The use of unfitted finite element methods (FEMs) is an appealing
approach for different reasons. They are interesting in coupled problems
or to avoid the generation of body-fitted meshes. One of the bottlenecks
of the simulation pipeline is the body-fitted mesh generation step and
the unstructured mesh partition. The use of unfitted methods on
background octree Cartesian meshes avoids the need to define body-fitted
meshes, and can exploit efficient and scalable space-filling curve
algorithms. In turn, such schemes complicate the numerical integration,
imposition of Dirichlet boundary conditions, and the linear solver
phase. The condition number of the resulting linear system does depend
on the characteristic size of the cut elements, the so-called small cut
cell problem.
In this work, we will present an parallel unfitted framework that relies
on adaptive octree background meshes and space-filling curve
partitioners. In order to solve the small cut cell problem, we will
pursue two different lines. The first one is a re-definition of the
finite element spaces that solves this issue, leading to condition
number bounds as the ones for body-fitted schemes without any kind of
perturbation/stabilization of the Galerkin formulation. Another approach
will be to define appropriate iterative linear solvers based on domain
decomposition preconditioning that are robust with respect to the small
cut cell problem. Finally, we will apply the resulting framework to the
numerical simulation of metal additive manufacturing.
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