M. Behr, Rice MEMS - Space-Time Finite Element Formulations

Space-Time Formulations for Problem Involving Moving Boundaries

Figure 3: Vertically oscillating container: Elevation and pressure field at two instants during the simulation.

Space-time finite element formulations are emerging as an elegant and mathematically well-understood way of handling fluid flow problems involving moving boundaries and interfaces. A subset of Lagrangian-Eulerian interface tracking techniques, the space-time methods formulate the problem using finite element discretization in both space and time. The deformation of the domain is accommodated by the deformation of the space-time elements and the time-dependent nature of interpolation functions. In most applications, the discretization of the space-time domain utilizes slabs, which are analogous to time steps in a standard semi-discrete approach, but that does not have to be the case. The continuity of the solution in time is enforced weakly at slab interfaces. Within the slab, interpolation function typically used are either constant or linear in time. The mesh in the slab interior can be unstructured, allowing for varying time resolution in different parts of the domain. As with any interface tracking technique, the implementation of the interface effects, such as surface tension, is relatively straightforward. An example application of the space-time methodology is the simulation of sloshing in a water-filled rectangular container, which is subjected to vertical vibrations. The horizontal cross-section of the container is close to square, leading to the onset of chaotic wave mode competition, analyzed both theoretically and experimentally. These results are reproduced here using a long-term time dependent simulation, employing 6,000 hexahedral space-time elements. Figure 3 shows, at three instants during the computation, the position of the free surface and color-coded pressure distribution in the container.
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