Hermite梯度重构核近似配点法及其在功能梯度材料板的静力分析中的应用

Hermite Gradient Reproducing Kernel Collocation Method and Its Application in Static Analysis of Functionally Gradient Material Plates

由于直接配点法在求解边值问题时边界上的求解精度较低,论文提出了Hermite梯度重构核近似配点法(HGCM)来改进边界求解精度.重构核近似是无网格法中一种常用的近似函数,但是其在求解高阶导数时格式复杂且非常耗时.HGCM采用梯度重构核近似构建形函数的任意高阶导数,提高了计算效率;通过Hermite配点法构建离散方程,提高了边界求解精度.这种方法在求解对应变系数四阶偏微分方程的功能梯度材料板的静力问题时精度高,计算效率高,并可进一步推广应用于高阶偏微分方程描述的边值问题.

Collocation type meshfree methods are one of the typical types of meshfree methods which have easy implementation and high efficiency. However, direct collocation method has low solution accuracy on the boundary when solving boundary value problems. Therefore, this paper proposes a Hermite Gradient Reproducing Kernel Collocation Method(HGCM) to improve the boundary solution accuracy. Hermite collocation method can improve the accuracy of the solutions by introducing more degrees of freedom on the boundaries. Reproducing kernel approximation is a commonly used approximation function in meshfree methods, but calculation of the high gradients is complex and time-consuming. HGCM adopts the gradient reproducing kernel(GRK) functions to create any high-order derivatives of the approximation function. High order derivations of RK function require high order differentiation of moment matrix inversion while GRK only needs the high order differentiation of basis functions. Utilizing GRK can remarkably decrease the complexity and computation costs, which is much easier and more efficient to calculate the gradients of the shape function. This improves the computational efficiency. Functionally graded materials(FGMs) are a type of advanced composite materials whose material properties areof continuous distribution. FGMs can achieve maximum benefits from the design of material composition based on their inhomogeneity. Different from the governing equations of constant coefficients for other homogeneous materials, the governing equations for FGMs are of variable coefficients partial differential equations(PDEs). It’s hard to use the conventional finite element method(FEM) based on low order shape functions to solve such variable coefficients PDEs. By employing high order shape functions, HGCM has high accuracy and high computational efficiency in solving the static problems of functionally graded material plates which is governed by a fourth-order partial differential equation of variant coefficients. Numerical examples demonstrate the good performance of the proposed method. This method can be further applied to boundary value problems described by high-order partial differential equations.