论文标题
无网状径向基函数搭配方法与解决泊松和斯托克问题的有限元方法之间的比较
A Comparison Between Meshless Radial Basis Function Collocation Method and Finite Element Method for Solving Poisson and Stokes Problems
论文作者
论文摘要
使用网格依赖的有限元方法和无网状径向基函数搭配方法来求解稳定且不稳定的泊松和Stokes方程，以比较几个标准的这两种数值技术的性能。通过实现形状参数优化算法，可以增强使用多Quadicrics的径向基函数搭配方法的准确性。对于时间依赖性问题，使用向后的Euler方法进行时间离散化。在精度，运行时，条件编号和易于实施标准的情况下评估表演。计算了三种错误；最小平方误，均方根误差和最大相对误差。为了在使用无网状径向基函数置法时计算最小的正方形误差，实现了一种新技术。创建了假想的数值溶液表面，然后计算那些虚构表面和分析溶液表面之间的体积，从而实现了公平的错误计算。最后，将所有解决方案放在一起，并在解决方案节点的数量与运行时，准确性与运行时以及准确性与节点的数量上观察到解决方案趋势。该评估表明有限元方法的性能更好的标准，而当径向基函数搭配方法的表现优于其网格依赖性对应物的标准。
Steady and unsteady Poisson and Stokes equations are solved using mesh dependent Finite Element Method and meshless Radial Basis Function Collocation Method to compare the performances of these two numerical techniques across several criteria. The accuracy of Radial Basis Function Collocation Method with multiquadrics is enhanced by implementing a shape parameter optimization algorithm. For the time-dependent problems, time discretization is conducted using Backward Euler Method. The performances are assessed over the accuracy, runtime, condition number, and ease of implementation criteria. Three kinds of errors were calculated; least square error, root mean square error and maximum relative error. To calculate the least square error while using meshless Radial Basis Function Collocation Method, a novel technique is implemented. Imaginary numerical solution surfaces are created and then the volume between those imaginary surfaces and the analytic solution surfaces is calculated, enabling a fair error calculation. Lastly, all solutions are put together and solution trends are observed over the number of solution nodes vs. runtime, accuracy vs. runtime, and accuracy vs. the number of nodes. The assessment indicates the criteria under which Finite Element Method perform better and those when Radial Basis Function Collocation Method outperforms its mesh dependent counterpart.
