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 引用本文: 李树忱, 程玉民. 基于单位分解法的无网格数值流形方法[J]. 力学学报, 2004, 36(4): 496-500.
Meshless numerical manifold method based on unit partition[J]. Chinese Journal of Theoretical and Applied Mechanics, 2004, 36(4): 496-500.
 Citation: Meshless numerical manifold method based on unit partition[J]. Chinese Journal of Theoretical and Applied Mechanics, 2004, 36(4): 496-500.

## Meshless numerical manifold method based on unit partition

• 摘要: 在数值流形方法和单位分解法的基础上，提出了无网格数值流形方法. 无网格数值流形方法在分析时采用了双重覆盖系统，即数学覆盖和物理覆盖. 数学覆盖提供的节点形成求解域的有限覆盖和单位分解函数；而物理覆盖描述问题的几何区域及其域内不连续性. 与原有的数值流形方法相比，无网格数值流形方法的数学覆盖形状更加灵活，可以用一系列节点的影响域来建立数学覆盖和单位分解函数，具有无网格方法的特性，从而摆脱了传统的数值流形方法中网格所带来的困难. 与无网格方法相比，由于采用了有限覆盖技术，试函数的构造不受域内不连续的影响，克服了原有的无网格方法在处理不连续问题时所遇到的困难.详细推导了无网格数值流形方法的试函数和求解方程，最后给出了算例，验证了该方法的正确性.

Abstract: In this paper the meshless numerical manifold method ispresented based on the numerical manifold method and the partition of unitymethod. In meshless numerical manifold method, two cover systems areemployed. The mathematical cover system provides the nodes for formingfinite covers of the solution domain and the partition of unityfunctions, and the physical cover system describes geometry of the domain of theproblem and the discontinuous surfaces in the domain. The shape function inthis method is formed by the partition of unity and the finite covertechnology, so the shape functions cannot be affected by discontinuousdomain, and crack problems can be treated better. To local problems, theshape functions are more effective than other method. So the method canavoid the disadvantages in other meshless methods in which the tip of thediscontinuous crack is not considered. Comparing with the conventionalnumerical manifold method, the shape of the finite cover can be selectedeasily. And the finite covers and the partition of unity functions areformed with the influence domains of a series of nodes. So the meshlessmanifold method has some advantages of the meshless and gets rid of thedisadvantage of the mesh in the numerical manifold method. Comparing withthe conventional meshless method, finite cover technology is used in themethod, and then the test functions cannot be influenced by thediscontinuity in the solving domain. And this method can conquer somedifficulties in the conventional meshless methods for the problems with adiscontinuous domain. The test function and the equations of the meshlessnumerical manifold method are obtained in detail. And a numerical example isgiven and it shows the method in this paper is correct.

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