Application of an a priori Jacobian-Based Error Estimation Metric to the Accuracy Assessment of 3D Finite Element Simulations

Abstract

The determinant of the Jacobian matrix is frequently used in the Finite Element Method as a measure of mesh quality. A new metric is defined, called the Standard Error, based on the distribution of the determinants of the Jacobian matrices of all elements of a finite element mesh. Where the Jacobian norm can be used to compare the quality of one element to another of the same type, the Standard Error compares the mesh quality of different versions of a finite element model where each version uses a different element type. To motivate this new Standard Error, we investigate the geometric meaning of the Jacobian norm on 3D Finite Elements. This mesh quality metric is applied to 8, 20, and 27 node hexahedra, 6 and 15 node prisms, 4 and 10 node tetrahedra, 5 and 13 node pyramid, and 3, 4, 6, 8, and 9 node shell elements. The shape functions for these 14 element types, or more precisely their first partial derivatives, are used to construct the Jacobian Matrix. The matrix is normalized to compensate for size. The determinant of the Jacobian is calculated at Gaussian points within each element. Statistics are gathered to form the Standard Error of the mesh. To illustrate the applicability of this a priori metric, we present two simple example problems having exact answers, and two industry-type problems, a pipe elbow with a crack and a magnetic resonance imaging (MRI) birdcage RF coil resonance, both having no analytical solution. Significance and limitations of using this a priori metric to assess the accuracy of finite element simulations of different mesh designs are presented and discussed.