薄板大挠度弯曲问题的IEFG方法
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摘要
本文从薄板大挠度弯曲问题基本理论出发,依据改进的移动最小二乘近似和Galerkin全局弱式无网格方法的基本思想,形成了改进的Galerkin无网格(IEFG)方法,并应用到薄板大挠度弯曲问题中.
IEFG方法是一种基于传统的EFG的改进方法.它以Schmidt带权正交函数作为基函数对移动最小二乘近似进行了改进,使需要求逆的矩阵化为对角阵,这样不但继承了MLS近似的一致性优点而且具有很好的稳定性,同时也可避免出现病态方程,提高了计算精度及效率.由于薄板大挠度弯曲问题的控制方程是高阶非线性耦合方程组,难以直接求解,故本文采用Galerkin全局弱式方法对能量控制方程进行离散,得到性能稳定的全局非线性离散系统方程.
本文最后根据IEFG方法编制程序计算薄板在多种边界条件的大挠度弯曲,计算结果表明了本文方法在薄板大挠度弯曲问题上的适用性和精确性.
IEFG method for large deflection of thin plates is proposed in this paper. It is based on IMLS and Galerkin method and combined with basic theory about large deflection.
IEFG is an improved method based on EFG method. It takes orthogonal functions formed by Schmidt method as basis functions of the moving least square approximation. It is not only inherited the MLS’consistency, stability, etc. but also avoiding ill-conditioned equations,so calculation is easier and more accurate. For solving high level nonlinear coupled control equation system directly is very difficult, the method with global Galerkin weak form is used in this paper to discrete control equation.
To verify the reliability of the method I programmed for three examples with different boundary conditions.The examples show that the IEFG method can also solve nonlinear problems of thin plates bending with higher efficiency and stability.
IEFG方法是一种基于传统的EFG的改进方法.它以Schmidt带权正交函数作为基函数对移动最小二乘近似进行了改进,使需要求逆的矩阵化为对角阵,这样不但继承了MLS近似的一致性优点而且具有很好的稳定性,同时也可避免出现病态方程,提高了计算精度及效率.由于薄板大挠度弯曲问题的控制方程是高阶非线性耦合方程组,难以直接求解,故本文采用Galerkin全局弱式方法对能量控制方程进行离散,得到性能稳定的全局非线性离散系统方程.
本文最后根据IEFG方法编制程序计算薄板在多种边界条件的大挠度弯曲,计算结果表明了本文方法在薄板大挠度弯曲问题上的适用性和精确性.
IEFG method for large deflection of thin plates is proposed in this paper. It is based on IMLS and Galerkin method and combined with basic theory about large deflection.
IEFG is an improved method based on EFG method. It takes orthogonal functions formed by Schmidt method as basis functions of the moving least square approximation. It is not only inherited the MLS’consistency, stability, etc. but also avoiding ill-conditioned equations,so calculation is easier and more accurate. For solving high level nonlinear coupled control equation system directly is very difficult, the method with global Galerkin weak form is used in this paper to discrete control equation.
To verify the reliability of the method I programmed for three examples with different boundary conditions.The examples show that the IEFG method can also solve nonlinear problems of thin plates bending with higher efficiency and stability.
引文
[1]铁木辛柯,沃诺斯基著,板壳理论,科学出版社, 1977.
[2] Courant R, Isaacson E and Rees M, On the solution of non-linear hyperbolic differential equations by finite differences, Comm. Pure. Appl. Math V, 243-255.
[3] Belytschko, Tetal, Meshless method, An overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 1996, 139, 3-47.
[4] Li Shao fan, Liu WK, Meshfree and particle methods and their application–s, Applied Mechanics Review, 2002, 55, 1-34.
[5] Liu W K. Reproducing Kernel Particle Methods, International Journal for Numerical Methods in Fluids, 1995, 1081-1106.
[6] Aluru N R, A point collocation method based on reproducing kernel approximations, Int.J.Numer. Meth. Engng., 2000, 47, 1083-1121.
[7] Frazer RA, Jones WP and Skan SW, Apporoximations to functions and to solutions of Differential Equations, Great Britain Aero Counc. London. Rep. and Memo. No.1799.
[8] Nayroles B, Touzot G, Villon P, Generalizing the finite element method, diffuse approximation and diffuse elements, Comput. Mech., 1992, 307- 318.
[9] Belytschko T., Lu Y Y, Gu L, Element free Galerkin methods, Int J Number Methods Engrg., 1994, 229-256.
[10] Chung H J, Belytschko T, An error estimating the EFG method, Comput. Mech., 1998, 21, 91-100.
[11] Y.Y.Lu, T.Belytschko and M.Tabbara, Element-free Galerkin method for wave propagation and dynamic fracture, Compute Methods Apal. Mech. Engrg., 1995, 131-153.
[12] Dolbow J, Belytschko T, Numerical integration of the Galerkin weak form in mesh free methods, Comput. Mech., 1999, 23, 219-230.
[13] Belytschko T, Krongauz Y, Organ D, etal.Smoothing and acclerated computations in the element free Galerkin method, J Comput. Appl. Math., 1996, 74, 111-126.
[14] Krongauz Y, Belytschko T, EFG approximation with discontinuous derivatives. Int J Numer Methods Engrg., 1998, 41, 1215-1233.
[15] Krysl P, Belytschko T. Analysis of thin shells by element2free Galerkin method, Comput. Mech., 1995, 17, 26-35.
[16] Liu G R, Gu Y T, Coupling of element free Galerkin and hybrid boundary element methods using modified variation formulation, Comput. Mech., 2000, 26, 166-173.
[17] Zhang Zan, K.M.Liew,Yumin Cheng,Y.Y.Lee, Analyzing 2D fracture problems with improved element-free Galerkin method, Engineering Analysis with boundary Elements, 2008, 32, 241–250.
[18]徐芝纶,弹性力学(第四版),高等教育出版社, 2006.
[19]张雄,刘岩,无网格法.清华大学出版社, 2004.8.
[20]凌道盛,徐兴,非线性有限元及程序,浙江大学出版社,2004.
[21]雷光宇,郑照北,薄板大挠度广义协调有限元分析,工程力学, 2003, 04, 11-14.
[22] Pica A, Wood R D, Hinton E, Finite element analysis of geometrically nonlinear plate behavior using a midlin formulation, Comput Struct, 1980, 11, 203, 205.
[23] Husain J. Al-Gahtani, Mahmoud Naffa’a, RBF meshless method for large de?ection of thin plates with immovable edges, Engineering Analysis with Boundary Elements, 2009, 33, 176–183.
[24]张福范,均布载荷下矩形悬臂板的弯曲,应用数学和力学, 1980, 03, 349-362.
[25]曲庆璋,梁兴复,张权,矩形悬臂板的非线性弯曲,强度与环境, 1999, 3, 5-9.
[2] Courant R, Isaacson E and Rees M, On the solution of non-linear hyperbolic differential equations by finite differences, Comm. Pure. Appl. Math V, 243-255.
[3] Belytschko, Tetal, Meshless method, An overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 1996, 139, 3-47.
[4] Li Shao fan, Liu WK, Meshfree and particle methods and their application–s, Applied Mechanics Review, 2002, 55, 1-34.
[5] Liu W K. Reproducing Kernel Particle Methods, International Journal for Numerical Methods in Fluids, 1995, 1081-1106.
[6] Aluru N R, A point collocation method based on reproducing kernel approximations, Int.J.Numer. Meth. Engng., 2000, 47, 1083-1121.
[7] Frazer RA, Jones WP and Skan SW, Apporoximations to functions and to solutions of Differential Equations, Great Britain Aero Counc. London. Rep. and Memo. No.1799.
[8] Nayroles B, Touzot G, Villon P, Generalizing the finite element method, diffuse approximation and diffuse elements, Comput. Mech., 1992, 307- 318.
[9] Belytschko T., Lu Y Y, Gu L, Element free Galerkin methods, Int J Number Methods Engrg., 1994, 229-256.
[10] Chung H J, Belytschko T, An error estimating the EFG method, Comput. Mech., 1998, 21, 91-100.
[11] Y.Y.Lu, T.Belytschko and M.Tabbara, Element-free Galerkin method for wave propagation and dynamic fracture, Compute Methods Apal. Mech. Engrg., 1995, 131-153.
[12] Dolbow J, Belytschko T, Numerical integration of the Galerkin weak form in mesh free methods, Comput. Mech., 1999, 23, 219-230.
[13] Belytschko T, Krongauz Y, Organ D, etal.Smoothing and acclerated computations in the element free Galerkin method, J Comput. Appl. Math., 1996, 74, 111-126.
[14] Krongauz Y, Belytschko T, EFG approximation with discontinuous derivatives. Int J Numer Methods Engrg., 1998, 41, 1215-1233.
[15] Krysl P, Belytschko T. Analysis of thin shells by element2free Galerkin method, Comput. Mech., 1995, 17, 26-35.
[16] Liu G R, Gu Y T, Coupling of element free Galerkin and hybrid boundary element methods using modified variation formulation, Comput. Mech., 2000, 26, 166-173.
[17] Zhang Zan, K.M.Liew,Yumin Cheng,Y.Y.Lee, Analyzing 2D fracture problems with improved element-free Galerkin method, Engineering Analysis with boundary Elements, 2008, 32, 241–250.
[18]徐芝纶,弹性力学(第四版),高等教育出版社, 2006.
[19]张雄,刘岩,无网格法.清华大学出版社, 2004.8.
[20]凌道盛,徐兴,非线性有限元及程序,浙江大学出版社,2004.
[21]雷光宇,郑照北,薄板大挠度广义协调有限元分析,工程力学, 2003, 04, 11-14.
[22] Pica A, Wood R D, Hinton E, Finite element analysis of geometrically nonlinear plate behavior using a midlin formulation, Comput Struct, 1980, 11, 203, 205.
[23] Husain J. Al-Gahtani, Mahmoud Naffa’a, RBF meshless method for large de?ection of thin plates with immovable edges, Engineering Analysis with Boundary Elements, 2009, 33, 176–183.
[24]张福范,均布载荷下矩形悬臂板的弯曲,应用数学和力学, 1980, 03, 349-362.
[25]曲庆璋,梁兴复,张权,矩形悬臂板的非线性弯曲,强度与环境, 1999, 3, 5-9.