基于泰勒展式的混合阶次流形方法

对于流形方法,其高阶位移场函数的构造大多是采用完全多项式函数,但这种处理使得升阶后的各个广义自由度完全丧失物理意义。为避免出现上述问题,采用泰勒展开法,将覆盖位移函数看作是某点的泰勒展开。基于此泰勒展式,建立了位移函数与节点位移、应变和转角之间的函数关系,使得升阶后的各个广义自由度都具有明确的物理意义。选取矩形格子作为数学网格,减少了物理片的生成,使前后处理变得更简单;在结构求解区域使用混合阶次的覆盖位移场函数来提高解题效率,能实现解析解与数值解的完美结合;采用改进的罚函数与广义节点法相结合的方式来处理边界条件,严格符合边界条件的物理意义;最后结合数值算例验证了该方法的高效性,与此同时数值解精度也得到了极大提高。

Mixed-order Manifold Method Based on Taylor Expansion

For manifold method, high-order displacement function is established by using complete polynomial function, which deprives the generalized freedom degrees of physical meaning. To avoid this problem, the cover displacement function can be regarded as Taylor expansion. Based on the Taylor expansion, the relations among displacement function, nodal displacement and strain are obtained, and generalized degrees of freedom have definite physical meaning after degree elevation. Furthermore, rectangular grid is used as mathematical grid to reduce physical piece and simplify preprocessing and post-processing. In the computational domain, cover displacement function with mixed-order is employed to improve computation efficiency and to combine analytical solution and numerical solution. Moreover, improved penalty function and generalized node function are used together to deal with boundary conditions, which is in strict accordance with the physical meaning of boundary condition. Finally, this method is proved to be highly efficient by numerical examples, and the accuracy of numerical solution is also improved.