基于互补理论的扩展有限元接触问题实现

Implementation of XFEM's contact problem based on complementary law

在处理含有裂隙这类不连续问题时,常规有限元方法需要对裂隙尖端部位进行局部网格加密,当裂隙扩展时还需要进行网络重构。基于单位分解思想的扩展有限元成功解决了常规有限元难以处理的裂隙类不连续问题。在研究复合裂隙时,通常需要考虑裂隙的接触问题。基于互补理论,建立了裂隙面上相对位移和接触力的互补方程,并采用牛顿法求解,无需开闭迭代,且能够快速收敛。最后,对含裂隙平板进行受压数值试验,计算结果表明,基于互补理论的扩展有限元接触算法能够有效地阻止裂隙两端网格的相互嵌入,且获得裂隙面上的应力分布与实际一致。

When dealing with discontinuous issues such as fracture and crack in geo-engineering,conventional finite element method（CFEM） need to refine mesh in the local zone including crack tip;furthermore,the mesh must be reconfigured and re-partitioned once the crack propagation happened.The extended finite element method（XFEM）,based on the idea of partition of unit method,can successfully solve these problems easily,which may hardly be deposed by CFEM.Usually,contact problem of crack surface must be considered when studying some compound fracture.Based on the complementary theory,the complementary equation between relative displacement and contact force of the crack surface can be established,and solved by Newton＇s method without considering opening and closing iteration.Finally,as a numerical example,a plate with crack is compressed to indicate the effectiveness of this method.The results indicate that the method can prevent mesh penetration from one side of the crack into the other side effectively;and also obtain the stress distribution that is consistent with the actual on the crack surfaces.