Nondegeneracy tests for hexahedral cells

The aims of the paper are to consider the nondegeneracy requirement for computational grids and to analyze eight tests used to check the nondegeneracy of hexahedral cells. The paper starts with consideration of nondegeneracy requirement and formulation of definitions and common theorems utilized for estimation of nondegeneracy of grids both structured and unstructured. Then hexahedral cells are introduced and sufficient nondegeneracy conditions (Ushakova, 2000) for them are given. Sufficient nondegeneracy conditions are 27 inequalities for 32 tetrahedral volumes. Besides sufficient nondegeneracy conditions other conditions are applied as nondegeneracy tests in grid generation theory and practice. Considered nondegeneracy tests are the checks for positivity of different values. Tests 1, 2, 3, 4, 5, 6 check the positivity of 8, 10, 24, 32, 58, 48 tetrahedral volumes, correspondingly. Test 7 verifies the positivity of the volume of a cell. Test 8 checks the positivity of the Jacobian of the mapping used for generation of a cell. The check is performed at the corners of a cell and hex center. Tests 1, 7, 8 are often used in commercial packages. For the most part, nondegeneracy tests are not sufficient nondegeneracy conditions, however they are used for the purpose of constructing nondegenerate grids and, some times, instead of sufficient nondegeneracy conditions. The effectiveness and reliability of such substitutions are investigated in special numerical experiments with random numbers. In the numerical experiment for each test, hexahedral cells are generated randomly. Results of such experiments are the following. Among eight tests, test 2 is considered the best since it verifies the volumes of only 10 tetrahedra for positiveness, guarantees the nondegeneracy in most of cases (68.7% randomly generated hexahedral cells satisfying test 2) and covers a wide class of cells (about 60% of nondegenerate cells). Tests 1, 3, 4, 5, 6, 7, 8 have success in 31.7%, 83.1%, 100%, 100%, 39.5%, 0.2%, 34% of cases and cover 100%, 7.9%, 7.9%, 4.2%, 59.5%, 100%, 100% of nondegenerate cells, correspondingly. Because of high rate of success, tests 3, 4, 5 also can be used for grid generation purpose. All tests are illustrated by the examples of structured grids.