基于比例边界坐标高阶完备多边形单元的二维线弹性方程数值求解

Numerical Method of Polygonal Elements for Two-Dimensional Linear Elasticity Equations Based on Scaled Boundary Coordinates

摘要: 针对多边形网格下二维线弹性问题的高阶离散需求，本文研究基于比例边界坐标的高阶完备多边形单元数值方法。首先，介绍比例边界坐标变换下具有高阶多项式完备的多边形单元插值基函数；其次，推导二维线弹性方程的弱形式，建立相应的半离散Galerkin格式与全离散θ差分格式，并给出质量矩阵、刚度矩阵和载荷向量的离散表达；进一步地，对非稳态问题引入θ差分格式，并在相应假设下给出全离散格式的稳定性估计。最后，通过稳态与非稳态数值实验，比较2DSBd2和2DSBd3单元在L^∞范数、L²范数和H¹半范数下的误差与收敛行为。数值结果表明，所构造方法具有良好的精度和收敛性。上述结果说明，基于比例边界坐标的高阶完备多边形单元可为二维线弹性方程的数值求解提供一种有效的离散途径。

Abstract: For the high-order discrete solution of two-dimensional linear elasticity problems on polygonal meshes, this paper studies a high-order complete polygonal element numerical method based on scaled boundary coordinates. First, polygonal element interpolation basis functions with high-order polynomial completeness under the scaled boundary coordinate transformation are introduced. Then, the weak form of the two-dimensional linear elasticity equations is derived, and the corresponding semi-discrete Galerkin formulation and fully discrete θ-difference scheme are established. The discrete expressions of the mass matrix, stiffness matrix, and load vector are also presented. Furthermore, for transient problems, a θ-difference scheme is introduced, and a stability estimate for the fully discrete formulation is given under suitable assumptions. Finally, steady-state and transient numerical experiments are carried out to compare the errors and convergence behavior of the 2DSBd2 and 2DSBd3 elements in the L^∞-norm, L²-norm, and H¹-seminorm. The numerical results show that the constructed method has good accuracy and convergence properties. These results indicate that high-order complete polygonal elements based on scaled boundary coordinates provide an effective discretization approach for the numerical solution of two-dimensional linear elasticity equations.