A Unified Framework for the Modeling and Simulation of Fluid Surfaces

ALEJANDRO TORREZ-SÁNCHEZ; MARINO ARROYO; DANIEL MILLÁN

New York

Congreso; WCCM XIII: 13th World Congress on Computational Mechanics and PANACM II: 2nd Pan American Congress on Computational Mechanics; 2018

Columbia University

We develop a novel framework for the three-dimensional modeling and simulation of fluid surfaces from a continuum mechanics viewpoint. Fluid surfaces are ubiquitous in cell and tissue biology, with examples including lipid bilayers, the acto-myosin cortex, or epithelial monolayers. These surfaces usually involve a non-linear coupling between mechanical and chemical signals, which play a key role in important biophysical processes such as cell division, migration or morphogenesis. Thus, there is a growing interest in modeling and simulating these fluid interfaces. This, however, requires of a general framework that tackles the chemo-mechanical coupling transparently and deals with the geometric aspects of a time-evolving surface. To handle the multi-physics aspects of these surfaces, we base our approach on Onsager's variational principle, which provides a variational formulation for the dissipative dynamics of soft-matter systems. In addition to coupling different physical ingredients, modeling fluid surfaces inevitably requires the tools and language of differential geometry to describe a deforming surface evolving in Euclidean space. For instance, the classical rate-of-deformation tensor couples interfacial flows with shape changes in the presence of curvature. Furthermore, the fluid nature of these surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Indeed, due to the fluid nature of the surface, Lagrangian parametrizations generate very large distortions that require a large amount of remeshing. On the other hand, since we need to track the position of the interface in Euclidean space, the meaning of an Eulerian description is unclear. Arbitrary Lagrangian-Eulerian formulations, well established for bulk media, appear as a natural choice but such a formulation for a deforming surface needs careful consideration. Finally, the three-dimensional simulation of lipid bilayers requires unconventional numerical methods since the resulting equations involve higher-order derivatives of the parametrization, lead to a mixed system of elliptic and hyperbolic partial differential equations and are stiff and difficult to integrate in time. Indeed, surface shape enters into the energy and dissipation expressions through curvature, which involves second-order derivatives of the parametrization. From a finite element method perspective, this implies that the basis functions used to represent the parametrization need to be smooth. Here, we propose a discretization based on subdivision surfaces. While the Galerkin FEM deals naturally with elliptic equations, hyperbolic systems such as the continuity equation modeling fluid transport require special treatment.