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Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational…

Applied Physics · Physics 2019-10-22 Elias Karabelas , Gundolf Haase , Gernot Plank , Christoph M. Augustin

In this contribution, a variational diffuse modeling framework for cracks in heterogeneous media is presented. A static order parameter smoothly bridges the discontinuity at material interfaces, while an evolving phase-field captures the…

Materials Science · Physics 2021-04-07 Arne Claus Hansen-Dörr , Jörg Brummund , Markus Kästner

We derive parameter-robust quasi-optimal error estimates for mixed finite element methods for the nonlinear Darcy--Forchheimer equations with mixed boundary conditions. Using the framework of operator preconditioning, we also design…

Numerical Analysis · Mathematics 2026-02-25 Rishi Das , Harsha Hutridurga , Amiya K. Pani , Ricardo Ruiz-Baier

A continuum model of crack propagation is presented and discussed. We obtain steady state solutions with a self-consistently selected propagation velocity and shape of the crack, provided that elastodynamic and viscoelastic effects are…

Materials Science · Physics 2020-02-26 M. Fleck , D. Pilipenko , R. Spatschek , E. A. Brener

We extend a phase-field/gradient damage formulation for cohesive fracture to the dynamic case. The model is characterized by a regularized fracture energy that is linear in the damage field, as well as non-polynomial degradation functions.…

Applied Physics · Physics 2019-03-27 Rudy J. M. Geelen , Yingjie Liu , Tianchen Hu , Michael R. Tupek , John E. Dolbow

While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted via variational or material-force-based phase field fracture models, these models often assume that the underlying elastic response of the…

Computational Engineering, Finance, and Science · Computer Science 2020-08-13 Hyoung Suk Suh , WaiChing Sun , Devin O'Connor

The phase field approach to modeling fracture uses a diffuse damage field to represent a crack. This addresses the singularities that arise at the crack tip in computations with sharp interface models, mollifying some of the difficulties…

Variational phase-field models of fracture are widely used to simulate nucleation and propagation of cracks in brittle materials. They are based on the approximation of the solutions of free-discontinuity fracture energy by two smooth…

Numerical Analysis · Mathematics 2023-02-14 Frederic Marazzato , Blaise Bourdin

A field theory is presented for predicting damage and fracture in quasi-brittle materials. The approach taken here is new and blends a non-local constitutive law with a two-point phase field. In this formulation, the material displacement…

Materials Science · Physics 2025-10-07 Semsi Coskun , Davood Damircheli , Robert Lipton

Computational modeling of faulting processes is an essential tool for understanding earthquake mechanics but remains challenging due to the structural and material complexities of fault zones. The phase-field method has recently enabled…

Geophysics · Physics 2025-03-11 Fan Fei , Md Shumon Mia , Ahmed E. Elbanna , Jinhyun Choo

This study addresses ductile fracture of single grains in metals by modeling of the formation and propagation of transgranular cracks. A proposed model integrates gradient extended hardening, phase-field modeling for fracture, and crystal…

Numerical Analysis · Mathematics 2024-11-01 Kim Louisa Auth , Jim Brouzoulis , Magnus Ekh

An adapted deflation preconditioner is employed to accelerate the solution of linear systems resulting from the discretization of fracture mechanics problems with well-conditioned extended/generalized finite elements. The deflation space…

Computational Engineering, Finance, and Science · Computer Science 2022-05-11 Konstantinos Agathos , Tim Dodwell , Eleni Chatzi , Stephane P. A. Bordas

The random feature method (RFM), a mesh-free machine learning-based framework, has emerged as a promising alternative for solving PDEs on complex domains. However, for large three-dimensional nonlinear problems, attaining high accuracy…

Numerical Analysis · Mathematics 2025-10-07 Longze Tan

The paper addresses the problem of a Mode III interfacial crack advancing quasi-statically in a heterogeneous composite material, that is a two-phase material containing elastic inclusions, both soft and stiff, and defects, such as…

Mathematical Physics · Physics 2011-10-25 Andrea Piccolroaz , Gennady Mishuris , Alexander Movchan , Natasha Movchan

Over the past seven years, full-field analyses of a wide range of classical as well as modern quasi-static fracture experiments on nominally elastic brittle materials -- ranging from hard ceramics to soft elastomers -- have repeatedly…

Materials Science · Physics 2024-11-26 Yangyuanchen Liu , Oscar Lopez-Pamies , John E. Dolbow

The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton-Raphson method. In order to…

Numerical Analysis · Mathematics 2022-04-27 Ritukesh Bharali , Somdatta Goswami , Cosmin Anitescu , Timon Rabczuk

In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix based approach within the Finite Element Method requires…

Numerical Analysis · Mathematics 2020-10-28 Daniel Jodlbauer , Ulrich Langer , Thomas Wick

In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an…

Analysis of PDEs · Mathematics 2013-12-09 Elena Bonetti , Christian Heinemann , Christiane Kraus , Antonio Segatti

In this work, we consider compressible single-phase flow problems in a porous media containing a fracture. In the latter, a non-linear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we…

In this paper, we are interested in an efficient numerical method for the mixed-dimensional approach to modeling single-phase flow in fractured porous media. The model introduces fractures and their intersections as lower-dimensional…

Numerical Analysis · Mathematics 2019-06-03 Ana Budiša , Xiaozhe Hu