A continuous energy-based immersed boundary method for elastic shells

Abstract

The immersed boundary method is a mathematical formulation and numerical method for solving fluidstructure interaction problems. For many biological problems, such as models that include the cell membrane, the immersed structure is a two-dimensional infinitely thin elastic shell immersed in an incompressible viscous fluid. When the shell is modeled as a hyperelastic material, forces can be computed by taking the variational derivative of an energy density functional. A new method for computing a continuous force function on the entire surface of the shell is presented here. The new method is compared to a previous formulation where the surface and energy functional are discretized before forces are computed. For the case of Stokes flow, a method for computing quadrature weights is provided to ensure the integral of the elastic spread force density remains zero throughout a dynamic simulation. Tests on the method are conducted and show that it yields more accurate force computations than previous formulations as well as more accurate geometric information such as mean curvature. The method is then applied to a model of a red blood cell in capillary flow and a 3D model of cellular blebbing.Studies on surface representations have generally been conducted separately from those on force computations. In [15,16], the authors focus on representing thin surfaces continuously with spherical harmonic or radial basis function (RBF) interpolants. In both cases, force computations are treated through simple explicit expressions for surface tension [15] or fiber elasticity [16]. The authors do not consider finding forces through variational derivatives.Boundary integral methods (BIMs) have been progressing concurrently with the IB method and have been used by others to model fluid-structure interactions in zero Reynolds number flow [17][18][19][20]. In BI methods, the "hydrodynamic traction jump" across the membrane [17], which is equivalent to the Lagrangian elastic force density per unit current area [20], must be computed independently prior to integration. [21] and [22] both demonstrate the use of a spectrally convergent spherical harmonic representation for surfaces that allows for the calculation of force in BIMs. However, the computations in [21] and [22] both follow the traditional formulation of BI methods in computing the traction jump from a system of equations involving the Cauchy stress tensor, which itself comes from a series of tensor expansions. Additionally, BI methods require numerical schemes to resolve integrals that are singular at the immersed surfaces.The aim of this work is to provide a "bridge" between the fields of surface representation and energybased force functions within the IB method. Our goal is to use the continuous surface representations from [15] to formulate a continuous function that represents the force density on the membrane as a function of its curvilinear coordinates. In this manner, we avoid the discretization error from surface representation. The advantages of the approach presen...

A Finite Element Method for a Fourth Order Surface Equation With Application to the Onset of Cell Blebbing

“…One could consider a function ψ that, like k, depends on the position y on Ŵ 0 . For the finite element approximation discussed in the next section, we make use of an approximation k h of k (see around (10)). Something similar could be done for ψ as well.…”

“…Something similar could be done for ψ as well. This would result in some additional terms that would need to be estimated using a consistency assumption similar to (10). As the procedure is similar to the one for k we omit the details for conciseness.…”

“…In 3D, the fluid flow becomes significantly more expensive, and tracking points on a surface whilst maintaining a good representation becomes much more involved. In Maxian et al [10], a spectral method for the flow is coupled with two approaches for the surface, one based on a parameterization with spherical harmonics and one based on a piecewise linear representation. In Campbell and Bagchi [11,12] a surface finite element method for reaction-diffusion equations on the cell membrane is coupled with a projection method for the flow.…”

3D Computational Modeling of Bleb Initiation Dynamics

“…A model of bleb expansion that included reaction-diffusion of membrane-cortex adhesion proteins with limited membrane deformation was presented in [24]. Lastly, a full 3D model of bleb expansion with a viscous fluid cytoplasm (an extension of the model in [7]) was presented in [25] to illustrate a numerical method for computing forces on an elastic shell using the immersed boundary method [26].…”

Thin and sharp edges bodies-fluid interaction simulation using cut-cell immersed boundary method

“…Zhu and Peskin (2002) (Favier, Revell, & Pinelli, 2016;Zhu & Peskin, 2002) painted filaments on numerous grid cells that transformed them into a dense fluid and artificially applied the right side of a Navier Stokes equation. Same as the behavior of the Penalty method for inflexible rigid conditions, the forces will be affected just the selected fluid response, thereby frequently requiring small time steps to achieve the desired accuracy without comprising the stability (Bailoor et al, 2017;de Tullio & Pascazio, 2016;Maxian, Kassen, & Strychalski, 2018).…”