Milestones of Direct Variational Calculus and its Analysis from the 17th Century until today and beyond- Mathematics meets Mechanics- with restriction to linear elasticity

Stein, E.

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Tytuł artykułu

Milestones of Direct Variational Calculus and its Analysis from the 17th Century until today and beyond- Mathematics meets Mechanics- with restriction to linear elasticity

Autorzy

Stein E.

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http://cames.ippt.gov.pl/

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Abstrakty

This treatise collects and re?ects the major developments of direct (discrete) variational calculus since the end of the 17th century until about 1990, with restriction to classical linear elastomechanics, such as 1D-beam theory, 2D-plane stress analysis and 3D-problems, governed by the 2nd order elliptic Lam´e-Navier partial di?erential equations. The extension of the historical review to non-linear elasticity, or even more, to inelastic deformations would need an equal number of pages and, therefore, should be published separately. A comprehensive treatment of modern computational methods in mechanics can be found in the Encyclopedia of Computational Mechanics [83]. The purpose of the treatise is to derive the essential variants of numerical methods and algorithms for discretized weak forms or functionals in a systematic and comparable way, predominantly using matrix calculus, because partial integrations and transforming volume into boundary integrals with Gauss’s theorem yields simple and vivid representations. The matrix D of 1st partial derivatives is replaced by the matrix N of direction cosines at the boundary with the same order of non-zero entries in the matrix; ?/?xi corresponds to cos(n, ei),x = xiei , n = cos(n, ei)ei, i = 1, 2, 3 for ? ? R3. A main goal is to present the interaction of mechanics and mathematics for getting consistent discrete variational methods and from there – in a comparative way – the properties of primal, dual and dual-mixed ?nite element methods in their historical development. Of course the progress of proper mathematical analysis since the 1970s and 1980s is outlined, concerning consistency, convergence and numerical stability as well as a priori and a posteriori error estimates in the frame of Sobolev spaces for the mostly C 0 -continuous test and trial ansatz functions at element interfaces. This mathematical research followed after the more intuitive engineering developments since the 1950s, using the principle of virtual work and the principle of minimum of total potential energy. A proper ?nite element method has to regard the motto of this treatise: “mathematics meets mechanics”. Comparative numerical results are not included because numerous new calculations would have been necessary for getting usable comparisons for the various cited articles.

Słowa kluczowe

finite element method history mechanics computation

obliczanie metoda elementów skończonych historia mechanika

Wydawca

Instytut Podstawowych Problemów Techniki PAN

Czasopismo

Computer Assisted Methods in Engineering and Science

Rocznik

2012

Tom

Vol. 19, no. 1

Strony

7--91

Opis fizyczny

Bibliogr. 101 poz., rys., tab., wykr.

Twórcy

autor

Stein E.

Leibniz Universit¨at Hannover, stein@ibnm.uni-hannover.de

Bibliografia

[1] M. Ainsworth, J.T. Oden. A posteriori error estimators in finite element analysis. John Wiley & Sons, NewYork, 2000.
[2] J.H. Argyris. Energy theorems and structural analysis. Aircraft Engineering, 26: 347–356, 383–387, 394, 1954.
[3] J.H. Argyris. Energy theorems and structural analysis. Aircraft Engineering, 27: 42–58, 80–94, 125–134, 154–158, 1955.
[4] J.H. Argyris, K.E. Buck, D.W. Scharpf, H.M. Hilber, G. Mareczek. Some new elements for the matrix displacement method. In Proc. 2nd Conf. on Matrix methods in Structural mechanics. Wright-Patterson Air Force Base, Ohio, 1968.
[5] J.H. Argyris, S. Kelsey, H. Kamel. Matrix methods of structural Analysis. AGARDograph 72. Pergamon Press, London, 1963.
[6] D.N. Arnold, F. Brezzi, J. Douglas Jr. PEERS: A new mixed finite element for plane elasticity. Japan. J. Appl. Math., 1: 347–367, 1984.
[7] D.N. Arnold, J. Douglas Jr., C.P. Gupta. A family of higher order finite element methods for plane elasticity. Numer. Math., 45: 1–22, 1984.
[8] J.P. Aubin. Behaviour of the error of the approximate solution of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Scuola Norm. Sup. Pisa, 21: 599–637, 1967.
[9] I. Babuška. The finite element method with Lagrangian multipliers. Numer. Math., 20: 179–192, 1973.
[10] I. Babuška, A. Miller. A feedback finite element method with a posteriori error estimation: Part i. the finite element method and some basic properties of the a posteriori error estimator. Comput. Methods Appl. Mech. Engng., 61: 1–40, 1987.
[11] I. Babuˇska, W.C. Rheinboldt. A-posteriori error estimates for the finite element method. Int. J. Num. Meth. Engng., 12: 1597–1615, 1978.
[12] I. Babuˇska, B. Szabó, I.N. Katz. The p-version of the finite element method. SIAM J. Numer. Anal., 18: 515–545, 1981.
[13] R.E. Bank, A. Weiser. Some a posteriori error estimators for elliptic partial differential equations. Math. Comp., 44: 283–301, 1985.
[14] T. Belytschko, T. Black. Elastic growth in finite elements with minimal remeshing. Int. J. Num. Meth. Eng., 45: 601–620, 1999.
[15] E. Betti. Teorema generale intorno alle deformazioni che fanno equilibrio a forze che agiscono alla superficie. Il Nuovo Cimento, series 2, 7–8: 87–97, 1872.
[16] M. Bischoff, E. Ramm, D. Braess. A class of equivalent enhanced assumed strain and hybrid stress finite elements. Comp. Mech., 22: 443–449, 1999.
[17] H. Borouchaki, P.L. George, F. Hecht, P. Laug, E. Saltel. Delaunay mesh generation governed by metric specifications. Part I. Algorithms. Finite Elem. Anal. Des., 25: 61–83, 1997.
[18] D. Braess. Finite elements. Cambridge University Press, 2nd edition, 2001, (1st edition in German language, 1991).
[19] F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. Rev. Fr. Automat. Inf. Rech. Op´erat. S´er. Rouge, 8: 129–151, 1974.
[20] F. Brezzi, M. Fortin. Mixed and hybrid finite element methods. Springer New York, 1991.
[21] F. Brezzi, J. Douglas Jr., M. Fortin, L.D. Marini. Efficient rectangular mixed finite elements in two and three space variables. RAIRO M2AN, 21: 581–604, 1987.
[22] F. Brezzi, J. Douglas Jr., L.D. Marini. Two families of mixed finite elements for second-order elliptic problems. Numer. Math., 47: 217–235, 1985.
[23] I. Bubnov. Structural Mechanics of Ships (translated from Russian). 1914.
[24] K.E. Buck, D.W. Scharpf. Einf¨uhrung in die Matrizen-Verschiebungsmethode. In K.E. Buck, D. W. Scharpf, E. Stein, W. Wunderlich, editors, Finite Elemente in der Statik, pages 1–70. Wilhelm Ernst & Sohn, Berlin, 1973.
[25] K.E. Buck, D.W. Scharpf, E. Schrem, E. Stein. Einige allgemeine Programmsysteme f¨ur finite Elemente. In K.E. Buck, D.W. Scharpf, E. Stein, W. Wunderlich, editors, Finite Elemente in der Statik, pages 399–454. Wilhelm Ernst & Sohn, Berlin, 1973.
[26] K.E. Buck, D.W. Scharpf, E. Stein, W. Wunderlich, editors. Finite Elemente in der Statik. Wilhelm Ernst & Sohn, Berlin, 1973.
[27] J. C´ea. Approximation variationnelle des problˇcmes aux limites (Ph.D. Thesis). Annales de l’institut Fourier 14, 2: 345–444, 1964.
[28] C. Carstensen, S.A. Funken. Averaging technique for FE-a posteriori error control in elasticity. Comput. Methods Appl. Mech. Engng., 190: 2483–2498, 4663–4675; 191: 861–877, 2001.
[29] C.A. Castigliano. Th´eorie de l’´equilibre des systˇcmes ´elastiques et ses applications. Nero, Turin, 1879.
[30] R. Courant. Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. Soc., 49: 1–23, 1943.
[31] M. Crouzeix, P.A. Raviart. Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Num´er., R3: 33–76, 1973.
[32] P.A. Cundall, O.D.L. Strack. Discrete numerical model for granular assemblies. Geotechnique, 29: 47–65, 1979.
[33] P. de Fermat. Analysis ad refractiones. Oeuvres, I. 1661.
[34] B.F. de Veubeke, editor. Matrix methods of structural analysis. Pergamon Press, Oxford, 1964.
[35] R. Descartes. Discours de la m´ethode pour bien conduire sa raison et chercher la v´erit´e dans les sciences. Leiden (French) (1637), Amsterdam (Latin), 1656.
[36] I.S. Duff, J.K. Reid. A set of FˇRRTRAN-subroutines for sparse symmetric sets of linear equations. Report R-10533, Computer Science and Systems Devisions, AERE Harwell, HMSO, London, 1982.
[37] L. Euler. Methodus inveniendi lineas curvas maximi minimive proprietate gaudens sive solutio problematis isoperimetrici latissimo sensu accepti, volume 25 of Opera omnia, Series I. Lausanne and Genevae, 1744.
[38] B.G. Galerkin. Beams and plates, series for some problems of elastic equilibrium of beams and plates. Wjestnik Ingenerow, 10: 897–908, 1915.
[39] G. Galilei. Discorsi e dimonstrazioni matematiche, intorno a due nuove scienze. Leiden, 1638.
[40] R.H. Gallagher, I. Rattinger, J.S. Archer. A correlation study of methods of matrix structural analysis. AGARDograph 69. Pergamon Press, Oxford, 1964.
[41] W. Hackbusch. Iterative solvers for large sparse systems of equations (English translation from German language). Springer Verlag, Berlin, 1994.
[42] E. Hellinger. Die allgemeinen Ans¨atze der Mechanik der Kontinua. In F. Klein and C. M¨uller, editors, Enzyklopadie der Mathematischen Wissenschaften 4 (Teil 4), pages 601–694. Teubner, Leipzig, 1914.
[43] C. Hermite. Sur un nouveau developpment en s´erie de fonctions. C. R. Acad. Sci., Paris, 58: 93–100, 1864.
[44] L.R. Herrmann. Finite-element bending analysis for plates. J. Engg. Mech. EM 5, 93: 13–26, 1967.
[45] D. Hilbert. Die Grundlagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universitat, Band VI. 1928.
[46] H.H. Hu, D.D. Joseph, M.J. Crochet. Direct simulation of fluid particle motions. Theor. Comp. Fluid Dyn., 3: 285–306, 1992.
[47] Th.J.R. Hughes, J.A. Cottrell, Y. Bazilevs. Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engng., 194: 4135–4195, 2005.
[48] Ch. Huygens. Horologicum oscillatorium sive de motu pendularium. Apud F. Muguet, Paris, 1673.
[49] B.M. Irons, O.C. Zienkiewicz. The isoparametric finite element system – a new concept in finite element analysis. In Proc. Conf.: Recent advances in stress analysis, Royal Aeronautical Society, 1968.
[50] J.W. Strutt, Baron Rayleigh, M.A., F.R.S. The theory of sound, volume I. Macmillan and Co., London, 1877.
[51] I. Kant. Gedanken von der wahren Sch¨atzung der lebendigen Kr¨afte. K¨onigsberg, 1746.
[52] K. Kl¨oppel, D. Reuschling. Zur Anwendung der Theorie der Graphen bei der Matrizenformulierung statischer Probleme. Der Stahlbau, 35: 236–245, 1966.
[53] J.S. Koenig. De universali principio aequilibrii et motus in vi viva reperto, deque nexu inter vim vivam et actionem, utriusque minimo dissertatio. Nova Acta Eruditorum, March 1751.
[54] P. Ladevˇcze, J.-P. Pelle. Mastering calculations in linear and nonlinear mechanics. Springer Science+Business Media, Inc., 2005.
[55] R. Lagrange. Polynomes et fonctions de Legendre. Gauthier-Villars, Paris, 1939.
[56] A.M. Legendre. Sur l’attraction des sph´eroides. M´em. Math. et Phys. pr´esent´es ´r l’Ac. r. des. sc. par divers savants, 10, 1785.
[57] G.W. Leibniz. Brevis demonstratio erroris memorabilis Cartesii... Acta Eruditorum, pages 161–163, 1686.
[58] G.W. Leibniz. Communicatio suae pariter, duarumque alienarum ad adendum sibi primum a Dn. Jo. Bernoullio, deinde a Dn. Marchione Hospitalio communicatarum solutionum problematis curvae celerrimi descensus a Dn. Jo. Bernoullio geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea propositi. Acta Eruditorum, pages 201–206, May 1697.
[59] J.C. Maxwell. Phil. Mag., 27: 294, 1864.
[60] R.J. Melosh. Basis for derivation of matrices for the direct stiffness method. AIAA J., 1: 1631–1637, 1963.
[61] F.L. Menabrea. Sul principio di elasticit´r, delucidazioni di L.F.M. 1870.
[62] K.-H. M¨oller, C.-H. Wagemann. Die Formulierungen der Einheitsverformungs- und der Einheitsbelastungszustande in Matrizenschreibweise mit Hilfe der Graphen. Der Stahlbau, 35: 257–269, 1966.
[63] H. M¨uller-Breslau. H¨utte – Des Ingenieurs Taschenbuch, chapter Elastizit¨at und Festigkeit und Baumechanik. Ernst & Korn, Berlin, 1877.
[64] G.E. Moore. Cramming more components onto integrated circuits. Electronics, 38: 4 pages, 1965.
[65] R. Niekamp, E. Stein. An object-oriented approach for parallel two- and three-dimensional adaptive finite element computations. Computers & Structures, 80: 317–328, 2002.
[66] J.A. Nitsche. l∞-convergence of finite element approximations. In Mathematical aspects of finite element methods, volume 606 of Lecture Notes in Mathematics, pages 261–274. Springer, New York, 1977.
[67] A.S. Ostenfeld. Teknisk Statik I. 3rd edition, 1920.
[68] E.C. Pestel, F.A. Leckie. Matrix methods in elastomechanics. McGraw-Hill, New York, 1963.
[69] T.H.H. Pian. Derivation of element stiffness matrices by assumed stress distributions. AIAA J., 2: 1533–1536, 1964.
[70] T.H.H. Pian, P. Tong. Basis of finite element methods for solid continua. Int. J. Num. Meth. Eng., 1: 3–28,1964.
[71] W. Prager. Variational principles for elastic plates with relaxed continuity requirements. Int. J. Solids & Structures, 4: 837–844, 1968.
[72] G. Prange. Die Variations- und Minimalprinzipe der Statik der Baukonstruktion. Habilitationsschrift. Technische Hochschule Hannover, 1916.
[73] E. Reissner. On a variational theorem in elasticity. J. Math. Phys., 29: 90–95, 1950.
[74] W. Ritz. ¨Uber eine neue Methode zur Lösung gewisser Probleme der mathematischen Physik. Journal f¨ur die reine und angewandte Mathematik, 135: 1–61, 1909.
[75] M.C. Rivara. Mesh refinement processes based on the generalized bisection of simplices. SIAM Journal on Numerical Analysis, 21: 604–613, 1984.
[76] J. Robinson. Structural Matrix Analysis for the Engineer. John Wiley & Sons, New York, 1966.
[77] R. Rodr´ıguez. Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Diff. Equations, 10: 625–635, 1994.
[78] K.H. Schellbach. Probleme der Variationsrechnung. Crelle’s Journal f¨ur die reine und angewandte Mathematik, 41: 293–363 + 1 table, 1851.
[79] J.C. Simo, M.S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Num. Meth. Eng., 29: 1595–1638, 1990.
[80] S.L. Sobolev. Applications of Functional Analysis in Mathematical Physics, volume 7 of Mathematical Monographs. AMS, Providence (RI), 1963.
[81] E. Stein. Gottfried Wilhelm Leibniz, seiner Zeit weit voraus als Philosoph, Mathematiker, Physiker, Techniker. – ein Extrakt der gleichnamigen Ausstellungen. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 54: 131–171, 2005.
[82] E. Stein. Theoria cum praxi: Leibniz als technischer Erfinder. In Thomas A.C. Reyden, H. Heit, P. Hoyningen- Huene, editors, Der universale Leibniz, pages 155–183. Franz Steiner Verlag, Stuttgart, 2009.
[83] E. Stein, R. de Borst, T.J.R. Hughes, editors. Encyclopedia of Computational Mechanics, vol. 1: Fundamentals, vol 2: Solids and Structures, vol 3: Fluids. John Wiley & Sons, Chichester, 2004, (2nd edition in Internet, 2007).
[84] E. Stein, M. R¨uter. Finite element methods for elasticity with error-controlled discretization and model adaptivity. In E. Stein, R. de Borst, T.J.R. Hughes, editors, Encyclopedia of Computational Mechanics, volume 2: Solids and Structures, pages 5–58. John Wiley & Sons, Chichester, 2004.
[85] E. Stein, M. R¨uter, S. Ohnimus. Implicit upper bound error estimates for combined expansive model and discretization adaptivity. Comput. Methods Appl. Mech. Engrg., 200: 2626–2638, 2011.
[86] E. Stein, W. Wunderlich. Finite-Element-Methoden als direkte Variationsverfahren in der Elastostatik. In K.E. Buck, D.W. Scharpf, E. Stein, W. Wunderlich, editors, Finite Elemente in der Statik, pages 71–125. Wilhelm Ernst & Sohn, Berlin, 1973.
[87] R. Stenberg. Analysis of mixed finite element methods for the Stokes problem: A unified approach. Math. Comp., 42: 9–23, 1984.
[88] H. Stolarski, T. Belytschko. Limitation principles for mixed finite elements based on the Hu-Washizu variational formulation. Comput. Methods Appl. Mech. Engng., 60: 195–216, 1987.
[89] B. Szabó, I. Babuˇska. Finite Element Analysis. John Wiley & Sons, New York, 1991.
[90] B.A. Szabó, A.K. Mehta. p-convergent finite element approximations in fracture mechanics. Int. J. Num. Meth. Engng., 12: 551–560, 1978.
[91] I. Szabó. Geschichte der mechanischen Prinzipien. Birkh¨auser, Basel, 3rd edition, 1987.
[92] R.L. Taylor, O.C. Zienkiewicz, J.C. Simo, A.H.C. Chan. The patch test – a condition for assessing f.e.m. convergence. Int. J. Num. Meth. Eng., 22: 39–62, 1986.
[93] E. Torricelli. De Motu gravium, volume 2 of Opere. Faenca, 1919 (reprint).
[94] M.J. Turner, R.W. Clough, H.C. Martin, L.J. Topp. Stiffness and deflection analysis of complex strucutres. Journal of the Aeronautical Sciences, 23: 805–823, 1956.
[95] R. Verfürth. A review of a posteriori error estimation and adaptive mesh refinement technis. Wiley-Teubner, Chichester, 1996.
[96] K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, 1968.
[97] N. Willems, W.M. Lucas Jr. Matrix analysis for structural engineers. Prentice-Hall, INC. / Englewood Cliffs, N.J., 1968.
[98] G. Zavarise, P. Wriggers. Trends in Computational Contact Mechanics. LNACM, vol. 58, Springer, Berlin.
[99] O.C. Zienkiewicz, S. Qu, R.L. Taylor, S. Nakazawa. The patch test for mixed formulations. Int. J. Num. Meth. Eng., 23: 1873–83, 1986.
[100] O.C. Zienkiewicz, J.Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Num. Meth. Eng., 24: 337–357, 1987.
[101] O.C. Zienkiewicz, J.Z. Zhu. The superconvergent patch recovery (SPR) and adaptive finite element refinements.Comput. Methods Appl. Mech. Engng., 101: 207–224, 1992

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