Four-Dimensional Anisotropic Mesh Adaptation

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.