隐格式近似不可压四面体4节点大变形单元

由于在处理体积自锁方面的优势,近似不可压问题的大变形求解多采用六面体单元/网格,但对于复杂工程问题,由于网格剖分上的限制,往往更需要一种可以很好解决体积自锁的四面体单元。Bonet和Burton的平均节点压力4节点四面体单元是为数不多能够较好处理体积自锁问题的四面体单元之一,但是该单元目前主要用于显式计算。利用单元平均压力对位移增量的精确方向导数,得到了严格的一致切线阵,保证了Newton-Raphson迭代的二阶收敛,从而使得该单元可以用于隐式计算。该单元的压力平均计算会耦合相邻单元的节点自由度,从而增加切线刚度阵的非零带宽,但不增加自由度总数。分别采用线性六面体选择缩减积分单元、标准线性四面体单元和本文的单元计算了3个近似不可压的典型算例。算例表明,本文推导的单元可以有效克服体积自锁,达到与常用六面体单元相近的效果,使得四面体网格可以方便地用于不可压问题的大变形隐式求解。

A large-strain nearly incompressible implicit 4-node tetrahedron

This paper presents a new 4-node tetrahedral element for large-strain nearly incompressible implicit applications.The proposed element is based on the work of Bonet and Burton (1998).The contribution of the paper is the derivation of the exact consistent tangent matrix,the key component of an implicit finite element,by accurate directional derivatives of the averaged element pressure with regard to displacement increment.In contrast to the standard linear tetrahedron element,there is a coupling among degrees of freedom (DOFs) of neighboring elements.This leads to an increased number of nonzero entries in the global tangent matrix,but it has no effect on the total number of DOFs and hence the cost in solving linear equations.Numerical tests were carried out to compare the element with its two popular counterparts,the reduced integrated 8-node hexahedron element and the standard 4-node tetrahedron element.It has been demonstrated that the proposed tetrahedron can avoid volumetric locking and show comparable performance with the afore-mentioned hexahedron element in all the examples tested.