High order isoparametric elements in boundary element method-Smooth elliptical element

摘要

The smooth elliptic element, a high order isoparametric element, is constructed and applied in the boundary element method for modeling closed geometries of ellipses as well as the field variables using a single element. The smoothness of the element is realized by repeated use of real nodes as auxiliary nodes along the circumferential direction of ellipse to remove the end node effect. By writing Lagrange polynomials into the summation of monomials in descending order then into the nested products, the coefficients of shape functions can be generated automatically to get rid of manual work encountered in the construction of a variety of high order elements with different node numbers and distributions. Special techniques of shape function manipulations are suggested and presented in dealing with the integrals with hyper- or near hyper-singularities in a uniform and simple way. A combined technique, the distance transformation with possible element subdivision is proposed to improve the evaluation of near singular integrals. The accuracy and efficiency of using high order elements are demonstrated through a number of numerical examples, by checking the fitting accuracy of geometrical parameters, the computation costs and some simple bench mark tests in elasticity.