Solution Strategies for Nonlinear Equations and Computation of Singular Points

  • P. Wriggers
Part of the International Centre for Mechanical Sciences book series (CISM, volume 342)


Investigation of nonlinear static and dynamic response of solids and structures requires knowledge about the stability behaviour. In general we have to detect singular points (e.g. limit or bifurcation points) or the global loss of stability and furthermore one has to be able to follow nonlinear solution paths.


Singular Point Limit Point Bifurcation Point Solution Strategy Extended System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Abbot, J. P. (1978): An Efficient Algorithm for the Determination of certain Bifurcation Points, J. Comp. Appl. Math., 4, 19–27.CrossRefGoogle Scholar
  2. Batoz, I.L., Dhatt, G. (1979): Incremental Displacement Algorithms for Non-linear Problems, Int. J. Num. Meth. Engng., 14, 1262–1267.CrossRefMATHMathSciNetGoogle Scholar
  3. Bergan, P. G., Horrigmoe, G., Krakeland, B., Soreide, T. H. (1978): Solution Techniques for Non-linear Finite Element Problems, Int. J. Num. Meth. Engng., 12, 1677–1696.CrossRefMATHGoogle Scholar
  4. Brendel, B., Ramm, E. (1982): Nichtlineare Stabilitätsuntersuchungen mit der Methode der Finiten Elemente, Ing. Archiv 51, 337–362.MATHGoogle Scholar
  5. Budiansky, B., E. S. Roth, (1962), Axisymmetric dynamic buckling of clamped shallow spherical shells. Collected Papers on Instability of Shell Structures, NASA TND-1510.Google Scholar
  6. Burmeister, A., E. Ramm, (1990), Dynamic Stability Analysis of Shell Structures, in Computational Mechanics of nonlinear response of shells, eds. W. Krätzig, E. Quate, Springer, Berlin 1990.Google Scholar
  7. Bushnell, D., (1985): Computerized Buckling Analysis of Shells, Mechanics of Elastic Stability, Vol. 9, Martinus Nijhoff Publishers, Boston.CrossRefGoogle Scholar
  8. Carstensen, C:, Wriggers, P. (1993): On Perturbation Behaviour in Nonlinear Dynamics, Communications Appl. Num. Meth., 9, 165–175.Google Scholar
  9. Chan, T. F. (1984): Deflation Techniques and Block-Elimination Algorithms for Solving Bordered Singular Systems, SIAM, J. Sci. Stat. Comput. 5, 121–134.CrossRefMATHGoogle Scholar
  10. Ciarlet, P.G. (1988): Mathematical Elasticity, Volume I, North-Holland, Amsterdam.MATHGoogle Scholar
  11. Crisfield, M. A. (1981): A Fast Incremental/Iterative Solution Procedure that Handles Snap Through, Computers & Structures, 13, 55–62.CrossRefMATHGoogle Scholar
  12. Decker, D. W., Keller, H.B. (1980): Solution Branching — A Constructive Technique, in P.J. Holmes, ed., New Approaches to Nonlinear Problems in Dynamics ( SIAM, Philadelphia ) 53–69.Google Scholar
  13. Dennis J.E., Schnabel R.B., (1983): Numerical Methods for Unconstrained Optimiza- tion and Nonlinear Equations, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  14. Felippa, C. A. (1987): Traversing Critical Points with Penalty Springs, in: Transient/Dynamic Analysis and Constitutive Laws for Engineering Materials, ed. Pande, Middleton (Niihoff, Dordrecht, Boston, Lancaster), C2/1-C2/8.Google Scholar
  15. Hallquist, J. (1979): NIKE2D: An Implicit, Finite Deformation, Finite Element Code for Analysing the Static and Dynamic Response of Two-Dimensional Solids, University of California, LLNL, Rep. UCRL-52678.Google Scholar
  16. Hill, R. (1958): A General Theory of Uniqueness and Stability in Elastic-Plastic Solids, J. Mech. Phys. Solids, 6, 236–249.CrossRefMATHGoogle Scholar
  17. Hirsch M.W., Smale S. (1974): Differential equations, dynamical systems and linear algebra Academic Press.Google Scholar
  18. Hughes, T. R. J. (1987): The Finite Element Method, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  19. Jepson A. D., Spence, A. (1985): Folds in Solutions of two Parameter Systems and their Calculation, SIAM, J. Numer. Anal. 22, 347–369.CrossRefMATHMathSciNetGoogle Scholar
  20. Johnson, C. (1987): Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge, U.K.Google Scholar
  21. Keller, H.B. (1977): Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems. In: Rabinowitz, P. (ed.): Application of Bifurcation Theory, New York: Academic Press 1977, 359–384.Google Scholar
  22. Kleiber, M., W. Kotula, M. Saran, (1987), Numerical Analysis of Dynamic Quasi-Bifurcation, Eng. Computation, 4, 48–52.CrossRefGoogle Scholar
  23. Koiter W. T. (1967): On the Stability of Elastic Equilibrium, Translation of ‘Over de Stabiliteit von het Elastisch Evenwicht’, Polytechnic Institute Delft, H. J. Paris Publisher Amsterdam 1945, NASA TT F-10, 833.Google Scholar
  24. Krätzig, W. B., (1989), Eine einheitliche statische und dynamische Stabilitätstheorie für Pfadverfolgungsalgorithmen in der numerischen Festkörpermechanik, ZAMM, 69, 203–207.CrossRefMATHGoogle Scholar
  25. Mittelmann, H.-D.; Weber, H. (1980): Numerical Methods for Bifurcation Problems–a Survey and Classification. In: Mittelmann, Weber (ed.): Bifurcation Problems and their Numerical Solution, ISNM 54, ( Basel, Boston, Stuttgart: Birkhäuser ), 1–45.CrossRefGoogle Scholar
  26. Moore, G., Spence, A. (1980): The Calculation of Turning Points of Nonlinear Equations, SIAM J. Num. Anal., 17, 567–576.CrossRefMATHMathSciNetGoogle Scholar
  27. Needleman, A. (1972): A Numerical Study of Necking in Circular Cylindrical Bars, J. Mech. Phys. Solids, 20, 111–127.CrossRefMATHGoogle Scholar
  28. Ogden, R.W. (1972): Large Deformation Isotropic Elasticity on the Correlation of Theory and Experiment for Incompressible Rubberlike Solids, Proc. R. Soc., London, A(326), 565–584.Google Scholar
  29. Papadrakakis, M. (1987): Analysis Methods for Spatial Structures, in: Shell and Spatial Structures: Computational Aspects; ed. Roeck, Samartin, van Laethem and Backx, (Springer, Berlin/Heidelberg/New-York ) 121–148.CrossRefGoogle Scholar
  30. Ramm, E., (1981): Strategies for Tracing the Nonlinear Response Near Limit Points, in: Nonlinear Finite Element Analysis in Structural Mechanics, ed. Wunderlich, Stein Bathe, Springer, Berlin-Heidelberg-New-York.Google Scholar
  31. Rheinboldt, W.C. (1981): Numerical Analysis of Continuation Methods for Nonlinear Structural Problems. Comp. & Struct., 13, 103–113.CrossRefMATHMathSciNetGoogle Scholar
  32. Riks, E. (1972): The Application of Newtons Method to the Problem of Elastic Stability. J. Appl. Mech., 39, 1060–1066.CrossRefMATHGoogle Scholar
  33. Riks, E., (1984), Some computational aspects of stability analysis of nonlinear structures, Comp. Meth. Appl. Mech. Engng., 47, 219–260.CrossRefMATHGoogle Scholar
  34. Riks, E., F. A. Brogan, C. C. Rankin, (1990), Numerical Aspects of Shell Stability Analysis, in Computational Mechanics of nonlinear response of shells, eds. W. Krätzig, E. Quate, Springer, Berlin 1990.Google Scholar
  35. Schwartz, H. R., (1974), The Eigenvalue Problem (A–A B) x = 0 for Symmetric Matrices of High Order, Comp. Meth. Appl. Mech. Engng., 3, 11–28.CrossRefGoogle Scholar
  36. Schweizerhof, K., Wriggers, P. (1986): Consistent Linearizations for Path Following Methods in Nonlinear FE Analysis, Computer Methods in Applied Mechanics and Engineering, 59, 261–279.CrossRefMATHGoogle Scholar
  37. Seydel, R. (1979): Numerical Computation of Branch Points in Nonlinear Equations, Numer. Math., 33, 339–352.MATHMathSciNetGoogle Scholar
  38. Shilkrut, D., (1983), Investigation of Axisymmetric Deformation of Geometrically Nonlinear, Rotationally, Orthotropic, Circular PLates, Int. J. Non-Linear Mechanics, 18, 95–118.CrossRefMATHGoogle Scholar
  39. Simitses, G. J., (1990), Dynamic Stability of Suddenly Loaded Structures, Springer, New York, Berlin; Heidelberg.Google Scholar
  40. Simo, J.C. (1988): A Framework for Finite Strain Elastoplasticity Based on the Multiplicative Decomposition and Hyperelastic Relations. Part II: Computational Aspects, Comp. Meth. Appl. Mech. Engng., 67, 1–31.CrossRefGoogle Scholar
  41. Simo, J. C., Wriggers, P., Schweizerhof, K.H., Taylor, R.L. (1986): Post-buckling Analysis involving Inelasticity and Unilateral Constraints, Int. J. Num. Meth. Engng., 23, 779–800.CrossRefMATHGoogle Scholar
  42. Simo, J.C., D.D. Fox &- M.S. Rifai, (1990): On a Stress Resultant Geometrically Exact Shell Model. Part III: Computational Aspects of the Nonlinear Theory, Computer Methods in Applied Mechanics and Engineering, 79, 21–70.CrossRefMATHMathSciNetGoogle Scholar
  43. Spence, A., Jepson A. D. (1985): Folds in the Solution of Two Parameter Systems and their Calculation. Part I, SIAM J. Numer. Anal., 22, 347–368.CrossRefMATHMathSciNetGoogle Scholar
  44. Timoshenko S.P., and J.M. Gere, (1961): Theory of Elastic Stability, Mc-Graw Hill, New York.Google Scholar
  45. Wagner, W., Wriggers, P. (1988): A Simple Method for the Calculation of Post-critical Branches, Engineering Computations, 5, 103–109.CrossRefGoogle Scholar
  46. Wagner, W. (1991): Zur Behandlung von Stabilitätsproblemen der Elastostatik mit der Methode der Finiten Elemente, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover, F91 /1.Google Scholar
  47. Weber, H. (1981): On the Numerical Approximation of Secondary Bifurcation Problems, in: Numerical Solution of Nonlinear Equations, ed. Allgower, Glashoff, Peitgen, Lecture Notes in Mathematics 878, Springer, Berlin, 407–425.CrossRefGoogle Scholar
  48. Werner, B., Spence, A. (1984): The Computation of Symmetry-Breaking Bifurcation Points, SIAM J. Num. Anal., 21, 388–399.CrossRefMATHMathSciNetGoogle Scholar
  49. Werner, B. (1984): Regular Systems for Bifurcation Points with Underlying Symmetries, in: Bifurcation Problems and their Numerical Solution, ISNM 70, ed. Mittelmann, Weber, Birkhäuser, Basel, Boston, Stuttgart, 562–574.Google Scholar
  50. Weinitschke, H. J. (1985): On the Calculation of Limit and Bifurcation Points in stability Problems of Elastic Shells, Int. J. Solids Structures, 21, 79–95.CrossRefMATHMathSciNetGoogle Scholar
  51. Willems J.L. Stability theory of dynamical systems Nelson 1970Google Scholar
  52. Wriggers, P.; Wagner, W.; Miehe, C. (1988): A Quadratically Convergent Procedure for the Calculation of Stability Points in Finite Element Analysis. Comp. Meth. Appl. Mech. Engng., 70, 329–347.CrossRefMATHGoogle Scholar
  53. Wriggers, P., Simo, J. C. (1990): A General Procedure for the Direct Computation of Turning and Bifurcation Points, Int. J. Num. Meth. Engng., 30, 155–176.CrossRefMATHGoogle Scholar
  54. Zienkiewicz, O. C., Taylor, R. L. (1989): The Finite Element Method, Vol. 1, Mc Graw-Hill,London.Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • P. Wriggers
    • 1
  1. 1.T.H. DarmstadtDarmstadtGermany

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