四阶两点边值问题单元能量投影法的数学分析
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摘要
有限元法是一种非常有效且通用的数值分析计算方法。随着有限元法的迅速发展,人们对常规有限元中的应力精度比位移精度呈数量级的下降越来越不满意,因而如何提高有限元解导数的精度成为近年来有限元研究的热点之一。
2004年袁驷教授对于二阶方程两点边值问题基于力学解释提出了单元能量投影法,其基本思想来源于结构力学中的矩阵位移法和有限元数学理论中的投影定理。数学例子显示了良好的效果。2006年单元能量投影法被推广到四阶两点边值问题的有限元计算中,同样获得了令人满意的效果。2007年赵庆华博士对部分结果进行了严格的数学分析。
本文对四阶两点边值问题的单元能量投影法进行数学分析,获得了一系列好的结果。我们的主要贡献是:
1.证明了内点位移恢复具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
2.证明了内点应力恢复具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
3.证明了内点弯矩恢复具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
4.证明了内点剪力恢复至少具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
The finite element method is one of the efficient and common numerical analysis and calculation methods.With the rapid development of the finite element,people increasingly dissatisfied with the finite element method because the accuracy of stresses is much lower than the accuracy of displacements.To improve the accuracy of derivative of the finite element solution became one of the most important research subject recently.
In 2004 for second order two-point boundary value problems Professor Yuan proposed the element-energy-projection(EEP) method based on mechanical interpretation.The fundamental idea comes from the strategy in conventional matrix displacement method for skeletal structures and projection theorem in finite element mathematical theory.The numerical examples show high accuracy of the new method.In 2006,Professor Yuan applied the element energy projection method to the finite element computation of fourth order two-point boundary value problems.The numerical examples show that the EEP method also works well for the fourth order problems.In 2007,Dr Zhao Qinghua give the strict mathematical analysis for some of the results.
In this thesis we give a mathematical analysis on the element energy projection method for fourth order two-point boundary value equations.Our main results are as follows:
1.We proved the O(h~(2k-2))(k≥3) super convergence for pointwise recovery displament, and also provided numerical examples to support the result..
2.We proved the O(h~(2k-2))(k≥3)super convergence for pointwise recovery stress,and also provided numerical examples to support the result.
3.We proved the O(h~(2k-2))(k≥3)super convergence for pointwise recovery bending moments,and also provided numerical examples to support the result.
4.We proved the O(h~(2k-3))(k≥3) super convergence for pointwise recovery shear forces, and also provided numerical examples to support the result.
2004年袁驷教授对于二阶方程两点边值问题基于力学解释提出了单元能量投影法,其基本思想来源于结构力学中的矩阵位移法和有限元数学理论中的投影定理。数学例子显示了良好的效果。2006年单元能量投影法被推广到四阶两点边值问题的有限元计算中,同样获得了令人满意的效果。2007年赵庆华博士对部分结果进行了严格的数学分析。
本文对四阶两点边值问题的单元能量投影法进行数学分析,获得了一系列好的结果。我们的主要贡献是:
1.证明了内点位移恢复具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
2.证明了内点应力恢复具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
3.证明了内点弯矩恢复具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
4.证明了内点剪力恢复至少具有O(h~(2k-2))阶的超收敛阶,同时还提供了数值算例用以验证所得的结论。
The finite element method is one of the efficient and common numerical analysis and calculation methods.With the rapid development of the finite element,people increasingly dissatisfied with the finite element method because the accuracy of stresses is much lower than the accuracy of displacements.To improve the accuracy of derivative of the finite element solution became one of the most important research subject recently.
In 2004 for second order two-point boundary value problems Professor Yuan proposed the element-energy-projection(EEP) method based on mechanical interpretation.The fundamental idea comes from the strategy in conventional matrix displacement method for skeletal structures and projection theorem in finite element mathematical theory.The numerical examples show high accuracy of the new method.In 2006,Professor Yuan applied the element energy projection method to the finite element computation of fourth order two-point boundary value problems.The numerical examples show that the EEP method also works well for the fourth order problems.In 2007,Dr Zhao Qinghua give the strict mathematical analysis for some of the results.
In this thesis we give a mathematical analysis on the element energy projection method for fourth order two-point boundary value equations.Our main results are as follows:
1.We proved the O(h~(2k-2))(k≥3) super convergence for pointwise recovery displament, and also provided numerical examples to support the result..
2.We proved the O(h~(2k-2))(k≥3)super convergence for pointwise recovery stress,and also provided numerical examples to support the result.
3.We proved the O(h~(2k-2))(k≥3)super convergence for pointwise recovery bending moments,and also provided numerical examples to support the result.
4.We proved the O(h~(2k-3))(k≥3) super convergence for pointwise recovery shear forces, and also provided numerical examples to support the result.
引文
[1]冯康.基于变分原理的差分格式.应用数学与计算数学,1965,2(4):238-262.
[2]Zienkiewicz,O.C.,Cheung,Y.K.The finite element method in structural and continuum mechanics.McGraw-Hill,1967.
[3]Douglas,Jr.J.,Dupont,T.Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary value problem.In:Topics in Numerical Analysis(Proc.Roy.Irish Acad.Conf.,univ.coll.,Dublin,1972),London,Academic Press,1973,89-92.
[4]Oganesjan,L.A.,Ruhovets,L.A.Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary.I.Vyeisl.Mat.Fiz.,1969,9:1102-1120.
[5]Lin,Q.,Yan,N.N.,Zhou,A.H.A rectangle test for interpolated finite elements.In:Proc.Syst.Sci.& Syst.Eng.,Great wall Culture Pub.Co.,Hong Kong,1991,217-229.
[6]Zlamal,M.Some superconvergence results in the finite element method.In Mathematical aspects of finite element methods(Pro.Conf.Consiglio Naz.Delle Ricerche(C.N.R.),Rome,1975).Lecture Notes in Math.,606,Springer,Berlin,1977,353-362.
[7]陈传淼,朱起定.有限元法的一种新估计及应力佳点定理.湘潭大学自然科学通讯,1978,1:10-20.
[8]Zhu,Q.D.Natural inner superconvergence for the finite element method.Proceedings of the China-France Symposinm on Finite Element methods(Beijing,1982),Science Press,Beijing,Gorden and breach,1983,935-960.
[9]Babuska,I.,Strouboulis,T.Upadhyay,C and Gangaraj,S.Computer-Based proof of the existence of superconvergence points in the finite element method:superconvergence of the derivatives in finite element solutions of Lapplace's,Poisson's and the Elasticity equations.Numeirical Methods for Partial Differential Equations,1996,12(3):372-392.
[10]赵庆华.单元能量投影法的数学分析.[湖南大学博士学位论文].长沙:湖南大学数学与计量经济学院,2007,1-3.
[11]孟令雄.有限元超收敛后处理技术:[湖南师范大学博士学位论文].长沙:湖南师范大学数学与计算机科学学院,2006,1-5.
[12]陈传淼,黄云清.有限元高精度理论.湖南科学技术出版社.1995,1-4.
[13]王烈衡,许学军.有限元方法的数学基础.北京:科学出版社,2005.
[14]Ciarlet,P.G.The Finite Element Method for Elliptic Problems.Amsterdam:North-Holland.1978.
[15]Brenner,S.C.,Scott,L.R.The Mathematical Theory of Finite Element Methods.New York:Springer-Verlag.1994.
[16]袁驷,王枚.一维有限元后处理超收敛解答的EEP法.工程力学,2004,21(2):1-9.
[17]Wang Mei,Yuan Si.Computation of super-convergent nodal stresses of Timoshenko beam elements by EEP method.Applied Mathematics and Mechanics,2004,25(11):1228-1240.
[18]袁驷,王枚,和雪峰.一维C~1有限元超收敛解答计算的EEP法.工程力学,2006,23(2):1-9.
[19]袁驷,和雪峰.基于EEP算法的一维有限元自适应求解.应用数学和力学.2006,27(1):1280-1291.
[20]袁驷,王枚,王旭.二维有限元线法超收敛解答计算的EEP法.工程力学,2007,24(1):1-10
[21]Stramg G,Fix G.An analysis of the finite element method[M].Englewood,Cliffs,N.J.:Prentice-Hall,1973.
[22]Tong P.Exact solution of certain problems by finite-element method[J].AIAA Journal,1969,(7):178-180.
[23]陈传淼.矩形奇妙族有限元的超收敛.中国科学,A辑,2002,32(7):587-595.
[24]陈传淼.有限元超收敛构造理论.湖南科学技术出版社.2001,345-355.
[25]陈传淼.板壳矩形元的弯矩和剪力佳点.湘潭大学学报(自然版).1979,3(1):9.15.
[26]黄达人,吴瑞恭.四阶微分方程两点边值问题样条有限元解及其二阶导数的超收敛性.浙江大学学报,1982(3):92-99.
[27]张林.固支梁有限元解的超收敛性及最大模估计.复旦大学学报(自然版),1996,35(4):421-428.
[28]赵新中,陈传淼.粱问题有限元逼近的新估计及超收敛.湖南师范大学自然科学学报.2000,23(4):6-11.
[29]王奇生,邓康.四阶方程两点边值问题Hermite有限元解的渐进展示与外推.高等学校 计算数学学报,2006,28(4):299-306.
[2]Zienkiewicz,O.C.,Cheung,Y.K.The finite element method in structural and continuum mechanics.McGraw-Hill,1967.
[3]Douglas,Jr.J.,Dupont,T.Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary value problem.In:Topics in Numerical Analysis(Proc.Roy.Irish Acad.Conf.,univ.coll.,Dublin,1972),London,Academic Press,1973,89-92.
[4]Oganesjan,L.A.,Ruhovets,L.A.Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary.I.Vyeisl.Mat.Fiz.,1969,9:1102-1120.
[5]Lin,Q.,Yan,N.N.,Zhou,A.H.A rectangle test for interpolated finite elements.In:Proc.Syst.Sci.& Syst.Eng.,Great wall Culture Pub.Co.,Hong Kong,1991,217-229.
[6]Zlamal,M.Some superconvergence results in the finite element method.In Mathematical aspects of finite element methods(Pro.Conf.Consiglio Naz.Delle Ricerche(C.N.R.),Rome,1975).Lecture Notes in Math.,606,Springer,Berlin,1977,353-362.
[7]陈传淼,朱起定.有限元法的一种新估计及应力佳点定理.湘潭大学自然科学通讯,1978,1:10-20.
[8]Zhu,Q.D.Natural inner superconvergence for the finite element method.Proceedings of the China-France Symposinm on Finite Element methods(Beijing,1982),Science Press,Beijing,Gorden and breach,1983,935-960.
[9]Babuska,I.,Strouboulis,T.Upadhyay,C and Gangaraj,S.Computer-Based proof of the existence of superconvergence points in the finite element method:superconvergence of the derivatives in finite element solutions of Lapplace's,Poisson's and the Elasticity equations.Numeirical Methods for Partial Differential Equations,1996,12(3):372-392.
[10]赵庆华.单元能量投影法的数学分析.[湖南大学博士学位论文].长沙:湖南大学数学与计量经济学院,2007,1-3.
[11]孟令雄.有限元超收敛后处理技术:[湖南师范大学博士学位论文].长沙:湖南师范大学数学与计算机科学学院,2006,1-5.
[12]陈传淼,黄云清.有限元高精度理论.湖南科学技术出版社.1995,1-4.
[13]王烈衡,许学军.有限元方法的数学基础.北京:科学出版社,2005.
[14]Ciarlet,P.G.The Finite Element Method for Elliptic Problems.Amsterdam:North-Holland.1978.
[15]Brenner,S.C.,Scott,L.R.The Mathematical Theory of Finite Element Methods.New York:Springer-Verlag.1994.
[16]袁驷,王枚.一维有限元后处理超收敛解答的EEP法.工程力学,2004,21(2):1-9.
[17]Wang Mei,Yuan Si.Computation of super-convergent nodal stresses of Timoshenko beam elements by EEP method.Applied Mathematics and Mechanics,2004,25(11):1228-1240.
[18]袁驷,王枚,和雪峰.一维C~1有限元超收敛解答计算的EEP法.工程力学,2006,23(2):1-9.
[19]袁驷,和雪峰.基于EEP算法的一维有限元自适应求解.应用数学和力学.2006,27(1):1280-1291.
[20]袁驷,王枚,王旭.二维有限元线法超收敛解答计算的EEP法.工程力学,2007,24(1):1-10
[21]Stramg G,Fix G.An analysis of the finite element method[M].Englewood,Cliffs,N.J.:Prentice-Hall,1973.
[22]Tong P.Exact solution of certain problems by finite-element method[J].AIAA Journal,1969,(7):178-180.
[23]陈传淼.矩形奇妙族有限元的超收敛.中国科学,A辑,2002,32(7):587-595.
[24]陈传淼.有限元超收敛构造理论.湖南科学技术出版社.2001,345-355.
[25]陈传淼.板壳矩形元的弯矩和剪力佳点.湘潭大学学报(自然版).1979,3(1):9.15.
[26]黄达人,吴瑞恭.四阶微分方程两点边值问题样条有限元解及其二阶导数的超收敛性.浙江大学学报,1982(3):92-99.
[27]张林.固支梁有限元解的超收敛性及最大模估计.复旦大学学报(自然版),1996,35(4):421-428.
[28]赵新中,陈传淼.粱问题有限元逼近的新估计及超收敛.湖南师范大学自然科学学报.2000,23(4):6-11.
[29]王奇生,邓康.四阶方程两点边值问题Hermite有限元解的渐进展示与外推.高等学校 计算数学学报,2006,28(4):299-306.