基于EEP法的一维非线性有限元自适应分析
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摘要
常微分方程（Ordinary Differential Equation，简称ODE）的数值求解对力学研究和工程计算均具有重要意义，而非线性常微分方程的数值求解更是其中的难点和热点。本文针对非线性常微分方程，提出了一套新型的自适应求解的有限元方法（FEM）。该方法通过对非线性问题进行线性化，将基于单元能量投影（Element Energy Projection，简称EEP）法的线性问题自适应求解方法直接引入非线性问题的求解，无需对非线性问题本身单独建立超收敛公式及其自适应算法，从而构成一个一般性的、统一的非线性问题自适应求解算法，进而开发了非线性ODE求解器的雏形。全文主要工作如下：
1.提出了基于弱形式的非线性有限元自适应迭代的基本策略。基于有限元弱形式推导了Newton迭代格式，提出“理想线性问题”的概念，以其为桥梁直接引入现有的线性问题自适应求解算法；将非线性迭代和自适应求解有机结合起来，提出了有限元求解非线性ODE问题的基本策略。该思路简明清晰，通用性强。
2.将基本策略成功地推广到一维C0和C1非线性问题的自适应求解，提出了相应的整套算法。数值算例表明本文算法高效、稳定、可靠，能够得到逐点满足给定误差限的FEM解，且误差分布均匀，精度冗余较小。
3.将基本策略成功地推广到非线性一阶方程组的自适应求解，提出了相应的整套算法。一阶方程组的自适应求解研究具有基础性意义，任意高于一阶的初值或边值问题的求解都可以等效地转化为一阶方程组的求解。非线性一阶方程组自适应求解的成功拓宽了本文算法的求解范围，并对非线性问题的求解形成了统一的模式。
4.对非线性边界条件、初始解选择、解路径追踪和临界点求解等关键问题给出具体处理算法。利用Newton格式，给出了对非线性边界条件进行线性化的处理方法，进一步完善求解器功能；结合ODE转化技巧，以简明直观的方式实现非线性求解中常用的延拓法；对非线性问题中的临界点求解问题，可直接精确计算出临界点的位置及其对应的解答。
大量算例表明，本文算法高效、稳定、可靠，解答可逐点以最大模度量满足用户给定的误差限，可作为先进高效的非线性ODE问题求解器的核心理论和算法。
The numerical solution of ODEs (Ordinary Differential Equations) plays animportant role in the modern mechanics and engineering computing, the numericalsolution of nonlinear ODEs is the central challenge among various difficulties. Thepresent dissertation proposed a new self-adaptive finite element (FE) strategy fornonlinear ODE problems. In this method, the existing linear self-adaptive strategybased on the EEP (Element Energy Projection) method is incorporated directly into thesolution of nonlinear ODEs to avoid constructing super-convergent formula and self-adaptive algorism for each specific and individual nonlinear problem. As a result, ageneral and unified self-adaptive algorism was proposed and the prototype of nonlinearODE solver was formed based on the algorism. The main work of this dissertation is asfollows:
1. A fundamental nonlinear iteration strategy of Newton type was proposed basedon the week form of nonlinear ODEs. The concept of “ideal linear problem” wasproposed so that the linear self-adaptive strategy can be introduced directly into thesolution of nonlinear ODEs. Combining the above nonlinear iteration and self-adaptivity techniques, a clear, concise and general fundamental strategy was proposed.
2. The fundamental strategy was successfully extended to solving nonlinear C0andC1problems self-adaptively. Mathematical analysis and a number of given numericalexperiments show that the algorism based on the fundamental strategy is able to obtaina final adaptive mesh on which the conventional FEM solutions satisfy the user-specified tolerance point-wise with little accuracy redundancy.
3. The fundamental strategy was successfully extended to solving nonlinear first-order ODE systems self-adaptively. The solution of first-order ODE systems hasfundamental significance, because any initial and boundary value problems of highorder ODEs can be equivalently converted to first-order ODE systems. The success ofthe self-adaptive strategy for solving first-order ODE systems broadens the range ofsolving nonlinear problems and forms a unified mode of solving nonlinear ODEproblems.
4. Some key issues in nonlinear ODE problems, such as the treatment of nonlinearboundary conditions, the choice of initial solution, and the tracking of solution path and critical points on the solution path, were discussed respectively. A Newton type methodwas proposed to treat nonlinear boundary conditions, further improving the function ofthe nonlinear ODE solver; the continuation method was implemented with some ODEconversion techniques; and the solution of critical points on the solution path wasdirectly solved by solving a converted nonlinear ODE problem.
A large number of numerical experiments show that the proposed method in thisdissertation is highly efficient, stable and reliable with the results satisfying the user-preset error tolerance by maximum norm, and hence can serve as the core theory andalgorithm of an advanced and efficient FE solver for nonlinear ODEs.
1.提出了基于弱形式的非线性有限元自适应迭代的基本策略。基于有限元弱形式推导了Newton迭代格式，提出“理想线性问题”的概念，以其为桥梁直接引入现有的线性问题自适应求解算法；将非线性迭代和自适应求解有机结合起来，提出了有限元求解非线性ODE问题的基本策略。该思路简明清晰，通用性强。
2.将基本策略成功地推广到一维C0和C1非线性问题的自适应求解，提出了相应的整套算法。数值算例表明本文算法高效、稳定、可靠，能够得到逐点满足给定误差限的FEM解，且误差分布均匀，精度冗余较小。
3.将基本策略成功地推广到非线性一阶方程组的自适应求解，提出了相应的整套算法。一阶方程组的自适应求解研究具有基础性意义，任意高于一阶的初值或边值问题的求解都可以等效地转化为一阶方程组的求解。非线性一阶方程组自适应求解的成功拓宽了本文算法的求解范围，并对非线性问题的求解形成了统一的模式。
4.对非线性边界条件、初始解选择、解路径追踪和临界点求解等关键问题给出具体处理算法。利用Newton格式，给出了对非线性边界条件进行线性化的处理方法，进一步完善求解器功能；结合ODE转化技巧，以简明直观的方式实现非线性求解中常用的延拓法；对非线性问题中的临界点求解问题，可直接精确计算出临界点的位置及其对应的解答。
大量算例表明，本文算法高效、稳定、可靠，解答可逐点以最大模度量满足用户给定的误差限，可作为先进高效的非线性ODE问题求解器的核心理论和算法。
The numerical solution of ODEs (Ordinary Differential Equations) plays animportant role in the modern mechanics and engineering computing, the numericalsolution of nonlinear ODEs is the central challenge among various difficulties. Thepresent dissertation proposed a new self-adaptive finite element (FE) strategy fornonlinear ODE problems. In this method, the existing linear self-adaptive strategybased on the EEP (Element Energy Projection) method is incorporated directly into thesolution of nonlinear ODEs to avoid constructing super-convergent formula and self-adaptive algorism for each specific and individual nonlinear problem. As a result, ageneral and unified self-adaptive algorism was proposed and the prototype of nonlinearODE solver was formed based on the algorism. The main work of this dissertation is asfollows:
1. A fundamental nonlinear iteration strategy of Newton type was proposed basedon the week form of nonlinear ODEs. The concept of “ideal linear problem” wasproposed so that the linear self-adaptive strategy can be introduced directly into thesolution of nonlinear ODEs. Combining the above nonlinear iteration and self-adaptivity techniques, a clear, concise and general fundamental strategy was proposed.
2. The fundamental strategy was successfully extended to solving nonlinear C0andC1problems self-adaptively. Mathematical analysis and a number of given numericalexperiments show that the algorism based on the fundamental strategy is able to obtaina final adaptive mesh on which the conventional FEM solutions satisfy the user-specified tolerance point-wise with little accuracy redundancy.
3. The fundamental strategy was successfully extended to solving nonlinear first-order ODE systems self-adaptively. The solution of first-order ODE systems hasfundamental significance, because any initial and boundary value problems of highorder ODEs can be equivalently converted to first-order ODE systems. The success ofthe self-adaptive strategy for solving first-order ODE systems broadens the range ofsolving nonlinear problems and forms a unified mode of solving nonlinear ODEproblems.
4. Some key issues in nonlinear ODE problems, such as the treatment of nonlinearboundary conditions, the choice of initial solution, and the tracking of solution path and critical points on the solution path, were discussed respectively. A Newton type methodwas proposed to treat nonlinear boundary conditions, further improving the function ofthe nonlinear ODE solver; the continuation method was implemented with some ODEconversion techniques; and the solution of critical points on the solution path wasdirectly solved by solving a converted nonlinear ODE problem.
A large number of numerical experiments show that the proposed method in thisdissertation is highly efficient, stable and reliable with the results satisfying the user-preset error tolerance by maximum norm, and hence can serve as the core theory andalgorithm of an advanced and efficient FE solver for nonlinear ODEs.
引文
[1] Birkhoff G, Rota G C. Ordinary Differential Equations (3rd ed.).New York: Wiley,1978.
[2] Yuan S. The Finite Element Method of Lines. Beijing-New York: Science Press,1993.
[3]王勖成.有限单元法.北京:清华大学出版社,2003.
[4]监凯维奇,泰勒著;庄茁,岑松译.有限元方法.第2卷.固体力学.北京:清华大学出版社,2006.
[5] Turner MR, Clough R, Martin H, Topp L. Stiffness and deflection analysis of complexstructures. J Aero Sci,1956,23(9):805-823.
[6] Argyris J H. Elasto-plastic matrix displacement analysis of three-dimensional continua. J. RoyalAeronautical Society,1965,69:633-635.
[7] Marcal PV, King IP. Elastic-plastic analysis of two dimensional stress system by the finiteelement method. Int J Mechanical Sciences,1967,9:143-155.
[8] Yamda Y, Yishimura N. Plastic stress-strain matrix and its application for the solution of elasto-plastic problem by the finite element method. Int J Meth Sci,1968,10:343-354.
[9] Zienkiewicz O C, Pande G N. Some useful forms of isotropic yield surfaces for soil and rockmechanics. In.G.Gudehus(ed.), Finite element in Geomechanics, John Wiley,1977.
[10] Valliappan S, Nath P. Tensile crack propagation in reinforced concrete beam by finite elementtechniques. In Int Conf on Shear, Torsion and Bond in Reinforced Concrete, Coimbatore, India,January1969.
[11] Zienkiewicz O C, Cormeau I C. Visco-plasticity, plasticity and creep in elastic solids-a unifiednumerical solution approach. Int J Num Meth Eng,1974,8:821-45.
[12] Banovec J. An efficient finite element method for elastic-plastic analysis of plane frames.Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag,1981.
[13] Telles J C F, Brebbia C A.. Simplified calculation models applied to postbuckling analysis ofthin plates. Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag1981.
[14] Eidsheim O M, Larson P K. A study of some generalized constitutive models for elasto-plasticshells. Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag1981.
[15] Brink K, Kratzig W B. Geometrically correct formulation for curved finite bar elements underlarge deformation. Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag,1981.
[16]袁驷.关于薄膜大挠度方程的极限解及其特性.数值计算与计算机应用,1996.
[17] Wood R D, Zieniewicz O C. Geometrically non-linear finite element analysis of beam-frames-circles and axisymmetric shells. Compute and Structures,1977,7:723-35.
[18]范重，龙驭球.几何非线性样条元.航空学报,1990.
[19] Bai X, Zhao X. Analysis of large deformation elastoplastic contact through finite gap elements.Compute and Structures,1988,30:975-978.
[20] Zhong W X, Sun S. A finite element method for elasto-plastic structures and contact problemsby parametric quadratic programming. Int J Num Meth Eng,1988,26:2723-2738.
[21] Bathe K J, Chaudhary, A. On finite element analysis of large deformation frictional contactproblems. Unification Finite Element Methods, North-Holland, Amsterdam,1984.
[22] Lee G B, Kwak B M. Formulation and implementation of beam contact problems under largedisplacement by a mathematical programming. Compute and Structures,1989,31:365-376.
[23] Yagawa G, Hirayama H. A finite element method for contact problems related to fracturemechanics. Int J Num Meth Eng,1984,20:2175-2195.
[24] Kuczma M S, Switzka R. Bending of elastic beams on Winkler-type viscoelastic foundationswith unilateral constraints. Compute and Structures,1990,34:125-136.
[25]刘晶波,杜修力.结构动力学.北京:机械工业出版社,2005.
[26]陈传淼.有限元超收敛构造理论.长沙:湖南科学技术出版社,2002.
[27]袁驷.从矩阵位移法看有限元应力精度的损失与恢复.力学与实践,1998,20(4):1-6.
[28] Oden J T, Brauchli H J. On the calculation of consistent stress distribution in finite elementapplications. Int J Numer Methods Eng,1971,3(3):317-325.
[29] Hinton E, Campbell J S. Local and global smoothing of discontinuous finite element functionsusing a Least Square Method. Int J Num Meth Eng,1974,8(3):461-480.
[30] Zhu J Z, Zienkiewicz O C. Super-convergence recovery technique and a posteriori errorestimator. Int J Num Meth Eng,1990,30(7):1321-1339.
[31] Zienkiewicz O C, Zhu J Z. A simple error estimator and adaptive procedure for practicalengineering analysis. Int J Num Meth Eng,1987,24(2):337-357.
[32] Zienkiewicz O C, Zhu J Z. The superconvergence patch recovery (SPR) and a posteriori errorestimates, Part1: The recovery technique, Part2: error estimates and adaptivity, Int J NumMeth Eng,1992,33(7):1331-1382.
[33] Zienkiewicz O C, Zhu J Z. The superconvergence patch recovery (SPR) and adaptive finiteelement refinement. Comp Meth Appl Mech Eng,1992,101(1-3):207-224.
[34]袁驷,王枚.一维有限元后处理超收敛解答计算的EEP法.工程力学，2004,21(2):1-9.
[35] Strang G, Fix G. An analysis of the finite element method, Prentice-Hall,1973.
[36]袁驷,王旭,邢沁妍,叶康生.具有最佳超收敛阶的EEP法计算格式: I算法公式.工程力学,2007,24(10):1-5.
[37] Douglas J, Dupont T. Galerkin approximations for the two point boundary problems usingcontinuous piecewise polynomial spaces. Numer Math,1974,22:99-109.
[38]赵庆华.单元能量投影法的数学分析[博士学位论文].长沙:湖南大学数学与计量经济学院,2007.
[39]袁驷,王枚,和雪峰.一维C1有限元超收敛解答计算的EEP法.工程力学,2006,23(2):1-9.
[40]袁驷,林永静.二阶非自伴两点边值问题Galerkin有限元后处理超收敛解答计算的EEP法.计算力学学报,2007,24(2):142-147.
[41] Wang M, Yuan S. Computation of super-convergent nodal stresses of Timoshenko beamelements by EEP method. Applied Mathematics and Mechanics,2004,25(11):1228-1240.
[42]袁征.二维有限元后处理超收敛的研究[硕士学位论文].北京:清华大学土木工程系,2003.
[43]王枚.一维有限单元法及二维有限元线法后处理超收敛计算的研究[博士学位论文].北京:清华大学土木工程系,2003.
[44]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.
[45]袁驷,邢沁妍,王旭,叶康生.具有最佳超收敛阶的EEP法计算格式: II数值算例.工程力学,2007,24(11):1-6.
[46]袁驷,赵庆华.具有最佳超收敛阶的EEP法计算格式: III数学证明.工程力学,2007,24(12):1-5.
[47] Oliverira E R A. Optimaization of finite element solution. Proceedings of the3rd conference onmatrix methods in structural mechanics, Wright-Patterson, Ohio,1971.
[48] Babuska I, Feedback. Adaptivity and a posteriori estimates in finite elements: aims, theory andexperience//Accuracy estimates and adaptive refinements in finite element computations, AWiley-Interscience Publication,1986.
[49] Babuska I, Rheinboldt W C. A-posteriori estimates for the finite element method. Int J NumMeth Eng,1978,12:1597-1615.
[50] Babuska I, Rheinboldt W C. Adaptive approaches and reliability estimates in finite elementanalysis. Comp Meth Appl Mech Eng,1979,17/18:519-540.
[51]Babuska I, Rheinboldt W C. Analysis of optimal finite element meshes in R1. Math Comp,1979,146(33):435-463.
[52] Zienkiewicz O C, Gago J P, Kelly D W. The Hierarchical concept in finite element analysis.Computers and Structures,1983,16(1-4):53-65.
[53] Zienkiewicz O C, Craig A. Adaptive refinement, error estimates, multigrid solution, andHierarchical Finite Element Method concepts//Accuracy estimates and adaptive refinements infinite element computations, A Wiley-Interscience Publication,1986.
[54] Szabo B A. Estimation and control of error based on p convergence//Accuracy estimates andadaptive refinements in finite element computations, A Wiley-Interscience Publication,1986.
[55] Szabo B A, Babuska I. Finite element analysis. Wiley-Interscience Publication,1991.
[56]Gallimard L, Ladeveze P, Pelle J P. Error estimation and adaptivity in elastoplasticity. Int J Num.Meth Eng,1996,39(2):189-217.
[57] Manolis Papadrakakis, George P Babilis. Solution techniques for the p-version of the adaptivefinite element method. Int. J Num Meth Eng,1994,37:1413-1431.
[58] Babuska I, Suri M. The p-and h-p versions of the finite element method, an overview. ComputMeth Appl Mech Engrg,1990,80(1-3):5-26.
[59] Babuska I, Suri M. The p-and h-p versions of the finite element method, basic principles andproperties. SIAM Review,1994,36(4):578-632.
[60] Gui W, Babuska I. The h, p and h-p version of the finite element method in one-dimensional,Part I: The error analysis of the p-version. Numer Math,1986,49:577-612.
[61] Gui W, Babuska I. The h, p and h-p version of the finite element method in one-dimensional,Part II: The error analysis of the h-and h-p version. Numer Math,1986,49:613-657.
[62] Gui W, Babuska I. The h, p and h-p version of the finite element method in one-dimensional,Part III: The adaptive h-p version. Numer Math,1986,49:659-683.
[63]Oden J T. Optimal h-p finite element methods. Comput Meth Appl Mech Engrg,1994,112(1-4):309-331.
[64]袁驷,邢沁妍,王旭,叶康生.基于最佳超收敛阶EEP法的一维有限元自适应求解.应用数学和力学,29(5):533-543,2008.
[65]袁驷,和雪峰.一个高效的一维有限元自适应求解的方案.第十三届全国结构工程学术大会特邀报告,工程力学(增刊),2004, S1:214-220.
[66]邢沁妍.基于EEP法的一维Galerkin有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2008
[67]隋昕.基于EEP法的一维C1有限元超收敛解答和自适应分析[博士学位论文].北京:清华大学土木工程系,2010.
[68]肖川.基于EEP法的一阶常微分方程组有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2009.
[69]肖嘉.基于EEP法的线法二阶常微分方程组有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2009.
[70]邢沁妍.有限元EEP超收敛计算和自适应分析的深入研究[博士后研究报告].北京:清华大学土木工程系,2010.
[71] Deuflhard P. Newton Methods for Nonlinear Problems. Berlin, Heidelberg, NewYork, Springer-Verlag,2004.
[72]Ascher U. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations.Prentice-Hall,1988.
[73]林永静.三维延拓康托洛维奇法及一维伽辽金有限元的超收敛计算[博士学位论文].北京:清华大学土木工程系,2005.
[74]王旭.基于EEP法的一维有限元与二维有限元线法自适应分析[博士学位论文].北京:清华大学土木工程系,2007.
[75]宋天霞.非线性结构有限元计算.武汉:华中理工大学出版社,1996.
[76] Wan F. The dimpling of spherical caps. Tech. Rep.78-6, Inst. of Applied Mathematics, Univ.British Columbia, Vancouver, Canada,1978.
[77] Ascher U, Christiansen J, Russell R D. A collocation solver for mixed order systems ofboundary value problem. Math Comp,1979,33:659-679.
[78] David F, Parker, Frederic Y M. Finite polar dimpling of shallow caps under sub-bucklingaxisymmetric pressure distributions. SIAM J Appl Math1984,44(2):301-326.
[79]刘鸿文.材料力学(上册,第3版).北京:高等教育出版社,1992.
[80] Jang T S,Baek H S, Paik J K. A new method for the non-linear deflection analysis of an infinitebeam resting on a non-linear elasticfoundation. International Journal of Non-Linear Mechanics,2011,46(1):339-346.
[81] Pruess S and Fulton CT. Performance of the Sturm-Liouville Software Package SLEDGE. MPat Los Alamos National Lab,1991.
[82] Love A E H. A treatise on the Mathematical Theory of Elasticity (4th edition), Dover,NewYork,1944.
[83]徐芝纶.弹性力学(第二版)，第二卷.北京:高等教育出版社,1981.
[84] DaDeppo D A, Schmidt R. Instability of clamped-hinged circular arches subjected to a pointload. ASME Journal of Applied Mechanics,1975,97:894-896.
[85] Borri M, Bottasso C. An intrinsic beam model based on a helicoidal approximation-Part II:linearization and finite element implementation. Int J Num Meth Eng,.1994,37:2291-2309.
[86] Jayachandran A., White D W. Variable order secant matrix technique applied to the elasticpostbuckling of arches. Communication in Numerical Methods and Engineering,2001,190:4195-4230.
[87] Sage M, Turk G, Kalagasidu A, Vratanar B. A kinematically exact finite element formulation ofelastic-plastic curved beams. Computer and Structures,1998,67:197-214.
[88] Zupan D, Saje M. The three-dimensional beam theory: Finite element formulation based oncurvature. Computer and Structures,2003,81:1875-1888.
[89] Dym C L, Shames I H. Solid Mechanics: a Variational Approach, McGraw-Hill,1973.
[90]祁泉泉.基于振动信号的结构参数识别系统方法研究[博士学位论文].北京:清华大学土木工程系,2011.
[91]张亿菓.若干非线性模型问题的FEMOL分析及其统一求解[硕士学位论文].北京:清华大学土木工程系,1992.
[92] Weibel E S. On the confinement of a plasma by magnetostatic fields. Phys. Fluids.,1959,2(1):52-56.
[93] Markin V S, Chernenko A A, Chizmadehev Y A, Chirkov Y G. Aspects of the theory of gasporous electrodes, in: V.S. Bagotskii, Y.B. Vasilev (Eds.), Fuel Cells: Their ElectrochemicalKinetics, Consultants Bureau, New York,1966,21-33.
[94] Gidaspow D, Baker B S. A model for discharge of storage batteries. J. Electrochem. Soc.,1973,120:1005-1010.
[95] Feng X L, Mei L Q, He G L. An efficient algorithm for solving Troesch’s problem. AppliedMathematics and Computation,2007,189:500-507.
[96] Chang S H. A variational iteration method for solving Troesch’s problem. Journal ofComputaional and Applied Mathematics,2010,234:3043-3047.
[97] Chang S H. Numercial solution of Troesch’s problem by simple shooting method. AppliedMathematics and Computation,2010,216:3303-3306.
[98] Holt J F. Numerical solution of nonlinear two-point boundary problems by finite differencemethods. Commun. ACM7,1964,6:366-373.
[99] King W S, Lewellen W S. Boundary-layer similarity solutions for rotating flows with andwithout magnetic interaction. Ref. ATN-63(9227)-6, Aerodynamics and Propulsion Res. Lab.,Aerospace Corp., Los Angeles, Calif.,July1963.
[100]方楠.基于EEP法的FEMOL有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2011.
[2] Yuan S. The Finite Element Method of Lines. Beijing-New York: Science Press,1993.
[3]王勖成.有限单元法.北京:清华大学出版社,2003.
[4]监凯维奇,泰勒著;庄茁,岑松译.有限元方法.第2卷.固体力学.北京:清华大学出版社,2006.
[5] Turner MR, Clough R, Martin H, Topp L. Stiffness and deflection analysis of complexstructures. J Aero Sci,1956,23(9):805-823.
[6] Argyris J H. Elasto-plastic matrix displacement analysis of three-dimensional continua. J. RoyalAeronautical Society,1965,69:633-635.
[7] Marcal PV, King IP. Elastic-plastic analysis of two dimensional stress system by the finiteelement method. Int J Mechanical Sciences,1967,9:143-155.
[8] Yamda Y, Yishimura N. Plastic stress-strain matrix and its application for the solution of elasto-plastic problem by the finite element method. Int J Meth Sci,1968,10:343-354.
[9] Zienkiewicz O C, Pande G N. Some useful forms of isotropic yield surfaces for soil and rockmechanics. In.G.Gudehus(ed.), Finite element in Geomechanics, John Wiley,1977.
[10] Valliappan S, Nath P. Tensile crack propagation in reinforced concrete beam by finite elementtechniques. In Int Conf on Shear, Torsion and Bond in Reinforced Concrete, Coimbatore, India,January1969.
[11] Zienkiewicz O C, Cormeau I C. Visco-plasticity, plasticity and creep in elastic solids-a unifiednumerical solution approach. Int J Num Meth Eng,1974,8:821-45.
[12] Banovec J. An efficient finite element method for elastic-plastic analysis of plane frames.Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag,1981.
[13] Telles J C F, Brebbia C A.. Simplified calculation models applied to postbuckling analysis ofthin plates. Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag1981.
[14] Eidsheim O M, Larson P K. A study of some generalized constitutive models for elasto-plasticshells. Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag1981.
[15] Brink K, Kratzig W B. Geometrically correct formulation for curved finite bar elements underlarge deformation. Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag,1981.
[16]袁驷.关于薄膜大挠度方程的极限解及其特性.数值计算与计算机应用,1996.
[17] Wood R D, Zieniewicz O C. Geometrically non-linear finite element analysis of beam-frames-circles and axisymmetric shells. Compute and Structures,1977,7:723-35.
[18]范重，龙驭球.几何非线性样条元.航空学报,1990.
[19] Bai X, Zhao X. Analysis of large deformation elastoplastic contact through finite gap elements.Compute and Structures,1988,30:975-978.
[20] Zhong W X, Sun S. A finite element method for elasto-plastic structures and contact problemsby parametric quadratic programming. Int J Num Meth Eng,1988,26:2723-2738.
[21] Bathe K J, Chaudhary, A. On finite element analysis of large deformation frictional contactproblems. Unification Finite Element Methods, North-Holland, Amsterdam,1984.
[22] Lee G B, Kwak B M. Formulation and implementation of beam contact problems under largedisplacement by a mathematical programming. Compute and Structures,1989,31:365-376.
[23] Yagawa G, Hirayama H. A finite element method for contact problems related to fracturemechanics. Int J Num Meth Eng,1984,20:2175-2195.
[24] Kuczma M S, Switzka R. Bending of elastic beams on Winkler-type viscoelastic foundationswith unilateral constraints. Compute and Structures,1990,34:125-136.
[25]刘晶波,杜修力.结构动力学.北京:机械工业出版社,2005.
[26]陈传淼.有限元超收敛构造理论.长沙:湖南科学技术出版社,2002.
[27]袁驷.从矩阵位移法看有限元应力精度的损失与恢复.力学与实践,1998,20(4):1-6.
[28] Oden J T, Brauchli H J. On the calculation of consistent stress distribution in finite elementapplications. Int J Numer Methods Eng,1971,3(3):317-325.
[29] Hinton E, Campbell J S. Local and global smoothing of discontinuous finite element functionsusing a Least Square Method. Int J Num Meth Eng,1974,8(3):461-480.
[30] Zhu J Z, Zienkiewicz O C. Super-convergence recovery technique and a posteriori errorestimator. Int J Num Meth Eng,1990,30(7):1321-1339.
[31] Zienkiewicz O C, Zhu J Z. A simple error estimator and adaptive procedure for practicalengineering analysis. Int J Num Meth Eng,1987,24(2):337-357.
[32] Zienkiewicz O C, Zhu J Z. The superconvergence patch recovery (SPR) and a posteriori errorestimates, Part1: The recovery technique, Part2: error estimates and adaptivity, Int J NumMeth Eng,1992,33(7):1331-1382.
[33] Zienkiewicz O C, Zhu J Z. The superconvergence patch recovery (SPR) and adaptive finiteelement refinement. Comp Meth Appl Mech Eng,1992,101(1-3):207-224.
[34]袁驷,王枚.一维有限元后处理超收敛解答计算的EEP法.工程力学，2004,21(2):1-9.
[35] Strang G, Fix G. An analysis of the finite element method, Prentice-Hall,1973.
[36]袁驷,王旭,邢沁妍,叶康生.具有最佳超收敛阶的EEP法计算格式: I算法公式.工程力学,2007,24(10):1-5.
[37] Douglas J, Dupont T. Galerkin approximations for the two point boundary problems usingcontinuous piecewise polynomial spaces. Numer Math,1974,22:99-109.
[38]赵庆华.单元能量投影法的数学分析[博士学位论文].长沙:湖南大学数学与计量经济学院,2007.
[39]袁驷,王枚,和雪峰.一维C1有限元超收敛解答计算的EEP法.工程力学,2006,23(2):1-9.
[40]袁驷,林永静.二阶非自伴两点边值问题Galerkin有限元后处理超收敛解答计算的EEP法.计算力学学报,2007,24(2):142-147.
[41] Wang M, Yuan S. Computation of super-convergent nodal stresses of Timoshenko beamelements by EEP method. Applied Mathematics and Mechanics,2004,25(11):1228-1240.
[42]袁征.二维有限元后处理超收敛的研究[硕士学位论文].北京:清华大学土木工程系,2003.
[43]王枚.一维有限单元法及二维有限元线法后处理超收敛计算的研究[博士学位论文].北京:清华大学土木工程系,2003.
[44]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.
[45]袁驷,邢沁妍,王旭,叶康生.具有最佳超收敛阶的EEP法计算格式: II数值算例.工程力学,2007,24(11):1-6.
[46]袁驷,赵庆华.具有最佳超收敛阶的EEP法计算格式: III数学证明.工程力学,2007,24(12):1-5.
[47] Oliverira E R A. Optimaization of finite element solution. Proceedings of the3rd conference onmatrix methods in structural mechanics, Wright-Patterson, Ohio,1971.
[48] Babuska I, Feedback. Adaptivity and a posteriori estimates in finite elements: aims, theory andexperience//Accuracy estimates and adaptive refinements in finite element computations, AWiley-Interscience Publication,1986.
[49] Babuska I, Rheinboldt W C. A-posteriori estimates for the finite element method. Int J NumMeth Eng,1978,12:1597-1615.
[50] Babuska I, Rheinboldt W C. Adaptive approaches and reliability estimates in finite elementanalysis. Comp Meth Appl Mech Eng,1979,17/18:519-540.
[51]Babuska I, Rheinboldt W C. Analysis of optimal finite element meshes in R1. Math Comp,1979,146(33):435-463.
[52] Zienkiewicz O C, Gago J P, Kelly D W. The Hierarchical concept in finite element analysis.Computers and Structures,1983,16(1-4):53-65.
[53] Zienkiewicz O C, Craig A. Adaptive refinement, error estimates, multigrid solution, andHierarchical Finite Element Method concepts//Accuracy estimates and adaptive refinements infinite element computations, A Wiley-Interscience Publication,1986.
[54] Szabo B A. Estimation and control of error based on p convergence//Accuracy estimates andadaptive refinements in finite element computations, A Wiley-Interscience Publication,1986.
[55] Szabo B A, Babuska I. Finite element analysis. Wiley-Interscience Publication,1991.
[56]Gallimard L, Ladeveze P, Pelle J P. Error estimation and adaptivity in elastoplasticity. Int J Num.Meth Eng,1996,39(2):189-217.
[57] Manolis Papadrakakis, George P Babilis. Solution techniques for the p-version of the adaptivefinite element method. Int. J Num Meth Eng,1994,37:1413-1431.
[58] Babuska I, Suri M. The p-and h-p versions of the finite element method, an overview. ComputMeth Appl Mech Engrg,1990,80(1-3):5-26.
[59] Babuska I, Suri M. The p-and h-p versions of the finite element method, basic principles andproperties. SIAM Review,1994,36(4):578-632.
[60] Gui W, Babuska I. The h, p and h-p version of the finite element method in one-dimensional,Part I: The error analysis of the p-version. Numer Math,1986,49:577-612.
[61] Gui W, Babuska I. The h, p and h-p version of the finite element method in one-dimensional,Part II: The error analysis of the h-and h-p version. Numer Math,1986,49:613-657.
[62] Gui W, Babuska I. The h, p and h-p version of the finite element method in one-dimensional,Part III: The adaptive h-p version. Numer Math,1986,49:659-683.
[63]Oden J T. Optimal h-p finite element methods. Comput Meth Appl Mech Engrg,1994,112(1-4):309-331.
[64]袁驷,邢沁妍,王旭,叶康生.基于最佳超收敛阶EEP法的一维有限元自适应求解.应用数学和力学,29(5):533-543,2008.
[65]袁驷,和雪峰.一个高效的一维有限元自适应求解的方案.第十三届全国结构工程学术大会特邀报告,工程力学(增刊),2004, S1:214-220.
[66]邢沁妍.基于EEP法的一维Galerkin有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2008
[67]隋昕.基于EEP法的一维C1有限元超收敛解答和自适应分析[博士学位论文].北京:清华大学土木工程系,2010.
[68]肖川.基于EEP法的一阶常微分方程组有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2009.
[69]肖嘉.基于EEP法的线法二阶常微分方程组有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2009.
[70]邢沁妍.有限元EEP超收敛计算和自适应分析的深入研究[博士后研究报告].北京:清华大学土木工程系,2010.
[71] Deuflhard P. Newton Methods for Nonlinear Problems. Berlin, Heidelberg, NewYork, Springer-Verlag,2004.
[72]Ascher U. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations.Prentice-Hall,1988.
[73]林永静.三维延拓康托洛维奇法及一维伽辽金有限元的超收敛计算[博士学位论文].北京:清华大学土木工程系,2005.
[74]王旭.基于EEP法的一维有限元与二维有限元线法自适应分析[博士学位论文].北京:清华大学土木工程系,2007.
[75]宋天霞.非线性结构有限元计算.武汉:华中理工大学出版社,1996.
[76] Wan F. The dimpling of spherical caps. Tech. Rep.78-6, Inst. of Applied Mathematics, Univ.British Columbia, Vancouver, Canada,1978.
[77] Ascher U, Christiansen J, Russell R D. A collocation solver for mixed order systems ofboundary value problem. Math Comp,1979,33:659-679.
[78] David F, Parker, Frederic Y M. Finite polar dimpling of shallow caps under sub-bucklingaxisymmetric pressure distributions. SIAM J Appl Math1984,44(2):301-326.
[79]刘鸿文.材料力学(上册,第3版).北京:高等教育出版社,1992.
[80] Jang T S,Baek H S, Paik J K. A new method for the non-linear deflection analysis of an infinitebeam resting on a non-linear elasticfoundation. International Journal of Non-Linear Mechanics,2011,46(1):339-346.
[81] Pruess S and Fulton CT. Performance of the Sturm-Liouville Software Package SLEDGE. MPat Los Alamos National Lab,1991.
[82] Love A E H. A treatise on the Mathematical Theory of Elasticity (4th edition), Dover,NewYork,1944.
[83]徐芝纶.弹性力学(第二版)，第二卷.北京:高等教育出版社,1981.
[84] DaDeppo D A, Schmidt R. Instability of clamped-hinged circular arches subjected to a pointload. ASME Journal of Applied Mechanics,1975,97:894-896.
[85] Borri M, Bottasso C. An intrinsic beam model based on a helicoidal approximation-Part II:linearization and finite element implementation. Int J Num Meth Eng,.1994,37:2291-2309.
[86] Jayachandran A., White D W. Variable order secant matrix technique applied to the elasticpostbuckling of arches. Communication in Numerical Methods and Engineering,2001,190:4195-4230.
[87] Sage M, Turk G, Kalagasidu A, Vratanar B. A kinematically exact finite element formulation ofelastic-plastic curved beams. Computer and Structures,1998,67:197-214.
[88] Zupan D, Saje M. The three-dimensional beam theory: Finite element formulation based oncurvature. Computer and Structures,2003,81:1875-1888.
[89] Dym C L, Shames I H. Solid Mechanics: a Variational Approach, McGraw-Hill,1973.
[90]祁泉泉.基于振动信号的结构参数识别系统方法研究[博士学位论文].北京:清华大学土木工程系,2011.
[91]张亿菓.若干非线性模型问题的FEMOL分析及其统一求解[硕士学位论文].北京:清华大学土木工程系,1992.
[92] Weibel E S. On the confinement of a plasma by magnetostatic fields. Phys. Fluids.,1959,2(1):52-56.
[93] Markin V S, Chernenko A A, Chizmadehev Y A, Chirkov Y G. Aspects of the theory of gasporous electrodes, in: V.S. Bagotskii, Y.B. Vasilev (Eds.), Fuel Cells: Their ElectrochemicalKinetics, Consultants Bureau, New York,1966,21-33.
[94] Gidaspow D, Baker B S. A model for discharge of storage batteries. J. Electrochem. Soc.,1973,120:1005-1010.
[95] Feng X L, Mei L Q, He G L. An efficient algorithm for solving Troesch’s problem. AppliedMathematics and Computation,2007,189:500-507.
[96] Chang S H. A variational iteration method for solving Troesch’s problem. Journal ofComputaional and Applied Mathematics,2010,234:3043-3047.
[97] Chang S H. Numercial solution of Troesch’s problem by simple shooting method. AppliedMathematics and Computation,2010,216:3303-3306.
[98] Holt J F. Numerical solution of nonlinear two-point boundary problems by finite differencemethods. Commun. ACM7,1964,6:366-373.
[99] King W S, Lewellen W S. Boundary-layer similarity solutions for rotating flows with andwithout magnetic interaction. Ref. ATN-63(9227)-6, Aerodynamics and Propulsion Res. Lab.,Aerospace Corp., Los Angeles, Calif.,July1963.
[100]方楠.基于EEP法的FEMOL有限元自适应分析[博士学位论文].北京:清华大学土木工程系,2011.