate problem is improved for the change of truss shape. Then, GA is used to optimize the individuals which include discrete 0/1 topology variables representing the deletion or retention of each bar and the integer-valued shape variables corresponding to nodal coordinates. Within each GA generation, a nesting strategy is applied in calculating the fitness value of each member [5, 12]. That is, for each member, a second-level approximation method is used to optimize the continuous size variables [5, 12]. In terms of GA, hybrid gene coding strategy is introduced, as well as the improvement of genetic operators. The controlled mutation of shape variables is considered in the generation of initial population. The uniform crossover is implemented for discrete topology variables and shape variables independently. Meanwhile, the first-level approximation problem update strategy was also improved in this paper. The proposed method is examined with typical truss structures and is shown to be quite effective and reliable.

This paper is organized as follows. In Section 2, we describe the optimization formulation for truss sizing/shape/topology optimization. In Section 3, we describe the optimization method GATA and in Section 4 the details of improvements for the optimization method are stated. In Section 5, we present our numerical examples and the algorithm performance and conclusion remarks are given in Sections 6 and 7, respectively.

#### 2. Problem Formulation

The truss sizing/shape/topology optimization problem is formulated in (1). Here, three kinds of design variables are defined as follows.(1)*Sizing Variables*. is the size variable vector, with denoting the cross-sectional area of bar members in th group and denoting the number of groups.(2)*Shape Variables*. is the shape variable vector, and is the number of shape variables; denotes the identifier number within the possible coordinates set , and is the number of possible coordinates of th shape variable.(3)*Topology Variables*. is the topology variable vector. If , members in th group are removed, and is set to a very small value , which is generally calculated as multiplied by the initial value of ; if , members in th group are retained, and is optimized between the upper bound and the lower bound :where is the total weight of the truss structure and denotes the weight of th group. represents th constraint in the model, which could be constraints of element stresses, node displacements, mode frequency, or buckling factor. denotes the total number of constraints, and is the number of frequency or bulking constraints. If some bar members are removed, the corresponding constraints are eliminated, such as the stress constraints of the removed members. Thus, indicates whether the respective constraint is eliminated; if , th constraint is eliminated; otherwise if , th constraint is retained.

To facilitate describing the optimization model, a truss structure is taken as an example, which is shown in Figure 1. coordinate of node 1 could be 100, 120, 140, and 160, while coordinate of node 2 could be 200, 240, and 280. It is required that the coordinate variation of node 1 and node 2 is independent and that node 1 should be symmetric with node 3 along the dotted line, which means that there are two independent shape variables. In addition, the cross-sectional dimension of each bar element is required to be optimized independently and each bar element is allowed to be deleted or retained, which means that there are 7 size variables and 7 topology variables.