Hybrid finite difference/finite element immersed boundary method

• Published in: International journal for numerical methods in biomedical engineering 2017-12, Vol.33 (12), p.n/a
• Main Authors: E. Griffith, Boyce, Luo, Xiaoyu
• Format: Article
• Language: English
• Subjects: Boundary element method; Coarsening; Computer Simulation; Coupling; Deformation; Discretization; Fibers; Finite difference method; Finite Element Analysis; Finite element method; Fluid-structure interaction; Heart; Heart Ventricles; Humans; immersed boundary method; Incompressibility; incompressible elasticity; incompressible flow; Independent variables; Mathematical models; Models, Cardiovascular; Ventricle; Viscosity
• ISSN: 2040-7939; 2040-7947
• DOI: 10.1002/cnm.2888

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 uses 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. We employ a nodal FE discretization of the structural equations while retaining a finite difference discretization of the Eulerian equations, and we apply this new scheme to benchmark problems involving elastic, rigid, and actively contracting structures of left ventricle. The primary contribution of this work is that it introduces a novel approach to coupling the Lagrangian and Eulerian variables that enables Independent Lagrangian and Eulerian spatial discretizations and thus the effective use of coarse structural meshes with the IB method.