Vorträge in der Woche 05.08.2019 bis 11.08.2019

Dienstag, 06.08.2019: Eine Netzwerkgleichung mit dynamischer Randbedingung

M. Schumacher (Tübingen)

Donnerstag, 08.08.2019: Unfitted Finite Element Method: Discretizing Geometry and Partial Differential Equations

André Massing (NTNU Trondheim, Umeå University)

Many advanced engineering problems require the numerical solution of multidomain, multidimension, multiphysics and multimaterial problems with interfaces. When the interface geometry is highly complex or evolving in time, the generation of conforming meshes may become prohibitively expensive, thereby severely limiting the scope of conventional discretization methods. For instance, the simulation of blood flow dynamics in vessel geometries requires a series of highly non-trivial steps to generate a high quality, full 3D finite element mesh from biomedical image data. Similar challenging and computationally costly preprocessing steps are required to transform geological image data into conforming domain discretizations which respect complex structures such as faults and large scale networks of fractures. Even if an initial mesh is provided, the geometry of the model domain might change substantially in the course of the simulation, as in, e.g., fluid-structure interaction and free surface flow problems, rendering even recent algorithms for moving meshes infeasible. Similar challenges arise in more elaborated optimization problems, e.g. when the shape of the problem domain is subject to the optimization process and the optimization procedure must solve a series of forward problems for different geometric configuration. In this talk, we focus on recent unfitted finite element technologies as one possible remedy. The main idea is to design discretization methods which allow for flexible representations of complex or rapidly changing geometries by decomposing the computational domain into several, possibly overlapping domains. Alternatively, complex geometries only described by some surface representation can easily be embedded into a structured background mesh. In the first part of this talk, we briefly review how finite element schemes on cut and composite meshes can be designed by either using a Nitsche-type imposition of interface and boundary conditions or, alternatively, a partition of unity approach. Some theoretical and implementational challenges and their rectifications are highlighted. In the second part we demonstrate how unfitted finite element techniques can be employed to address various challenges from mesh generation to fluid-structure interaction problems, solving PDE systems on embedded manifolds of arbitrary co-dimension and PDE systems posed on and coupled through domains of different topological dimensionality.