A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis

Abstract

SUMMARYA mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in an FEM context is presented. From the optimization point of view, standard arc-length strain-driven elastoplastic analyses, 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 non-linear programming methods, simpler convergence proofs and duality to be exploited. Owing to the unified approach in terms of total stresses, the strain-driven algorithms become more effective and less non-linear with respect to a self-equilibrated stress formulation and easier to implement in the existing codes performing elastoplastic analysis. The elastic domain is represented avoiding any linearization of the yield function so improving both the accuracy and the performance. Better results are obtained using two different finite elements, one with a good behavior in the elastic range and the other suitable for performing elastoplastic analysis. The proposed formulation is compared with a specialized implementation of the primal-dual interior point method suitable to solve the problems at hand.

Nonlinear FEM analysis for beams and plate assemblages based on the implicit corotational method

“…The convergence of the iterative process (5) has been widely discussed in [Garcea et al 1998;. It is highly sensitive to the format of the equations, that is to the actual manner in which the equations are split and organized (also see [Garcea and Leonetti 2011;Bilotta et al 2012]). Different formats behave very differently and, in particular, compatible formats expressed in displacement unknowns and equilibrium equations can lead to convergence failures as a consequence of locking (which we call extrapolation locking in [Garcea et al 1998]) due to the interaction of large axial/flexural stiffness ratios with even small element rotations.…”

Composite Fem Models for Limit and Shakedown Analysis

“…The limit analysis aims to identify the plastic collapse state, characterized by the collapse multiplier λ c and the associated plastic mechanism. A way to solve this problem is represented by the evolutive analysis which furnishes the entire structural response through the solution of a sequence of incremental elasto-plastic problems [6]. The collapse multiplier λ c is evaluated as the limit value for the equilibrium path.…”

“…It is worth noting that matrix, the single step of the elastoplastic analysis [6,21,10,22] can be seen as the solution of the mathematical problem (12) by adding the compliance matrix F that is block diagonal at the element (stress) level due to the constant interpolation adopted and its inverse F −1 can be directly assembled. In this way an inexpensive construction of the tangent stiffness matrix [6,10] is obtained.…”

“…In this paper improved S-FEMs formulations with an enriched displacement field, making use of modified Allman's shape functions, is presented in its assumed stress version. This mixed interpolation strategy doesn't cause any difficulties in the linear elastic analysis and is the natural context in performing lower bound strategy for shakedown, limit analysis and elastoplastic analysis [6]. In addition, as the introduction of nonlinearities in the model leads to very complicate nonlinear contributions, an essential and working well element with few addressed requirement is of great interest.…”

Effective treatment of complex statical and dynamical load combinations within shakedown analysis of 3D frames

“…The availability of high optimized IPM solvers, such as MOSEK [12], makes this approach interesting. Alternative specialized formulations to evaluate the shakedown safety factors in a FE context of analysis, have also been proposed in [13][14][15][16][17][18][19].…”