位势问题的平均源无网格法

发布时间：2018-05-03 23:21

本文选题：位势问题 + 完全规则化边界积分方程　；参考：《山东理工大学》2017年硕士论文

【摘要】：众所周知,在数值模拟技术中有限元法占统治地位,在科学与工程计算领域中得到了广泛的应用。然而,有限元法需剖分整个计算区域,对于某些复杂计算区域的问题,可能需要付出巨大的计算代价,有时甚至带来数学上的麻烦。在求解无限域问题时,有限元法需要将人工边界无限区域截断为有限区域,即用人为的有限区域问题代替原来的无限区域问题,这样的截断完全依赖于研究者的经验和各种误差试验。边界元法作为可供选择的数值方法,正好弥补了有限元法的这些缺点,并具有降低所求问题维数的优点。但是,边界元法也有其局限性,对于具有复杂几何边界的三维问题,边界单元的生成仍然不是一件容易的事。此外,对于大规模计算问题,数值积分计算耗时太多。本文提出一种新的边界型无网格法,即平均源无网格法。该法建立在耦合“完全”规则化边界积分方程和平均源技术。这里“完全”是指弱奇异积分、强奇异积分均被规则化。通过规则化方程,避免了核函数的奇异性,使得源点和场点相遇的困难被消除;通过使用平均源技术,分布源可以简化为集中源,边界积分不再需要。因此平均源无网格法不仅继承了边界元法降低求解问题维数的优点,而且仅需边界配点,避免了有限元法和边界元法中费时费力的网格划分,特别便于模拟复杂的几何形状问题。此外,平均源无网格法不涉及单元和积分的概念,具有数学简单,编程容易,计算效率,高精度好等优点。具体工作是:第一章介绍了无网格方法的研究背景意义和主要研究内容。第二章给出建立边界型无网格法用到的相关知识。第三章建立二维位势问题的“完全”规则化边界积分方程。在此基础上,通过引入平均源技术的概念,将耦合二者,建立一种新的边界型无网格法——平均源无网格法。第四章研究建立三维位势问题的平均源无网格法。为此,首先导出三维位势问题的“完全”规则化边界积分方程,然后引入平均源技术,获得平均源无网格法。第五章是对无网格方法的总结与展望。
[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.
【学位授予单位】：山东理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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