A six-node co-rotational curved triangular shell finite element with a novel rotation treatment for folded and multi-shell structures is presented. Different from other co-rotational triangular element formulations, rotations are not represented by axial (pseudo) vectors, but by components of polar (proper) vectors, of which additivity and commutativity lead to symmetry of the tangent stiffness matrices in both local and global coordinate systems. In the co-rotational local coordinate system, the two smallest components of the shell director are defined as the nodal rotational variables. Similarly, the two smallest components of each director in the global coordinate system are adopted as the global rotational variables for nodes located either on smooth shells or away from non-smooth shell intersections. At intersections of folded and multi-shells, global rotational variables are defined as three selected components of an orthogonal triad initially oriented along the global coordinate system axes. As such, the vectorial rotational variables enable simple additive update of all nodal variables in an incremental-iterative procedure, resulting in significant enhancement in computational efficiency for large deformation analysis. To alleviate membrane and shear locking phenomena, an assumed strain method is employed in obtaining the element tangent stiffness matrices and the internal force vector. The effectiveness of the presented co-rotational triangular shell element formulation is verified by analyzing several benchmark problems of smooth, folded and multi-shell structures undergoing large displacements and large rotations.

提出了一种六节点同向旋转三角壳有限元，对折叠和多壳结构进行了新颖的旋转处理。与其他同向旋转三角元素公式不同，旋转不是由轴向（伪）向量表示，而是由极性（适当）向量的分量表示，其极性的相加和交换导致局部和全局坐标上切线刚度矩阵的对称系统。在同向旋转局部坐标系中，壳体指向矢的两个最小分量定义为节点旋转变量。同样，全局坐标系中每个指向矢的两个最小分量被用作位于光滑壳体上或远离非光滑壳体相交处的节点的全局旋转变量。在折叠壳和多壳的相交处，全局旋转变量定义为最初沿全局坐标系轴定向的正交三重轴的三个选定分量。这样，矢量旋转变量就可以在增量迭代过程中对所有节点变量进行简单的累加更新，从而大大提高了大变形分析的计算效率。为了减轻膜和剪切锁定现象，采用假定的应变方法来获得单元切线刚度矩阵和内力矢量。通过分析承受大位移和大旋转的光滑，折叠和多壳结构的几个基准问题，验证了提出的同向旋转三角形壳单元公式的有效性。全局旋转变量定义为最初沿全局坐标系轴定向的正交三重轴的三个选定分量。这样，矢量旋转变量使增量迭代过程中的所有节点变量都能简单地累加更新，从而大大提高了大变形分析的计算效率。为了减轻膜和剪切锁定现象，采用假定的应变方法来获得单元切线刚度矩阵和内力矢量。通过分析承受大位移和大旋转的光滑，折叠和多壳结构的几个基准问题，验证了提出的同向旋转三角形壳单元公式的有效性。全局旋转变量定义为最初沿全局坐标系轴定向的正交三重轴的三个选定分量。这样，矢量旋转变量就可以在增量迭代过程中对所有节点变量进行简单的累加更新，从而大大提高了大变形分析的计算效率。为了减轻膜和剪切锁定现象，采用假定的应变方法来获得单元切线刚度矩阵和内力矢量。通过分析承受大位移和大旋转的光滑，折叠和多壳结构的几个基准问题，验证了提出的同向旋转三角形壳单元公式的有效性。