Multi-Resolution Phase-Field Formulations Using Higher Order Immersed Boundary Methods
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Immersed Boundary Methods are characterized by an independent description of a geometric model and the mesh on which a finite-element-like approximation is defined. While the mesh of cells is very often a simple, grid-like structure, the geometric model can be formulated in various ways. Boundary Representations, Constructive Solid Geometries and voxel-grid based formulations have successfully been applied. All these models are characterized by a geometric resolution, which is typically finer than that of the cells, e.g. allowing domain boundaries passing through the interior of cells, or material being inhomogeneous inside one cell. Considering voxel-based geometric formulations it has been observed in particular for higher order p-elements that many voxels can be resolved within one cell (see e.g. [1]). This fact is applied successfully in FCM-based topology optimization [2] where a sub-grid of n voxels per space dimension with n=p+2 was found favourable w.r.t. accuracy and efficiency. In this paper, we will investigate multi-resolution approaches for phase-field based fracture simulation as well as for Full Waveform Inversion [3], which can be interpreted as an inverse phase-field problem to identify unknown internal structural components. p -version finite element as well as isogeometric element approximations will be applied, and a pure sub-cell resolution by a voxel-grid will be compared with a true local refinement of the Ansatz space. We will demonstrate on various numerical examples opportunities and limitations of these sub-scale formulations.
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