Leonardo Leonetti

A mixed node-based smoothed finite element method (MNS-FEM) for elasticity

In this paper, an alternative formulation of the NS-FEM based on an assumed stress field is presented to include drilling rotations. Within each triangular element the displacement field is described by a revised Allman triangle interpolation, while the stress field is assumed as linear or linear reduced on the conflict domain of the background grid. The elastic solution is constructed through the stationarity condition of a constrained mixed Hellinger–Reissner principle. The numerical experiments show that the proposed model performs well in elastic problems, in particular in the case of incompressibility, and takes advantage of the enrichment of the interpolation functions from quadratic contributions to the displacement field. The paper also shows a way to improve the description of the stress field.

An Efficient Algorithm for Shakedown Analysis Based on Equality Constrained Sequential Quadratic Programming

A new iterative algorithm to evaluate the elastic shakedown multiplier is proposed. On the basis of a three field mixed finite element, a series of mathematical programming problems or steps, obtained from the application of the proximal point algorithm to the static shakedown theorem, are obtained. Each step is solved by an Equality Constrained Sequential Quadratic Programming (EC-SQP) technique that retain all the equations and variables of the problem at the same level so allowing a consistent linearization that improves the computational efficiency. The numerical tests performed for 2D problems show the good performance and the great robustness of the proposed algorithm.

Decomposition Methods and Strain Driven Algorithms for Limit and Shakedown Analysis

A mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in a FEM context, is presented. From the optimization point of view, standard arc–length strain driven elastoplastic analysis, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other nonlinear programming methods, simpler convergence proofs and duality to be exploited. Due to the unified approach in terms of total stresses, the strain driven algorithms become more effective and less nonlinear with respect to a self equilibrated stress formulation and easier to implement in existing codes performing elastoplastic analysis.

MIXED SOLID MODELS IN NUMERICAL ANALYSIS OF SLENDER STRUCTURES

The reasons of the better performances of mixed, stress–displacements, 3D solid finite elements in the analysis of slender elastic structures are explained. It will be shown that mixed or compatible description, also when derived from the same finite element and then completely equivalent from the discretization point of view, behave very differently when implemented in both asymptotic and path–following solution strategies due to the occurrence of a pathological locking phenomenon in the compatible formulation. The notable advantages of the used of a 3D mixed solid finite element in Koiter asymptotic analysis are also highlighted.

COMPOSITE FEM MODELS FOR LIMIT AND SHAKEDOWN ANALYSIS

The paper improved S-FEMs formulations with an enriched displacement field, making use of modified Allman’s shape functions. This mixed interpolation is the natural context in performing lower bound strategy for shakedown, limit analysis and elastoplastic analysis. The model takes advantages from the simplicity and few addressed requirements for good performances in nonlinear analysis. The simple assumption made for the stress field regards the convenience of using self-equilibrated stress interpolations in Cartesian coordinates. In the proposed composite elements the stress is discontinuous on the element and across their sides and the mesh of the elements is coincident with the discretization of the geometry. This stress interpolation is able to address the discontinuities in the plastic strain and, in such a way, to define in their description a finer mesh with respect the basic grid.

Shakedown Analysis of 3D Frames with an Effective Treatment of the Load Combinations

Using the Melan static theorem and an algorithm based on dual decomposition, a formulation for the shakedown analysis of 3D frames is proposed. An efficient treatment of the load combinations and an accurate and simple definition of the cross-section yield function are employed to increase effectiveness and to make shakedown analysis an affordable design tool. The section yield function, obtained by its support function values associated with presso-flexural mechanisms, is defined as the Minkowski sum of ellipsoids. The return mapping process, resulting from the dual decomposition, is solved at the element level by means of an algorithm based again on the dual decomposition. It allows the separation of the problem at the ellipsoid level and the use of a simple and inexpensive radial return mapping process for its solution. A series of numerical tests are presented to show both the accuracy and the effectiveness of the proposed formulation.