非协调元的新进展及各向异性元的理论分析与数值实验
|
摘要
本文应用双参数法针对不同问题构造了几个新的有限元，分析其经典收敛性及各向异性收敛性；也分析了几个著名元的各向异性收敛性，并进行了大量的数值试验。
对薄板弯曲问题，已提出了很多适用的三角形元和矩形元，有协调的，也有非协调的。但关于任意四边形板元的研究尚不多见。对于矩形元，从参考元(?)到一般单元K上的变换很简单，只需在参考元(?)上构造形函数。而对任意四边形元，情况就大不一样了。一方面，要在参考元(?)上构造形函数，因为不可能在任意四边形上构造形函数；另一方面，为了保证收敛性，K上的形函数(而非(?)上的)跨越K的边界应具有某种连续性。但是，由于从参考元(?)到一般单元K上的变换是非线性的，形函数在参考元K的连续性在一般单元K得不到保持。因此，要构造参考元(?)上适定的形函数并使其在K上具有保证收敛所需的连续性是很困难的。我们用双参数法构造了一个13-自由度任意四边形板元，证明了其收敛性，并进行了数值实验。数值结果表明此元有较好的收敛性。
求解四阶椭圆奇异摄动问题的有限元应是收敛的板元，并且要关于ε(?)致收敛。但并非所有收敛的板元都关于ε一致收敛。那么，哪些板元对奇异摄动问题关于参数ε是一致收敛的哪?我们将分别给出判别c~0非协调板元和非c~0非协调板元对这一问题收敛的一般判定定理，由此可判定已知的板元对该问题的收敛性。我们应用双参数法构造了几个对此问题收敛的有限元。数值实验也验证了这些新有限元对此问题的有效性。
传统有限元收敛性分析要求单元剖分满足非退化条件，即要求多面体单元的各条边的长度不应相差太大。这极大地限制了有限元方法在一些工程领域的应用，如具有各向异性结构的问题。对于这些问题，人们自然希望能去掉这一限制，根据问题的需要，单元在各个方向上的尺寸可以有较大的不同，并仍能保持其收敛性。我们把这种不要求单元剖分满足非退化条件仍保持收敛的有限元称为各向异性元。
关于如何判定一个有限元是否是各向异性元，最近十多年来已有一些研究成果。如T.Apel等学者给出了Lagrange元的各向异性插值误差估计，但对非协调元的相容误差的各向异性误差估计在文献中尚无报道。我们改进了各
向异性插值误差估计的判定方法，并给出非协调元的相容误差的各向异性误
差估计的方法.
Wilson元和Adini元是著名的矩形非协调元.我们给出Wilso，，元和A〔111，，
元的插值误差和相容误差的各向异性估计
类Wilson元是任意四边形上的非协调元，我们证明了类Wils()l，元的各向
异性收敛性，给出能量模和护一模的各向异性误差估计.
利用“双参数法”我们构造了一个8自由度12参矩形板元，证明该元具
有各向异性，给出包括插值误差和相容误差的完整的误差估计，
我们进行了大量的数值试验对理论结果进行实际验证.数值试验所使用
的程序是我们用C语言开发的，基本形成了一个有限元软件包.
In this paper several new finite elements are constructed for differrent kinds of probelms by the Double Set Parameter Method and their classical and anisotropic convergences are analyzed. Anisotropic convergences of some famous finite elements are also analyzed, and a lot of numerical experiments have been carried out-Many applicable triangular and rectangular elements have been developed for the plate bending problems, some of them are conforming and some of them arc unconforming, but the researches on quadrilateral piar.e elements have been seldom reported. For the rectangular elements, the mapping from the reference element. l to a general element K is very simple. It is enough to construct, the shape function on the K. But for arbitrary quadrilateral elements, the situation is different. On the one hand, the shape function should be constructed on K because it is impossible to construct it directly on a arbitrary quadrilateral, on the other hand tin1 shape function on K, not on K, should have some continuity across the boundaries of K to ensure the convergence. But the continuities of the shape function on A dn not remain on K because the mapping from K to K is nonlinear. It. is difficult to construct the shape function on K being well posed and have the continuities on K required by convergence. An efficient 13-parameter arbitrary quadrilateral plate element is proposed by the double set parameter method and its convergence is proven. Some numerical tests are given which show the the numerical validity of this element.
A finite element solving the elliptic fourth order singular perturbation problem need to be a convergent plate element and to be convergent, uniformly with respect to the parameter e. But riot everyone of the convergent plate elements is convergent uniformly with respect to the parameter e. Then, which of the convergent plate elements are convergent uniformly with respect to the parameter We will present a general convergence theorem for c?nonconforming plate elements and a general convergence theorem for non c?nonconforming plate elements for the problem re-spectively. Using the double set parameter method we have constructed several finite elements which are convergent for this problem, and the numerical tests that
we have made have proved the efficiency of these elements.
The classical finite element approximation theory relies on the regular or non-degenerate condition. This means that the length of every edges of the elments can not change greatly and this has limited applications of the finite element method in some enginering problem such as those of anisotropic structures. For such problems it is naturally wanted taht this condition is not necessary, the sizes of the elements in all directions are allowed to have great differrencs as the problems theirselves require and the convergences of the elements are still remained. This kind of finite elements are called anisotropic elements.
Anisotropic finite elements have been developed recently. Among them A pel and Dobrowoiski have published a series of papers deal with anisotropic Lagvangc interpolations. However, the studies on the estimations of consistent errors for nonconforming elements have been seldom reported. We have improved the method of anisotropic interpolation error estimations and presented a method to estimate the consistent errors of nonconforming elements.
Wilson's and Adini's elements are famous nonconforming rectangular elements. Using our method we have presented the anisotropic estimations of both interpola-tion errors and consistency errors for these two elements.
Quasi-Wilson's element is a nonconforming element for arbitrary quadrilateral meshes. We have proved that the quasi-Wilson element is anisotropic-convergent for narrow quadrilateral meshes. Anisotropic error estimates of interpolation error and consistency error in the energy norm and the L~(2)-norm have been given.
Using the double set parameter method a noncomforming rectangular plate element with 8 degrees and 12 nodal parameters is
对薄板弯曲问题，已提出了很多适用的三角形元和矩形元，有协调的，也有非协调的。但关于任意四边形板元的研究尚不多见。对于矩形元，从参考元(?)到一般单元K上的变换很简单，只需在参考元(?)上构造形函数。而对任意四边形元，情况就大不一样了。一方面，要在参考元(?)上构造形函数，因为不可能在任意四边形上构造形函数；另一方面，为了保证收敛性，K上的形函数(而非(?)上的)跨越K的边界应具有某种连续性。但是，由于从参考元(?)到一般单元K上的变换是非线性的，形函数在参考元K的连续性在一般单元K得不到保持。因此，要构造参考元(?)上适定的形函数并使其在K上具有保证收敛所需的连续性是很困难的。我们用双参数法构造了一个13-自由度任意四边形板元，证明了其收敛性，并进行了数值实验。数值结果表明此元有较好的收敛性。
求解四阶椭圆奇异摄动问题的有限元应是收敛的板元，并且要关于ε(?)致收敛。但并非所有收敛的板元都关于ε一致收敛。那么，哪些板元对奇异摄动问题关于参数ε是一致收敛的哪?我们将分别给出判别c~0非协调板元和非c~0非协调板元对这一问题收敛的一般判定定理，由此可判定已知的板元对该问题的收敛性。我们应用双参数法构造了几个对此问题收敛的有限元。数值实验也验证了这些新有限元对此问题的有效性。
传统有限元收敛性分析要求单元剖分满足非退化条件，即要求多面体单元的各条边的长度不应相差太大。这极大地限制了有限元方法在一些工程领域的应用，如具有各向异性结构的问题。对于这些问题，人们自然希望能去掉这一限制，根据问题的需要，单元在各个方向上的尺寸可以有较大的不同，并仍能保持其收敛性。我们把这种不要求单元剖分满足非退化条件仍保持收敛的有限元称为各向异性元。
关于如何判定一个有限元是否是各向异性元，最近十多年来已有一些研究成果。如T.Apel等学者给出了Lagrange元的各向异性插值误差估计，但对非协调元的相容误差的各向异性误差估计在文献中尚无报道。我们改进了各
向异性插值误差估计的判定方法，并给出非协调元的相容误差的各向异性误
差估计的方法.
Wilson元和Adini元是著名的矩形非协调元.我们给出Wilso，，元和A〔111，，
元的插值误差和相容误差的各向异性估计
类Wilson元是任意四边形上的非协调元，我们证明了类Wils()l，元的各向
异性收敛性，给出能量模和护一模的各向异性误差估计.
利用“双参数法”我们构造了一个8自由度12参矩形板元，证明该元具
有各向异性，给出包括插值误差和相容误差的完整的误差估计，
我们进行了大量的数值试验对理论结果进行实际验证.数值试验所使用
的程序是我们用C语言开发的，基本形成了一个有限元软件包.
In this paper several new finite elements are constructed for differrent kinds of probelms by the Double Set Parameter Method and their classical and anisotropic convergences are analyzed. Anisotropic convergences of some famous finite elements are also analyzed, and a lot of numerical experiments have been carried out-Many applicable triangular and rectangular elements have been developed for the plate bending problems, some of them are conforming and some of them arc unconforming, but the researches on quadrilateral piar.e elements have been seldom reported. For the rectangular elements, the mapping from the reference element. l to a general element K is very simple. It is enough to construct, the shape function on the K. But for arbitrary quadrilateral elements, the situation is different. On the one hand, the shape function should be constructed on K because it is impossible to construct it directly on a arbitrary quadrilateral, on the other hand tin1 shape function on K, not on K, should have some continuity across the boundaries of K to ensure the convergence. But the continuities of the shape function on A dn not remain on K because the mapping from K to K is nonlinear. It. is difficult to construct the shape function on K being well posed and have the continuities on K required by convergence. An efficient 13-parameter arbitrary quadrilateral plate element is proposed by the double set parameter method and its convergence is proven. Some numerical tests are given which show the the numerical validity of this element.
A finite element solving the elliptic fourth order singular perturbation problem need to be a convergent plate element and to be convergent, uniformly with respect to the parameter e. But riot everyone of the convergent plate elements is convergent uniformly with respect to the parameter e. Then, which of the convergent plate elements are convergent uniformly with respect to the parameter We will present a general convergence theorem for c?nonconforming plate elements and a general convergence theorem for non c?nonconforming plate elements for the problem re-spectively. Using the double set parameter method we have constructed several finite elements which are convergent for this problem, and the numerical tests that
we have made have proved the efficiency of these elements.
The classical finite element approximation theory relies on the regular or non-degenerate condition. This means that the length of every edges of the elments can not change greatly and this has limited applications of the finite element method in some enginering problem such as those of anisotropic structures. For such problems it is naturally wanted taht this condition is not necessary, the sizes of the elements in all directions are allowed to have great differrencs as the problems theirselves require and the convergences of the elements are still remained. This kind of finite elements are called anisotropic elements.
Anisotropic finite elements have been developed recently. Among them A pel and Dobrowoiski have published a series of papers deal with anisotropic Lagvangc interpolations. However, the studies on the estimations of consistent errors for nonconforming elements have been seldom reported. We have improved the method of anisotropic interpolation error estimations and presented a method to estimate the consistent errors of nonconforming elements.
Wilson's and Adini's elements are famous nonconforming rectangular elements. Using our method we have presented the anisotropic estimations of both interpola-tion errors and consistency errors for these two elements.
Quasi-Wilson's element is a nonconforming element for arbitrary quadrilateral meshes. We have proved that the quasi-Wilson element is anisotropic-convergent for narrow quadrilateral meshes. Anisotropic error estimates of interpolation error and consistency error in the energy norm and the L~(2)-norm have been given.
Using the double set parameter method a noncomforming rectangular plate element with 8 degrees and 12 nodal parameters is
引文
[1] A. Acosta, R.G. Duran, The maximum angle condition for mixed and nonconforming elements: Application to the Stokes equations, SIAM.J. Numer. Anal. 37(1999) , 18-36．
[2] R.A. Adams, Sobolev Space, Academic Press, New York, 1975．
[3] T. Apel , M. Dobrowolski , Anisotropic interpolation with applications to the finite element method, Computing, Vol 47, (1992) , 277-293．
[4] T. Apel and G. Lube, Anisotropic mesh refinement in stabilized Galerkin mehtods. Numer. Math. 3(1996) , 261-282．
[5] T. Apel, Anisotropic finite element: local estimates and applications. Stuttgart Tenb-ner, 1999
[6] T. Apel, Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements. Computing, 60(1998) 157-174．
[7] D.G. Bergan, M.K. Nygard, Finite elements with increased freedom in choosing shape functions, Int. J. Numer. Math. Eng., 20(1984) 643-6G4．
[8] P.G. Bergan , L. Hansen, A new approach for deriving "good" finite dement stiffness matricex, in J.R. Whiteman ed: The mathematics of finite elements and applications. Vol.II, Academic Press, London, 1976,483-398．
[9] P.G. Bergan, Finite elements based on energy orthogonal functions, I.J. Numer. Math Engin., Vol.15, (1980) 1541-1555．
[10] Brenner, S.C., Scott, L.R., The mathematical theory of finite element methods. New York, Springer-Verlag, 1994．
[11] 陈绍春,石钟慈,构造单元刚度矩阵的双参数法,计算数学, 3(1991) ,286-296．
[12] S.C. Chen and Z.C. Shi, Double set parameter method of constructing a stiffness matrix. Chinese J. Numer. Math. Appl,4(1991) ,55-69．
[13] S.C. Chen , L.X. Luo, A set of 12 parameter rertanglar plate clement with high aeen racy, Appl. Math J. Chinese Uni. Ser. B, Vol.l4,No.4,(1999) 433-439．
[14] S.C. Chen and D.Y. shi, 12-parameter rectangular plate elements with geometric sys mmetry. Numer. Math. J. Chinese Uni. (in Chinese). 3(1996) 233-238．
[15] Shaochun Chen and Dongyang Shi, Accuracy analysis for quasi-Wilson element. Acta Math. Scientia, 1(2000) 44-48．
[16] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. math, soc., 49, 1, 1943．
[17] P.G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Company, 1978．
[18] 冯康,基于变分原理的差分格式,应用数学与计算数学,第2卷,第4期, 238, 1965．
[19] B.M.Irons and A.Razzaque, Experience with the patch tests for convergence of finite elements method, The Mathematical Foundations of Finite Element Methods with Applications to Partial Differential Equations, A.K.Aziz, ed, Academic Press, Now York, London, (1972) , 557-587．
[20] Jinsheng Jiang and Xiaoliang Cheng, A nonconforming element like Wilson's for second order problems. Acta. Numer. Math. (Chinese), 3(1992) 274-278．
[21] D. Lascanx , P. Lesiant, Some nonconforming finite elements for the plate bending problem, RAIRO. Anal. Numer., R-1, (1995) 9-53．
[22] P. Lascaux and P. Lesaint, Some noncomforming finite element for the plate bending problem, RAIRO, Anal. Numer. R-1(1975) , 9-53
[23] P. Lascaux and P. Lesaint, Some noncomforming finite element for the plate bending problem, RAIRO, Anal. Numer. R-1(1975) , 9-53．
[24] P. Lesaint and M. Zlamal, Convergence for the nonconforming Wilson element for arbitrary quadrilateral meshes. Numer. Math. 36(1980) 33-52．
[25] Y.Q. Long , K.Q. Xin, Generalized conforming elements, J. Civil Engin., No.1． (1987) 1-14．
[26] Yuqiu Long and Yin Xu, Generalized conforming quadrilateral memhrance element. Comp. Struc. 4(1994) 749-755．
[27] T.K. Nilssen, X.C. Tai and R. Winther, A robust nonconforming H2-element.. Math. Comp. 70(2001) ,No. 234, 489-505
[28] Dongyang Shi and Shaochun Chen, A kind of Improved Wilson arbitrary quadrilateral elements. J. Numer. Math, of Chinese Uni. (Chinese), 2(1994) 161-167．
[29] Dongyang Shi and Shaochun Chen, A class of nonconforming arbitrary quadrilateral elements. J. Appl. Math, of Chinese Uni. (Chinese), 2(1996) 231-238．
[30] Z.c. Shi, The F-E-M test for convergence of nonconforming finite element methods, Math. Comp., 49(1987) , 391-405．
[31] Z.c. Shi,The generalized patch-test for .Zienkiewicz's triangles, J. comp. math., 2(1984) ,279-286．
[32] 石钟慈,两种不协调元的统一型函数形式,计算数学, 8:4(1986) ,428-434．
[33] Z.C. Shi, Error Estimates of Mbrley element, Math. Numer. Sinica, 12(1990) ,113-118
[34] Z.C. Shi and S.C. Chen, Analysis of a nine degree plate bending element of Specht, Chinese J. Numer. Math. Appl., 4(1989) , 73-79
[35] Zhongci Shi, A convergence condition for the quadrilateral Wilson element. Numer. Math. 44(1984) 349-361．
[36] B. Specht, Modified shape functions for the three node plate bending, element passing the patch test, Int. J. Numer. Maths. Eng., Vol.26,(1988) 705-715．
[37] B. Specht, Modified shape functions for the three node plate bending element passing the patch test, I. J. Numer. Maths. Eng., Vol.26,(1988) 705-715．
[38] G. Strang and G. J. Fix, An analysis of the finite element method, Prentice-Hall,1973．
[39] F.Stummel, The limitation of the patch test, Inter. J. Numer. Anal., 15(1980) ,177-188．
[40] F.Stummel, The generalized patch test, SIAM .J. Numer. Anal., 16(1979) ,449-471．
[41] L.M.Tang, W.J. Chen and Y.X. liu, Quasi-conforming elements in finite dement anal ysis, J. Dalian Inst. Tech. (in Chinese), Vol. 19, No. 2, (1980) 19-35．
[42] R.L.Taylor, P.J.Beresford and E.L.Wilson, A nonconforming element, for stress analysis. Int. J. Numer. Math. Eng. 10(1980) 1211-1219．
[43] S.Timoshenko , S. Woinowsky-Krieger, Theory of plates and shells. 2nd ed. . McGraw Hill,1959．
[44] F.D. veubeke, Variational principles and the patch-test, Int. J. Numer. Math. Eng.,8(1974) ,783-801．
[45] E.L.Wachspress, Incompatible quadrilateral basis function. Int. J. Numer. Math. Eng 12(1978) 589-595．
[46] E.L.Wilson, R.L.Taylor, W.Doherty and J.Ghaboussi, Incompressible displacement modes, in numerical and Computer Models in Structure Mechanics. S.J. Fenves et al, eds. Academic Press, New York, 1973．
[47] A. Zenisek, M. Vanmaele, The interpolation theorem for narrow quadrilateral isopara metric finite elements, Numer. Math. .63(1992) 521-539．
[48] A. Zenisek, M. Vanmaele, The interpolation theory for narrow quadrilateral isoparametric finite elements. Numer. Math. Vol. 72, (1995) 123-141．
[49] A. Zenisek, M. Vanmaele, Applicability of Bramble-Hibert Lemma in interpolation problems of narrow quadrilateral isoparametric finite elements. J. Comp. Arml. Math. Vol. 63, (1995) 109-122．
[2] R.A. Adams, Sobolev Space, Academic Press, New York, 1975．
[3] T. Apel , M. Dobrowolski , Anisotropic interpolation with applications to the finite element method, Computing, Vol 47, (1992) , 277-293．
[4] T. Apel and G. Lube, Anisotropic mesh refinement in stabilized Galerkin mehtods. Numer. Math. 3(1996) , 261-282．
[5] T. Apel, Anisotropic finite element: local estimates and applications. Stuttgart Tenb-ner, 1999
[6] T. Apel, Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements. Computing, 60(1998) 157-174．
[7] D.G. Bergan, M.K. Nygard, Finite elements with increased freedom in choosing shape functions, Int. J. Numer. Math. Eng., 20(1984) 643-6G4．
[8] P.G. Bergan , L. Hansen, A new approach for deriving "good" finite dement stiffness matricex, in J.R. Whiteman ed: The mathematics of finite elements and applications. Vol.II, Academic Press, London, 1976,483-398．
[9] P.G. Bergan, Finite elements based on energy orthogonal functions, I.J. Numer. Math Engin., Vol.15, (1980) 1541-1555．
[10] Brenner, S.C., Scott, L.R., The mathematical theory of finite element methods. New York, Springer-Verlag, 1994．
[11] 陈绍春,石钟慈,构造单元刚度矩阵的双参数法,计算数学, 3(1991) ,286-296．
[12] S.C. Chen and Z.C. Shi, Double set parameter method of constructing a stiffness matrix. Chinese J. Numer. Math. Appl,4(1991) ,55-69．
[13] S.C. Chen , L.X. Luo, A set of 12 parameter rertanglar plate clement with high aeen racy, Appl. Math J. Chinese Uni. Ser. B, Vol.l4,No.4,(1999) 433-439．
[14] S.C. Chen and D.Y. shi, 12-parameter rectangular plate elements with geometric sys mmetry. Numer. Math. J. Chinese Uni. (in Chinese). 3(1996) 233-238．
[15] Shaochun Chen and Dongyang Shi, Accuracy analysis for quasi-Wilson element. Acta Math. Scientia, 1(2000) 44-48．
[16] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. math, soc., 49, 1, 1943．
[17] P.G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Company, 1978．
[18] 冯康,基于变分原理的差分格式,应用数学与计算数学,第2卷,第4期, 238, 1965．
[19] B.M.Irons and A.Razzaque, Experience with the patch tests for convergence of finite elements method, The Mathematical Foundations of Finite Element Methods with Applications to Partial Differential Equations, A.K.Aziz, ed, Academic Press, Now York, London, (1972) , 557-587．
[20] Jinsheng Jiang and Xiaoliang Cheng, A nonconforming element like Wilson's for second order problems. Acta. Numer. Math. (Chinese), 3(1992) 274-278．
[21] D. Lascanx , P. Lesiant, Some nonconforming finite elements for the plate bending problem, RAIRO. Anal. Numer., R-1, (1995) 9-53．
[22] P. Lascaux and P. Lesaint, Some noncomforming finite element for the plate bending problem, RAIRO, Anal. Numer. R-1(1975) , 9-53
[23] P. Lascaux and P. Lesaint, Some noncomforming finite element for the plate bending problem, RAIRO, Anal. Numer. R-1(1975) , 9-53．
[24] P. Lesaint and M. Zlamal, Convergence for the nonconforming Wilson element for arbitrary quadrilateral meshes. Numer. Math. 36(1980) 33-52．
[25] Y.Q. Long , K.Q. Xin, Generalized conforming elements, J. Civil Engin., No.1． (1987) 1-14．
[26] Yuqiu Long and Yin Xu, Generalized conforming quadrilateral memhrance element. Comp. Struc. 4(1994) 749-755．
[27] T.K. Nilssen, X.C. Tai and R. Winther, A robust nonconforming H2-element.. Math. Comp. 70(2001) ,No. 234, 489-505
[28] Dongyang Shi and Shaochun Chen, A kind of Improved Wilson arbitrary quadrilateral elements. J. Numer. Math, of Chinese Uni. (Chinese), 2(1994) 161-167．
[29] Dongyang Shi and Shaochun Chen, A class of nonconforming arbitrary quadrilateral elements. J. Appl. Math, of Chinese Uni. (Chinese), 2(1996) 231-238．
[30] Z.c. Shi, The F-E-M test for convergence of nonconforming finite element methods, Math. Comp., 49(1987) , 391-405．
[31] Z.c. Shi,The generalized patch-test for .Zienkiewicz's triangles, J. comp. math., 2(1984) ,279-286．
[32] 石钟慈,两种不协调元的统一型函数形式,计算数学, 8:4(1986) ,428-434．
[33] Z.C. Shi, Error Estimates of Mbrley element, Math. Numer. Sinica, 12(1990) ,113-118
[34] Z.C. Shi and S.C. Chen, Analysis of a nine degree plate bending element of Specht, Chinese J. Numer. Math. Appl., 4(1989) , 73-79
[35] Zhongci Shi, A convergence condition for the quadrilateral Wilson element. Numer. Math. 44(1984) 349-361．
[36] B. Specht, Modified shape functions for the three node plate bending, element passing the patch test, Int. J. Numer. Maths. Eng., Vol.26,(1988) 705-715．
[37] B. Specht, Modified shape functions for the three node plate bending element passing the patch test, I. J. Numer. Maths. Eng., Vol.26,(1988) 705-715．
[38] G. Strang and G. J. Fix, An analysis of the finite element method, Prentice-Hall,1973．
[39] F.Stummel, The limitation of the patch test, Inter. J. Numer. Anal., 15(1980) ,177-188．
[40] F.Stummel, The generalized patch test, SIAM .J. Numer. Anal., 16(1979) ,449-471．
[41] L.M.Tang, W.J. Chen and Y.X. liu, Quasi-conforming elements in finite dement anal ysis, J. Dalian Inst. Tech. (in Chinese), Vol. 19, No. 2, (1980) 19-35．
[42] R.L.Taylor, P.J.Beresford and E.L.Wilson, A nonconforming element, for stress analysis. Int. J. Numer. Math. Eng. 10(1980) 1211-1219．
[43] S.Timoshenko , S. Woinowsky-Krieger, Theory of plates and shells. 2nd ed. . McGraw Hill,1959．
[44] F.D. veubeke, Variational principles and the patch-test, Int. J. Numer. Math. Eng.,8(1974) ,783-801．
[45] E.L.Wachspress, Incompatible quadrilateral basis function. Int. J. Numer. Math. Eng 12(1978) 589-595．
[46] E.L.Wilson, R.L.Taylor, W.Doherty and J.Ghaboussi, Incompressible displacement modes, in numerical and Computer Models in Structure Mechanics. S.J. Fenves et al, eds. Academic Press, New York, 1973．
[47] A. Zenisek, M. Vanmaele, The interpolation theorem for narrow quadrilateral isopara metric finite elements, Numer. Math. .63(1992) 521-539．
[48] A. Zenisek, M. Vanmaele, The interpolation theory for narrow quadrilateral isoparametric finite elements. Numer. Math. Vol. 72, (1995) 123-141．
[49] A. Zenisek, M. Vanmaele, Applicability of Bramble-Hibert Lemma in interpolation problems of narrow quadrilateral isoparametric finite elements. J. Comp. Arml. Math. Vol. 63, (1995) 109-122．