Four-Dimensional Anisotropic Mesh Adaptation

• Published in: Computer aided design Vol. 129; p. 102915
• Main Authors: Caplan, Philip Claude, Haimes, Robert, Darmofal, David L., Galbraith, Marshall C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.12.2020; Elsevier BV
• Subjects: Adaptation; Algorithms; Anisotropy; Computer simulation; Computing costs; Domains; Finite element method; Four-dimensional; Function approximation; High-order finite elements; Mesh adaptation; Metric-conforming; Time dependence; Mesh adaptation; Four-dimensional; Metric-conforming; High-order finite elements; Function approximation
• ISSN: 0010-4485; 1879-2685
• DOI: 10.1016/j.cad.2020.102915

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. [Display omitted] •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.