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On path-following methods for structural failure problems. (English) Zbl 1398.74407
Summary: We revisit the consistently linearized path-following method that can be applied in the nonlinear finite element analysis of solids and structures in order to compute a solution path. Within this framework, two constraint equations are considered: a quadratic one (that includes as special cases popular spherical and cylindrical forms of constraint equation), and another one that constrains only one degree-of-freedom (DOF), the critical DOF. In both cases, the constrained DOFs may vary from one solution increment to another. The former constraint equation is successful in analysing geometrically nonlinear and/or standard inelastic problems with snap-throughs, snap-backs and bifurcation points. However, it cannot handle problems with the material softening that are computed e.g. by the embedded-discontinuity finite elements. This kind of problems can be solved by using the latter constraint equation. The plusses and minuses of the both presented constraint equations are discussed and illustrated on a set of numerical examples. Some of the examples also include direct computation of critical points and branch switching. The direct computation of the critical points is performed in the framework of the path-following method by using yet another constraint function, which is eigenvector-free and suited to detect critical points.

74S05 Finite element methods applied to problems in solid mechanics
74G60 Bifurcation and buckling
74S30 Other numerical methods in solid mechanics (MSC2010)
74K25 Shells
74K99 Thin bodies, structures
Full Text: DOI
[1] Alfano, G; Crisfield, MA, Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues, Int J Numer Methods Eng, 50, 1701-1736, (2001) · Zbl 1011.74066
[2] Alfano, G; Crisfield, MA, Solution strategies for the delamination analysis based on a combination of local-control arc-length and line searches, Int J Numer Methods Eng, 58, 999-1048, (2003) · Zbl 1032.74660
[3] Armero, F; Ehrlich, D, Numerical modeling of softening hinges in thin eulerbernoulli beams, Comput Struct, 84, 641-656, (2006)
[4] Bathe, K-J; Dvorkin, E, A four-node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation, Int J Numer Methods Eng, 21, 367-383, (1985) · Zbl 0551.73072
[5] Betsch, P; Menzel, A; Stein, E, On the parametrization of finite rotations in computational mechanics: a classification of concepts with application to smooth shells, Comput Methods Appl Mech Eng, 155, 273-305, (1998) · Zbl 0947.74060
[6] Brank, B; Perić, D; Damjanić, F, On large deformations of thin elasto-plastic shells: implementation of a finite rotation model for quadrilateral shell element, Int J Numer Methods Eng, 40, 689-726, (1997) · Zbl 0892.73055
[7] Brank, B; Carrera, E, A family of shear-deformable shell finite elements for composite structures, Comput Struct, 76, 287-297, (2000)
[8] Brank, B; Ibrahimbegović, A, On the relation between different parametrizations of finite rotations for shells, Eng Comput, 18, 950-973, (2001) · Zbl 1017.74069
[9] Brank, B; Korelc, J; Ibrahimbegović, A, Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation, Comput Struct, 80, 699-717, (2002)
[10] Brank, B, Nonlinear shell models with seven kinematic parameters, Comput Methods Appl Mech Eng, 194, 2336-2362, (2005) · Zbl 1082.74050
[11] Brank, B, Assessment of 4-node EAS-ANS shell elements for large deformation analysis, Comput Mech, 42, 39-51, (2008) · Zbl 1161.74049
[12] Carrera, E, A study on arc-length-type methods and their operation failures illustrated by a simple model, Comput Struct, 50, 217-229, (1994) · Zbl 0800.73522
[13] Caseiro, JF; Alves de Sousa, RJ; Valente, RAF, A systematic development of EAS three-dimensional finite elements for the alleviation of locking phenomena, Finite Elem Anal Des, 73, 30-40, (2013)
[14] Crisfield, MA, A fast incremental/iterative solution procedure that handles ‘snap-through’, Comput Struct, 13, 55-62, (1981) · Zbl 0479.73031
[15] Crisfield MA (1991) Non-linear finite element analysis of solids and structures, vol. 1: essentials. Wiley, Chichester · Zbl 0809.73005
[16] Crisfield, MA; Peng, X, Instabilities induced by coarse meshes for a nonlinear shell problem, Eng Comput, 13, 110-114, (1996) · Zbl 0983.74575
[17] Crisfield MA (1997) Non-linear finite element analysis of solids and structures, vol. 2: advanced topics. Wiley, Chichester · Zbl 0890.73001
[18] Borst, R, Computation of post-bifurcation and post-failure behavior of strain-softening solids, Comput Struct, 25, 211-224, (1987) · Zbl 0603.73046
[19] Souza Neto, EA; Feng, YT, On the determination of the path direction for arc-length methods in the presence of bifurcations and “snap-backs”, Comput Methods Appl Mech Eng, 179, 81-89, (1999) · Zbl 1003.74025
[20] Dujc, J; Brank, B; Ibrahimbegovic, A, Multi-scale computational model for failure analysis of metal frames that includes softening and local buckling, Comput Methods Appl Mech Eng, 199, 1371-1385, (2010) · Zbl 1227.74067
[21] Dujc, J; Brank, B; Ibrahimbegovic, A, Quadrilateral finite element with embedded strong discontinuity for failure analysis of solids, Comput Model Eng Sci, 69, 223-259, (2010) · Zbl 1231.74378
[22] Dujc, J; Brank, B, Stress resultant plasticity for shells revisited, Comput Methods Appl Mech Eng, 247, 146-165, (2012) · Zbl 1352.74172
[23] Dujc, J; Brank, B; Ibrahimbegovic, A, Stress-hybrid quadrilateral finite element with embedded strong discontinuity for failure analysis of plane stress solids, Int J Numer Methods Eng, 94, 1075-1098, (2013) · Zbl 1231.74378
[24] Eriksson, A, On linear constraints for Newton-raphson corrections and critical point searches in structural F.E. problems, Int J Numer Methods Eng, 28, 1317-1334, (1989) · Zbl 0711.73224
[25] Eriksson, A, Structural instability analyses based on generalised path-following, Comput Methods Appl Mech Eng, 179, 265-305, (1998) · Zbl 0960.74047
[26] Eriksson, A; Pacoste, C; Zdunek, A, Numerical analysis of complex instability behaviour using incremental-iterative strategies, Comput Methods Appl Mech Eng, 179, 265-305, (1999) · Zbl 0967.74062
[27] Feng, YT; Perić, D; Owen, DRJ, Determination of travel directions in path-following methods, Math Comput Model, 21, 43-59, (1995) · Zbl 0819.73073
[28] Feng, YT; Perić, D; Owen, DRJ, A new criterion for determination of initial loading parameter in arc-length methods, Comput Struct, 58, 479-485, (1996) · Zbl 0900.73950
[29] Geers, MGD, Enhanced solution control for physically and geometrically non-linear problems. part I-the subplane control approach, Int J Numer Methods Eng, 46, 177-204, (1999) · Zbl 0957.74034
[30] Geers, MGD, Enhanced solution control for physically and geometrically non-linear problems. part II-comparative performance analysis, Int J Numer Methods Eng, 46, 205-230, (1999) · Zbl 0957.74034
[31] Thai, H-T; Kim, S-E, Large deflection inelastic analysis of space trusses using generalized displacement control method, J Constr Steel Res, 65, 1987-1994, (2009)
[32] Ibrahimbegović, A; Wilson, EL, A modified method of incompatible modes, Commun Appl Numer Methods, 7, 187-194, (1991) · Zbl 0723.73097
[33] Ibrahimbegović, A; Taylor, RL, On the role of frame-invariance in structural mechanics models at finite rotations, Comput Methods Appl Mech Eng, 191, 5159-5176, (2002) · Zbl 1023.74048
[34] Ibrahimbegovic A (2009) Nonlinear solid mechanics. Theoretical formulations and finite element solution methods. Springer, New York · Zbl 1168.74002
[35] Ibrahimbegovic, A; Brank, B; Courtois, P, Stress resultant geometrically exact form of classical shell model and vector-like parametrization of constrained finite rotations, Int J Numer Methods Eng, 52, 1235-1252, (2001) · Zbl 1112.74420
[36] Jelenić, G; Saje, M, A kinematically exact space finite strain beam model: finite element formulation by generalized virtual work principle, Comput Methods Appl Mech Eng, 120, 131-161, (1995) · Zbl 0852.73062
[37] Jelenić, G; Crisfield, MA, Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics, Comput Methods Appl Mech Eng, 171, 141-171, (1999) · Zbl 0962.74060
[38] Jun, SM; Hong, CS, Buckling behavior of laminated composite cylindrical panels under axial compression, Comput Struct, 29, 479-490, (1988) · Zbl 0636.73031
[39] Jukić, M; Brank, B; Ibrahimbegović, A, Embedded discontinuity finite element formulation for failure analysis of planar reinforced concrete beams and frames, Eng Struct, 50, 115-125, (2013)
[40] Jukić, M; Brank, B; Ibrahimbegović, A, Failure analysis of reinforced concrete frames by beam finite element that combines damage, plasticity and embedded discontinuity, Eng Struct, 75, 507-527, (2014)
[41] Kegl, M; Brank, B; Harl, B; Oblak, M, Efficient handling of stability problems in shell optimization by asymmetric ‘worst-case’ shape imperfection, Int J Numer Methods Eng, 73, 1197-1216, (2008) · Zbl 1159.74028
[42] Korelc, J, Direct computation of critical points based on crout’s elimination and diagonal subset test function, Comput Struct, 88, 189-197, (2010)
[43] Korelc J (2014) AceGen manual, AceFEM manual. Available at http://www.fgg.uni-lj.si/symech/
[44] Kouhia, R; Mikkola, M, Tracing the equilibrium path beyond compound critical points, Int J Numer Methods Eng, 46, 1049-1074, (1999) · Zbl 0967.74065
[45] Lopez, S, Post-critical analysis of structures with a nonlinear pre-buckling state in the presence of imperfections, Comput Methods Appl Mech Eng, 191, 4421-4440, (2002) · Zbl 1042.74017
[46] Manzoli, OL; Shing, PB, A general technique to embed non-uniform discontinuities into standard solid finite elements, Comput Struct, 84, 742-757, (2006)
[47] Noguchi, H; Fujii, F, Eigenvector-free indicator, pinpointing and branch-switching for bifurcation, Commun Numer Methods Eng, 19, 445-457, (2003) · Zbl 1154.74342
[48] Ohsaki M, Ikeda K (2007) Stability and optimization of structures. Generalized sensitivity analysis. Springer, New York · Zbl 1127.74002
[49] Junior EP, de Holanda AS, da Silva SMBA (2006) Tracing nonlinear equilibrium paths of structures subjected to thermal loading. Comput Mech 38(6):505-520 · Zbl 1168.74397
[50] Pian, T; Sumihara, K, Rational approach for assumed stress finite elements, Int J Numer Methods Eng, 20, 1685-1695, (1985) · Zbl 0544.73095
[51] Piculin, S; Brank, B, Weak coupling of shell and beam computational models for failure analysis of steel frames, Finite Elem Anal Des, 97, 20-42, (2015)
[52] Pohl, T; Ramm, E; Bischoff, M, Adaptive path following schemes for problems with softening, Finite Elem Anal Des, 86, 12-22, (2014)
[53] Ramm, E; Wunderlich, W (ed.); Stein, E (ed.); Bathe, KJ (ed.), Strategies for tracing nonlinear response near limit points, 63-89, (1981), New York
[54] Rheinboldt WC (1986) Numerical analysis of parametrized nonlinear equations. Wiley, New York · Zbl 0582.65042
[55] Rigobello, R; Coda, HB; Neto, JM, Inelastic analysis of steel frames with a solid-like finite element, J Constr Steel Res, 86, 140-152, (2013)
[56] Riks, E, An incremental approach to the solution of snapping and buckling problems, Int J Solids Struct, 15, 529-551, (1979) · Zbl 0408.73040
[57] Ritto-Correa, M; Camotim, D, On the arc-length and other quadratic control methods: established, less known and new implementation procedures, Comput Struct, 86, 1353-1368, (2008)
[58] Sabir, AB; Lock, AC; Brebbia, CA (ed.); Tottenham, H (ed.), The application of finite elements to the large-deflection geometrically nonlinear behaviour of cylindrical shells, (1972), Southampton
[59] Schweizerhof, KH; Wriggers, P, Consistent linearization for path following methods in nonlinear FE analysis, Comput Methods Appl Mech Eng, 59, 261-279, (1986) · Zbl 0588.73138
[60] Simo, JC; Fox, DD; Rifai, MS, On a stress resultant geometrically exact shell model. part III: computational aspects of the nonlinear theory, Comput Methods Appl Mech Eng, 79, 21-70, (1990) · Zbl 0746.73015
[61] Simo, JC; Vu-Quoc, L, A three-dimensional finite-strain rod model. part II: computational aspects, Comput Methods Appl Mech Eng, 58, 79-116, (1986) · Zbl 0608.73070
[62] Simo, JC; Armero, F, Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Int J Numer Methods Eng, 33, 1413-1449, (1992) · Zbl 0768.73082
[63] Sze, KY; Liu, XH; Lo, SH, Popular benchmark problems for geometric nonlinear analysis of shells, Finite Elem Anal Des, 40, 1551-1569, (2004) · Zbl 1068.20065
[64] Verhoosel, CV; Remmers, JJC; Gutierrez, MA, A dissipation-based arc-length method for robust simulation of brittle and ductile failure, Int J Numer Methods Eng, 77, 1290-1321, (2008) · Zbl 1156.74397
[65] Zhou, Y; Stanciulescu, I; Eason, T; Spottswood, M, Nonlinear elastic buckling and postbuckling analysis of cylindrical panels, Finite Elem Anal Des, 96, 41-50, (2015)
[66] Wagner, W; Wriggers, P, A simple method for the calculation of postcritical branches, Eng Comput, 5, 103-109, (1988)
[67] Wriggers, P; Simo, JC, A general procedure for the direct computation of turning and bifurcation points, Int J Numer Methods Eng, 30, 155-167, (1990) · Zbl 0728.73069
[68] Wriggers P (2008) Nonlinear finite element methods. Springer, New York · Zbl 1153.74001
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