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**Enhanced solution control for physically and geometrically non-linear problems. I: The subplane control approach. II: Comparative performance analysis.**
*(English)*
Zbl 0957.74034

From the summary: Geometrically or physically nonlinear problems are often characterized by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to inefficiency of standard numerical solution techniques. Special numerical procedures are therefore required to pass critical points. This paper presents a solution technique which is based on a constraint equation that is defined on a subplane of the degree-of-freedom hyperspace or a hyperspace constructed from specific functions of the degrees of freedom. This unified approach includes many existing methods which have been proposed by various authors. Part I fully elaborates the proposed solution strategy, including a fully automatic load control, i.e. load estimation, adaptation and correction. Part II presents a comparative analysis in which several choices for the control function in the subplane method are confronted with classical update algorithms.

### MSC:

74M05 | Control, switches and devices (“smart materials”) in solid mechanics |

74K99 | Thin bodies, structures |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

### Keywords:

subplane of degree-of-freedom hyperspace; local subplane method; weighted subplane method; path following technique; arc-length control; critical points; constraint equation; fully automatic load control; load estimation
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\textit{M. G. D. Geers}, Int. J. Numer. Methods Eng. 46, No. 2, 177--230 (1999; Zbl 0957.74034)

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### References:

[1] | Solution algorithms for nonlinear structural problems. In International Conference on Engineering Applications of the F.E. Method, Norway, A. S. Computas, 1979. |

[2] | Wright, Journal of Structural Division pp 1143– (1968) |

[3] | Continua and discontinua. In Proceedings of the 1st Conference on Matrix Methods in Structural Mechanics, Wright-Patterson A.F.B. OH, 1965; 11-189. |

[4] | Batoz, International Journal for Numerical Methods in Engineering 14 pp 1262– (1979) |

[5] | Riks, Journal of Applied Mechanics 39 pp 1060– (1972) · Zbl 0254.73047 |

[6] | Wempner, International Journal for Numerical Methods in Engineering 7 pp 1581– (1971) |

[7] | Crisfield, Computers and Structures 13 pp 55– (1981) |

[8] | Strategies for tracing the nonlinear response near limit points. In Nonlinear Finite Element Analysis in Structural Mechanics, et al. (eds). Springer: Berlin, 1981; 63-89. |

[9] | Bellini, Computers and Structures 26 pp 99– (1987) |

[10] | Hellweg, Computers and Structures 66 pp 705– (1998) |

[11] | Kuo, International Journal for Numerical Methods in Engineering 38 pp 4053– (1995) |

[12] | Qi, Applied Mathematics and Mechanics 16 pp 851– (1995) |

[13] | Forde, Computers and Structures 27 pp 625– (1987) |

[14] | Schweizerhof, Computer Methods in Applied Mechanics and Engineering 59 pp 261– (1986) |

[15] | Bathe, Computers and Structures 17 pp 871– (1983) |

[16] | Chen, International Journal for Numerical Methods in Engineering 36 pp 909– (1993) |

[17] | Widjaja, Computers and Structures 66 pp 201– (1998) |

[18] | Clarke, International Journal for Numerical Methods in Engineering 29 pp 1365– (1990) |

[19] | Crisfield, International Journal for Numerical Methods in Engineering 19 pp 1269– (1983) |

[20] | Schweizerhof, Communications in Applied Numerical Methods in Engineering 9 pp 773– (1993) |

[21] | Carrera, Computers and Structures 50 pp 217– (1994) · Zbl 0800.73522 |

[22] | Powell, International Journal for Numerical Methods in Engineering 17 pp 1455– (1981) |

[23] | Automated incremental-iterative solution methods in structural mechanics. In Recent Advances in Non-Linear Computational Mechanics, et al. (eds). chapter 2. Pineridge Press: Swansea, U.K., 1982; 41-62. |

[24] | A self-adaptive load estimator based on strain energy. In Computational Plasticity, Fundamentals and Applications, (eds). CIMNE, Barcelona, Pineridge Press: Swansea, U.K., April 1992; 187-198. |

[25] | Bergan, International Journal for Numerical Methods in Engineering 12 pp 1677– (1978) |

[26] | Non-linear analysis of frictional materials. Ph.D. Thesis, Delft University of Technology, The Netherlands, 1986. |

[27] | Chen, Computers and Structures 37 pp 1043– (1990) |

[28] | Chen, Computer Methods in Applied Mechanics and Engineering 90 pp 869– (1991) |

[29] | Napoleão, Computers and Structures 42 pp 833– (1992) |

[30] | Wriggers, Computer Methods in Applied Mechanics and Engineering 70 pp 329– (1988) |

[31] | Wagner, Engineering Computations 5 pp 103– (1988) |

[32] | Wriggers, International Journal for Numerical Methods in Engineering 30 pp 155– (1990) |

[33] | Fujii, Engineering Structures 19 pp 385– (1997) |

[34] | Magnusson, International Journal for Numerical Methods in Engineering 41 pp 955– (1998) |

[35] | Riks, Computer Methods in Applied Mechanics and Engineering 136 pp 59– (1996) |

[36] | Applications of adapted, nonlinear solution strategies. In Advances in Finite Element Technology, (ed.). CIMNE: Barcelona, 1995; 212-236. |

[37] | Bergan, Computers and Structures 12 pp 497– (1980) |

[38] | Meek, Computer Methods in Applied Mechanics and Engineering 47 pp 261– (1984) |

[39] | Peerlings, International Journal for Numerical Methods in Engineering 39 pp 3391– (1996) |

[40] | Geers, Computer Methods in Applied Mechanics and Engineering 160 pp 133– (1998) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.