杂交有限元构造的正交假设应力场方法

On efficient construction of hybrid finite elements with orthogonal assumed stress field method

分别对各向同性和正交各向异性材料的假设应力场进行正交化并形成相应的杂交元,由于避免了柔度矩阵求逆运算,从而提高了杂交元分析效率。对各向同性材料杂交元直接利用柔度矩阵特征向量导出了正交假设应力场,其正交性不依赖于材料,因而具有更好的适用性。此外由于不需要借助本征变形模式进行迭代而避免了复杂的数值计算。对于正交各向异性材料提出了一种材料矩阵分裂法对假设应力场进行正交化,研究结果表明,所得的正交应力场只与材料两个主方向弹性模量的比值有关,因而不受横向泊松效应的影响。采用本文方法对2D-4节点单元和3D-8节点单元的常用应力场进行正交化,给出了十分简洁的结果。

A set of orthogonally assumed stress fields are presented to efficiently formulate the hybrid elements of isotropic and orthotropic materials. In the proposed approach, the inverse of flexibility matrix is avoided, so the computational efficiency is significantly improved. The orthogonal stress field for isotropic material is obtained by using the transformation matrix formed by the eigenvectors of element flexibility matrix. It is noted that there is no numerical computation involved here and the orthogonality within the stress field is fully independent of the material properties. For orthotropic materials, a method of material matrix decomposition is proposed to derive the orthogonal stress field. This orthogonal relationship depends on the ratio between the elastic modulus in the two principal directions but is totally free of the Poison response due to transverse deformation. Several 2-D and 3-D examples show that the proposed method is very straightforward and effective.