A continuous energy-based immersed boundary method for elastic shells

• Published in: Journal of computational physics 2018-10, Vol.371, p.333-362
• Main Authors: Maxian, Ondrej, Kassen, Andrew T., Strychalski, Wanda
• Format: Article
• Language: English
• Subjects: Biological models (mathematics); Blebbing; Capillary flow; Computational fluid dynamics; Computational physics; Computer simulation; Continuity (mathematics); Curvature; Derivatives; Elastic shells; Energy; Erythrocytes; Fluid flow; Fluid–structure interaction; Flux density; Hyperelasticity; Incompressible flow; Integrals; Linear algebra; Materials elasticity; Mathematical analysis; Numerical analysis; Numerical methods; Red blood cells; Stokes flow; Test procedures; Three dimensional flow; Three dimensional models; Viscous fluids
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.05.045

The immersed boundary method is a mathematical formulation and numerical method for solving fluid–structure 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. •New variational method for computing forces on thin elastic shells within the IB method is presented.•Method gives a continuous force function on the entire surface of a hyperelastic shell.•Comparison to a previous formulation where the surface and energy functional are first discretized is provided.•Evidence of improved accuracy of elastic forces is presented.•Method is applied to 3D models of a red blood cell in capillary flow and cellular blebbing.