关于平板屈曲重调和特征值问题的H~2协调谱元法
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摘要
通过使用H~2协调谱元法,具体求解了平板屈曲重调和特征值问题。首先给出H~2协调谱元法的误差估计,然后利用广义雅可比多项式和节点基函数构造二维谱元空间的基函数,最后报道了L形区域和方形区域上的数值实验,实验结果表明谱元法所计算的特征值受网格直径和多项式次数的影响,在区域选择上较谱方法更为灵活,适用于平板屈曲重调和特征值问题。
By using H~2-conforming spectral element method,the biharmonic eigenvalue problem of plate buckling is solved. Firstly,the error estimates of H~2-conforming spectral element method are given. Then a set of basis functions on the two-dimensional spectral element space is constructed by generalized Jacobi polynomials and nodal basis functions. Finally,the numerical experiments are carried out on the L-shaped domain and the square domain. The experimental results show that eigenvalues obtained by spectral element method are affected by grid diameter and degree of polynomial. The spectral element method is more flexible in domain than spectral method and is applicable to the biharmonic eigenvalue problem of plate buckling.
By using H~2-conforming spectral element method,the biharmonic eigenvalue problem of plate buckling is solved. Firstly,the error estimates of H~2-conforming spectral element method are given. Then a set of basis functions on the two-dimensional spectral element space is constructed by generalized Jacobi polynomials and nodal basis functions. Finally,the numerical experiments are carried out on the L-shaped domain and the square domain. The experimental results show that eigenvalues obtained by spectral element method are affected by grid diameter and degree of polynomial. The spectral element method is more flexible in domain than spectral method and is applicable to the biharmonic eigenvalue problem of plate buckling.
引文
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[5]ANTUNES P R S.On the buckling eigenvalue problem[J].Journal of Physics A:Mathematical and Theoretical,2011,44:215205.
[6]BRENNER S C,MONK P,SUN J G.C0interior penalty galerkin method for biharmonic eigenvalue problems[M].Kirby R M,et al.(eds.).Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014,Lecture Notes in Computational Science and Engineering106.Switzerland:Springer International Publishing,2015:3-15.
[7]JOST J,LI-JOSTL X,WANG Q L,XIA C Y.Universal inequalities for eigenvalues of the buckling problem of arbitrary order[J].Communications in Partial Differential E-quations,2010,35:1563-1589.
[8]CHENG Q M,YANG H C.Universal bounds for eigenvalues of a buckling problem II[J].Transactions of the A-merican Mathematical Society,2012,364(11):6139-6158.
[9]CHENG Q M,QI X R,WANG Q L,XIA C Y.Inequalities for eigenvalues of the buckling problem of arbitrary order[J].Annali di Matematica,2018,197(1):211-232.
[10]GUO B Y,JIA H L.Spectral method on quadrilaterals[J].Mathematics of Computation,2010,79(272):2237-2264.
[11]SHEN J,WANG L L,LI H Y.A triangular spectral element method using fully tensorial rational basis functions[J].Siam Journal on Numerical Analysis,2009,47(3):1619-1650.
[12]SAMSON M D,LI H,WANG L L.A new triangular spectral element method I:implementation and analysis on a triangle[J].Numerical Algorithms,2012,64:519-547.
[13]YU X H,GUO B Y.Spectral element method for mixed inhomogeneous boundary value problems of fourth order[J].Journal of Scientific Computing,2014,61(3):673-701.
[14]HAN J Y,YANG Y D.An Hm-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues[J].Science China Mathematics,2017,60(8):1529-1542.
[15]BABUSKA I,OSBORN J E.Eigenvalue Problems[M].in:CIARLET P G,LIONS J L(eds.).Finit Element Methods(Part 1),Handbook of Numberical Analysis,vol.2.North-Holand:Elsevier Science Publishers,1991:640-787.
[16]CHEN L.i FEM:an integrated finite element method package in MATLAB,Technical Report,University of California at Irvine,2009.
[17]李艳琴,安静.四阶椭圆特征值问题的有效谱Galerkin方法[J].四川师范大学学报(自然科学版),2015,38(2):249-254.
[2]ISHIHARA K.On the mixed finite element approximation for the buckling of plates[J].Numerische Mathematik,1979,33:195-210.
[3]KATHRIN S,WAGNER A.Optimality conditions for the buckling of a clamped plate[J].Journal of Mathematical Analysis and Applications,2015,432:254-273.
[4]CHENG Q M,YANG H C.Universal bounds for eigenvalues of a buckling problem[J].Communications in Mathematical Physics,2006,262:663-675.
[5]ANTUNES P R S.On the buckling eigenvalue problem[J].Journal of Physics A:Mathematical and Theoretical,2011,44:215205.
[6]BRENNER S C,MONK P,SUN J G.C0interior penalty galerkin method for biharmonic eigenvalue problems[M].Kirby R M,et al.(eds.).Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014,Lecture Notes in Computational Science and Engineering106.Switzerland:Springer International Publishing,2015:3-15.
[7]JOST J,LI-JOSTL X,WANG Q L,XIA C Y.Universal inequalities for eigenvalues of the buckling problem of arbitrary order[J].Communications in Partial Differential E-quations,2010,35:1563-1589.
[8]CHENG Q M,YANG H C.Universal bounds for eigenvalues of a buckling problem II[J].Transactions of the A-merican Mathematical Society,2012,364(11):6139-6158.
[9]CHENG Q M,QI X R,WANG Q L,XIA C Y.Inequalities for eigenvalues of the buckling problem of arbitrary order[J].Annali di Matematica,2018,197(1):211-232.
[10]GUO B Y,JIA H L.Spectral method on quadrilaterals[J].Mathematics of Computation,2010,79(272):2237-2264.
[11]SHEN J,WANG L L,LI H Y.A triangular spectral element method using fully tensorial rational basis functions[J].Siam Journal on Numerical Analysis,2009,47(3):1619-1650.
[12]SAMSON M D,LI H,WANG L L.A new triangular spectral element method I:implementation and analysis on a triangle[J].Numerical Algorithms,2012,64:519-547.
[13]YU X H,GUO B Y.Spectral element method for mixed inhomogeneous boundary value problems of fourth order[J].Journal of Scientific Computing,2014,61(3):673-701.
[14]HAN J Y,YANG Y D.An Hm-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues[J].Science China Mathematics,2017,60(8):1529-1542.
[15]BABUSKA I,OSBORN J E.Eigenvalue Problems[M].in:CIARLET P G,LIONS J L(eds.).Finit Element Methods(Part 1),Handbook of Numberical Analysis,vol.2.North-Holand:Elsevier Science Publishers,1991:640-787.
[16]CHEN L.i FEM:an integrated finite element method package in MATLAB,Technical Report,University of California at Irvine,2009.
[17]李艳琴,安静.四阶椭圆特征值问题的有效谱Galerkin方法[J].四川师范大学学报(自然科学版),2015,38(2):249-254.