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

Abstract

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.

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

“…Together with [15], this paper demonstrates that a numerical limit analysis procedure using a strain smoothing technique is capable of providing accurate solutions using a relatively small number of elements (nodes). It was also shown that most properties of CS-FEM found in elastic analysis, for instance superconvergent behavior, working well for extremely distorted elements and locking free behaviour (using one cell version), are retained here in plastic limit analysis.…”

“…It is known as the stabilized conforming nodal integration (SCNI) scheme. The SCNI scheme has been applied successfully to various problems, for instance, elastic analysis [12][13][14], plastic limit analysis [15], error estimation [16] and a stabilized mesh-free equilibrium model for limit analysis [17]. It is shown that, when the SCNI scheme is applied, the solutions obtained are accurate and stable, and locking problems can also be prevented.…”

“…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].…”

Limit analysis of plates and slabs using a meshless equilibrium formulation

“…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.…”

“…The main idea of the scheme is that nodal values are determined by spatially averaging field values using the divergence theorem. The scheme has been applied successfully to various analysis problems [9,[12][13][14]. It is shown that, when the SCNI scheme is applied, the solutions obtained are accurate and stable, and the computational cost is much lower than when using Gauss integration.…”