Lower and upper bound limit analysis via the alternating direction method of multipliers

Computational limit analysis methods invariably lead to the need to solve a mathematical programming problem. The alternating direction method of multipliers (ADMM) is one versatile and robust technique to solve non-linear convex optimization problems that has recently found applications in a wide r...

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Bibliographic Details
Published inComputers and geotechnics Vol. 124; p. 103571
Main Authors Vicente da Silva, M., Deusdado, N., Antão, A.N.
Format Journal Article
LanguageEnglish
Published New York Elsevier Ltd 01.08.2020
Elsevier BV
Subjects
Algorithms
Alternating direction method of multipliers
Approximation
Augmented Lagrangian
Computational geometry
Computer applications
Convexity
Direction
Limit analysis
Lower bound
Lower bounds
Mathematical analysis
Mathematical programming
Multipliers
Nonsmooth optimization
Optimization
Plastic deformation
Robustness (mathematics)
Strain rate
Stress distribution
Upper bound
Upper bounds
Velocity distribution
Lower bound
Alternating direction method of multipliers
Upper bound
Limit analysis
Nonsmooth optimization
Augmented Lagrangian
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Summary:Computational limit analysis methods invariably lead to the need to solve a mathematical programming problem. The alternating direction method of multipliers (ADMM) is one versatile and robust technique to solve non-linear convex optimization problems that has recently found applications in a wide range of fields. Its solution scheme, based on an operator splitting algorithm, is not only easy to implement but also suitable to efficiently solve large-scale variational problems. Starting from the ADMM framework, we derive a strict upper bound finite element formulation using a two-(primal)-field approximation, one for the velocity field and the other for the plastic strain rate field. Next, following a similar approach, we develop a novel strict lower bound formulation. Here, the two-(primal)-field model is based on a redundant approximation of the stress field. Duality principles are then explored in order to unify these two formulations.The effectiveness of this approach is demonstrated on test problems and, to conclude, some considerations are made about the performance results.
ISSN:0266-352X
1873-7633
DOI:10.1016/j.compgeo.2020.103571