A Bezier-Based Moving Mesh Framework for Simulation with Elastic Membranes

In this paper we present an application of our Bezier-based approach to moving meshes [1] to Navier-Stokes simulations with several immersed elastic membranes. By a moving mesh we mean one that moves with the material and is adapted to maintain good aspect ratio triangles of minimal size. The adaptations we employ include point insertion and removal, as well as edge smoothing. This work is being done as part of the Sangria project [2] whose goal is to develop geometric and numerical algorithms and software for the simulation of blood flow at the microstructural level. In our approach, we adopt the Lagrangian paradigm where domain boundaries and object interfaces move together with the fluid in which they are immersed. This approach has the advantage that boundaries and object interfaces are easy to track. A moving mesh also poses diffcult geometric problems since very distorted elements can be created as the simulation evolves. This can lead to several undesirable or catastrophic situations such as inverted or overlapping elements. From the computational geometry perspective, the challenge presented by the Lagrangian paradigm is the ability to maintain a good quality mesh as the simulation evolves in time. We tackle this problem by using non-linear elements and by locally modifying the mesh using a few primitive operations. The use of non-linear elements allows us to represent the mesh with fewer elements in our simulations, and the use of local operators allows us to avoid remeshing the simulation domain at every time step.