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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…

Distributed model fitting refers to the process of fitting a mathematical or statistical model to the data using distributed computing resources, such that computing tasks are divided among multiple interconnected computers or nodes, often…

Computation · Statistics 2024-06-04 Xiaofei Wu , Rongmei Liang , Fabio Roli , Marcello Pelillo , Jing Yuan

Fault tolerant algorithms for the numerical approximation of elliptic partial differential equations on modern supercomputers play a more and more important role in the future design of exa-scale enabled iterative solvers. Here, we combine…

Mathematical Software · Computer Science 2015-06-23 Markus Huber , Björn Gmeiner , Ulrich Rüde , Barbara Wohlmuth

In this paper we derive a new two-dimensional brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small…

Numerical Analysis · Mathematics 2020-04-21 Stefano Almi , Sandro Belz , Stefano Micheletti , Simona Perotto

The favored phase field method (PFM) has encountered challenges in the finite strain fracture modeling of nearly or truly incompressible hyperelastic materials. We identified that the underlying cause lies in the innate contradiction…

Numerical Analysis · Mathematics 2022-05-04 Fucheng Tian , Jun Zeng , Mengnan Zhang , Liangbin Li

We present a fully implicit formulation of coupled fluid flow and geomechanics for fluid injection/withdrawal in fractured reservoirs in the context of CO2storage. Utilizing a Galerkin finite-element approach, both flow and poroelasticity…

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the…

Numerical Analysis · Mathematics 2018-01-17 Fei Zhang , Weizhang Huang , Xianping Li , Shicheng Zhang

A phase-field approach is proposed for interface failure between two possibly dissimilar materials. The discrete adhesive interface is regularised over a finite width. Due to the use of a regularised crack model for the bulk material, an…

Materials Science · Physics 2019-09-06 Arne Claus Hansen-Dörr , René de Borst , Paul Hennig , Markus Kästner

Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and…

Numerical Analysis · Mathematics 2019-12-05 Hanyu Li , Wing Tat Leung , Mary F. Wheeler

In this work, we couple a high-accuracy phase-field fracture reconstruction approach iteratively to fluid-structure interaction. The key motivation is to utilize phase-field modelling to compute the fracture path. A mesh reconstruction…

Numerical Analysis · Mathematics 2024-08-19 Henry von Wahl , Thomas Wick

This contribution presents a diffuse framework for modeling cracks in heterogeneous media. Interfaces are depicted by static phase-fields. This concept allows the use of non-conforming meshes. Another phase-field is used to describe the…

Materials Science · Physics 2020-05-11 Arne Claus Hansen-Dörr , Franz Dammaß , René de Borst , Markus Kästner

A robust $hp$-adaptive finite element framework is presented for the investigation of static cracks in materials characterized by complex, pointwise density variations. Within such heterogeneous media, the equilibrium equation governed by…

Numerical Analysis · Mathematics 2025-12-29 S. M. Mallikarjunaiah

We present a novel, generalised formulation to treat coupled structural integrity problems by combining phase field and multi-physics modelling. The approach exploits the versatility of the heat transfer equation and is therefore well…

Computational Engineering, Finance, and Science · Computer Science 2025-06-27 Y. Navidtehrani , C. Betegón , E. Martínez-Pañeda

In this contribution, a novel framework for simulating mixed-mode failure in rock is presented. Based on a hybrid phase-field model for mixed-mode fracture, separate phase-field variables are introduced for tensile (mode I) and shear (mode…

Computational Engineering, Finance, and Science · Computer Science 2023-12-05 Lisa Hug , Martin Potten , Georg Stockinger , Kurosch Thuro , Stefan Kollmannsberger

The phase-field model (PFM) represents the crack geometry in a diffusive way without introducing sharp discontinuities. This feature enables PFM to effectively model crack propagation compared with numerical methods based on discrete crack…

Computational Engineering, Finance, and Science · Computer Science 2019-02-18 Shuwei Zhou , Timon Rabczuk , Xiaoying Zhuang

Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. Most common is to use linear…

Numerical Analysis · Mathematics 2019-04-30 Juan Michael Sargado , Eirik Keilegavlen , Inga Berre , Jan Martin Nordbotten

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

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviours of these variables, the meshes can…

Numerical Analysis · Mathematics 2010-05-27 Thomas Witkowski , Axel Voigt

We propose and investigate a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution…

Optimization and Control · Mathematics 2017-02-01 F. Yang , C. Venkataraman , V. Styles , A. Madzvamuse

This work introduces an adaptive mesh refinement technique for hierarchical hybrid grids with the goal to reach scalability and maintain excellent performance on massively parallel computer systems. On the block structured hierarchical…

Numerical Analysis · Mathematics 2025-08-11 Benjamin Mann , Ulrich Rüde