本文選題：光滑有限元方法 + G空間��； 參考：《太原理工大學(xué)》2017年碩士論文

【摘要】：隨著計算機的高速發(fā)展,人們提出了許多有效的數(shù)值方法去解決實際中的工程問題,如有限元方法(FEM)。但是,由于網(wǎng)格的形狀會直接影響計算精度,所以FEM對網(wǎng)格的質(zhì)量要求非常高。為克服這一難題,僅基于低質(zhì)量網(wǎng)格的光滑有限元方法(S-FEMs)和一些無網(wǎng)格的方法被建立,并廣泛地應(yīng)用于固體力學(xué)、熱傳導(dǎo)學(xué)和結(jié)構(gòu)聲學(xué)等復(fù)雜的領(lǐng)域中。本文將對這兩類數(shù)值算法進行相應(yīng)的理論分析。第2章的G~s空間理論是依賴于弱弱形式(W2)模型建立的,因此以該空間為基礎(chǔ)的S-FEMs和光滑點插值(S-PIMs)等數(shù)值方法能很好地處理低質(zhì)量或嚴(yán)重變形的網(wǎng)格問題。我們首先在Liu等人建立的G~s h空間理論的基礎(chǔ)上,從數(shù)學(xué)角度精確地闡述了不依賴于形函數(shù)選取的G~s空間及其范數(shù)的一般化定義。和希爾伯特空間H~1相比,G~s空間中的范數(shù)具有下界性,并收斂于H~1范數(shù),這為W2形式中解的收斂性奠定了理論基礎(chǔ)。除此之外,我們進一步探討了G~s范數(shù)的等價性,以確�；诳臻g的W2形式的數(shù)值方法是穩(wěn)定的。這些結(jié)論極其重要,對今后在G空間上建立的數(shù)值方法提供了有力的理論依據(jù)。在第3章中,我們提出了一種直接的強形式無網(wǎng)格配點方法(直接的Kansa方法),用于求解黎曼流形上的橢圓偏微分方程。該流形是任意余維數(shù)的,且需滿足光滑、閉合、連通和完備的條件。這種方法采用了強形式的多配點方法和最小二乘法。除了采用一些約束在流形上的一般嵌入空間的核函數(shù)外,該方法的計算過程和一般區(qū)域型的方法相同。本文主要應(yīng)用解析和近似的方法來處理流形上的變換微分算子,且僅在流形上進行配點。我們在給定了一些基本的光滑性假設(shè)的基礎(chǔ)上,證明了直接無網(wǎng)格配點方法的高階收斂性。最后,為驗證前兩章中分析的理論成果,我們分別使用相應(yīng)的數(shù)值方法求解數(shù)值實例。(1)與基于弱形式的FEM對比,我們采用典型的S-FEM,即基于節(jié)點的NS-FEM和αS-FEM,來求解數(shù)值實例,以證實G~s空間的性質(zhì)。另外,我們應(yīng)用NS-FEM求解二維固體力學(xué)問題,提出了有效的修正方法來計算其固有值的下界。(2)我們在各種余維數(shù)的流形上進行數(shù)值模擬,分別比較了在數(shù)值和理論的配點設(shè)置下、以及采用兩種直接的Kansa方法得到的結(jié)果,以驗證流形上無網(wǎng)格配點方法的收斂性;并用解析的方法求解了曲面上的淺水方程。
[Abstract]:With the rapid development of computer, many effective numerical methods have been proposed to solve practical engineering problems, such as finite element method (FEM). However, due to the shape of the mesh will directly affect the accuracy of the calculation, the FEM is very demanding for the quality of the mesh. In order to overcome this problem, the smooth finite element method (S-FEMs) based on low mass meshes and some meshless methods have been established and widely used in complex fields such as solid mechanics, heat conduction and structural acoustics. In this paper, the corresponding theoretical analysis of these two kinds of numerical algorithms will be carried out. In Chapter 2, the GHS space theory is based on the weak form of W2) model, so the S-FEMs and smooth point interpolation S-PIMs) based on this space can be used to deal with the low mass or severe deformation mesh problems. On the basis of the theory of GCS h space established by Liu et al., we present the generalized definition of GCS space and its norm which are not dependent on shape function selection from the mathematical point of view. Compared with Hilbert space H ~ (1), the norm in G ~ (2) space has lower bound and converges to H ~ (1) norm, which lays a theoretical foundation for the convergence of solutions in W _ (2) form. In addition, we further discuss the equivalence of the GCS norm to ensure the stability of the space-based W2-form numerical method. These conclusions are extremely important and provide a strong theoretical basis for the numerical methods to be established in G space in the future. In Chapter 3, we propose a direct strong form meshless collocation method (direct Kansa method) for solving elliptic partial differential equations on Riemannian manifolds. The manifold is of arbitrary codimension and must satisfy the conditions of smoothness, closure, connectivity and completeness. This method adopts the strong form multi-collocation method and the least square method. With the exception of kernel functions in some general embedded spaces with constraints on manifolds, the calculation process of this method is the same as that of general region-type methods. In this paper, the analytic and approximate methods are mainly used to deal with the transformation differential operators on the manifolds, and only the matching points are carried out on the manifolds. On the basis of some basic smoothness assumptions, we prove the higher order convergence of direct meshless collocation methods. Finally, in order to verify the theoretical results of the previous two chapters, we use the corresponding numerical method to solve the numerical example. Compared with the FEM based on weak form, we use the typical S-FEM-based NS-FEM and 偽 S-FEM-based to solve the numerical examples. In order to prove the properties of the G ~ s space. In addition, we use NS-FEM to solve two-dimensional solid mechanics problems, and propose an effective correction method to calculate the lower bound of its intrinsic value. The convergence of the meshless collocation method on the manifold is verified by using two direct Kansa methods, and the shallow water equation on the surface is solved by analytic method.
【學(xué)位授予單位】：太原理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.82
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本文編號：2033743