Level‐set topology optimization for maximizing fracture resistance of brittle materials using phase‐field fracture model

Abstract

Summary Fracture is one of the most common failure modes in brittle materials. It can drastically decrease material integrity and structural strength. To address this issue, we propose a level‐set (LS) based topology optimization procedure to optimize the distribution of reinforced inclusions within matrix materials subject to the volume constraint for maximizing structural resistance to fracture. A phase‐field fracture model is formulated herein to simulate crack initiation and propagation, in which a staggered algorithm is developed to solve such time‐dependent crack propagation problems. In line with diffusive damage of the phase‐field approach for fracture; topological derivatives, which provide gradient information for the topology optimization in a LS framework, are derived for fracture mechanics problems. A reaction‐diffusion equation is adopted to update the LS function within a finite element framework. This avoids the reinitialization by overcoming the limitation to time step with the Courant‐Friedrichs‐Lewy condition. In this article, three numerical examples, namely, a L‐shaped section, a rectangular slab with predefined cracks, and an all‐ceramic onlay dental bridge (namely, fixed partial denture), are presented to demonstrate the effectiveness of the proposed LS based topology optimization for enhancing fracture resistance of multimaterial composite structures in a phase‐field fracture context.

Topology optimization of structures undergoing brittle fracture

“…If divu ≥ 0 the material is said to be in traction, otherwise it is in compression. The degradation function (52) is constructed in such a way that damage occurs only under tension. In other words, when divu < 0, whatever the value of α, one has C(u, α) = C 0 .…”

“…In the level-set framework we are only aware of [52] which optimizes the conguration of composite materials in a phase-eld based fracture model. We dier from [52] in many aspects: they do not use a level-set equation but rather a reaction-diusion equation, they do not use a shape derivative but instead a topological gradient for a simplied model with a xed damage eld and nally they do not optimize the overall shape but just the inclusion's shape inside a composite structure. There are more works on topology optimization using SIMP (solid isotropic material with penalization) applied to various fracture models.…”

A SIMP-phase field topology optimization framework to maximize quasi-brittle fracture resistance of 2D and 3D composites

“…The first works to our knowledge combining phase field and TO was introduced in Xia et al and Da et al in [21,66], where the BESO TO [27] was used to optimize the fracture resistance of two-phase structures with respect to inclusions shapes, including cracks in both bulk and interfaces, and applied to periodic composites and multiple loads in [20]. In [54,55], Russ and Waisman developed a SIMP TO combined with phase field to optimize the fracture energy in one-phase material structures, and Wu et al [64] developed a Level-Set TO-phase field approach to optimize the fracture resistance of composites.…”

Machine learning based topology optimization of fiber orientation for variable stiffness composite structures

“…4 In literature, Ghiasi et al 5,6 reviewed a range of optimization algorithms associated with both CSC and VSC. In so-called density-based [7][8][9] and boundary-based (e.g., level-set) design frameworks, [9][10][11][12] gradient-driven algorithms have been extensively implemented for the capability of handling a large number of design variables in comparison with heuristic algorithms such as genetic algorithms (GA). Inspired by the parametrization of multiphase topology optimization, Stegmann and Lund 13 proposed the discrete material optimization (DMO) method to tackle the orientation selection problem for composite laminated structures, which was based upon the conventional mathematical programming technique with gradient information.…”