TY - JOUR
T1 - Lower and upper bound limit analysis via the alternating direction method of multipliers
AU - Vicente da Silva, Mário
AU - Deusdado, Nuno
AU - Antão, Armando N.
PY - 2020/8
Y1 - 2020/8
N2 - 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.
AB - 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.
KW - Alternating direction method of multipliers
KW - Augmented Lagrangian
KW - Limit analysis
KW - Lower bound
KW - Nonsmooth optimization
KW - Upper bound
UR - http://www.scopus.com/inward/record.url?scp=85085974502&partnerID=8YFLogxK
U2 - 10.1016/j.compgeo.2020.103571
DO - 10.1016/j.compgeo.2020.103571
M3 - Article
AN - SCOPUS:85085974502
SN - 0266-352X
VL - 124
JO - Computers and Geotechnics
JF - Computers and Geotechnics
M1 - 103571
ER -