Stress-based upper-bound method and convex optimization: case of the Gurson material

Loute

Numerical Meth Engineering

2008

Self Cite

International audienceThis paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776 - value to be compared with the best published lower bound 3.772 - by succeeding in solving non-linear optimization problems with millions of variables and constraints

Section: Implementing Velocity Discontinuitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Limit analysis and convex programming: A decomposition approach of the kinematic mixed method

Loute

Numerical Meth Engineering

2008

Self Cite

“…After defining a frame (n 0 , t 1 , t 2 ) where n 0 is the normal to the discontinuity triangle, for each vertex of the discontinuity side, relation (22) reads…”

Section: Implementation Of the Ka Conditionsmentioning

confidence: 99%

“…Another extension to discontinuous plane strain velocity fields was also given in , on the basis of the LP duality properties assumed to remain valid in the nonlinear case. A more general mixed formulation, based on convexity properties to obtain strict upper bounds, was proposed in and detailed in . This original mixed formulation allowed to solve very large scale problems, also in the plane strain case, see .…”

Section: Introductionmentioning

confidence: 99%

3D‐FEM formulations of limit analysis methods for porous pressure‐sensitive materials

Pastor¹,

Kondo

Numerical Meth Engineering

2013

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SUMMARYThe first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure‐sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker–Prager, Mises–Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non‐classic 3D problems, that is, the Gurson‐like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker–Prager, Mises–Schleicher, and Green materials. Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng

“…Another important topic in limit analysis is the development of FEs that produce strict upper or lower bounds; see, for example, Sloan and Kleeman , Pastor et al , Pastor et al , Makrodimopoulos and Martin , and Muñoz et al . To address this topic, in the set of interpolations considered in this work, we include one FE that produces true upper bounds and also one that produces true lower bounds.…”

Section: Introductionmentioning

confidence: 99%

Quadratic velocity-linear stress interpolations in limit analysis

Zouain

Borges

Silveira

2014

We compare a family of mixed interpolations on triangles with straight edges as applied to limit analysis. The aim of this paper is to prove theoretically that the approximate collapse factors obtained with these finite elements always comply with certain inequalities that exist among them. Two of these interpolations are used in limit analysis for the first time in this article. The inequalities in the proposition are also demonstrated via numerical applications. To this end, the most frequently used benchmark problems in limit analysis are revisited and discussed with respect to the results computed with these finite elements.