Mesh adaptive computation of upper and lower bounds in limit analysis

Ciria, Héctor Manuel Camué; Peraire, J.; Bonet, Javier

doi:10.1002/nme.2275

Cited by 209 publications

(160 citation statements)

References 31 publications

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“…However, when FEM is applied some of the well-known characteristics of mesh-based methods can lead to problems: the solutions are often highly sensitive to the geometry of the original mesh, particularly in the region of stress or displacement/velocity singularities; furthermore, volumetric locking may occur in plane strain and 3D problems [11]. Although adaptive schemes with the h-version [12][13][14][15][16] or p-version FEM [17,18] have been used to try to overcome such disadvantages, and show immense promise, the schemes quickly become complex and a large number of elements are generally required to obtain accurate solutions. On the other hand, the objective function in the associated optimization problem is convex, but not everywhere differentiable.…”

Section: Introductionmentioning

confidence: 99%

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Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2009) SUMMARYThe meshless Element-Free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation which involves approximating the displacement/velocity field using the moving least squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing directly displacements at the nodes. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using Second-Order Cone Programming (SOCP). The procedure is applied to several benchmark problems and is found in practice to generate good upper bound solutions for benchmark problems.

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Section: Introductionmentioning

confidence: 99%

“…One of the most efficient algorithms to overcome this difficulty is the primal-dual interior-point method presented in [19,20] and implemented in commercial codes such as the Mosek software package. The limit analysis problem involving conic constraints can then be solved by this efficient algorithm [16,21,22].…”

Section: Introductionmentioning

confidence: 99%

Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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“…This problem was introduced to illustrate locking phenomena by Nagtegaal et al [54] and became a popular benchmark test for plastic analysis procedures, particularly for rigid-plastic limit analysis [2,3,5,55,56]. The test problem consists of a rectangular specimen with two external thin symmetric cuts under in-plane tensile stresses τ 0 , as shown in Figure 9.…”

Section: Double Notched Tensile Specimenmentioning

confidence: 99%

“…Various numerical procedures based on the finite element method have been developed to solve real-world problems in engineering practice [1][2][3][4][5]. Owing to their simplicity, low-order finite elements are often used in these procedures.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, most commonly used yield criteria can be cast in the form of conic constraints, and optimization problems involving such constraints can be solved using highly efficient primal-dual interiorpoint solvers [41]. Consequently, several numerical limit analysis procedures which involve the use of cone programming techniques have been reported recently [5,15,[42][43][44].…”

Section: Introductionmentioning

confidence: 99%

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A cell‐based smoothed finite element method for kinematic limit analysis

Nguyen‐Xuan

Askes

et al. 2010

Numerical Meth Engineering

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Published paperLe, Canh V., Nguyen-Xuan, H., Askes, H., Bordas , Stéphane P. A., Rabczuk, T. and Nguyen-Vinh, H. (2010) SUMMARYThis paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method (CS-FEM) with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged.

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Shakedown and Safety Assessment

Zouain

2017

Encyclopedia of Computational Mechanics Second Edition

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The main subjects of this chapter are the variational formulations for the computation of a safety factor in shakedown analysis, the numerical procedures available to implement this analysis, and an overview on the wide field of applications and conceptual extensions. In structural engineering and solid mechanics, the word shakedown , introduced in the 1940s by Prager, became, since then, synonymous with elastic adaptation in the presence of variable loadings, a safe stabilized phenomenon, often due to some limited plastic deformation (or equivalently, due to the residual stress distribution associated with inelastic strains). Concerning the failure analysis of ductile structures, the terms alternating plasticity (AP) and incremental collapse (IC), besides plastic collapse , are also widely used to identify failure modes under variable loadings. A structure undergoes AP when the fluctuating loading program produces, after any arbitrarily large time, some plastic deformation as well as a subsequent vanishing of the net plastic deformation. This induces failure due to low‐cycle fatigue. Likewise, the structure fails by IC when plastic deformations accumulate in the form of a compatible strain distribution that leads to excessive inelastic deformation. Shakedown analysis allows working under the realistic assumption that only the range of variable loadings is known, unlike the usual prescription of a particular loading history. That is, we deal in this chapter with direct methods that are based solely on the knowledge of a range of load variations (or a reference loading, in limit analysis). It is worth emphasizing that the differences between incremental and direct methods go beyond the computational advantages frequently obtained with the latter ones. In fact, the fundamental questions about whether or not critical loads or cycles do exist, independently from loading histories, can only be answered by the theories of direct methods. The mathematical foundations of shakedown theory strongly rest on optimization theory and the related areas of convex analysis and mathematical programming. This chapter contains simplified derivations of the static, kinematic, and mixed variational principles of shakedown theory by means of the basic techniques of convex analysis. Moreover, numerical procedures suitable to solve the shakedown analysis problem are discussed here in the framework of mathematical programming techniques combined with common procedures in the field of finite element methods. Applications in fundamental examples are then presented, namely (i) a restrained block under thermomechanical loadings; (ii) the classical Bree problem of a tube under variable pressure and temperature, with a more realistic model and including an exact solution; (iii) a plate with a circular hole subjected to variable tractions; (iv) a plate with simulated imperfections showing critical mechanisms with strongly localized deformations; and (v) a pipe elbow under internal pressure and in plane bending moment.

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Mesh adaptive computation of upper and lower bounds in limit analysis

Cited by 209 publications

References 31 publications

Limit analysis of plates using the EFG method and second‐order cone programming

Limit analysis of plates using the EFG method and second‐order cone programming

A cell‐based smoothed finite element method for kinematic limit analysis

Shakedown and Safety Assessment

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