Structural and Multidisciplinary Optimization https://doi.org/10.1007/s00158-018-2170-5 RESEARCH PAPER An automatically connected graph representation based on B-splines for structural topology optimization Dieu T. T. Do 1 · Jaehong Lee 1 Received: 4 April 2018 / Revised: 23 November 2018 / Accepted: 28 November 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract This paper introduces an automatically connected graph representation for structural topology optimization. Structural members of optimal topologies are constructed based on a graph whose each edge is represented by a B-spline curve with varying thickness. A square matrix including connective coefficients with either 0 or 1 is first proposed to automatically determine the way of linking two vertices of an edge. This significantly improves the flexibility of searching optimal topologies. Additionally, resulting designs are completely free from the checkerboard effect and grayscale without any filtering techniques. Control point coordinates of B-spline curves are considered as continuous design variables, while connective coefficients and thickness parameters are taken as discrete ones. Accordingly, this approach helps to reduce the total number of design variables considerably. Modified differential evolution (mDE), which is modified from original differential evolution (DE) to reduce computational cost but still guarantee the quality of solution, is proposed. mDE with the possibility of easily handling such variables and the capability of finding a global solution is utilized as an optimizer. The proposed method is essentially an intuitive method without a strong mathematical or theoretical background. Gained results are compared with other studies to verify the effectiveness of the present method. Keywords Topology optimization · Automatically connected graph representation · B-splines · Modified differential evolution (mDE) 1 Introduction Topology optimization is a computational technique for effec- tively distributing material within a certain design domain in order to enhance its stiffness and save its mass in resulting designs. In recent years, this field has attracted considerable attention of many researchers and been extensively applied to various areas, such as aircraft, automotive, and other structural design industries (Akin and Arjona-Baez 2001; Rozvany 2001; Zhu et al. 2016; Nanthakumar et al. 2015; Dapogny et al. 2017; Luo et al. 2017). A large number of Responsible Editor: Ji-Hong Zhu  Jaehong Lee jhlee@sejong.ac.kr Dieu T. T. Do dttdieu@sju.ac.kr 1 Department of Architectural Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 05006, Republic of Korea effective tools have thus been developed to deal with such problems (Bendsøe and Kikuchi 1988; Bendsøe 1989; Xie and Steven 1993; Yang et al. 1999; Lau et al. 2001; Chian- dussi 2006; Wang et al. 2006; Xia and Wang 2008; Huang and Xie 2009; Xia et al. 2014, 2019). Among the abovementioned methods, the homogeniza- tion method Bendsøe and Kikuchi (1988) is one of the most early proposed approaches. Its outstanding variants have also been found in Refs. Chen et al. (2004), Yi et al. (2015), Okada et al. (2001), Valisetty et al. (2005), Coelho et al. (2016), and Allaire and Kohn (1993). That approach is a material distribution method, based on the use of an artificial composite material with microscopic voids. Accordingly, topologies based on the searching pro- cess of optimal porosities can be achieved according to the optimality criteria (OC). Although the performances of the method are quite simple, it may generate optimized designs with infinitesimal pores. And this leads some particular dif- ficulties for the manufacturing of such ones. Fortunately, the above issues can be overcome using the solid isotropic material with penalization (SIMP) approach (Bendsøe 1989; Mlejnek 1992; Rozvany et al. 1992; Bendsøe and Sigmund