Sensitivity analysis for any functions of static elastic systems using combination of adjoint variable method and automatic differentiation for topology optimization

Abstract

This study proposes a methodology to generalize and reduce the computational cost of sensitivity analysis for static linear elastic systems in topology optimization. The process is fully generalized by applying sensitivity analysis in conjunction with the adjoint variable method and automatic differentiation. The design sensitivities can be computed independently of any evaluation function and the material interpolation method for static elastic systems. Furthermore, the issue of excessive computational memory required by automatic differentiation is resolved by utilizing the adjoint variable method. To demonstrate the generality of the proposed method, we focus on multi-material topology optimization for isotropic linear elastic systems, evaluating compliance and maximum stress based on the Lp-norm. Additionally, the extended SIMP and Discrete Material Optimization (DMO) methods are investigated as interpolation schemes for material properties in multi-material problems. Finally, the finite difference method and automatic differentiation are benchmarked against other sensitivity analysis methods, and the proposed method is evaluated in terms of accuracy, computational cost, and memory usage. Moreover, examples of topology optimization for a 3-dimensional problem are presented to demonstrate the applicability of the proposed method to large-scale analyses.

Keywords

• topology optimization
• multi-material
• sensitivity analysis
• automatic differentiation
• adjoint variable method
• finite element method