Discontinuous Galerkin Simulation of Multiphase Flows with curved elements

To compute any simulation the generating of a discretization domain is always the start point of the numerical model. This process well known as meshing should assure a good representation of the exact geometry with an appropriate level of approximation to ensure the convergence and accuracy of the resulting solution. In typical biomedical applications, however, the real geometry is often so complex so that most of the engineering analysis techniques - which use piecewise linear or piecewise quadratic approximations of the boundary- can not lead to accurate results, especially in curved domains. Indeed, an often overlooked aspect is that without properly curved mesh entities the discretization error for such curvilinear geometries remains critical, despite the use of high-order spatial discretizations and adaptive techniques (e.g. hp-refinement). Otherwise, in many biomedical applications such as viscous flow simulations for aorta with a stenosis bypassed by a graft not only a suitable geometric discretization of the curvilinear boundaries is important but also carefully defined and controlled layered elements in close proximity to viscous walls are required. Such boundary layers with curved elements are crucial for the fidelity of the computed results. Curvilinear meshes are also desirable by moving boundaries like the ones of red blood cells by blood flow simulations.

As a general classification, different curvilinear meshing schemes can be devided in two groups: direct and a-posteriori approach. In the first one a valid curvilinear boundary discretization is to be set from the beginning. In the last one, the edges and respectively the surfaces on the boundaries of an already existing linear mesh will be curved and any local invalidities of the mesh arising out of interference between curved boundary mesh entities and interior mesh entities are eliminated by local mesh modification tools (e.g. swapping, deleting…).

Due to the lack of robust curvilinear mesh generation software, it is often necessary to write own meshing algorithm (e.g. Bézier-based higher-order mesh) when using the direct approach. For the a-posteriori approach some invalid elements are unavoidable (e.g. negative determinant of Jacobian in the closures of curved elements) and should be removed by hand. Here it worth saying that Gmsh (an open source mesh generation package using the a-posteriori approach) offers promising results (up to fifth-order curved meshes). Despite the various attempts to advance high-order mesh generation technology, an integrated solution capable of robustly creating optimal curved element meshes especially in 3D does not exist and much future work is needed before high-order meshing schemes can be used routinely to solve complex problems.

higher-order numerical methods (DG) for partial differential equations, Multiphase flows, curved elements

Emna Hassani

Dipl.-Ing.

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