从边界元法研究亲身体验数值方法研究的一些误区
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摘要
作者基于37年边界元法研究的亲身经历,体验到边界元法研究的最大误区是将自身的精度验证依赖于和简单问题解析解或相应问题边界元解的比较,从而导致在工程应用中越来越被边缘化。为了摆脱这种依赖,作者从误差分析出发,提出了一种新的高精度边界元法,并正在引人快速算法发展新的高性能边界元法。作者还体验到,在数值方法研究中存在一个普遍的误区,就是只通过少数简单的考题来验证方法的有效性、精度与效率,并对不同方法进行比较。梁板壳的有限元分析通过了许多标准考题的验证,但是当最大应力出现在梁的端部或板壳边界的情况下,所得的局部应力就是很不准确的。因此作者将真实梁板壳的局部应力分析作为高精度与高性能边界元法的突破点来加以研究。本世纪以来快速算法的引入,使边界元法研究者看到了边界元法用于复杂工程问题的广阔前景,但是很少注意到快速算法附加误差对结果可靠性和精度的重要影响,这是研究中的又一个重要误区。只有克服了这个误区,才能使边界元法真正成为有限元法不可或缺的重要补充。
引文
1姚振汉,王海涛.边界元法[M].北京:高等教育出版社,2010[Yao Zhenhan,Wang Haitao.Boundary element method[M].Beijing:Higher Education Press,2010(in Chinese)]
2 Yao Zhenhan,Wang Haitao.Some benchmark problems and basic ideas on the accuracy of boundary element analysis[J].Eng Anal Boundary Elem,2013;37:1674-1692
3 姚振汉.真实梁板壳局部应力分析的高性能边界元法[J].工程力学,2015,32(8):8–15[Zhenhan Yao.High-performance boundary element method for the local stress analysis of real beam,plate and shell[J].Engineering Mechanics,2015,32(8):8–15(in Chinese)]
4 Yao Zhenhan.A new type of high-accuracy BEM and local stress analysis of real beam,plate and shell structures[J].Eng Anal Boundary Elem,2016;65:1-17
5 Nishimura Naoshi.Fast multipole accelerated boundary integral equation methods[J].Appl Mech Rev,2002;55:299-324
6 Yao Zhenhan,Wang Haitao,Wang Pengbo,Lei Ting.Investigations on fast multipole BEM in solid mechanics[J].Journal of University of Science and Technology of China,2008;38(1):1–17
7 Yao Zhenhan.Some investigation on the accuracy of fast BE computation using ACA[C].Computational Methods in Engineering,Proc.of the Global Chinese Workshop in conjunction with 10th National Conference on Computational Methods in Engineering,May 25-28,2012,Changsha
2 Yao Zhenhan,Wang Haitao.Some benchmark problems and basic ideas on the accuracy of boundary element analysis[J].Eng Anal Boundary Elem,2013;37:1674-1692
3 姚振汉.真实梁板壳局部应力分析的高性能边界元法[J].工程力学,2015,32(8):8–15[Zhenhan Yao.High-performance boundary element method for the local stress analysis of real beam,plate and shell[J].Engineering Mechanics,2015,32(8):8–15(in Chinese)]
4 Yao Zhenhan.A new type of high-accuracy BEM and local stress analysis of real beam,plate and shell structures[J].Eng Anal Boundary Elem,2016;65:1-17
5 Nishimura Naoshi.Fast multipole accelerated boundary integral equation methods[J].Appl Mech Rev,2002;55:299-324
6 Yao Zhenhan,Wang Haitao,Wang Pengbo,Lei Ting.Investigations on fast multipole BEM in solid mechanics[J].Journal of University of Science and Technology of China,2008;38(1):1–17
7 Yao Zhenhan.Some investigation on the accuracy of fast BE computation using ACA[C].Computational Methods in Engineering,Proc.of the Global Chinese Workshop in conjunction with 10th National Conference on Computational Methods in Engineering,May 25-28,2012,Changsha