基于比例边界S网四边形样条单元的热传导与对流扩散问题数值求解

Numerical Solution of Heat Conduction and Convection-Diffusion Problems Based on Scaled Boundary S-Mesh Quadrilateral Spline Elements

摘要: 比例边界有限元方法是一类基于比例边界坐标的数值方法，在多边形单元构造和复杂区域离散中具有一定优势。本文将基于比例边界坐标的四边形样条单元QS-r2d2用于热传导方程和对流扩散方程的数值求解。在空间离散上，采用比例边界坐标，并在三角片上构造径向和环向均为二次的退化张量积Bernstein基函数；在时间离散上，采用θ差分格式，建立相应的全离散Galerkin有限元格式。首先，给出了热传导方程和对流扩散方程的半离散与全离散形式；其次，写出了离散系统的矩阵表达，并对热传导问题的全离散格式进行了误差分析；最后，通过数值算例考察了该方法在不同网格尺度下的误差和收敛表现。结果表明，基于比例边界坐标的QS-r2d2单元能够较好地用于热传导问题和对流扩散方程的数值求解，所得数值结果具有较好的收敛性。与同阶Lagrange有限元方法相比，该方法在部分测试算例中表现出更高的计算精度。上述结果说明，比例边界S网样条有限元方法可作为求解这类方程的一种有效空间离散方式。

Abstract: The scaled boundary finite element method is a numerical method based on scaled boundary coordinates and has advantages in the construction of polygonal elements and the discretization of complex computational domains. In this paper, the quadrilateral spline element QS-r2d2 based on scaled boundary coordinates is applied to the numerical solution of heat conduction equations and convection-diffusion equations. For the spatial discretization, scaled boundary coordinates are employed, and degenerated tensor-product Bernstein basis functions of degree two in both the radial and circumferential directions are constructed on each triangular subregion. For the temporal discretization, a θ-difference scheme is adopted, and the corresponding fully discrete Galerkin finite element formulation is established. First, the semi-discrete and fully discrete formulations for the heat conduction equation and the convection-diffusion equation are derived. Then, the matrix form of the discrete system is presented, and an error analysis is carried out for the fully discrete scheme of the heat conduction problem. Finally, several numerical examples are provided to investigate the error behaviour and convergence performance of the proposed method under different mesh sizes. The numerical results show that the QS-r2d2 element based on scaled boundary coordinates can be effectively applied to the numerical solution of heat conduction and convection-diffusion problems, and that the obtained solutions exhibit good convergence properties. Compared with the Lagrange finite element method of the same order, the proposed method achieves higher computational accuracy in some test cases. These results indicate that the scaled-boundary S-net spline finite element method provides an effective spatial discretization approach for this class of time-dependent partial differential equations.