Jiao Jia^1,*, Xin He², Zhenchen Liu³, Shiqing Wu⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.143, No.3, pp. 3215-3231, 2025, DOI:10.32604/cmes.2025.066489 - 30 June 2025

Abstract As primary load-bearing components extensively utilized in engineering applications, beam structures necessitate the design of their microstructural configurations to achieve lightweight objectives while satisfying diverse mechanical performance requirements. Combining topology optimization with fully coupled homogenization beam theory, we provide a highly efficient design tool to access desirable periodic microstructures for beams. The present optimization framework comprehensively takes into account for key deformation modes, including tension, bending, torsion, and shear deformation, all within a unified formulation. Several numerical results prove that our method can be used to handle kinds of microstructure design for beam-like structures, e.g., More >

Qingrong Zeng, Xiaochen Liu, Xuefeng Zhu^*, Xiangkui Zhang, Ping Hu

CMES-Computer Modeling in Engineering & Sciences, Vol.141, No.3, pp. 2065-2085, 2024, DOI:10.32604/cmes.2024.052620 - 31 October 2024

Abstract Traditional topology optimization methods often suffer from the “dimension curse” problem, wherein the computation time increases exponentially with the degrees of freedom in the background grid. Overcoming this challenge, we introduce a real-time topology optimization approach leveraging Conditional Generative Adversarial Networks with Gradient Penalty (CGAN-GP). This innovative method allows for nearly instantaneous prediction of optimized structures. Given a specific boundary condition, the network can produce a unique optimized structure in a one-to-one manner. The process begins by establishing a dataset using simulation data generated through the Solid Isotropic Material with Penalization (SIMP) method. Subsequently, we More >

Rafael Marin Ferro^1,2,*, Renato Pavanello²

CMES-Computer Modeling in Engineering & Sciences, Vol.136, No.2, pp. 1371-1397, 2023, DOI:10.32604/cmes.2023.026043 - 06 February 2023

Abstract This work analyzes the implementation of a continuous method of structural topology optimization (STO) using open-source software for all stages of the topology optimization problem: modeling, sensitivity analysis and optimization. Its implementation involves three main components: numerical analysis using the Finite Element Method (FEM), sensitivity analysis using an Adjoint method and an optimization solver. In order to allow the automated numerical solution of Partial Differential Equations (PDEs) and perform a sensitivity analysis, FEniCS and Dolfin Adjoint software are used as tools, which are open-source code. For the optimization process, Ipopt (Interior Point OPTimizer) is used, More >

Jun Yan^1,3, Qi Xu¹, Zhirui Fan¹, Zunyi Duan^2,*, Hongze Du¹, Dongling Geng¹

CMES-Computer Modeling in Engineering & Sciences, Vol.128, No.3, pp. 1179-1196, 2021, DOI:10.32604/cmes.2021.016950 - 11 August 2021

Abstract This study investigates structural topology optimization of thermoelastic structures considering two kinds of objectives of minimum structural compliance and elastic strain energy with a specified available volume constraint. To explicitly express the configuration evolution in the structural topology optimization under combination of mechanical and thermal load conditions, the moving morphable components (MMC) framework is adopted. Based on the characteristics of the MMC framework, the number of design variables can be reduced substantially. Corresponding optimization formulation in the MMC topology optimization framework and numerical solution procedures are developed for several numerical examples. Different optimization results are More >

Haishan Lu, Shuguang Gong^*, Jianping Zhang, Guilan Xie, Shuohui Yin

CMES-Computer Modeling in Engineering & Sciences, Vol.128, No.3, pp. 1151-1178, 2021, DOI:10.32604/cmes.2021.016165 - 11 August 2021

Abstract We proposed an improved graphics processing unit (GPU) acceleration approach for three-dimensional structural topology optimization using the element-free Galerkin (EFG) method. This method can effectively eliminate the race condition under parallelization. We established a structural topology optimization model by combining the EFG method and the solid isotropic microstructures with penalization model. We explored the GPU parallel algorithm of assembling stiffness matrix, solving discrete equation, analyzing sensitivity, and updating design variables in detail. We also proposed a node pair-wise method for assembling the stiffness matrix and a node-wise method for sensitivity analysis to eliminate race conditions More >

Xiaolei Deng^a,b,c,*,† , Jin Wang^d , Jinyu Zhou^a, Hongcheng Shen^a, Zefeng Sheng^a, Jianglin Zhang^a, Xiaowen L^a, Changxiong Xie^a

Frontiers in Heat and Mass Transfer, Vol.12, pp. 1-6, 2019, DOI:10.5098/hmt.12.13

Abstract A hybrid cellular automaton model combined with finite element method for structural topology optimization with mechanical and heat constraints is developed. The effect of thermal stress on structural optimization is taken into account. Higher order 8-node element and von Neumann strategy are employed for the finite element and the cellular element, respectively. The validating studies of standard testing structure for topological optimization are carried out. The structure evolution, stress evolution and thermal evolution of topology optimization with mechanical and heat constraints are investigated. The results show the developed hybrid method is more efficient for structural More >

Shaohua Zhang^1,2, Pei Li¹, Yongteng Zhong¹, Jiawei Xiang^1,3

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.1, pp. 17-31, 2014, DOI:10.3970/cmes.2014.101.017

Abstract In order to obtain smooth boundary and improve computational efficiency, a new topology optimization scheme based on the level set method is presented. Using the level set function as design variable and the volume ratio of the solid material as volume constraint, respectively, this scheme can easily implement compliance minimization structure topology optimization in associated with the reaction-diffusion equation in commercial software COMSOL. Compared with the results of solid isotropic material with penalization (SIMP) and traditional level set method, this scheme obtained a smooth geometry boundary. In the present computational scheme, the computational cost could More >

Shiwei Zhou¹, Michael Yu Wang²

CMES-Computer Modeling in Engineering & Sciences, Vol.16, No.2, pp. 83-102, 2006, DOI:10.3970/cmes.2006.016.083

Abstract This paper describes a self-mass-conservative Cahn-Hilliard (C-H) model with elastic strain energy (mean compliance) for the optimization of multi-material structure topology. The total free energy of the generalized C-H system can be represented as a Lyapunov functional so that the elastic strain energy, as a part of the total free energy, decreases gradually to attain optimal material distribution. The interface energy relating to phase gradient in the total free energy plays an important role in regularizing the original ill-posed problem by restricting the structure's boundaries. On the other hand, interface coalescence and break-up due to More >

Michael Yu Wang^1,2, Shiwei Zhou²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.6, pp. 547-566, 2004, DOI:10.3970/cmes.2004.006.547

Abstract In this paper we present a variational method to address the topology optimization problem -- the phase transition method. A phase-field model is employed based on the phase-transition theory in the fields of mechanics and material sciences. The topology optimization is formulated as a continuous problem with the phase-field as design variables within a fixed reference domain. All regions are described in terms of the phase field which makes no distinction between the solid, void and their interface. The Van der Waals-Cahn-Hilliard theory is applied to define the variational topology optimization as a dynamic process… More >

Michael Yu Wang¹, Xiaoming Wang²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.4, pp. 373-396, 2004, DOI:10.3970/cmes.2004.006.373

Abstract This paper addresses the problem of structural shape and topology optimization. A level set method is adopted as an alternative approach to the popular homogenization based methods. The paper focuses on four areas of discussion: (1) The level-set model of the structure’s shape is characterized as a region and global representation; the shape boundary is embedded in a higher-dimensional scalar function as its “iso-surface.” Changes of the shape and topology are governed by a partial differential equation (PDE). (2) The velocity vector of the Hamilton-Jacobi PDE is shown to be naturally related to the shape… More >