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**Multi-grid solutions to the elastic plastic torsion problem in multiply connected domains.**
*(English)*
Zbl 0703.73091

Summary: The elastic plastic torsion problem for an elastic, perfectly plastic cylinder with multiply connected cross section twisted around its longitudinal axis is formulated as an obstacle problem for an associated stress potential, the obstacle being defined in terms of a generalized distance function. Based upon the reformulation of the obstacle problem as an equivalent linear complementarity problem, the latter is discretized by means of finite difference techniques, and a monotonically convergent iterative scheme for its numerical solution is developed. At each step of the iteration the solution of a reduced system of discrete Poisson equations is required which is done by applying multi-grid techniques with respect to a hierarchy of grid-point sets. Combined with a suitably chosen nested iteration process this results in a computationally very efficient algorithm for the approximate solution of the elastic plastic torsion problem.

### MSC:

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

74P10 | Optimization of other properties in solid mechanics |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

74S20 | Finite difference methods applied to problems in solid mechanics |

65F10 | Iterative numerical methods for linear systems |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

49M25 | Discrete approximations in optimal control |

### Keywords:

elastic, perfectly plastic cylinder; obstacle problem; generalized distance function; linear complementarity problem; discretized; finite difference techniques; monotonically convergent iterative scheme; reduced system of discrete Poisson equations; hierarchy of grid-point sets; nested iteration process
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\textit{R. H. W. Hoppe}, Int. J. Numer. Methods Eng. 26, No. 3, 631--646 (1988; Zbl 0703.73091)

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