Hybrid finite difference/finite element immersed boundary method

Griffith, B. E. and Luo, X. (2017) Hybrid finite difference/finite element immersed boundary method. International Journal for Numerical Methods in Biomedical Engineering, 33(12), e2888. (doi:10.1002/cnm.2888) (PMID:28425587)

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The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach employs a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Luo, Professor Xiaoyu
Authors: Griffith, B. E., and Luo, X.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:International Journal for Numerical Methods in Biomedical Engineering
ISSN (Online):2040-7947
Published Online:20 April 2017
Copyright Holders:Copyright © 2017 The Authors
First Published:First published in International Journal for Numerical Methods in Biomedical Engineering 33(12):e2888
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
560111Finite element-immersed boundary method and its application to mitral valvesXiaoyu LuoEngineering & Physical Sciences Research Council (EPSRC)EP/I029990/1M&S - MATHEMATICS
694461EPSRC Centre for Multiscale soft tissue mechanics with application to heart & cancerRaymond OgdenEngineering & Physical Sciences Research Council (EPSRC)EP/N014642/1M&S - MATHEMATICS
689601The first fully coupled mitral valve - left ventricle computational modelXiaoyu LuoLeverhulme Trust (LEVERHULME)RF-2015-510M&S - MATHEMATICS