A numerical approach for fluid deformable surfaces with conserved enclosed volume

We consider surface finite elements and a semi-implicit time stepping scheme to simulate fluid deformable surfaces. Such surfaces are modeled by incompressible surface Navier-Stokes equations with bending forces. Here, we consider closed surfaces and enforce conservation of the enclosed volume. The numerical approach builds on higher order surface parameterizations, a Taylor-Hood element for the surface Navier-Stokes part, appropriate approximations of the geometric quantities of the surface, mesh redistribution and a Lagrange multiplier for the constraint. The considered computational examples highlight the solid-fluid duality of fluid deformable surfaces and demonstrate convergence properties that are known to be optimal for different sub-problems. •The algorithm allows for reliable simulations of fluid deformable surfaces controlling numerical errors for surface geometry and flow velocity.•The simulation results demonstrate the strong influence of surface viscosity on the evolution of the surface geometry.•Appropriate mesh regularisation allows for large deformation maintaining numerical accuracy.