基于S-R和分解定理的三维几何非线性无网格法

S-R(strain-rotation)和分解定理克服了经典有限变形理论的一些缺点, 使其可以为几何非线性数值分析提供可靠的理论基础. 对于大变形问题, 由于无网格法(element-free method)避免了对单元网格的依赖, 从而从根本上避免了有限单元法(finite element method, FEM)的单元畸变问题, 保证了求解精度. 因此, 将无网格法和S-R和分解定理结合起来势必能建立一套更加合理可靠的几何非线性数值计算方法. 目前基于S-R 定理的无网格数值方法研究较少并且只能用于二维平面问题的求解, 但实际上绝大多数问题都必须以三维模型来进行处理, 因此建立适用于三维情况的S-R无网格法是非常有必要的. 本文给出了适用于三维情况的S-R 无网格法: 采用由更新拖带坐标法和势能率原理推导出来的增量变分方程, 利用基于全局弱式的无网格Galerkin 法(EFG)得到了用于求解三维空间问题的离散格式. 利用MATLAB编制三维S-R 无网格法程序, 对受均布载荷的三维悬臂梁和四边简支矩形板结构的非线性弯曲问题进行了计算. 最后将所得的数值结果与已有文献进行了比较, 验证了本文的三维S-R无网格数值算法的合理性、有效性和准确性. 本文的三维S-R无网格数值算法可以作为一种可靠的三维几何非线性数值分析方法. 

THREE-DIMENSIONAL GEOMETRIC NONLINEARITY ELEMENT-FREE METHOD BASED ON S-R DECOMPOSITION THEOREM

Due to its overcoming the deficiencies of classic finite deformation theories, Strain-Rotation (S-R) decomposition theorem can provide a reliable theoretical support for the geometrically nonlinear simulation. In addition, due to it’s independent of the elements and meshes, the element-free method has more advantages to solve large deformation problems compared to finite element method (FEM), so that the accuracy is guaranteed as a result of avoiding the element distortions. Therefore, a more reasonable and reliable geometric nonlinearity numerical method certainly will be established by combining the S-R decomposition theorem and element-free method. But the studies of element-free methods based on S-R decomposition theorem in current literature are limited to two-dimensional problems. In most cases, three-dimensional mathematical-physical models must be established for the practical problems. Therefore it is very necessary to establish a three-dimensional element-free method based on the S-R decomposition theorem. Present study extends the previously work by authors into three-dimensional case: The incremental variation equation is derived from updated co-moving coordinate formulation and principle of potential energy rate in this paper, and three-dimensional discretization equations are obtained by element-free Galerkin method (EFG). By using the MATLAB programs based on the proposed 3D S-R element-free method in present study, the nonlinear bending problems for three-dimensional cantilever beam and simply supported plates subjected to uniform load are numerical discussed. The reasonability, availability and accuracy of 3D S-R element-free method proposed by present paper are verified through comparison studies, and the numerical method in present work can provide a reliable way to analysis 3D geometric nonlinearity problems.