Large deformation of liquid capsules enclosed by thin shells immersed in the fluid

Le, Duc Vinh; Tan, Zhijun

doi:10.1016/j.jcp.2010.01.042

Cited by 38 publications

(43 citation statements)

References 48 publications

(120 reference statements)

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“…Several previous numerical studies for capsules [18,21,22,23,20,24,25,15] have dealt with more or less extensible capsules. Real capsules, made of polymers, are indeed extensible [26].…”

Section: Inextensible Capsulesmentioning

confidence: 99%

3D numerical simulations of vesicle and inextensible capsule dynamics

Farutin

Biben

Misbah

2014

Journal of Computational Physics

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Vesicles are locally-inextensible fluid membranes while inextensible capsules are in addition endowed with in-plane shear elasticity mimicking the cytoskeleton of red blood cells (RBCs). Boundary integral (BI) methods based on the Green's function techniques are used to describe their dynamics, that falls into the category of highly nonlinear and nonlocal dynamics. Numerical solutions raise several obstacles and challenges that strongly impact the results. Of particular complexity is (i) the membrane inextensibility, (ii) the mesh stability and (iii) numerical precisions for evaluation of the boundary integral equations. Despite intense research these questions are still a matter of debate.We regularize the single layer integral by subtraction of exact identities for the terms involving the normal and the tangential components of the force. In addition, the regularized kernel remains explicitly self-adjoint. The stability and precision of BI calculation is enhanced by taking advantage of additional quadrature nodes located in vertices of an auxiliary mesh, constructed by a standard refinement procedure from the main mesh. We extend the partition of unity technique to boundary integral calculation on triangular meshes: We split the calculation of the boundary integral between the original and the auxiliary mesh using a smooth weight function, which takes the distance between the source and the target as the argument and falls to zero beyond a certain cut-off distance. We provide an efficient lookup algorithm that allows us to discard most of the vertices of the auxiliary mesh lying beyond the cut-off distance from a given point without actually calculating the distances to them. The proposed algorithm offers the same treatment of near-singular integration regardless if the source and the target points belong to the same surface or not.Additional innovations are used to increase the stability and precision of the method: The bending forces are calculated by differential geometry expressions using local coordinates defined in vicinity of each vertex. The approximation of the surface in vicinity of a vertex is obtained by fitting with a second-degree polynomial of local coordinates.We solve for the Lagrange multiplier associated with membrane incompressibility using two penalization parameters per suspended entity: one for deviation of the global area from prescribed value and another for the sum of squares of local strains defined on each vertex. The proposed advancement is to vary the penalization parameters at each time step in such a way, that the global area of each membrane be conserved and the sum of squares of local strains be at minimum. This optimization is achieved by solving a linear system of rank three times the number of entities involved in the simulation. If no auxiliary mesh is used, the method reduces to steepest descent method thanks to the explicit self-adjointness of the regularized single-layer kernel in the boundary integral equation.Inextensible capsules, a model of RBC, are studied ...

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“…Several previous numerical studies for capsules [18,21,22,23,20,24,25,15] have dealt with more or less extensible capsules. Real capsules, made of polymers, are indeed extensible [26].…”

Section: Inextensible Capsulesmentioning

confidence: 99%

3D numerical simulations of vesicle and inextensible capsule dynamics

Farutin

Biben

Misbah

2014

Journal of Computational Physics

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“…where E s ¼ E h is the surface elastic modulus, c 11 is a large constant represents the ratio between the area dilatation modulus and the shear modulus, and the invariants I 1 and I 2 are given in [26]. In addition, we consider the Helfrich [22] bending energy function for biological membranes, given by…”

Section: Kinematic Description Of the Shellmentioning

confidence: 99%

“…where m accounts for the curvature of the shell in the volume integration [26]. The bending stress follows from (19) by the work conjugacy, with the result…”

Section: Kinematic Description Of the Shellmentioning

confidence: 99%

“…This model was later extended to take into account bending moments to study the effects of bending rigidity [27,39]. Recently, a thin-shell model [13,14] with variable thickness has been employed in [26] for computing stress and moment resultants on the same unstructured quadratic triangular mesh. The equations of shell mechanics in this thin-shell model were developed in surface curvilinear coordinates and implemented in terms of the principle of virtual displacements.…”

Section: Introductionmentioning

confidence: 99%

“…First, we propose a method for calculating the forces generated on the capsule surface by employing the thin-shell model using in [26] with the Catmull-Clark subdivision elements. Second, we incorporate the Helfrich bending energy function [22] for biological membranes into the weak form of equilibrium to take into account the bending moments generated due to a preferred three-dimensional unstressed configuration.…”

Section: Introductionmentioning

confidence: 99%

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