Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

Abstract

SUMMARYWe introduce a new numerical method to model the fluid-structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in-plane compression, which had been shown to be a destabilizing factor for other methods. The results are in very good agreement with the literature. When the viscous forces are increased with respect to the membrane elastic forces, three regimes are found for both flow cases. Our method allows a precise characterization of the critical parameters governing the transitions.

Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

“…Studies of capsules in fluid flow have been conducted with different membrane constitutive laws, including Hookean (Beale & Strain 2008), neo-Hookean or Skalak (Walter et al 2010), and area-incompressible or inextensible membranes (Veerapaneni et al 2009). Different numerical methods have also been used, such as boundary integral and immersed boundary methods.…”

“…For values of the capillary number with Ca < Ca L or Ca > Ca H , compressive stresses develop in at least one of the two principal elastic tensions on the interface (see also Barthès-Biesel 1980), and when this occurs the evolution problem becomes highly unstable and the numerical method fails. Later work (Walter et al 2010) using a finiteelement method for the membrane obtains stable steady solutions even when compressive stresses are present. The greater stability of the finite-element method is thought to be due to numerical regularization.…”

Boundary element analysis of deformation and movement of a capsule and a red blood cell close to the wall

“…The numerical method, a coupling of the boundary integral method and finite element method, was developed by Walter et al [8] to simulate the motion of capsules in an infinite shear flow.…”

Boundary element analysis of the multi-capsule flow using an ultra-high speed GPGPU computation

“…In the present study, we follow the discretized method developed by Walter et al [3]. The membrane mechanics is solved by discretizing the equation (1) to finite element (FE) matrix.…”

“…The BE method, which is used in the present work, is the most accurate method to simulate the motion of capsules, since the method directly treats jump condition at the membrane without any interpolation.Various studies have been conducted to simulate the detailed motion of capsules using this method, such as deformation of single capsule in shear flow [1] and interaction of two capsules [2]. Recently, Walter et al [3] proposed a coupling method of the BE method with the finite element (FE) method of membrane mechanics to improve the stability of the simulation. However, the analyzing scale of pervious studies was still limited in few capsules, because of heavy computational load of BE method.…”