Hierarchical universal matrices for triangular finite elements with varying material properties and curved boundaries

The integration required to find the stiffness matrix for a triangular fini te element is inexpensive if the polynomial order of the element is low. Hi gher-order elements can be handled efficiently by universal matrices provid ed they are straight-edged and the material properties are uniform. For cur ved elements and elements with varying material properties (e.g. non-linear B-H curves), Gaussian integration is generally used, but becomes expensive for high orders. Two new methods are proposed in which the high-order part of the integrand is integrated exactly and the results stored in pre-compu ted universal matrices. The effect of curved edges and varying material pro perties is approximated via interpolation. The storage requirement of the p rocedure is kept to a minimum by using specifically devised basis functions which are hierarchical and possess the three-fold symmetry of a triangular element. Care has been taken to maintain the conditioning of the basis. On e of the new methods is hierarchical in nature and suitable for use in an a daptive integration scheme. Results show that, for a given required accurac y, the new approaches are more efficient than Gauss quadrature for element orders of 4 or greater. The computational advantage increases rapidly with increasing order. Copyright (C) 1999 John Wiley & Sons, Ltd.