A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach
A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach
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DOI：
10.1007/s004660050297
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发表时间：
1998-04
影响因子：
4.1
通讯作者：
T. Zhu;J.-D. Zhang;S. Atluri
中科院分区：
文献类型：
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作者：
T. Zhu;J.-D. Zhang;S. Atluri
伽辽金有限元法（Galerkin finite element method， GFEM）的流行得益于节点基函数的局域性，即，当全局观察时，节点基函数仅在连接所讨论的节点与其直接相邻节点的一组元素上不为零。另一方面，边界元法（BEM）通过只在区域整体边界上的积分中涉及试函数及其导数，将问题的维数降低了1；而GFEM则涉及对节点周围的一组元素的试函数对应的“能量”进行积分。GFEM得到带状、稀疏和对称矩阵；基于全局边界积分方程（GBIE）的边界元法可以得到满矩阵和不对称矩阵。由于在GFEM中自动生成单元网格，特别是在三维问题中，似乎存在不可克服的困难，因此在最近的文献中，人们对无单元伽辽金方法（EFGM）产生了相当大的兴趣。然而，efgm仍然涉及阴影元素上的域积分，导致在执行基本边界条件和处理非线性问题方面存在困难。本文的目的是提出一种新的方法，它结合了三种方法的优点：GFEM、BEM和EFGM。这是一种无网格方法。然而，它只涉及边界积分，在以所讨论的节点为中心的局部边界上；它在满足基本边界条件方面没有困难；它导致带状和稀疏系统矩阵；它使用移动最小二乘（MLS）近似。该方法基于局部边界积分方程（LBIE）方法，具有较强的通用性，易于应用于非线性问题和非齐次域。引入了“伴解”的概念，使得在给定问题的域Ω内的源点上的试解值的LBIE只涉及在问题节点的局部边界Ωsof子域Ωscentered上的积分中的试函数。这与传统的GBIE形成了对比，后者涉及到试验函数及其在Ω的全局边界Γ上的梯度。另一方面，对于位于Γ上的源点，则涉及到试验函数及其梯度在Ωsinvolve上的积分。结果表明，该方法满足基本边界条件和自然边界条件非常简单，算法效率很高。在处理拉普拉斯和泊松方程的示例问题中，发现Sobolev范数||·||和||·||1具有高收敛率。从本质上讲，目前的efe - lbie（单元自由局部边界积分方程）方法被认为是一种简单、有效和有吸引力的方法，可以替代最近文献中广泛推广的EFG方法。
The Galerkin finite element method (GFEM) owes its popularity to the local nature of nodal basis functions, i.e., the nodal basis function, when viewed globally, is non-zero only over a patch of elements connecting the node in question to its immediately neighboring nodes. The boundary element method (BEM), on the other hand, reduces the dimensionality of the problem by one, through involving the trial functions and their derivatives, only in the integrals over the global boundary of the domain; whereas, the GFEM involves the integration of the “energy” corresponding to the trial function over a patch of elements immediately surrounding the node. The GFEM leads to banded, sparse and symmetric matrices; the BEM based on the global boundary integral equation (GBIE) leads to full and unsymmetrical matrices. Because of the seemingly insurmountable difficulties associated with the automatic generation of element-meshes in GFEM, especially for 3-D problems, there has been a considerable interest in element free Galerkin methods (EFGM) in recent literature. However, the EFGMs still involve domain integrals over shadow elements and lead to difficulties in enforcing essential boundary conditions and in treating nonlinear problems.The object of the present paper is to present a new method that combines the advantageous features of all the three methods: GFEM, BEM and EFGM. It is a meshless method. It involves only boundary integration, however, over a local boundary centered at the node in question; it poses no difficulties in satisfying essential boundary conditions; it leads to banded and sparse system matrices; it uses the moving least squares (MLS) approximations. The method is based on a Local Boundary Integral Equation (LBIE) approach, which is quite general and easily applicable to nonlinear problems, and non-homogeneous domains.The concept of a “companion solution” is introduced so that the LBIE for the value of trial solution at the source point, inside the domain Ω of the given problem, involves only the trial function in the integral over the local boundary Ωsof a sub-domain Ωscentered at the node in question. This is in contrast to the traditional GBIE which involves the trial function as well as its gradient over the global boundary Γ of Ω. For source points that lie on Γ, the integrals over Ωsinvolve, on the other hand, both the trial function and its gradient. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple and algorithmically very efficient in the present LBIE approach.In the example problems dealing with Laplace and Poisson's equations, high rates of convergence for the Sobolev norms ||·||0and ||·||1have been found.In essence, the present EF-LBIE (Element Free-Local Boundary Integral Equation) approach is found to be a simple, efficient, and attractive alternative to the EFG methods that have been extensively popularized in recent literature.