Struct Multidisc Optim (2010) 41:671–683 DOI 10.1007/s00158-010-0487-9 FORUM DISCUSSION A further review of ESO type methods for topology optimization Xiaodong Huang · Yi-Min Xie Received: 22 July 2008 / Revised: 11 November 2009 / Accepted: 27 January 2010 / Published online: 6 March 2010 c  Springer-Verlag 2010 Abstract Evolutionary Structural Optimization (ESO) and its later version bi-directional ESO (BESO) have gained widespread popularity among researchers in structural opti- mization and practitioners in engineering and architecture. However, there have also been many critical comments on various aspects of ESO/BESO. To address those criticisms, we have carried out extensive work to improve the original ESO/BESO algorithms in recent years. This paper summa- rizes latest developments in BESO for stiffness optimization problems and compares BESO with other well-established optimization methods. Through a series of numerical exam- ples, this paper provides answers to those critical comments and shows the validity and effectiveness of the evolutionary structural optimization method. Keywords Evolutionary Structural Optimization (ESO) · Bi-directional ESO (BESO) · Local optimum · Optimal design · Displacement constraint 1 Introduction Evolutionary structural optimization (ESO) method was firstly introduced by Xie and Steven (1992, 1993, 1997). The idea is based on a simple and empirical concept that a structure evolves towards an optimum by slowly removing (hard-killing) elements with lowest stresses. To maximize X. Huang (B ) · Y.-M. Xie School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476V, Melbourne 3001, Australia e-mail: huang.xiaodong@rmit.edu.au Y. M. Xie e-mail: mike.xie@rmit.edu.au the stiffness of the structure, stress criterion was replaced with elemental strain energy criterion by Chu et al. (1996). Bi-directional evolutionary structural optimization (BESO) (Querin et al. 1998, 2000) method is an extension of that idea which allows for new elements to be added in the locations next to those elements with highest stresses. For stiffness optimization problems using the strain energy cri- terion, Yang et al. (1999) estimated the strain energy of void elements by linearly extrapolating the displacement field. ESO/BESO has been used for a wide range of applica- tions and hundreds of publications have been produced by researchers around the world. Several landmark buildings designed using ESO/BESO have now been constructed in Japan and Qatar (Cui et al. 2005; Ohmori et al. 2005). Mean- while, some shortcomings have been pointed out by Sig- mund and Petersson (1998), Zhou and Rozvany (2001) and Rozvany (2009). Firstly, the original ESO/BESO methods fail to achieve a convergent optimal solution. As a result, we have to select the best solution by comparing a large number of solutions generated during the optimization process (Rozvany 2009). Secondly, a notable question about ESO/ BESO has arisen following the work of Zhou and Rozvany (2001) in which a highly inefficient solution to a cantilever tie-beam structure by the ESO method was pointed out. Thirdly, the ESO/BESO procedure cannot be easily extend- ed to other constraints such as displacement (Sigmund and Petersson 1998; Rozvany 2009). In order to answer these critical comments, this paper is organized as follows. In Section 2, we briefly sum- marize the recent improvements in the BESO method. In Section 3, we compare the results of the BESO method with those from other optimization approaches. In Section 4, we revisit the cantilever tie-beam example in Zhou and Rozvany (2001) and explore the essence of the inefficient solution. In Section 5, we extend the BESO method to an