两个抗畸变的四边形膜元

Two robust quadrilateral membrane elements

为了保证单元的可靠性,单元应具备抗畸变的良好性能。但现有不少单元对网格畸变十分敏感,如Serendipity等参元。在规则网格情况下,它们的精度不错;而当网格畸变时,其精度则急剧下降。为了克服这一缺陷,文献中提出了各种方案,使畸变敏感现象得到减轻,但目前这一缺陷尚未得到根治。该文旨在研究抗畸变的四结点四边形膜元。鉴于Serendipity等参元的上述缺点,该文不采用等参坐标而改用四边形面积坐标,并构造出两个抗畸变的四边形膜元AQ6I和AQ6II。数值试验结果表明,这两个单元不仅可以在畸变网格下给出纯弯问题的精确解,而且可以克服MacNeal畸变网格细长梁的梯形闭锁现象。弱式分片检验表明这两个单元是收敛的、可靠的。

A robust element used in the finite element method should be insensitive to mesh distortion. Some elements in the literature are very sensitive to mesh distortion, such as the Serendipity isoparametric elements. The precision of these elements is very high for regular meshes but very low for distorted meshes. Various methods have been proposed in the literature to overcome the sensitivity to mesh distortion. This paper presents two quadrilateral membrane elements that are insensitive to mesh distortion. The formulations use quadrilateral area coordinates instead of the isoparametric coordinates. Numerical examples show that both elements yield exact solutions for pure bending problems in distorted meshes and provide lockfree solutions for the MacNeal test problem of trapezoidal locking. The weak form of the patch test shows that both elements are convergent and reliable.