数学网格和物理网格分离的有限单元法(I):基本理论

常规有限单元法在复杂边界问题的网格剖分、可移动边界和非连续变形问题的数值模拟等方面存在困难.本文将常规的有限单元分离为几何上相互独立的数学单元和物理单元,基于数学单元构造近似函数,引入位移模式关联法则以确定物理单元的位移模式,提出了在现有有限单元法框架内、基于数学网格和物理网格分离的强化有限单元法(FEM++).与常规有限单元法(SFEM)比较表明,强化有限单元法不仅很好地克服了常规有限单元法网格剖分上的困难,而且提供了一条更简便、更自然的分析移动边界问题和非连续变形问题的新途径.最后,通过数值算例验证了强化有限单元法的适用性和有效性.

An enhanced finite element method with separate mathematical and physical mesh

Basing on separation between mathematical meshes and physical meshes, the enhanced finite element method (FEM⁺⁺) makes it easy to describe discontinuous deformation. The cohesive zone model (CZM) which can avoid the stress singularity at the crack tip is one of the most effective and simplest model to simulate the interaction within fracture process zone. FEM⁺⁺ and CZM are applied to simulate the propagation of arbitrary cohesive crack by enriching mathematical node in this paper. The rule of mathematical node enrichment and mathematical element definition are also summarized, and an 8-16 node CZM element is proposed for cohesive crack. At last, three point bending beam is analyzed to illuminate the efficiency of the proposed method.