$\partial^2 ( \mathrm{TO} ) $: A Dual Topological Derivative-Based Enriched Topology Optimization for Fracture Mitigation in 3-D Brittle Solids

2026-07-26

AI Summary

This work addresses the susceptibility to fracture in topology optimization of three-dimensional brittle solids by proposing an integrated framework that concurrently optimizes structural performance and fracture resistance. The approach combines the level-set method with radial basis function–based geometric parameterization, enhanced by interface-enriched finite element analysis. It introduces, for the first time, dual topological derivatives to simultaneously guide void nucleation and enable efficient evaluation of the total boundary energy release rate. Fracture prediction accuracy is further improved through the maximum circumferential stress criterion and non-local stress recovery techniques. A p-norm aggregation function is employed to balance competing objectives in the multi-objective formulation. Numerical examples, including an L-shaped bracket, demonstrate a significant reduction in boundary energy release rate, confirming the framework’s effectiveness and robustness in achieving fracture-resistant topologies for complex three-dimensional geometries.

Abstract

We propose a fracture-mitigation topology optimization framework for 3-D brittle solids. The topology is described by a level set function parameterized by radial basis functions, and the structural response is computed using an interface-enriched finite element formulation. Dual topological derivatives serve two purposes. First, they are used to nucleate holes within the solid during the optimization process. Second, they are used to evaluate energy release rates (ERRs) along the entire boundary, requiring only the stress field from a single enriched finite element analysis of the uncracked geometry. For this purpose, penny-shaped cracks are assumed to nucleate using the maximum hoop stress criterion, at the locations of enriched nodes introduced along the boundary for accurate finite element analysis. Because ERR estimates depend sensitively on stress accuracy, we compute a nodal stress field using a non-local stress-recovery procedure. The topology optimization objective aggregates the boundary ERRs using a $p$-mean function. Three-dimensional numerical examples, including the commonly studied L-bracket benchmark problem, demonstrate the capability of the proposed framework.

Problem

• fracture mitigation
• topology optimization
• energy release rate
• 3-D brittle solids
• crack nucleation

Innovation

• topological derivative
• enriched finite element method
• fracture mitigation
• energy release rate
• level set method