@article {VicentedaSilva2007,
	title = {A non-linear programming method approach for upper bound limit analysis},
	journal = {International Journal for Numerical Methods in Engineering},
	volume = {72},
	year = {2007},
	note = {n/a},
	pages = {1192-1218},
	abstract = {<p>This paper presents a finite element model based on mathematical non-linear programming in order to determine upper bounds of colapse loads of a mechanical structure.The proposed formulation is derived within a kinematical approach framework, employing two simultaneous and independent field approximations for the velocity and strain rate fields. The augmented Lagrangian is used to establish the compatibility between these two fields. In this model, only continuous velocity fields are used.Uzawa{\textquoteright}s minimization algorithm is applied to determine the optimal kinematical field that minimizes the difference between external and dissipated work rate. The use of this technique allows to bypass the complexity of the non-linear aspects of the problem, since non-linearity is addressed as a set of small local subproblems of optimization for each finite element.The obtained model is quite versatile and suitable for solving a wide range of collapse problems. This paper studies 3D strut-and-tie structures, 2D plane strain/stress and 3D solid problems.</p>
},
	keywords = {augmented Lagrangian, limit analysis, non-linear optimization, upper bound theorem},
	doi = {10.1002/nme.2061},
	url = {http://dx.doi.org/10.1002/nme.2061},
	author = {Vicente da Silva, M. and Ant{\~a}o, A. N.}
}
