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

Silva, M. Vicente da; Deusdado, N.; Antão, A. N.

doi:10.1016/j.compgeo.2020.103571

Cited by 12 publications

(3 citation statements)

References 27 publications

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“…Many scholars have done much work in this regard. For example, the two-stage quasi-Newton algorithm and alternating direction method of multipliers are effective methods for solving high degree of freedom nonlinear programming problems [16,17].…”

Section: Introductionmentioning

confidence: 99%

Bearing Capacity of Karst Cave Roof under Pile Foundation Load Using Limit Analysis

Liu

et al. 2023

Applied Sciences

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Reducing the computational workload by simplifying the analysis of karst foundations into a plane strain problem can yield significant advantages. Yet, such an approach fails in reproducing the engineering situation in a rigorous manner. In this regard, this paper proposes an upper-bound method that can effectively analyze the bearing capacity of three-dimensional karst foundations. This method is utilized to investigate the impact of various pile diameters, the ratio of roof thickness to pile diameter, and the ratio of cave width to pile diameter on the stability of karst foundations. The validity of this method is established through an illustrative example. The outcomes illustrate that when subjected to both tensile and compressive horizontal stresses if the maximum horizontal stress surpasses the tensile strength of the rock mass, the roof rock mass may suffer damage. Increasing the ratio of roof thickness to pile diameter can bring down the horizontal stress value. The stability factor is largely influenced by the ratio of roof thickness to pile diameter. The most prominent growth trend of the stability factor is observed when the ratio is less than 3. If the ratio of the roof thickness to pile diameter exceeds 3, the prediction of the bearing capacity estimation for the karst foundation in three-dimensional circumstances is more conservative than that in two-dimensional circumstances.

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Section: Introductionmentioning

confidence: 99%

Bearing Capacity of Karst Cave Roof under Pile Foundation Load Using Limit Analysis

Liu

et al. 2023

Applied Sciences

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“…However, as the displacement jumps vary quadratically, additional constraints 17,18 had to be applied in order to ensure that the flow rule is satisfied along the discontinuities. Either with or without discontinuities, the quadratic element has replaced the linear one in the most recent rigorous upper bound procedures 20,21 …”

Section: Introductionmentioning

confidence: 99%

A class of strain‐displacement elements in upper bound limit analysis

Makrodimopoulos

2022

Numerical Meth Engineering

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The main restriction in applying the upper bound theorem in limit analysis by means of the finite element method is that once the displacement field has been discretized, the flow rule and the boundary conditions should be satisfied everywhere throughout the discretized structure. This is to secure mathematically that the result will be a rigorous upper bound of the exact limit load multiplier. So far, only the linear and the quadratic displacement elements with straight sides, which result in constant and linear strains, respectively, satisfy this requirement.In this article, it is proven both theoretically and practically that there is a general class of strain-displacement triangular elements with straight sides which provide rigorous upper bounds. The interpolation scheme is based on the Bernstein polynomials, and there is no upper restriction of the polynomial order. Both continuous and discontinuous displacements can be considered. The efficiency is examined through plane strain and Kirchhoff plate examples. The generalization to 3D (i.e., tetrahedral elements with plane faces) and other structural conditions is straightforward.

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“…There were various categories of computational approaches: those of large-scale linear programming [7,8], general convex nonlinear programming [6,9,10] and interior point algorithms adapted to the special form of the von Mises criterion [11]. Another recent development is the algorithm of da Silva et al [12] which is based on alternating multipliers. The formulations such as the second-order cone programming [13][14][15][16][17] led to the solution of highly discretized problems, providing tight lower and upper bounds, without the need of preparing algorithms, as it was sufficient to use the existing optimization software.…”

Section: Introductionmentioning

confidence: 99%

A fundamental class of stress elements in lower bound limit analysis

Makrodimopoulos

2020

Proc. R. Soc. A.

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There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.

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

Cited by 12 publications

References 27 publications

Bearing Capacity of Karst Cave Roof under Pile Foundation Load Using Limit Analysis

Bearing Capacity of Karst Cave Roof under Pile Foundation Load Using Limit Analysis

A class of strain‐displacement elements in upper bound limit analysis

A fundamental class of stress elements in lower bound limit analysis

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