Mimetic Finite Difference methods for Hamiltonian wave equations in 2D

In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in time to integrate the semi-discrete Hamiltonian system. The main characteristic of MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach, associated with a symplectic method for the time integration yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper.Comment: 26 pages, 8 figure

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

Asymptotic energy behavior of two classical intermediate benchmark shell problems

We consider two classical problems which are widely used as benchmark tests for shell numerical methods: the Scordelis-Lo roof and the pinched roof. Due to the particular load and boundary conditions applied, neither belongs to the well known classes of purely bending or purely membrane dominated shells. Consequently the asymptotic energy norm behavior, which is useful not only because it represents the structure stiffness, but also for numerical comparison purposes, is not a priori known. In this work, using space interpolation techniques and a recently developed ``intermediate\u27\u27 shell theory, the asymptotic energy behavior of both problems is found analytically. The results are in agreement with the numerical estimates obtained in other papers

Uniform error estimates for a class of intermediate cylindrical shell problems

A uniform in thickness error estimate is obtained for a particular class of intermediate Koiter shell problems, solved with a classical conforming finite element method. The model problem is that of a cylinder under a class of irregular loads which, due to particular symmetries, allow a simplified reformulation on a one dimensional domain. The result is an almost (h^{s}) error behavior in the (H^{-1}) dual norm, were $s>0$ depends on the load regularity. Such estimate is believable to be sharp (this additional claim is supported by some numerical tests)

Asymptotic study of the solution for pinched cylindrical shells

We consider the displacement problem of a cylindrical roof under a pointwise Dirac load acting in the normal direction, and recognize two classes of possible responses in dependence of the boundary conditions: the bending dominated and the ``intermediate\u27\u27 case. We theoretically analyze the local form of the solution around the load application point in both cases; the latter (which is equivalent to the classical pinched cylinder benchmark) shows a layer of characteristic lenght $t^{1/4}$ in the angular direction, while the first one shows a smoother solution. Finally we exploit the results obtained in order to derive some good numerical strategies for the problem. In the appendix, we show the perfect correspondence between the theoretical results and the solution obtained through numerical means

Numerical evaluation of the asymptotic energy behavior of intermediate shells with application to two classical benchmark tests

We consider the class of shell problems which are neither purely bending neither purely membrane dominated. In such cases the asymptotic energy norm behavior (which is useful not only because it represents the structure stiffness, but also for numerical comparison purposes) is not a priori known. In this work we apply a numerical procedure in order to estimate the energy behavior of a general shell problem. In order to test its reliability, the method is applied to various problems for which the theoretical energy behavior is known and the results can be compared. Among the problems tested, we have two classical engineering shell benchmarks which are neither bending neither membrane dominated, and for which an analytical evaluation has been obtained in a recent work. All the energy behavior estimates obtained with the numerical method are in perfect agreement with the theoretical values

Reissner-Mindlin plates with free boundary conditions

It is well known that the solutions of Reissner-Mindlin equations can have, for small thickness $t$, severe boundary layers. In particular, near the part of the boundary where the so-called {it free plate} boundary conditions are prescribed, the layer can be so strong that rotations are not uniformly bounded in $H^2$, for $trightarrow 0$. This is clearly a major drawback for numerical methods, as one cannot achieve error estimates of order $h$ uniformly in $t$. Here we propose a new model for free plate boundary conditions that has less severe layers, and we propose a numerical method that provides a priori error estimates of order $O(h)$ uniformly in $t$