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In the present paper we investigate the capability of finite element methods to correctly reproduce the stability range of finite strain problems in the incompressible regime. To this end, we develop a numerical scheme, obtained combining a stream function formulation with an isogeometric NURBS approach, which is able to sharply estimate the stability limits of the continuous problem. Using such a method, we show a pair of benchmark problems on which various well-known finite element methods largely fail in approximating the correct stability range. © 2008 Elsevier B.V. All rights reserved.

We consider the free vibration problem of thin shells of revolution of constant type of geometry, focusing on the asymptotic behaviour of the lowest eigenfrequency, as the thickness tends to zero. Numerical experiments are computed using two discretization methods, collocation and finite elements, each corresponding to a different shell model. Our results are in agreement with theoretical results obtained using interpolation theory and cited in literature. © 2008 Springer-Verlag.

A family of plate elements introduced by Falk and Tu [25] is considered. A new stability and a-priori error analysis is given. In addition, an a-posteriori error estimate is proved. The analysis is confirmed by numerical benchmark computations. © 2008 Springer Science + Business Media B.V.

In this paper, we study plane incompressible elastic problems by means of a ‘‘stream-function’’ formulation such that a divergence-free displacement field can be computed from a scalar potential. The numerical scheme is constructed within the framework of NURBS-based isogeometric analysis and we take advantage of the high continuity guaranteed by NURBS basis functions in order to obtain the displacement field from the potential differentiation. As a consequence, the obtained numerical scheme is automatically locking-free in the presence of the incompressibility constraint. A Discontinuous Galerkin approach is proposed to deal with multiple mapped, possibly multiply connected, domains. Extensive numerical results are provided to show the method capabilities. � 2007 Elsevier B.V. All rights reserved.

The paper investigates mixed-enhanced strain finite elements developed within the context of the $u/p$ formulation for nearly incompressible linear elasticity problems. A rigorous convergence and stability analysis is detailed, providing also $L^2$-error estimates for the displacement field. Extensive numerical tests are developed, showing in particular the accordance of the computational results with the theoretical predictions

We review some results about the behaviour of a general Koiter shell, in the framework of linear elasticity. In particular, we investigate the asymptotics (for the thickness tending to zero) of the energy functional and of the percentage of the energy which is stored in the bending term. Such an analysis is motivated by the need to better understand how to numerically treat an arbitrary thin shell, when the discretization is performed using a finite element strategy. We present some instances to which our theory can be applied. Some numerical tests confirming our theoretical predictions are also provided. © 2002 Elsevier Science Ltd. All rights reserved.

We analyse the Virtual Element Methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter), can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the "classical" one introduced in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199-214], and a recent one presented in [Wriggers, P., Rust, W.T., and Reddy, B.D., A virtual element method for contact, submitted for publication]. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.