A constrained non-linear system approach for the solution of an extended limit analysis problem (original) (raw)

Upper bound limit analysis using linear finite elements and non-linear programming

International Journal for Numerical and Analytical Methods in Geomechanics, 2002

A new method for computing rigorous upper bounds on the limit loads for one-, two-and threedimensional continua is described. The formulation is based on linear finite elements, permits kinematically admissible velocity discontinuities at all interelement boundaries, and furnishes a kinematically admissible velocity field by solving a non-linear programming problem. In the latter, the objective function corresponds to the dissipated power (which is minimized) and the unknowns are subject to linear equality constraints as well as linear and non-linear inequality constraints.

A general non-linear optimization algorithm for lower bound limit analysis

International Journal for Numerical Methods in Engineering, 2003

The non-linear programming problem associated with the discrete lower bound limit analysis problem is treated by means of an algorithm where the need to linearize the yield criteria is avoided. The algorithm is an interior point method and is completely general in the sense that no particular ÿnite element discretization or yield criterion is required. As with interior point methods for linear programming the number of iterations is a ected only little by the problem size. Some practical implementation issues are discussed with reference to the special structure of the common lower bound load optimization problem, and ÿnally the e ciency and accuracy of the method is demonstrated by means of examples of plate and slab structures obeying di erent non-linear yield criteria. Copyright ? 2002 John Wiley & Sons, Ltd.

Lower bound limit analysis using non-linear programming

International Journal for Numerical Methods in Engineering, 2002

This paper describes a new formulation, based on linear ÿnite elements and non-linear programming, for computing rigorous lower bounds in 1, 2 and 3 dimensions. The resulting optimization problem is typically very large and highly sparse and is solved using a fast quasi-Newton method whose iteration count is largely independent of the mesh reÿnement. For two-dimensional applications, the new formulation is shown to be vastly superior to an equivalent formulation that is based on a linearized yield surface and linear programming. Although it has been developed primarily for geotechnical applications, the method can be used for a wide range of plasticity problems including those with inhomogeneous materials, complex loading, and complicated geometry.

A finite element, linear programming methods for the limit analysis of thin plates

International Journal for Numerical Methods in Engineering, 1973

Finite elements having linear moment distributions and use of linearized yield criteria allow one to determine lower bounds to the collapse load of thin plates as solutions of linear programs. The method is quite general and rigorously meets the requirements of the lower bound theorem of limit analysis for concentrated or line load distributions. Ways of treating distributed surface loads are also discussed and tested. Actual bounds are computed for a variety of plate problems governed by Tresca yield criterion and compared with previous solutions obtained from higher order stress elements and non-linear optimization techniques. The comparison shows that the present method can yield accurate bounds with considerably shorter computer times and relatively small number of elements. Additional tests show that numerical convergence to the limit loads is assured by suitable refinement of the mesh pattern. * Research supported by the National Research Council (CNR) of Italy. t Finite elements in a limit analysis problem were probably first used in Reference 6.

Large static problem in numerical limit analysis: A decomposition approach

2010

A general decomposition approach for the static method of limit analysis is proposed. It is based on piecewise linear stress fields, on a partition into finite element sub-problems and on a specific coordination of the subproblem stress fields through auxiliary interface problems. The final convex optimization problems are solved using nonlinear interior point programming methods. As validated for the compressed bar with Tresca/von Mises materials in plane strain, this method appears rapidly convergent, so that very large problems with millions of constraints and variables can be solved. Then the method is applied to the classical problem of the stability of a Tresca vertical cut: the static bound to the stability factor is improved to 3.7752, a value to be compared with the recent best upper bound 3.7776.

Upper and lower bound limit analysis of plates using FEM and second-order cone programming

Computers & Structures, 2010

This paper presents two novel numerical procedures to determine upper and lower bounds on the actual collapse load multiplier for plates in bending. The conforming Hsieh-Clough-Tocher (HCT) and enhanced Morley (EM) elements are used to discrete the problem fields. A Morley element with enhanced moment fields is used. The constant moment fields is added a quadratic mode in which the pressure is equilibrated by corner loads only, ensuring that exact equilibrium relations associated with a uniform pressure can be obtained. Once the displacement or moment fields are approximated and the bound theorems applied, limit analysis becomes a problem of optimization. In this paper, the optimization problems are formulated in the form of a standard second-order cone programming which can be solved using highly efficient interior point solvers. The procedures are tested by applying it to several benchmark plate problems and are found good agreement between the present upper and lower bound solutions and results in the literature.

Meshless Methods for Upper Bound and Lower Bound Limit Analysis of Thin Plates

Limit state techniques are used to design and assess the safety of engineering components and structures. Difficulties in performing incremental-iterative elasto-plastic analyses have motivated the development of numerical variants of classical 'limit analysis' methods, in which the limit load is identified directly. Current research in the field of limit analysis is focussed on the development of numerical tools which are sufficiently efficient and robust to be used in engineering practice. This places demands on the numerical discretisation strategy adopted, as well as on the mathematical programming tools applied. Numerical procedures based on the finite element method (FEM) are particularly well-established. However, when finite elements are used, the solutions obtained can be highly sensitive to the geometry of the original mesh, particularly in the region of stress or displacement singularities. It is therefore worthwhile exploring a range of alternative methods. Here the element-free Galerkin (EFG) method is applied to limit analysis problems. In the upper bound formulation, a moving least squares technique is used to approximate the displacement field, which involves only one displacement variable for each EFG node. The total number of variables in the resulting optimisation problem is therefore much smaller than when using finite element formulations involving compatible elements. On the other hand, in the lower bound formulation pure stress/moment fields are approximated using the moving least squares technique, ensuring that the resulting fields are smooth over the entire problem domain. This means that there is no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable equilibrium-based finite element formulation. In order to increase the efficiency of the EFG method, the stabilised conforming nodal integration (SCNI) scheme can be extended to stabilise curvature rates in the upper bound formulation and equilibrium equations in the lower bound formulation. When using a SCNI scheme it is found that far fewer variables and constraints are needed in the optimisation problem than when using a more standard Gauss integration scheme. Once the displacement or stress fields are approximated and the bound theorems applied, the underlying limit analysis problem becomes a problem of optimisation involving either linear or nonlinear programming. This research continues recent trends by combining second-order cone programming with displacement and equilibrium-based EFG models. The upper bound limit analysis problem for plates is formulated as a minimising the sum of norms problem, which is then cast as a second-order cone programming (SOCP) problem. In the lower bound formulation the von Mises yield criteria is enforced by introducing a second-order cone constraint, ensuring that the resulting optimization problem can be solved using efficient interior-point solvers.

A novel augmented Lagrangian-based formulation for upper-bound limit analysis

International Journal for Numerical Methods in Engineering, 2012

This paper describes a novel upper-bound formulation of limit analysis. This formulation is an innovative variant of an existing two-field mixed formulation based on the augmented Lagrangian method also developed by the authors. A natural approach is used to describe the deformation of each finite element. Furthermore, and in contrast to the previous formulation, two independent field approximations are now both used to define the velocity field, defined globally and at element level. It is shown that this feature allows a governing system of uncoupled linear equations to be obtained. Some numerical examples in plane strain conditions are presented in order to illustrate the current model performance. In conclusion, the potential and advantages of this new approach are discussed.