Four-Dimensional Anisotropic Mesh Adaptation. (December 2020) - Caplan, Philip Claude; Haimes, Robert; Darmofal, David L.; Galbraith, Marshall C.
Abstract: Anisotropic mesh adaptation is important for accurately simulating physical phenomena at reasonable computational costs. Previous work in anisotropic mesh adaptation has been restricted to studies in two- or three-dimensional computational domains. However, in order to accurately simulate time-dependent physical phenomena in three dimensions, a four-dimensional mesh adaptation tool is needed. This work develops a four-dimensional anisotropic mesh adaptation tool to support time-dependent three-dimensional numerical simulations. Anisotropy is achieved through the use of a background metric field and the mesh is adapted using a dimension-independent cavity framework. Metric-conformity – in the sense of edge lengths, element quality and element counts – is effectively demonstrated on four-dimensional benchmark cases within a unit tesseract in which the background metric is prescribed analytically. Next, the metric field is optimized to minimize the approximation error of a scalar function with discontinuous Galerkin discretizations on four-dimensional domains. We demonstrate that this four-dimensional mesh adaptation algorithm achieves optimal element sizes and orientations. To our knowledge, this is the first presentation of anisotropic four-dimensional meshes. Graphical abstract: Highlights: A four-dimensional anisotropic mesh adaptation algorithm was implemented. Anisotropy was achieved via a background metric field and the adaptation algorithm builds upon a local cavity operator framework. Metric-conformity was demonstrated on benchmark cases, whereby the metric field was prescribed analytically. Optimal mesh size and aspect ratio distributions were obtained in the approximation of four-dimensional functions.