發(fā)布時(shí)間：2018-05-04 04:11

  本文選題：無(wú)網(wǎng)格方法 + Trefftz方法��； 參考：《太原理工大學(xué)》2017年碩士論文

【摘要】：無(wú)網(wǎng)格方法是針對(duì)網(wǎng)格法例如有限元計(jì)算方法而生的。有限元法在工程上有著非常廣泛的應(yīng)用。但是有限元方法的插值都是基于網(wǎng)格的,當(dāng)網(wǎng)格扭曲或者網(wǎng)格質(zhì)量很差時(shí),計(jì)算精度會(huì)降低,而且很多不連續(xù)的問(wèn)題無(wú)法進(jìn)行網(wǎng)格劃分。無(wú)網(wǎng)格方法不需要進(jìn)行網(wǎng)格劃分,從而解決了上述問(wèn)題。但是無(wú)網(wǎng)格方法也不是萬(wàn)能的,也存在一些值得改進(jìn)的地方。本文介紹了三種改進(jìn)的無(wú)網(wǎng)格方法:帶有LOOCV程序的基本解方法,多尺度Trefftz方法和帶有多項(xiàng)式基函數(shù)的特解法�；窘夥椒ê团渲肨refftz方法是兩種邊界型的方法,它們可以只使用邊界點(diǎn)即可高效求解調(diào)和方程。盡管這兩種方法有眾多的優(yōu)點(diǎn),但是它們?cè)谟?jì)算過(guò)程中還存在一些缺陷有待改進(jìn)。近年來(lái),基本解方法在資源配置點(diǎn)的選擇上有了很大的改進(jìn),而Trefftz方法則在改進(jìn)奇異性和降低條件數(shù)的方面上有了很大的提升。在這里,我們使用LOOCV程序改進(jìn)了基本解方法,使得基本解方法在最優(yōu)資源點(diǎn)的選取成為可能;還使用了多尺度方法改進(jìn)了Trefftz方法,降低了Trefftz方法的病態(tài)問(wèn)題。本文將對(duì)這兩種改進(jìn)了的方法在不規(guī)則的問(wèn)題域上求解非調(diào)和方程的計(jì)算結(jié)果進(jìn)行對(duì)比。本文還介紹了使用多項(xiàng)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,然后用求解方程減去特解后轉(zhuǎn)化成為調(diào)和方程,再使用基本解方法或者Trefftz方法。本文采用了多項(xiàng)式作為基函數(shù)的特解法,不需要結(jié)合其他方法,即可直接求解三維問(wèn)題域上的軸對(duì)稱泊松方程。首先將三維問(wèn)題域上的軸對(duì)稱方程轉(zhuǎn)換為二維問(wèn)題域上的橢圓方程。為了改進(jìn)了多項(xiàng)式基帶來(lái)的不穩(wěn)定性,在特解法求解過(guò)程中,使用特征長(zhǎng)度對(duì)裝配矩陣進(jìn)行處理。本文中帶有多項(xiàng)式基函數(shù)的特解法分為正向特解法和逆向特解法。正向特解法是直接使用多項(xiàng)式的線性組合來(lái)近似方程的解。逆向特解法是將待求解的微分方程右端項(xiàng)先使用若干多項(xiàng)式線性組合來(lái)近似。然后找到在該微分方程算子下用來(lái)近似右端項(xiàng)的每一個(gè)多項(xiàng)式的特解。然后把這些特解作為特解法的基函數(shù)來(lái)數(shù)值近似方程的解。本文將Kansa’s-RBF在三維上求解軸對(duì)稱的泊松方程的結(jié)果與這兩種特解法在二維問(wèn)題域上求解該方程的結(jié)果進(jìn)行了對(duì)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.
【學(xué)位授予單位】：太原理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.82
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本文編號(hào)：1841528