本文關鍵詞：基本解方法與Trefftz方法基于三維拉普拉斯方程的比較　出處：《太原理工大學》2017年碩士論文　論文類型：學位論文

  更多相關文章： Trefftz方法 多重尺度技術 基本解方法 LOOCV

【摘要】：基本解方法和Trefftz方法都是解決齊次偏微分方程邊界值問題的兩種有效的無網(wǎng)格方法。在Trefftz方法中,近似解由一系列的T完備基函數(shù)逼近,而在基本解方法中,近似解由齊次線性微分方程的基本解來逼近。盡管這兩種方法都有很長的發(fā)展歷史,在物理學的各個領域都有廣泛的應用,但在數(shù)值實現(xiàn)方面都有各自的弊端。Trefftz方法的基函數(shù)本質上是多項式函數(shù),因此當用來逼近近似解的T完備基函數(shù)數(shù)量增多時,會導致基函數(shù)的次數(shù)呈現(xiàn)指數(shù)型增大,從而使所生成的線性系統(tǒng)方程的條件數(shù)呈指數(shù)型增大,則會造成線性系統(tǒng)方程的嚴重病態(tài)性。而基本解方法需要在問題域外部的譜邊界上分布資源點來消除基本解的奇異性,但資源點的最佳分布位置一直是一個很有挑戰(zhàn)性的問題。若資源點最佳位置能夠確定的話,那么基本解方法則是最有效的邊界無網(wǎng)格方法。近年來,Trefftz方法在減弱病態(tài)性方面有了很大的發(fā)展,特別是使用多重尺度技術在減小線性系統(tǒng)方程的條件數(shù)方面有很顯著的改善,這樣使得Trefftz方法在解決有挑戰(zhàn)性的問題時能夠更加有效。本文中同樣也使用多重尺度技術來研究Trefftz方法在求解三維拉普拉斯方程在不同的復雜三維問題域上的有效性。同時,基本解方法在在確定資源點最佳分布位置方面也有了很大的突破,尤其是近年來使用LOOCV算法使得基本解方法呈現(xiàn)出很高的近似解精確度。基本解方法在求解帶有調和邊界條件的微分方程時相當有效,但在求解帶有非調和邊界條件的微分方程時效果并不理想。在本文中,同樣也使用LOOCV算法來確定資源點最佳位置,同時提出了一個更簡單有效的方法,進一步改善了求解帶有非調和邊界條件的微分方程的精確度,并且在耗時上也有明顯改進�；谑褂眠@些新的方法,本文中對兩種方法在不規(guī)則復雜三維問題域下對精確性、穩(wěn)定性以及時間效率上進行了比較。
[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.
【學位授予單位】：太原理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.82
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本文編號：1407451