两种基于低质量网格的数值方法

发布时间：2018-06-18 03:04

  本文选题：光滑有限元方法 + G空间　； 参考：《太原理工大学》2017年硕士论文

【摘要】：随着计算机的高速发展,人们提出了许多有效的数值方法去解决实际中的工程问题,如有限元方法(FEM)。但是,由于网格的形状会直接影响计算精度,所以FEM对网格的质量要求非常高。为克服这一难题,仅基于低质量网格的光滑有限元方法(S-FEMs)和一些无网格的方法被建立,并广泛地应用于固体力学、热传导学和结构声学等复杂的领域中。本文将对这两类数值算法进行相应的理论分析。第2章的G~s空间理论是依赖于弱弱形式(W2)模型建立的,因此以该空间为基础的S-FEMs和光滑点插值(S-PIMs)等数值方法能很好地处理低质量或严重变形的网格问题。我们首先在Liu等人建立的G~s h空间理论的基础上,从数学角度精确地阐述了不依赖于形函数选取的G~s空间及其范数的一般化定义。和希尔伯特空间H~1相比,G~s空间中的范数具有下界性,并收敛于H~1范数,这为W2形式中解的收敛性奠定了理论基础。除此之外,我们进一步探讨了G~s范数的等价性,以确保基于空间的W2形式的数值方法是稳定的。这些结论极其重要,对今后在G空间上建立的数值方法提供了有力的理论依据。在第3章中,我们提出了一种直接的强形式无网格配点方法(直接的Kansa方法),用于求解黎曼流形上的椭圆偏微分方程。该流形是任意余维数的,且需满足光滑、闭合、连通和完备的条件。这种方法采用了强形式的多配点方法和最小二乘法。除了采用一些约束在流形上的一般嵌入空间的核函数外,该方法的计算过程和一般区域型的方法相同。本文主要应用解析和近似的方法来处理流形上的变换微分算子,且仅在流形上进行配点。我们在给定了一些基本的光滑性假设的基础上,证明了直接无网格配点方法的高阶收敛性。最后,为验证前两章中分析的理论成果,我们分别使用相应的数值方法求解数值实例。(1)与基于弱形式的FEM对比,我们采用典型的S-FEM,即基于节点的NS-FEM和αS-FEM,来求解数值实例,以证实G~s空间的性质。另外,我们应用NS-FEM求解二维固体力学问题,提出了有效的修正方法来计算其固有值的下界。(2)我们在各种余维数的流形上进行数值模拟,分别比较了在数值和理论的配点设置下、以及采用两种直接的Kansa方法得到的结果,以验证流形上无网格配点方法的收敛性;并用解析的方法求解了曲面上的浅水方程。
[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 伪 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.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
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本文编号：2033743