### Abstract

This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the von Mises condition. After discretization with the finite element method, using divergence-free elements for the plastic flow, the kinematic formulation reduces to the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. Recently an efficient solution algorithm has been developed for this particular convex optimization problem in large sparse form. The approach is applied to test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80% of the terms in the objective function are zero in the optimal solution, causing extreme ill conditioning. In the second case all terms are nonzero. In both cases the method works very well, and problems are solved which are larger by at least an order of magnitude than previously reported. The relative accuracy for the solution of the discrete problems, measured by duality gap and feasibility, is typically of the order 10^{-8}.

Original language | English (US) |
---|---|

Pages (from-to) | 1046-1062 |

Number of pages | 17 |

Journal | SIAM Journal on Scientific Computing |

Volume | 19 |

Issue number | 3 |

State | Published - May 1998 |

### Fingerprint

### Keywords

- Finite element method
- Limit analysis
- Nonsmooth optimization
- Plasticity

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*19*(3), 1046-1062.

**Computing limit loads by minimizing a sum of norms.** / Andersen, Knud D.; Christiansen, Edmund; Overton, Michael L.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 19, no. 3, pp. 1046-1062.

}

TY - JOUR

T1 - Computing limit loads by minimizing a sum of norms

AU - Andersen, Knud D.

AU - Christiansen, Edmund

AU - Overton, Michael L.

PY - 1998/5

Y1 - 1998/5

N2 - This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the von Mises condition. After discretization with the finite element method, using divergence-free elements for the plastic flow, the kinematic formulation reduces to the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. Recently an efficient solution algorithm has been developed for this particular convex optimization problem in large sparse form. The approach is applied to test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80% of the terms in the objective function are zero in the optimal solution, causing extreme ill conditioning. In the second case all terms are nonzero. In both cases the method works very well, and problems are solved which are larger by at least an order of magnitude than previously reported. The relative accuracy for the solution of the discrete problems, measured by duality gap and feasibility, is typically of the order 10-8.

AB - This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the von Mises condition. After discretization with the finite element method, using divergence-free elements for the plastic flow, the kinematic formulation reduces to the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. Recently an efficient solution algorithm has been developed for this particular convex optimization problem in large sparse form. The approach is applied to test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80% of the terms in the objective function are zero in the optimal solution, causing extreme ill conditioning. In the second case all terms are nonzero. In both cases the method works very well, and problems are solved which are larger by at least an order of magnitude than previously reported. The relative accuracy for the solution of the discrete problems, measured by duality gap and feasibility, is typically of the order 10-8.

KW - Finite element method

KW - Limit analysis

KW - Nonsmooth optimization

KW - Plasticity

UR - http://www.scopus.com/inward/record.url?scp=0001132110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001132110&partnerID=8YFLogxK

M3 - Article

VL - 19

SP - 1046

EP - 1062

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 3

ER -