On the solution of non-linear finite element equations

Abstract The paper deals with the basic requirements in the construction of a reliable continuation procedure. Adaptive step length determination and calculation of critical equilibrium states are discussed. For simple critical points an algorithm, which does not need classification between different types of bifurcations or even distinction between limit vs bifurcation point, is described. Situations where the extension of the parameter space could reveal vital information concerning the behaviour of the structure being analysed, are addressed.

[1]  R. Borst.  Computation of post-bifurcation and post-failure behavior of strain-softening solids , 1987 .

[2]  W. Rheinboldt.  Numerical methods for a class of nite dimensional bifur-cation problems , 1978 .

[3]  Taweep Chaisomphob,et al.  AN AUTOMATIC ARC LENGTH CONTROL ALGORITHM FOR TRACING EQUILIBRIUM PATHS OF NONLINEAR STRUCTURES , 1988 .

[4]  J. J. Moré,et al.  Quasi-Newton Methods, Motivation and Theory , 1974 .

[5]  Peter Wriggers,et al.  Consistent linearization for path following methods in nonlinear FE analysis , 1986 .

[6]  E. Riks.  An incremental approach to the solution of snapping and buckling problems , 1979 .

[7]  J. V. Huddleston.  Finite Deflections and Snap-Through of High Circular Arches , 1968 .

[8]  M. Crisfield.  A FAST INCREMENTAL/ITERATIVE SOLUTION PROCEDURE THAT HANDLES "SNAP-THROUGH" , 1981 .

[9]  W. Rheinboldt.  Numerical analysis of parametrized nonlinear equations , 1986 .

[10]  Bernard Budiansky,et al.  Theory of buckling and post-buckling behavior of elastic structures , 1974 .

[11]  K. Bathe,et al.  ON THE AUTOMATIC SOLUTION OF NONLINEAR FINITE ELEMENT EQUATIONS , 1983 .

[12]  Leszek Demkowicz,et al.  Variational approach to sensitivity analysis in nonlinear elasticity , 1987 .

[13]  A NON-ITERATIVE NONLINEAR ANALYSIS SCHEME OF FRAMES WITH THIN-WALLED ELASTIC MEMBERS , 1987 .

[14]  Siegfried F. Stiemer,et al.  Improved arc length orthogonality methods for nonlinear finite element analysis , 1987 .

[15]  A. H. Chilver,et al.  Multiple path generation at coincident branching points , 1971 .

[16]  G. Strang,et al.  The solution of nonlinear finite element equations , 1979 .

[17]  P. Bergan,et al.  Solution techniques for non−linear finite element problems , 1978 .

[18]  Isaac Fried,et al.  Orthogonal trajectory accession to the nonlinear equilibrium curve , 1984 .

[19]  M. Crisfield,et al.  A faster modified newton-raphson iteration , 1979 .

[20]  Markku Tuomala,et al.  Static and dynamic analysis of space frames using simple Timoshenko type elements , 1993 .

[21]  J. L. Meek,et al.  Geometrically nonlinear analysis of space frames by an incremental iterative technique , 1984 .

[22]  G. Wempner.  Discrete approximations related to nonlinear theories of solids , 1971 .

[23]  Gouri Dhatt,et al.  Incremental displacement algorithms for nonlinear problems , 1979 .

[24]  Giles W Hunt,et al.  A general theory of elastic stability , 1973 .

[25]  E. Riks.  The Application of Newton's Method to the Problem of Elastic Stability , 1972 .

[26]  M. J. Sewell.  A general theory of equilibrium paths through critical points. I , 1968, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[27]  Anders Eriksson,et al.  On linear constraints for Newton–Raphson corrections and critical point searches in structural F.E. problems , 1989 .

[28]  W. T. Koiter.  Current Trends in the Theory of Buckling , 1976 .

[29]  W. Rheinboldt.  On the computation of multi-dimensional solution manifolds of parametrized equations , 1988 .

[30]  C. Haselgrove,et al.  The Solution of Non-Linear Equations and of Differential Equations with Two-Point Boundary Conditions , 1961, Comput. J..

[31]  E. Allgower,et al.  An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold , 1985 .

[32]  Kuang-Han Chu,et al.  Large Deflection Buckling of Space Frames , 1972 .

[33]  Achintya Haldar,et al.  Elastoplastic nonlinear post-buckling analysis of partially restrained space structures , 1988 .

[34]  A. Eriksson.  USING EIGENVECTOR PROJECTIONS TO IMPROVE CONVERGENCE IN NON-LINEAR FINITE ELEMENT EQUILIBRIUM ITERATIONS. , 1987 .

[35]  Ekkehard Ramm,et al.  Strategies for Tracing the Nonlinear Response Near Limit Points , 1981 .

[36]  E. Riks.  Some computational aspects of the stability analysis of nonlinear structures , 1984 .

[37]  M. A. Crisfield,et al.  NEW SOLUTION PROCEDURES FOR LINEAR AND NON-LINEAR FINITE ELEMENT ANALYSIS , 1985 .