Inexact Newton combined approximations in the topology optimization of geometrically nonlinear elastic structures and compliant mechanisms

TL;DR

This paper introduces new strategies combining inexact Newton methods with iterative combined approximations to efficiently solve nonlinear topology optimization problems for elastic structures and mechanisms, improving speed and accuracy.

Contribution

It proposes five novel strategies for controlling Jacobian factorizations and integrates ICA schemes, enhancing the efficiency of topology optimization under geometric nonlinearity.

Findings

Strategies improve computational speed without sacrificing accuracy

Numerical experiments validate robustness across benchmark problems

Stopping criteria effectively balance iteration count and solution quality

Abstract

This work blends the inexact Newton method with iterative combined approximations (ICA) for solving topology optimization problems under the assumption of geometric nonlinearity. The density-based problem formulation is solved using a sequential piecewise linear programming (SPLP) algorithm. Five distinct strategies have been proposed to control the frequency of the factorizations of the Jacobian matrices of the nonlinear equilibrium equations. Aiming at speeding up the overall iterative scheme while keeping the accuracy of the approximate solutions, three of the strategies also use an ICA scheme for the adjoint linear system associated with the sensitivity analysis. The robustness of the proposed reanalysis strategies is corroborated by means of numerical experiments with four benchmark problems -- two structures and two compliant mechanisms. Besides assessing the performance of the strategies considering a fixed budget of iterations, the impact of a theoretically supported stopping criterion for the SPLP algorithm was analyzed as well.