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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。
The manifold method can have many unique advantages by using any form of analytic solution series or polynomials of various orders to improve the accuracy of numerical solution while keeping the computational grid unchanged. At present, the commonly used method of using 0-order to multi-order polynomials as cover function in physical coverage not only greatly increases the total unknowns of the analysis, but also brings more difficulties to the processing of the boundary conditions. Based on least square approximation method, the physical cover displacement function of manifold method can be constructed with the condition of δij interpolation while keeping the total unknowns constant, and can be the same as the finite element method The boundary conditions are conveniently applied to overcome the shortcomings of the manifold method and provide a new way for the construction of manifold displacement interpolation function. Cantilever beam, RD beam and other examples verify the correctness of the theory and methods and computational efficiency.