A virtual element method for the vibration problem of Kirchhoff plates

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose an $H^2(\Omega)$ conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting schemeprovides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented

A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements

Virtual Delamination Testing through Non-Linear Multi-Scale Computational Methods: Some Recent Progress

This paper deals with the parallel simulation of delamination problems at the meso-scale by means of multi-scale methods, the aim being the Virtual Delamination Testing of Composite parts. In the non-linear context, Domain Decomposition Methods are mainly used as a solver for the tangent problem to be solved at each iteration of a Newton-Raphson algorithm. In case of strongly nonlinear and heterogeneous problems, this procedure may lead to severe difficulties. The paper focuses on methods to circumvent these problems, which can now be expressed using a relatively general framework, even though the different ingredients of the strategy have emerged separately. We rely here on the micro-macro framework proposed in (Ladev\`eze, Loiseau, and Dureisseix, 2001). The method proposed in this paper introduces three additional features: (i) the adaptation of the macro-basis to situations where classical homogenization does not provide a good preconditioner, (ii) the use of non-linear relocalization to decrease the number of global problems to be solved in the case of unevenly distributed non-linearities, (iii) the adaptation of the approximation of the local Schur complement which governs the convergence of the proposed iterative technique. Computations of delamination and delamination-buckling interaction with contact on potentially large delaminated areas are used to illustrate those aspects

Nonlinear and Linearized Analysis of Vibrations of Loaded Anisotropic Beam/Plate/Shell Structures

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