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Abstractions and Automated Algorithms for Mixed Domain Finite Element Methods

Authors:

Cécile Daversin-Catty,

Chris N. Richardson,

Ada J. Ellingsrud,

Marie E. RognesAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 47, Issue 4

Article No.: 31, Pages 1 - 36

https://doi.org/10.1145/3471138

Published: 28 September 2021 Publication History

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Abstract

Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology, physiology, biology, and fracture mechanics. Mixed dimensional PDEs are also commonly encountered when imposing non-standard conditions over a subspace of lower dimension, e.g., through a Lagrange multiplier. In this article, we present general abstractions and algorithms for finite element discretizations of mixed domain and mixed dimensional PDEs of codimension up to one (i.e., nD-mD with |n-m| ≤ 1). We introduce high-level mathematical software abstractions together with lower-level algorithms for expressing and efficiently solving such coupled systems. The concepts introduced here have also been implemented in the context of the FEniCS finite element software. We illustrate the new features through a range of examples, including a constrained Poisson problem, a set of Stokes-type flow models, and a model for ionic electrodiffusion.

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Cited By

Rosilho de Souza GKrause RPezzuto S(2024)Boundary integral formulation of the cell-by-cell model of cardiac electrophysiologyEngineering Analysis with Boundary Elements10.1016/j.enganabound.2023.10.021158(239-251)Online publication date: Jan-2024
https://doi.org/10.1016/j.enganabound.2023.10.021
Ali BHeider YMarkert B(2024)Predicting residual stresses in SLM additive manufacturing using a phase-field thermomechanical modeling frameworkComputational Materials Science10.1016/j.commatsci.2023.112576231(112576)Online publication date: Jan-2024
https://doi.org/10.1016/j.commatsci.2023.112576
Jovanović ALoffhagen DBecker M(2023)Introduction and verification of FEDM, an open-source FEniCS-based discharge modelling codePlasma Sources Science and Technology10.1088/1361-6595/acc54b32:4(044003)Online publication date: 19-Apr-2023
https://doi.org/10.1088/1361-6595/acc54b
Show More Cited By

Index Terms

Abstractions and Automated Algorithms for Mixed Domain Finite Element Methods
1. Computing methodologies
  1. Modeling and simulation
    1. Model development and analysis
      1. Modeling methodologies
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Partial differential equations
  2. Mathematical software
    1. Solvers

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 47, Issue 4

December 2021

242 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/3485138

Editors:
Zhaojun Bai
University of California at Davis, USA
,
Wolfgang Bangerth
Colorado State University, USA

Issue’s Table of Contents

Copyright © 2021 Association for Computing Machinery.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Publication History

Published: 28 September 2021

Accepted: 01 June 2021

Revised: 01 February 2021

Received: 01 November 2019

Published in TOMS Volume 47, Issue 4

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European Union’s Horizon 2020 research and innovation programme

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Cited By

Rosilho de Souza GKrause RPezzuto S(2024)Boundary integral formulation of the cell-by-cell model of cardiac electrophysiologyEngineering Analysis with Boundary Elements10.1016/j.enganabound.2023.10.021158(239-251)Online publication date: Jan-2024
https://doi.org/10.1016/j.enganabound.2023.10.021
Ali BHeider YMarkert B(2024)Predicting residual stresses in SLM additive manufacturing using a phase-field thermomechanical modeling frameworkComputational Materials Science10.1016/j.commatsci.2023.112576231(112576)Online publication date: Jan-2024
https://doi.org/10.1016/j.commatsci.2023.112576
Jovanović ALoffhagen DBecker M(2023)Introduction and verification of FEDM, an open-source FEniCS-based discharge modelling codePlasma Sources Science and Technology10.1088/1361-6595/acc54b32:4(044003)Online publication date: 19-Apr-2023
https://doi.org/10.1088/1361-6595/acc54b
Jiang CXu PBai XZhao Zel Moctar OZhang G(2023)A review of advances in modeling hydrodynamics and hydroelasticity for very large floating structuresOcean Engineering10.1016/j.oceaneng.2023.115319285(115319)Online publication date: Oct-2023
https://doi.org/10.1016/j.oceaneng.2023.115319
Daversin-Catty CGjerde IRognes M(2022)Geometrically Reduced Modelling of Pulsatile Flow in Perivascular NetworksFrontiers in Physics10.3389/fphy.2022.88226010Online publication date: 11-May-2022
https://doi.org/10.3389/fphy.2022.882260
Colomés OVerdugo FAkkerman I(2022)A monolithic finite element formulation for the hydroelastic analysis of very large floating structuresInternational Journal for Numerical Methods in Engineering10.1002/nme.7140124:3(714-751)Online publication date: 10-Nov-2022
https://doi.org/10.1002/nme.7140

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