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

The integration required to Þnd the sti¤ness matrix for a triangular Þnite element is inexpensive if the polynomial order of the element is low. Higher-order elements can be handled e¦ciently by universal matrices provided they are straight-edged and the material properties are uniform. For curved elements and elements with varying material properties (e.g. non-linear BÐH 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-computed universal matrices. The e¤ect of curved edges and varying material properties is approximated via interpolation. The storage requirement of the procedure is kept to a minimum by using speciÞcally 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. One of the new methods is hierarchical in nature and suitable for use in an adaptive integration scheme. Results show that, for a given required accuracy, the new approaches are more e¦cient than Gauss quadrature for element orders of 4 or greater. The computational advantage increases rapidly with increasing order. Copyright