Lower‐bound limit analysis by using the EFG method and non‐linear programming

2009

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.

Section: Numerical Examplesmentioning

confidence: 99%

“…It therefore seems appropriate to investigate the performance of the EFG method when applied to limit analysis problems. Recently, a numerical procedure for lower-bound limit analysis was presented by [26]. In the paper, a self-equilibrium stress basis vector at each Gaussian point is computed using the EFG method.…”

Section: Introductionmentioning

confidence: 99%

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

Gilbert

2009

“…• FEMQ4-standard bilinear four-noded quadrilateral element using full integration (2 × 2 Gauss points) • FEMRI-standard bilinear four-noded quadrilateral element using reduced integration (1 The first example deals with a square plate with a central circular hole which is subjected to biaxial uniform loads p 1 and p 2 as shown in Figure 4a, where L = 10 m. This benchmark plane stress problem has been solved numerically for different loading cases by finite element models [4,6,[47][48][49], by the boundary element method (BEM) [50] and more recently by the element-free Galerkin (EFG) method [51]. Due to symmetry, only the upper-right quarter of the plate is modeled, see Figure 4b.…”

Section: Numerical Examplesmentioning

confidence: 99%

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

Nguyen‐Xuan

et al. 2010

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.

“…Recently Le et al [9] proposed a numerical kinematic formulation using the Element-Free Galerkin (EFG) method and second-order cone programming (SOCP) to furnish good (approximate) upper-bound solutions for Kirchhoff plate problems governed by the von Mises failure criterion. It has also been demonstrated [9,10] that the EFG method is in general well suited for limit analysis problems, allowing accurate solutions to be obtained with relatively few nodes. Following this line of research, the main objective of this paper is to develop an equilibrium formulation which combines the EFG method with SOCP to obtain accurate solutions for both plate and slab problems.…”

Section: Introductionmentioning

confidence: 99%

“…In Chen et al [10], a self-equilibrium stress basis vector at each Gaussian point is calculated by solving the equivalent weak form of the equilibrium equations. The self-equilibrium stress field is then obtained by a linear combination of several self-equilibrium stress basis vectors which are generated by considering the differences between intermediate stresses during the elasto-plastic equilibrium iteration.…”

Section: Introductionmentioning

confidence: 99%

Limit analysis of plates and slabs using a meshless equilibrium formulation

Gilbert

2010

Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2010) SUMMARYA meshless Element-Free Galerkin (EFG) equilibrium formulation is proposed to compute the limit loads which can be sustained by plates and slabs. In the formulation pure moment fields are approximated using a moving least squares technique, which means that the resulting fields are smooth over the entire problem domain. There is therefore no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable finite element formulation. The collocation method is used to enforce the strong form of the equilibrium equations and a stabilized conforming nodal integration scheme is introduced to eliminate numerical instability problems. The combination of the collocation method and the smoothing technique means that equilibrium only needs to be enforced at the nodes, and stable and accurate solutions can be obtained with minimal computational effort. The von Mises and Nielsen yield criteria which are used in the analysis of plates and slabs respectively are enforced by introducing second-order cone constraints, ensuring that the resulting optimization problem can be solved using efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plate and slab problems.