基于最小二乘近似构造流形方法的物理覆盖函数

流形方法可以在保持计算网格不变的前提下,通过在物理覆盖上采用任意形式的解析解级数或各阶多项式来提高数值解的精度,具有许多独特的优势。目前通常采用的、在物理覆盖上使用0阶到多阶多项式作为覆盖函数的方法,不但大幅度地增加了分析问题的总未知量数,而且给边界条件的处理带来了较多的困难。基于最小二乘近似方法来构造流形方法的物理覆盖位移函数,可以在保持总的未知量数不变的情况下构造出满足δij插值条件的高次流形单元,且可以象有限单元法一样方便地施加边界条件,克服了流形方法的上述缺点,为流形单元位移插值函数的构造提供了一条新的途径。悬臂梁、RD梁等算例验证了本文理论与方法的正确性和计算效率。

Construction of physical cover approximation in manifold method based on least square interpolation

In manifold method,the accuracy of numerical solutions can be increased by using an arbitrary expansion of analytical solution or high order polynomials on the physical cover free from changing the computational mesh.However,the use of high order polynomials introduces a large number of extra unknowns and makes it difficult to apply boundary conditions.To overcome these deficiencies,the least square method is employed to construct the displacement interpolation on the physical cover.It provides a new approach to construct high order manifold element without increasing the degrees of freedom and the essential boundary condition can be imposed as easy as it is in conventional FEM.Examples of cantilever beams and Cook beams are presented to demonstrate the effectiveness and efficiency of the present method.