高阶数值流形方法的初应力公式

Initial stress equation for high-order numerical manifold method

高阶数值流形方法和高阶DDA方法可以显著提高结构变形的计算精度,但目前涉及几何非线性问题的研究成果大都计算精度差甚至不收敛,这是由高阶初应力公式的不准确或不正确引起的。本文介绍数值流形方法的大变形计算格式,基于平面三角形数学网格和多项式覆盖函数,提出高阶流形法的两种初应力处理方法,首次导出了高阶初应力的准确公式。该公式在分步计算的初应力累加中考虑了大变形结构的构形变化,并将初应力表示成多项式函数形式以满足单纯形积分的要求。文中给出的悬臂梁大变形数值算例与理论解的对比结果证明了方法的正确性。本文的方法和公式也适用于三维四面体数学网格,稍加修改后将可应用于高阶DDA方法和常规的有限元方法。

With high-order Numerical Manifold Method （NMM） or high-order Discontinuous Deformation Analysis （DDA） method, computational accuracy of structure deformation can be improved greatly. However, poor accuracy was obtained and even computation was not convergent while treating geomet- ric nonlinear problems, due to inaccurate or incorrect high-order initial stress equation. Based on 2-D triangular mathematical meshes and polynomial cover functions, two methods are presented in this paper to solve the initial stress problems in high-order NMM. Exact equation for high-order initial stress is de- rived for the first time, reflecting configuration change of structures under large deformations when accumulating initial stresses at each step. The equation is expressed in polynomial form so as to be used in simplex integrations. Comparing with analytical solutions, good results for large deformation of a canti- lever prove the validity of the equation. The methods can also take effect with 3-D tetrahedral mathematical meshes, and can be applied in high-order DDA or regular FEM after some modifications in the future.