Numerical results for mimetic discretization of Reissner-Mindlin plate problems

A low-order mimetic finite difference (MFD) method for Reissner-Mindlin plate problems is considered. Together with the source problem, the free vibration and the buckling problems are investigated. Full details about the scheme implementation are provided, and the numerical results on several different types of meshes are reported

A new locking-free polygonal plate element for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields

A new $n-$ noded polygonal plate element is proposed for the analysis of plate structures comprising of thin and thick members. The formulation is based on the discrete Kirchhoff Mindlin theory. On each side of the polygonal element, discrete shear constraints are considered to relate the kinematical and the independent shear strains. The proposed element: (a) has proper rank; (b) passes patch test for both thin and thick plates; (c) is free from shear locking and (d) yields optimal convergence rates in $L^2-$norm and $H^1-$semi-norm. The accuracy and the convergence properties are demonstrated with a few benchmark examples

A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems

We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underling partition. We also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom. Numerical results that validate the theory are also presented

Membrane locking in discrete shell theories

This work is concerned with the study of thin structures in Computational Mechanics. This field is particularly interesting, since together with traditional finite elements methods (FEM), the last years have seen the development of a new approach, called discrete differential geometry (DDG). The idea of FEM is to approximate smooth solutions using polynomials, providing error estimates that establish convergence in the limit of mesh refinement. The natural language of this field has been found in the formalism of functional analysis. On the contrary, DDG considers discrete entities, e.g., the mesh, as the only physical system to be studied and discrete theories are being formulated from first principles. In particular, DDG is concerned with the preservation of smooth properties that break down in the discrete setting with FEM. While the core of traditional FEM is based on function interpolation, usually in Hilbert spaces, discrete theories have an intrinsic physical interpretation, independently from the smooth solutions they converge to. This approach is related to flexible multibody dynamics and finite volumes. In this work, we focus on the phenomenon of membrane locking, which produces a severe artificial rigidity in discrete thin structures. In the case of FEM, locking arises from a poor choice of finite subspaces where to look for solutions, while in the DDG case, it arises from arbitrary definitions of discrete geometric quantities. In particular, we underline that a given mesh, or a given finite subspace, are not the physical system of interest, but a representation of it, out of infinitely many. In this work, we use this observation and combine tools from FEM and DDG, in order to build a novel discrete shell theory, free of membrane locking

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

A stream virtual element formulation of the Stokes problem on polygonal meshes

In this paper we propose and analyze a novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem. The new formulation hinges upon the introduction of a suitable stream function space (characterizing the divergence free subspace of discrete velocities) and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beir\ue3o da Veiga et al., J. Comput. Phys., 228(2009), pp. 7215-7232] (up to a suitable reformulation into the VEM framework). Both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem. Several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

A Virtual Element Method for a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology

We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion system of the cardiac electric field. To this system, we analyze an $H^1(\Omega)$-conforming discretization by means of VEM which can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results