Computing limit loads by minimizing a sum of norms

Knud D. Andersen, Edmund Christiansen, Michael L. Overton

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1046-1062
Number of pages17
JournalSIAM Journal on Scientific Computing
Volume19
Issue number3
StatePublished - May 1998

Fingerprint

Convex optimization
Load limits
Plastic flow
Kinematics
Norm
Finite element method
Limit Analysis
Computing
Duality Gap
Ill-conditioning
Divergence-free
Plane Strain
Term
Linear Constraints
Convex Optimization
Efficient Solution
Minimization Problem
Test Problems
Plastics
Vanish

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: SIAM Journal on Scientific Computing, Vol. 19, No. 3, 05.1998, p. 1046-1062.

Research output: Contribution to journalArticle

Andersen, KD, Christiansen, E & Overton, ML 1998, 'Computing limit loads by minimizing a sum of norms', SIAM Journal on Scientific Computing, vol. 19, no. 3, pp. 1046-1062.
Andersen, Knud D. ; Christiansen, Edmund ; Overton, Michael L. / Computing limit loads by minimizing a sum of norms. In: SIAM Journal on Scientific Computing. 1998 ; Vol. 19, No. 3. pp. 1046-1062.
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