RELATIONSHIP BETWEEN TETRAHEDRON SHAPE MEASURES

Tetrahedron shape measures are used for evaluating the quality of tetrahedra in finite element meshes. Three shape measures, the minimum solid angle theta(min), the radius ratio rho, and the mean ratio eta, are discussed in this paper. A new formula for the computation of a solid angle of tetrahedron is derived. For different shape measures mu and nu (with values less-than-or-equal-to 1), we establish a relationship between mu and nu of the form c0mu(e0) less-than-or-equal-to nu less-than-or-equal-to c1mu(e1) where c0, c1, e0, and e1 are positive constants. This means that if one measure approaches zero for a poorly-shaped tetrahedron, so does the other. Combined with the property that each measure attains a maximum value only for the regular tetrahedron, this means that the shape measures are ''equivalent.''