On The Construction And Performance Of High Order Smooth Boundary Elements

Inclusions,second phases and pores in solids have important effects on the properties of materials,including particle reinforced composites.Huge numbers of elements in the use of FEM have to be employed to simulate particles accurately.Although the interfaces between the matrix and particles need only to be discretized in the use of the BEM,many elements are also indispensable in order to fit the highly curved boundary of particles accurately even with the quadratic boundary elements.Therefore,the scale of the final system matrix of algebraic equations will grow large dramatically whether for the FEM or for the traditional BEM,bringing difficulties to the solution of the problem.One of the effective ways to reduce the solution scale of the problem is to use the closed high-order element.However,as the end nodes and end lines exist in the traditional elements,including the closed high-order element,across which the element geometry is not smooth when discretizing particles.The accuracy of simulation cannot be guaranteed at these places.For these problems,based on the traditional high-order closed element,a novel high-order smooth isoparametric element is constructed in the paper by using the geometric characteristics of ellipses and ellipsoids,such as symmetry,smoothness and periodicity.The numerical methods are proposed and improved for evaluating various singular and near singular integrals encountered in the BEM.For the 2-D problem,by making use of the symmetry,smoothness and periodicity of elliptical particles,the nodes at the two ends of the element are used repeatedly as supplemental interpolation nodes to raise the order of the Lagrange interpolation polynomials and to eliminate the end effect of the original closed element without the increase of the number of nodes of the element.In this way,the novel high order smooth elliptic element is constructed with the much improved simulation accuracy.In order to further improve the efficiency,by writing Lagrangepolynomials into the summation of monomials in descending order then into the nested products,the coefficients of interpolation shape functions can be generated automatically to get rid of tedious manual work encountered in the construction of a variety of high order elements with different node numbers and distributions.For the spheroidal particles or holes in the 3-D problem,the novel high order smooth spheroidal element is constructed by making use of the symmetry,smoothness and periodicity of spheroids as well as the repeated use of nodes.The discretization of the spheroidal surface is carried out along the latitude and meridian directions.Noticed that latitudes are all closed circles therefore the same measure is taken as that in the 2-D to eliminate the end-line effect in the original closed element.Since the meridians are all open half circles or half ellipses with the two poles being the common endpoints of them,supplementary interpolation points are suitably selected cross over the poles to put the two poles inside the element.In this way,the orders of the Lagrange interpolation polynomials are raised in both the latitude and meridian directions with the much improved interpolation accuracy without the increase of the actual number of nodes of the element.The correctness and effectiveness of the high order smooth boundary elements are verified through a large number of numerical examples.The use of the high order smooth boundary elements can improve the computational efficiency by greatly reducing the number of nodes to be used with the decrease of the solution scale of the problem,comparing with the low order elements,and can greatly improve the computational accuracy,comparing with the traditional closed elements,very suitable for the large scale numerical simulation of solids with inclusions,second phases and pores.In the end of the paper,a brief introduction is given for the limits of the present model and the expectation of further work.