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Shape‐preserving finite elements in cylindrical and spherical geometries: The double Jacobian approach
Author(s) -
Morgan Jason P.,
Taramón Jorge M.,
Hasenclever Jörg
Publication year - 2020
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4799
Subject(s) - cartesian coordinate system , spherical coordinate system , jacobian matrix and determinant , log polar coordinates , polar coordinate system , geometry , bipolar coordinates , spherical geometry , cylindrical coordinate system , mathematics , tetrahedron , orthogonal coordinates , mathematical analysis , finite element method , spherical trigonometry , physics , thermodynamics
We present a new technique, the “double Jacobian,” to solve problems in cylindrical or spherical geometries, for example, the Stokes flow problem for convection in Earth's mantle. Our approach combines the advantages of working simultaneously in Cartesian and polar or spherical coordinates. The governing matrix equations are kept in Cartesian coordinates, thereby preserving their Cartesian symmetry. However, the element geometry is described as a linear simplex in polar or spherical coordinates, thereby preserving appropriate cylindrical or spherical surfaces and internal interfaces. Isoparametric representations can still be used to define complex surface shapes. Using linear polar or spherical elements allows search routines for triangular or tetrahedral simplexes to rapidly find arbitrary points in terms of their polar or spherical coordinates. The double Jacobian approach becomes especially powerful when element sizes vary strongly within the mesh, while the exact cylindrical or spherical surfaces or internal interfaces have to be preserved, as happens in several geophysical applications.