DIRECT COLLOCATION METHOD AND STABILIZED COLLOCATION METHOD BASED ON LAGRANGE INTERPOLATION FUNCTION
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摘要: 由于无网格法中大多数近似函数均为有理式, 不具有Kronecker delta性质, 因此难以精确地施加本质边界条件. 边界误差较大容易导致整个求解域求解结果精度低, 甚至引起数值不稳定现象. 文章在无网格直接配点法和稳定配点法中引入拉格朗日插值函数作为形函数, 构建了拉格朗日插值配点法(LICM)和拉格朗日插值稳定配点法(SLICM). 由于拉格朗日插值具有Kronecker delta性质, 可以像有限元法一样简单而精确地施加本质边界条件, 提高这两种方法的数值求解精度. 稳定配点法基于子域对强形式方程进行积分, 可以满足高阶积分约束, 即可以保证形函数在积分形式下也满足高阶一致性条件, 实现精确积分. 同时, 进行子域积分还可以减少离散矩阵的条件数, 从而提高算法的稳定性. 进一步提高拉格朗日插值稳定配点法的精度和稳定性. 通过数值算例验证这两种方法的精度、收敛性和稳定性, 结果表明基于拉格朗日插值的配点法的精度优于基于重构核近似的配点法, 拉格朗日插值稳定配点法的精度和稳定性均优于拉格朗日插值配点法.
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关键词:
- 拉格朗日插值配点法 /
- 拉格朗日插值稳定配点法 /
- Kronecker delta性质 /
- 精确积分 /
- 精度 /
- 稳定性
Abstract: Since most of the approximation functions in the meshfree method are rational and do not have the Kronecker delta property, it is difficult to accurately impose the essential boundary conditions. Large errors on the boundary can easily lead to low accuracy of the solution in the whole solution domain and may even introduce the numerical instability in solution process. In this paper, the Lagrange interpolation function is introduced as the shape function in the meshfree direct collocation method and the stabilized collocation method, and the Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM) are constructed. Since Lagrange interpolation has the Kronecker delta property, the essential boundary conditions can be imposed as simply and precisely as the finite element method, which promotes the numerical solution accuracy of the two methods. The stabilized collocation method is based on the subdomain integration, which can satisfy the high order integration constraints. That is, it can ensure that the shape function also meets the high-order consistency conditions in the integral form and achieve accurate integration. At the same time, the subdomain integration can also reduce the condition number of the discrete matrix, which improves the stability of the algorithm. By combining the Lagrange interpolation function and the stabilized collocation method, the accuracy and stability of the stabilized Lagrange interpolation collocation method is further improved. Numerical examples validate the accuracy, convergence and stability of the proposed Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM). The results show that the accuracy of the collocation methods based on the Lagrange interpolation function is higher than that of the collocation method based on the reproducing kernel function, and the accuracy and stability of the stabilized Lagrange interpolation collocation method are superior to those of the Lagrange interpolation collocation method. -
图 1 一维2节点和3节点拉格朗日插值形函数的节点分布图
Figure 1. Node distributions of the 1D 2-node and 3-node Lagrange interpolation shape functions
图 2 二维4节点、6节点和9节点拉格朗日插值形函数的节点分布图
Figure 2. Node distributions of the 2D 4-node, 6-node and 9-node Lagrange interpolation shape functions
图 12 二维泊松问题源点离散图
Figure 12. Discrete points of source points for the 2D Poisson problem
图 14 二维泊松问题边界位移解
Figure 14. Numerical solutions of boundary for the 2D Poisson problem
图 17 三维亥姆霍兹问题源点离散图
Figure 17. Discrete points of source points for the 3D Helmholtz problem
图 18 三维亥姆霍兹问题域内位移解
Figure 18. Numerical solutions of the inner domain for the 3D Helmholtz problem
图 19 三维亥姆霍兹问题边界位移解
Figure 19. Numerical solutions of boundary for the 3D Helmholtz problem
19 三维亥姆霍兹问题边界位移解 (续)
19. Numerical solutions of boundary for the 3D Helmholtz problem (continued)
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