An Introduction to the Immersed Boundary and the Immersed Interface Methods

Abstract

This paper attempts to give a brief survey or tutorial on the immersed boundary (IB) method and the immersed interface method (IIM). The immersed boundary method was originally introduced by Peskin for studying flow patterns around heart valves and for studying blood flow in the heart [118] and has since been applied to many other problems, particularly, in biophysics. The IB method is a mathematical modeling framework as well as a numerical method. The original motivation of the immersed interface method [76,80] was to improve the accuracy of the IB method, at least from first-order accuracy to second; and deal with discontinuous coefficients in the governing equations. The IIM is a sharp interface method based on Cartesian grids or triangulation. The IIM makes use of jump conditions across interfaces so that the finite difference/element discretization can be accurate. Some new developments in the immersed interface method include the augmented 1 Interface Problems and Methods in Biological and Physical Flows Downloaded from www.worldscientific.com by Mr Yaosheng Liang on 03/20/15. For personal use only. 2 R. H. Dillon and Z. Li immersed interface method; immersed finite element method; and singularity removal techniques. The immersed boundary and the interface methods have been coupled with different evolution schemes for free boundary and moving interface problems.Keywords: Immersed boundary method, immersed interface method, interface problem, irregular domain, discontinuous coefficient, singular source, delta function, discontinuous or non-smooth solution, augmented method, immersed finite element method, source singularity removal.

Implementation and Evaluation of Normal Ray Refinement Technique in Adaptive Cartesian Framework

“…In the ghost cell IBM, ghost cells are introduced inside solids and boundary conditions on a solid surface are obtained at cell centers outside the flow domain. 18,19,20 Extrapolation from some reference cells to the ghost cells is used to satisfy boundary conditions on the surface. The forcing IBM adds a surface "force" to Navier-Stokes equations either before or after their discretization.…”

Simulations of self-propelled anguilliform swimming using the immersed boundary method in OpenFOAM

“…The reduction of computational cost is contributed by one of the CONTACT Hui Feng E0013532@u.nus.edu reasons: there is almost no cost for the mesh regeneration in each individual step (Dillon & Li, 2009); it is much easier to describe a complex motion of the body relative to a fixed domain mesh (Mittal & Iaccarino, 2005); and many mature methods, solver packages, and algorithms developed for Cartesian grids are available. Following this strategy, a variety of refinements and modifications of the IBM have been developed during the past few decades (Diao et al, 2018;Dillon & Li, 2009;Hopkins, Vanderlei, & Fauci, 2012;Mittal & Iaccarino, 2005;Peskin, 2003;Tutkun & Edis, 2017). With proper methods being used to program the IBM solvers, several FSI problems have been solved (Ardekani & Brandt, 2019;Coclite, Ranaldo, de Tullio, Decuzzi, & Pascazio, 2019).…”