Isogeometric analysis of fluid cellular membranes: Application of discrete exterior calculus and isogeometric analysis to Stokes flow on time-evolving surfaces

An isogeometric finite element method for incompressible fluid film equations is presented. The method can be applied to numerically model the behaviour of thin cellular membranes, such as lipid bilayers. The membranes are represented by infinitely thin closed surfaces. Both the surface parametrization and analysis are based on state-of-the-art polar spline spaces. These spaces are defined such that a C1 continuous genus 0 surface can be constructed. At the discrete setting, point-wise conservation of mass is attained, using the framework of discrete exterior calculus. Therefore, the polar spline spaces are called divergence conforming. Time discretization of the highly non-linear system is done via the fixed-point iterations. It is found that for certain non-uniformly curved domains, the iterations converge and time stepping can be performed. However, for surfaces that are closely resembling a perfect sphere, the iterations are not stable for any ∆t. The solution is parameter-dependent and this indicates a possible bug in the Matlab code.
Animation of the basic domain: https://youtu.be/YBrJwtOAqZ8 Gitlab project, containing the Matlab code that was used for this research: https://gitlab.tudelft.nl/hmverhelst/lipid-bilayers
Applied Mathematics