基本解方法与Trefftz方法基于三维拉普拉斯方程的比较

发布时间：2018-01-11 00:31

  本文关键词：基本解方法与Trefftz方法基于三维拉普拉斯方程的比较　出处：《太原理工大学》2017年硕士论文　论文类型：学位论文

  更多相关文章： Trefftz方法 多重尺度技术 基本解方法 LOOCV

【摘要】：基本解方法和Trefftz方法都是解决齐次偏微分方程边界值问题的两种有效的无网格方法。在Trefftz方法中,近似解由一系列的T完备基函数逼近,而在基本解方法中,近似解由齐次线性微分方程的基本解来逼近。尽管这两种方法都有很长的发展历史,在物理学的各个领域都有广泛的应用,但在数值实现方面都有各自的弊端。Trefftz方法的基函数本质上是多项式函数,因此当用来逼近近似解的T完备基函数数量增多时,会导致基函数的次数呈现指数型增大,从而使所生成的线性系统方程的条件数呈指数型增大,则会造成线性系统方程的严重病态性。而基本解方法需要在问题域外部的谱边界上分布资源点来消除基本解的奇异性,但资源点的最佳分布位置一直是一个很有挑战性的问题。若资源点最佳位置能够确定的话,那么基本解方法则是最有效的边界无网格方法。近年来,Trefftz方法在减弱病态性方面有了很大的发展,特别是使用多重尺度技术在减小线性系统方程的条件数方面有很显著的改善,这样使得Trefftz方法在解决有挑战性的问题时能够更加有效。本文中同样也使用多重尺度技术来研究Trefftz方法在求解三维拉普拉斯方程在不同的复杂三维问题域上的有效性。同时,基本解方法在在确定资源点最佳分布位置方面也有了很大的突破,尤其是近年来使用LOOCV算法使得基本解方法呈现出很高的近似解精确度。基本解方法在求解带有调和边界条件的微分方程时相当有效,但在求解带有非调和边界条件的微分方程时效果并不理想。在本文中,同样也使用LOOCV算法来确定资源点最佳位置,同时提出了一个更简单有效的方法,进一步改善了求解带有非调和边界条件的微分方程的精确度,并且在耗时上也有明显改进。基于使用这些新的方法,本文中对两种方法在不规则复杂三维问题域下对精确性、稳定性以及时间效率上进行了比较。
[Abstract]:The basic solution method and the Trefftz method are two effective meshless methods for solving the boundary value problem of homogeneous partial differential equations. In the Trefftz method. The approximate solution is approximated by a series of T complete basis functions, while in the basic solution method, the approximate solution is approximated by the basic solution of homogeneous linear differential equation, although both methods have a long history of development. It has been widely used in various fields of physics, but it has its own drawbacks in numerical realization. The basis function of Trefftz method is essentially polynomial function. Therefore, when the number of T complete basis functions used to approximate the approximate solution increases, the number of basis functions will increase exponentially, and the condition number of the generated linear system equations will increase exponentially. The fundamental solution method needs to distribute resource points on the spectral boundary outside the problem domain to eliminate the singularity of the fundamental solution. However, the optimal location of resource points is always a challenging problem. If the optimal location of resource points can be determined, the basic solution method is the most effective boundary meshless method in recent years. The Trefftz method has made great progress in reducing the ill-condition, especially in reducing the condition number of linear system equations by using multi-scale technique. In this way, the Trefftz method can be more effective in solving challenging problems. In this paper, we also use multi-scale technique to study the Trefftz method in solving three-dimensional Laplacian equations. The validity of the same complex three-dimensional problem field. The basic solution method has also made a great breakthrough in determining the optimal distribution of resource points. Especially in recent years, LOOCV algorithm is used to make the basic solution method present a high accuracy of approximate solution. The basic solution method is very effective in solving differential equations with harmonic boundary conditions. However, the effect of solving differential equations with non-harmonic boundary conditions is not satisfactory. In this paper, LOOCV algorithm is also used to determine the optimal location of resource points, and a simpler and more effective method is proposed. The accuracy of solving differential equations with nonharmonic boundary conditions is further improved, and the time consuming is also improved obviously. Based on the use of these new methods. In this paper, we compare the accuracy, stability and time efficiency of the two methods in irregular and complex three-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：1407451