The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods
The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods
复制标题
DOI：
10.3970/cmes.2002.003.011
复制
发表时间：
2002-03
影响因子：
2.4
通讯作者：
S. Atluri;S. Shen
中科院分区：
文献类型：
--
作者：
S. Atluri;S. Shen
本文基于无网格局部Petrov-Galerkin（MLPG）方法的一般概念，对各种无网格试函数和检验函数的效率和精度进行了比较研究。探索了5类试函数和6类测试函数。不同的测试功能导致不同的MLPG方法，本文介绍了六种MLPG方法。在所有这六种MLPG方法中，无论是用于试函数和检验函数的插值，还是用于弱形式的积分，都绝对不需要网格;而其他无网格方法则需要背景单元。由于复杂的形状函数的试验功能是不可避免的，在现阶段，为了开发一个快速和强大的无网格方法，我们探索如何避免使用域积分的弱形式，通过选择一个合适的测试功能。MLPG 5方法（其中，在以节点为中心的局部子域上，基于局部的测试函数是Heaviside阶跃函数）避免了对伴随的对称弱形式的域积分以及奇异积分的需要。收敛性研究的数值例子表明，所有的MLPG方法具有良好的收敛速度，为未知变量及其衍生物。计算成本的分析表明，MLPG 5方法是更便宜的，无论是在计算成本，以及肯定在人力成本，比有限元法，或边界元法。因此，由于其速度，准确性和鲁棒性，MLPG 5方法可能会在不久的将来取代FEM。
A comparison study of the efficiency and ac- curacy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are pre- sented in this paper. In all these six MLPG methods, absolutely no meshes are needed either for the interpo- lation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells. Because complicated shape functions for the trial function are inevitable at the present stage, in order to develop a fast and robust meshless method, we explore ways to avoid the use of a domain integral in the weak-form, by choosing an appropriate test function. The MLPG5 method (wherein the local, nodal-based test function, over a local sub-domain Ω s (or Ω te) centered at a node, is the Heaviside step function) avoids the need for both a domain integral in the attendant symmetric weak-form as well as a singular integral. Convergence studies in the numerical examples show that all of the MLPG methods possess excellent rates of convergence, for both the unknown variables and their derivatives. An analysis of computational costs shows that the MLPG5 method is less expensive, both in computational costs as well as definitely in human-labor costs, than the FEM, or BEM. Thus, due to its speed, accuracy and robustness, the MLPG5 method may be expected to replace the FEM, in the near future.