本文選題：位勢問題 + 完全規(guī)則化邊界積分方程　； 參考：《山東理工大學(xué)》2017年碩士論文

【摘要】：眾所周知,在數(shù)值模擬技術(shù)中有限元法占統(tǒng)治地位,在科學(xué)與工程計(jì)算領(lǐng)域中得到了廣泛的應(yīng)用。然而,有限元法需剖分整個計(jì)算區(qū)域,對于某些復(fù)雜計(jì)算區(qū)域的問題,可能需要付出巨大的計(jì)算代價,有時甚至帶來數(shù)學(xué)上的麻煩。在求解無限域問題時,有限元法需要將人工邊界無限區(qū)域截斷為有限區(qū)域,即用人為的有限區(qū)域問題代替原來的無限區(qū)域問題,這樣的截斷完全依賴于研究者的經(jīng)驗(yàn)和各種誤差試驗(yàn)。邊界元法作為可供選擇的數(shù)值方法,正好彌補(bǔ)了有限元法的這些缺點(diǎn),并具有降低所求問題維數(shù)的優(yōu)點(diǎn)。但是,邊界元法也有其局限性,對于具有復(fù)雜幾何邊界的三維問題,邊界單元的生成仍然不是一件容易的事。此外,對于大規(guī)模計(jì)算問題,數(shù)值積分計(jì)算耗時太多。本文提出一種新的邊界型無網(wǎng)格法,即平均源無網(wǎng)格法。該法建立在耦合“完全”規(guī)則化邊界積分方程和平均源技術(shù)。這里“完全”是指弱奇異積分、強(qiáng)奇異積分均被規(guī)則化。通過規(guī)則化方程,避免了核函數(shù)的奇異性,使得源點(diǎn)和場點(diǎn)相遇的困難被消除;通過使用平均源技術(shù),分布源可以簡化為集中源,邊界積分不再需要。因此平均源無網(wǎng)格法不僅繼承了邊界元法降低求解問題維數(shù)的優(yōu)點(diǎn),而且僅需邊界配點(diǎn),避免了有限元法和邊界元法中費(fèi)時費(fèi)力的網(wǎng)格劃分,特別便于模擬復(fù)雜的幾何形狀問題。此外,平均源無網(wǎng)格法不涉及單元和積分的概念,具有數(shù)學(xué)簡單,編程容易,計(jì)算效率,高精度好等優(yōu)點(diǎn)。具體工作是:第一章介紹了無網(wǎng)格方法的研究背景意義和主要研究內(nèi)容。第二章給出建立邊界型無網(wǎng)格法用到的相關(guān)知識。第三章建立二維位勢問題的“完全”規(guī)則化邊界積分方程。在此基礎(chǔ)上,通過引入平均源技術(shù)的概念,將耦合二者,建立一種新的邊界型無網(wǎng)格法——平均源無網(wǎng)格法。第四章研究建立三維位勢問題的平均源無網(wǎng)格法。為此,首先導(dǎo)出三維位勢問題的“完全”規(guī)則化邊界積分方程,然后引入平均源技術(shù),獲得平均源無網(wǎng)格法。第五章是對無網(wǎng)格方法的總結(jié)與展望。
[Abstract]:As we all know, finite element method is dominant in numerical simulation technology and has been widely used in the field of science and engineering calculation. However, the finite element method needs to divide the whole computational area, and it may have to pay a huge computational cost for some complex computational areas, and sometimes even bring problems in mathematics. In solving the infinite domain problem, the finite element method needs to truncate the infinite area of artificial boundary into a finite region, that is, the limited domain problem instead of the original infinite region problem. Such truncation depends entirely on researchers' experience and various error tests. As an alternative numerical method, the BEM makes up for these shortcomings of the finite element method and has the advantage of reducing the dimension of the problem. However, the boundary element method has its limitations, and it is still not easy to generate boundary element for 3D problems with complex geometric boundaries. In addition, numerical integral computation takes too much time for large scale computation. In this paper, a new boundary meshless method, the mean source meshless method, is proposed. The method is based on coupled "complete" regularized boundary integral equation and mean source technique. Here "complete" means weakly singular integrals, and strongly singular integrals are regularized. By regularizing the equation, the singularity of kernel function is avoided, and the difficulty of meeting the source point and the field point is eliminated. By using the averaging source technique, the distributed source can be simplified to a concentrated source, and the boundary integral is no longer needed. So the average source-free meshless method not only inherits the advantages of BEM in reducing the dimension of the problem, but also needs only the boundary collocation, thus avoiding the time-consuming and laborious mesh generation in the finite element method and the BEM. It is especially convenient to simulate complex geometry problems. In addition, the average source-free meshless method does not involve the concepts of element and integration, and has the advantages of simple mathematics, easy programming, high calculation efficiency and high precision. The main work is as follows: the first chapter introduces the background significance and main research contents of meshless methods. In the second chapter, the relevant knowledge of establishing the boundary meshless method is given. In chapter 3, the "complete" regularized boundary integral equation for two-dimensional potential problem is established. On this basis, by introducing the concept of averaging source technique, a new boundary meshless method-average source-free meshless method is established by coupling the two methods. In chapter 4, the average source-free meshless method for three-dimensional potential problem is studied. For this reason, the "complete" regularized boundary integral equation of the three-dimensional potential problem is first derived, and then the mean source meshless method is obtained by introducing the averaging source technique. The fifth chapter is the summary and prospect of meshless methods.
【學(xué)位授予單位】：山東理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.8

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本文編號：1840548