数值流形方法中覆盖函数选用的建议

›› 2004, Vol. 25 ›› Issue (4): 624-627.

数值流形方法中覆盖函数选用的建议

彭自强^{1, 2}，葛修润¹

1. 中国科学院武汉岩土力学所，湖北武汉 430071； 2. 武汉理工大学设计研究院，湖北武汉 430070

收稿日期:2003-01-24 出版日期:2004-04-10 发布日期:2014-07-17
作者简介:彭自强，男，1975年生，博士生，主要从事数值分析方法的研究.

Suggestion to the choice of physical cover functions in numerical manifold method

PENG Zi-qiang^{1, 2}, GE Xiu-run¹

1.Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China; 2. The Design Institute, Wuhan University of Technology, Wuhan 430070, China

Received:2003-01-24 Online:2004-04-10 Published:2014-07-17

摘要/Abstract

摘要： 在数值流形方法中，常用的覆盖函数基并非是最佳选择。循着刚度矩阵的形成过程，分析了选用常用覆盖函数时，在非对角元上出现绝对值很大的元素之成因，且发现这会增加刚度矩阵的条件数，尤其是在用刚性弹簧约束位移的情况下，而这在数值流形方法中普遍而基本。建议采用局部化较好的覆盖函数，取代常用的关联于全局坐标的覆盖函数，可显著消除这一情况。建议方式简单明了，程序改动极小，对改善刚度矩阵性态却有很大作用。算例验证了这一建议的合理性，通过比较局部化的覆盖函数及全局性的覆盖函数所形成的刚度矩阵，表明前者形成了较小条件数的刚度矩阵。

关键词: 数值流形方法, 覆盖函数, 刚度矩阵, 条件数

Abstract: In the numerical manifold method (NMM), the most commonly used physical cover functions are not the best choices. Along the procedure of the formation of stiffness matrix, this paper analyses the causes of the non-diagonal entries of relatively huge absolute value in stiffness matrix, when ordinary physical cover functions are used. These extremely large entries can increase the condition number of stiffness matrix to a great extent, especially when stiff spring method is used, which is very universal and basic in the NMM. This paper suggests that well-defined local physical functions, orthogonal the better, be used instead of currently used ones related to global coordinates. This choice can help to eliminate the above-mentioned phenomenon and provide a relatively well-conditioned stiffness matrix. The suggestion is straightforward to implement, causing little changes to currently used procedure, while it affords a lot to improvement of the quality of stiffness matrix. Some examples are analyzed to test the suggested cover functions. Comparisons between the stiffness matrixes based on locally- and globally-defined cover functions show that the previous ones generate better stiffness matrix in terms of condition number.

Key words: numerical manifold method, cover functions, stiffness matrix, condition number

中图分类号:

TB 115

彭自强，葛修润，. 数值流形方法中覆盖函数选用的建议[J]. , 2004, 25(4): 624-627.

PENG Zi-qiang , GE Xiu-run，. Suggestion to the choice of physical cover functions in numerical manifold method[J]. , 2004, 25(4): 624-627.

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