改进的无网格方法求解带有拉普拉斯算子的方程

发布时间：2018-05-04 04:11

  本文选题：无网格方法 + Trefftz方法　； 参考：《太原理工大学》2017年硕士论文

【摘要】：无网格方法是针对网格法例如有限元计算方法而生的。有限元法在工程上有着非常广泛的应用。但是有限元方法的插值都是基于网格的,当网格扭曲或者网格质量很差时,计算精度会降低,而且很多不连续的问题无法进行网格划分。无网格方法不需要进行网格划分,从而解决了上述问题。但是无网格方法也不是万能的,也存在一些值得改进的地方。本文介绍了三种改进的无网格方法:带有LOOCV程序的基本解方法,多尺度Trefftz方法和带有多项式基函数的特解法。基本解方法和配置Trefftz方法是两种边界型的方法,它们可以只使用边界点即可高效求解调和方程。尽管这两种方法有众多的优点,但是它们在计算过程中还存在一些缺陷有待改进。近年来,基本解方法在资源配置点的选择上有了很大的改进,而Trefftz方法则在改进奇异性和降低条件数的方面上有了很大的提升。在这里,我们使用LOOCV程序改进了基本解方法,使得基本解方法在最优资源点的选取成为可能;还使用了多尺度方法改进了Trefftz方法,降低了Trefftz方法的病态问题。本文将对这两种改进了的方法在不规则的问题域上求解非调和方程的计算结果进行对比。本文还介绍了使用多项式基的特解法。之前流行的特解法是先找到要求解方程的特解,然后用求解方程减去特解后转化成为调和方程,再使用基本解方法或者Trefftz方法。本文采用了多项式作为基函数的特解法,不需要结合其他方法,即可直接求解三维问题域上的轴对称泊松方程。首先将三维问题域上的轴对称方程转换为二维问题域上的椭圆方程。为了改进了多项式基带来的不稳定性,在特解法求解过程中,使用特征长度对装配矩阵进行处理。本文中带有多项式基函数的特解法分为正向特解法和逆向特解法。正向特解法是直接使用多项式的线性组合来近似方程的解。逆向特解法是将待求解的微分方程右端项先使用若干多项式线性组合来近似。然后找到在该微分方程算子下用来近似右端项的每一个多项式的特解。然后把这些特解作为特解法的基函数来数值近似方程的解。本文将Kansa’s-RBF在三维上求解轴对称的泊松方程的结果与这两种特解法在二维问题域上求解该方程的结果进行了对比。
[Abstract]:The meshless method is based on mesh method such as finite element method. Finite element method is widely used in engineering. But the interpolation of finite element method is based on meshes. When the mesh is distorted or the quality of mesh is very poor, the accuracy will be reduced, and many discontinuous problems can not be meshed. The meshless method does not need to be meshed, thus solving the above problem. However, the meshless method is not omnipotent, and there are some improvements to be made. This paper introduces three improved meshless methods: the basic solution method with LOOCV program, the multiscale Trefftz method and the special solution with polynomial basis function. The basic solution method and the collocation Trefftz method are two kinds of boundary type methods, which can efficiently solve harmonic equations only by using boundary points. Although the two methods have many advantages, there are still some defects in the calculation process. In recent years, the basic solution method has been greatly improved in the selection of resource allocation points, while the Trefftz method has been greatly improved in improving singularity and reducing the number of conditions. Here, we use LOOCV program to improve the basic solution method, which makes it possible to select the basic solution method in the optimal resource point. We also use the multi-scale method to improve the Trefftz method and reduce the ill-posed problem of the Trefftz method. In this paper, the results of the two improved methods for solving irreconcilable equations in irregular problem domain are compared. The special solution using polynomial basis is also introduced in this paper. The popular special solution is to find the special solution which requires the solution of the equation, then subtract the special solution from the solution equation and transform it into a harmonic equation, and then use the basic solution method or the Trefftz method. In this paper, the special solution of polynomial as the basis function is adopted, and the axisymmetric Poisson equation on the three-dimensional problem domain can be solved directly without the combination of other methods. Firstly, the axisymmetric equations in the three-dimensional problem domain are transformed into the elliptic equations in the two-dimensional problem domain. In order to improve the instability caused by polynomial basis, the characteristic length is used to deal with the assembly matrix in the special solution. In this paper, the special solution with polynomial basis function is divided into forward special solution and reverse special solution. The forward special solution is a direct use of the linear combination of polynomials to approximate the solution of the equation. The inverse special solution is to approximate the right end term of differential equation by some polynomial linear combination. Then we find the special solution of every polynomial which is used to approximate the right end term under the operator of the differential equation. Then these special solutions are used as the basis functions of the special solution to approximate the solution of the equation. In this paper, the results of solving the axisymmetric Poisson equation by Kansa's-RBF in 3D are compared with the results obtained by these two special solutions in the two-dimensional problem domain.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
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本文编号：1841528