Finite Variation Sensitivity Analysis for Discrete Topology Optimization of Continuum Structures

TL;DR

This paper introduces two novel finite variation sensitivity analysis methods for discrete topology optimization in continuum structures, improving accuracy and understanding over traditional approaches, with practical numerical demonstrations.

Contribution

It presents the Woodbury and Conjugate Gradient Method approaches, offering exact and highly accurate sensitivity calculations for BESO-based topology optimization.

Findings

Woodbury approach yields exact sensitivity values.

CGM approach provides high-precision sensitivities efficiently.

Proposed methods outperform traditional strategies in accuracy.

Abstract

This paper proposes two novel approaches to perform more suitable sensitivity analyses for discrete topology optimization methods. To properly support them, we introduce a more formal description of the Bi-directional Evolutionary Structural Optimization (BESO) method, in which the sensitivity analysis is based on finite variations of the objective function. The proposed approaches are compared to a naive strategy; to the conventional strategy, referred to as First-Order Continuous Interpolation (FOCI) approach; and to a strategy previously developed by other researchers, referred to as High-Order Continuous Interpolation (HOCI) approach. The novel Woodbury approach provides exact sensitivity values and is a better alternative to HOCI. Although HOCI and Woodbury approaches may be computationally prohibitive, they provide useful expressions for a better understanding of the problem. The novel Conjugate Gradient Method (CGM) approach provides sensitivity values with arbitrary precision and is computationally viable for a small number of steps. The CGM approach is a better alternative to FOCI since, for appropriate initial conditions, it is always more accurate than the conventional strategy. The standard compliance minimization problem with volume constraint is considered to illustrate the methodology. Numerical examples are presented together with a broad discussion about BESO-type methods.