Mesh-Based and Meshfree Reduced Order Phase-Field Models for Brittle Fracture: One Dimensional Problems.

Ngoc-Hien Nguyen,Thi-Hong-Hieu Le,Yan Ding,Jian-Ying Wu,Vinh Phu Nguyen

doi:10.3390/ma12111858

Abstract

Modelling brittle fracture by a phase-field fracture formulation has now been widely accepted. However, the full-order phase-field fracture model implemented using finite elements results in a nonlinear coupled system for which simulations are very computationally demanding, particularly for parametrized problems when the randomness and uncertainty of material properties are considered. To tackle this issue, we present two reduced-order phase-field models for parametrized brittle fracture problems in this work. The first one is a mesh-based Proper Orthogonal Decomposition (POD) method. Both the Discrete Empirical Interpolation Method (DEIM) and the Matrix Discrete Empirical Interpolation Method ((M)DEIM) are adopted to approximate the nonlinear vectors and matrices. The second one is a meshfree Krigingmodel. For one-dimensional problems, served as proof-of-concept demonstrations, in which Young’s modulus and the fracture energy vary, the POD-based model can speed up the online computations eight-times, and for the Kriging model, the speed-up factor is 1100, albeit with a slightly lower accuracy. Another merit of the Kriging’s model is its non-intrusive nature, as one does not need to modify the full-order model code.

Highlights

Fracture is one of the most commonly-encountered failure modes of engineering materials and structures
Based on computations of a one-dimensional bar with varying the Young’s modulus and fracture energy, the Proper Orthogonal Decomposition (POD)-(M)Discrete Empirical Interpolation Method (DEIM) Reduced-Order Model (ROM) can speed up the online computations eight-times, whereas for the Kriging model, the speed up factor is 1100, albeit with a slightly lower accuracy
To illustrate the performance of the POD-(M)DEIM approach, we generated Ns = 60 sample points in Ω using the Latin Hypercube Sampling method (LHS), and built the ROM using the algorithm given in Box 3

Summary

Introduction

Fracture is one of the most commonly-encountered failure modes of engineering materials and structures. As with many other physical phenomena, computational modelling of fracture constitutes an indispensable tool to predict the failure of cracked structures, for which full-scale experiments are either too costly or even impracticable, and to shed light onto understanding the fracture processes of many materials such as concrete, rock, ceramics, metals, biological soft tissues, etc. Within the context of continuum modelling of brittle fracture, this paper presents mesh-based and mesh-free reduced-order phase-field models. Materials 2019, 12, 1858 models, and fracture constrained optimization problems; that is, all sorts of problems involving the repeated solution of differential equations that govern the phenomenon of brittle fracture [1]. The crack patterns are the natural outcome of a system of two coupled partial differential equations obtained from the minimization of a potential energy that consists of a stored bulk energy, the work of external forces, and the surface energy

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Conclusion