Conditions of Nondegeneracy of Three-Dimensional Cells. A Formula of a Volume of Cells

Authors:
Olga V. Ushakova
Affiliations:
-
Venue:
SIAM Journal on Scientific Computing
Year:
2001

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Abstract

Numerical solutions of partial differential equations in three dimensions often use hexahedral cells composing a computational grid. After the grid is constructed, it is first of all verified whether the grid is unfolded or not. For such a test in this paper, conditions of nondegeneracy are found for hexahedral cells which are given by eight corner points and generated by the trilinear map from a unit cube to a region defined by these points. The conditions include necessary conditions and two sets of sufficient conditions. They are found as conditions of positivity of the Jacobian of the trilinear map. Thus, the conditions which guarantee the invertibility of the trilinear map from a unit cube to a hexahedron are given. How general the nondegeneracy conditions are is shown by a numerical experiment. Formulas of the Jacobian of the trilinear map are obtained, and following from them, as a separate result, a formula of a volume of a cell is obtained.