一种Trefftz孔洞单元及其在接触问题中的应用

A Trefftz Hole Element and its Application to Contact Problems

作为一种高效的数值计算工具,Trefftz有限元法愈来愈引人瞩目。与传统有限元法相比,Trefftz有限元法在处理带有孔洞、裂纹、夹杂等局部效应的问题时,无需另外细分网格,只需调整单元域内(Trefftz)插值函数,就能达到比较理想的精度。基于弹性复变理论和保角变换,求出孔洞问题的完备系,以其作为Trefftz插值函数,然后将两套单元位移场和边界条件代入修正变分泛函,再根据驻值条件构造出一种改进的Trefftz孔洞单元。数值算例表明,该类单元对于分析孔洞问题既简单又有效。

As an efficient computational tool,Trefftz finite element method（FEM） has been attracting extensive attention.Compared with the conventional FEM,the Trefftz FEM offers high accuracy in treating the problems with local effects such as holes,cracks or inclusions,simply by using appropriate intra-element（Trefftz） interpolation functions without mesh refinement.Based on the elastic complex theory and conforming transformation,the complete solution system of the hole problems was first derived and used as the Trefftz interpolation functions.Then,the two set of displacement fields and boundary conditions were substituted into the modified variational functional to construct an improved hole element by stationary condition.Numerical examples indicate that the proposed element exhibits simple and effective when analyzing the hole problems.