Geometry-Aware FEM in Computational Mechanics
The interdisciplinary research area of Computational Mechanics combines concepts from computer science, applied mathematics, and engineering for the efficient simulation of mechanical structures. Applications of such simulations can be found in a wide range of areas, including engineering, biomechanics, or life sciences. They all require a suitable discrete representation of the mechanical structures under consideration: on the one hand, their possibly highly complex surfaces have to be represented in a sufficiently accurate way; on the other hand, a volume mesh of high quality is required in order to ensure the quality of the finite element approximation and the good convergence of iterative solvers. Although state-of-the-art mesh generation methods and tools allow for a relatively comfortable generation of meshes, creating high quality meshes for complex structures still requires considerable effort and is a time-consuming task. This is particularly true for hexahedral meshes, because they are much harder to generate but usually give rise to simulation results of higher quality in mechanics. Thus, it is desirable to utilize a created volume mesh as long as possible throughout the course of a simulation. At this point, however, two main difficulties show up. The first is related to surface representations and adaptivity. Using adaptive refinement strategies, a higher resolution at the boundary should be accompanied by a better approximation of the "real" surface of the structure under consideration. The second is related to the degradation of mesh quality during time-dependent simulations which involve large deformations. Despite the fact that in case of a highly deformed mesh a complete remeshing will be necessary, the quality of the mesh might decrease gradually during the simulation process, thus affecting the quality and the speed of the simulation results significantly. Most state-of-the-art simulation tools are based on a one-way connection between the geometry information and the simulation environment. Geometric information is taken into account during mesh generation, but not exploited afterwards. As a consequence, adaptive refinement is only seldom accompanied by an increase in geometry resolution, and for complex geometries, efficient solvers such as geometric multigrid methods are replaced by less efficient iterative methods. A notable exception from this one-way approach is isogeometric analysis (IGA), which allows for refining the used mesh and elevating the order of the underlying finite element basis "without changing the geometry or its parameterization". This approach, however, requires a particular choice of basis functions for the finite element space. Moreover, in case of large deformations, the mesh quality tends to degrade if the deformations are sufficiently large. In this case, remeshing has to be employed in order to preserve the accuracy of the simulation results, thus posing the problem of mesh generation again. The main idea of this project is to overcome these difficulties by developing and implementing a geometry-aware simulation environment for computational mechanics, which combines the handling of complex geometries and volume meshes with the discretization and the solution process in a modular fashion. Thus, instead of seeing geometry approximation as a "burden", which has to be taken into account when computing a finite element solution, we exploit concepts and methods from the field of Geometry Processing to guarantee a constantly high quality of the boundary approximation and the volume mesh throughout a transient simulation. In the first phase of this project, which has been running for 18 months now, we have successfully taken the first steps towards this goal and have already developed strategies and code for (i) maintaining a high mesh quality during time-dependent simulations; (ii) a consistent handling of mesh transfer with different projection operators; (iii) the construction of efficient and geometry-flexible multi-scale solvers. In addition, we are currently working on a geometry-adaptive boundary refinement strategy. For the second phase of this project, our aim is to continue these activities and to (i) explore alternative methods for transferring volume meshes in case of large deformations; (ii) utilize parametric elements to further avoid remeshing as long as possible; (iii) improve the performance of our solvers by parallelizing all algorithmic steps. As a consequence, our approach turns the common one-way connection between geometry and simulation (geometry -> mesh -> simulation) into a two-way connection, as geometry-specific information is used within the adaptive simulation process, which in turn changes the geometries under consideration.