RESEARCH PAPER Simultaneous size, shape, and topology optimization of truss structures using integrated particle swarm optimizer Ali Mortazavi 1 & Vedat Toğan 2 Received: 7 January 2015 /Revised: 1 March 2016 /Accepted: 10 March 2016 # Springer-Verlag Berlin Heidelberg 2016 Abstract The current investigation deals with the weight minimization of truss structures accomplishing the simultaneous shape, size, and topology optimization. In this regard, this study presents an effective algorithm called integrated particle swarm optimizer (iPSO) as an optimization tool. The iPSO combines favorable features of the standard PSO with an efficient concept of ‘weighted particle’ to improve its performance. In addi- tion, ‘improved fly-back’ technique is introduced to handle the problem constraints. The proposed methodol- ogy is tested on a series of benchmark problems and the obtained results are compared with those available in the technical literature. The iPSO achieves the results which are capable of competitive with those obtained by other techniques used for simultaneous optimization of truss structures and reported in the literature. Furthermore, the relative simplicity of the formulation can be considered as one of the significant features of this method. Keywords Particle swarm optimizer . Truss structures . Size . Shape and topology optimization 1 Introduction Structural design optimization problems available in lit- erature can be categorized into three groups. The first group determines the solution of design optimization problems where member cross sectional dimensions are treated as design variables. Such problems are called structural size optimization problem. In the second group in addition to cross sectional dimensions, joint coordinates are also treated as design variables. These design optimization problems are characterized as the structural shape optimization problem. The solution of such design problem yields optimum structural shape. In the third group members of the structure are also treated as design variables. Solution of such design optimiza- tion problem results in optimum configuration namely the optimum arrangement of members which constitute the structure. These design optimization problem is called optimum topology design of structures. Although, various traditional techniques are devel- oped for solving these kinds of problems, most of them are based on the multilevel optimization, which tries to optimize each category of variables of the problem, sep- arately (Dobbs and Felton 1969; Miguel Leandro et al. 2013). However, since the corresponding design vari- ables are not linearly independent, they should be taken into account together. So, a nonlinear approach to solv- ing this type of problems is required (Rajeev and Krishnamoorthy 1992; Haug and Arora 1989; Kirsch 1989; Ringertz 1985; Topping 1983; Vanderplaat and Moses 1972). Also, it should be noticed that the design variables of optimization problem may be discrete, con- tinuous or even mixed. So, simultaneous structural op- timization would be recognized as highly complex opti- mization problems (Ohsaki 1995, 1998). * Vedat Toğan togan@ktu.edu.tr 1 Department of Civil Engineering, Ege University, Izmir, Turkey 2 Department of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey Struct Multidisc Optim DOI 10.1007/s00158-016-1449-7