scholarly journals A note on the necessity of filtering mechanism for polynomial observability of time-discrete wave equation

2017 ◽  
Vol 9 (1) ◽  
pp. 98-103
Author(s):  
Z. Hajjej
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
High Frequency ◽  
Time Discretization ◽  
Continuous Model ◽  
Discrete Approximations ◽  
Discretization Schemes ◽  
High Frequency Modes

The problem of uniform polynomial observability was recently analyzed. It is shown that, when the continuous model is uniformly polynomially observable, it is sufficient to filter initial data to derive uniform polynomial observability inequalities for suitable time-discretization schemes. In this note, we prove that a filtering mechanism of high frequency modes is necessary to obtain uniform polynomial observability. More precisely, we give a counterexample which proves that this latter fails without filtering the initial data for time semi-discrete approximations of the wave equation.

HIGH FREQUENCY ANALYSIS OF FAMILIES OF SOLUTIONS TO THE EQUATION OF VISCOELASTICITY OF KELVIN–VOIGT

2004 ◽  
Vol 01 (04) ◽  
pp. 789-812 ◽  
Author(s):  
AMEL ATALLAH-BARAKET ◽  
CLOTILDE FERMANIAN KAMMERER
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Initial Data ◽  
Lower Bounds ◽  
Frequency Analysis ◽  
High Frequency ◽  
Weak Limit ◽  
Negative Viscosity

In this paper, we study the evolution of the energy density of a sequence of solutions to the Kelvin–Voigt viscoelasticity equation. We do not suppose lower bounds on the non-negative viscosity matrix. We prove that, in the zone where the viscosity matrix is invertible, this term prevents propagation of concentation and oscillation effects contrary to what happens in the wave equation. We calculate precisely the weak limit of the energy density in terms of microlocal defect measures associated with the initial data under the assumption that the oscillations of the data are not microlocally localized on directions which are in the kernel of the viscosity matrix.

scholarly journals Galerkin–collocation approximation in time for the wave equation and its post-processing

2020 ◽  
Vol 54 (6) ◽  
pp. 2099-2123 ◽  
Author(s):  
Mathias Anselmann ◽  
Markus Bause ◽  
Simon Becher ◽  
Gunar Matthies
Keyword(s):  
Wave Equation ◽  
Time Discretization ◽  
Collocation Methods ◽  
Optimal Order ◽  
Post Processing ◽  
Order Error ◽  
Computational Costs ◽  
Optimal Order Error Estimates ◽  
Discretization Schemes ◽  
The Galerkin Method

We introduce and analyze families of Galerkin–collocation discretization schemes in time for the wave equation. Their conceptual basis is the establishment of a connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex algebraic systems. Firstly, continuously differentiable in time discrete solutions are studied. Optimal order error estimates are proved. Then, the concept of Galerkin–collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct link between the two families by a computationally cheap post-processing is presented. A key ingredient of the proposed methods is the application of quadrature rules involving derivatives. The performance properties of the schemes are illustrated by numerical experiments.

GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

2004 ◽  
Vol 01 (04) ◽  
pp. 747-768
Author(s):  
CHRISTIAN ROHDE ◽  
MAI DUC THANH
Keyword(s):  
Phase Transition ◽  
Surface Tension ◽  
Initial Data ◽  
Time Discretization ◽  
Approximate Solutions ◽  
Implicit Time ◽  
Time Step ◽  
Dynamical Phase Transition ◽  
Transition Dynamics ◽  
Variational Scheme

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

Shock Mitigation by Energy Reversal to the High Frequency Modes

2014 ◽  
Author(s):  
Mohammad A. AL-Shudeifat ◽  
Alexander F. Vakakis ◽  
Lawrence A. Bergman
Keyword(s):  
High Frequency ◽  
Large Scale ◽  
Computational Study ◽  
Dynamic Structure ◽  
Frequency Mode ◽  
Frame Structure ◽  
Shock Energy ◽  
Modal Damping ◽  
Shock Mitigation ◽  
High Frequency Modes

In this computational study, a light-weight dynamic device is investigated for passive energy reversal from the lowest frequency mode to the high frequency modes of a large-scale frame structure for rapid shock mitigation. The device is based on the single-sided vibro-impact mechanism. It has two functions for passive energy transfer: a nonlinear energy sink (NES) for local energy dissipation and an energy pump to high frequency modes where a significant amount of the shock energy is rapidly dissipated. As a result, a significant portion of the shock energy induced into the linear dynamic structure can be passively reversed from the lowest frequency mode to the high frequency modes and rapidly dissipated by their modal damping. The amount of the energy dissipated by the modal damping of the high frequency modes can be controlled by the amount of inherent damping in the device. Ideally, the device can passively reverse up to 80% of the input shock energy from the lowest frequency mode to the high frequency modes when its damping is assumed to be zero and its impact coefficient of restitution is equal to unity. The shock energy redistribution between this device and the high frequency modes is found to be efficient for rapid shock mitigation in the considered 9-story dynamic structure.

An Impedance-Based Approach for Rods With Shear-Type Damping Layer Treatment

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
P. W. Wang ◽  
D. Q. Zhuang
Keyword(s):  
High Frequency ◽  
Continuous Model ◽  
Damping Coefficient ◽  
System Response ◽  
Traditional Model ◽  
Constrained Layer Damping ◽  
High Frequency Range ◽  
Frequency Range ◽  
Shear Type ◽  
Shear Spring

An impedance-based approach for analyzing an axial rod with shear-type damping layer treatment is proposed. The rod and shear-type damping layer are regarded as two subsystems and both impedances are calculated analytically. The system impedance can be obtained through the impedance coupling between the host rod and the damping layer. The shear-type damping layer is regarded as a shear spring with complex shear modulus. Under the traditional model, the damping coefficient diminishes with the increasing frequency. The paper develops two shear-type damping layer models, including the single degree-of-freedom (SDOF) model and continuous model to predict the behavior of the damping layer. Both damping layer models are compared with the traditional model and the system responses from these models are validated by finite element method (FEM) code COMSOL Multiphysics. Results show that the damping coefficients of both the traditional shear-spring model and SDOF model diminish as the increasing frequency so that the system responses are discrepant with that from COMSOL in the high frequency range. On the other hand, the system response from the continuous model is consistent with that from COMSOL in the full frequency range. Hence, the continuous damping layer model can predict a correct damping coefficient in the high frequency range and this property can be also employed to improve the analysis of the constrained-layer damping treated structures. Finally, the modal loss factor and fundamental frequency of the system with respect to different damping layer thicknesses are presented using the developed approach.

scholarly journals High frequency modes meshfree analysis of Reissner–Mindlin plates

2016 ◽  
Vol 1 (3) ◽  
pp. 400-412 ◽  
Author(s):  
Tinh Quoc Bui ◽  
Duc Hong Doan ◽  
Thom Van Do ◽  
Sohichi Hirose ◽  
Nguyen Dinh Duc
Keyword(s):  
High Frequency ◽  
Meshfree Analysis ◽  
High Frequency Modes

scholarly journals Decay of solutions of the wave equation in expanding cosmological spacetimes

2019 ◽  
Vol 16 (01) ◽  
pp. 35-58
Author(s):  
João L. Costa ◽  
José Natário ◽  
Pedro F. C. Oliveira
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Decay Rate ◽  
Time Derivative ◽  
Iteration Scheme ◽  
Higher Order ◽  
Accelerated Expansion ◽  
De Sitter ◽  
Decay Of Solutions ◽  
Partial Energy

We study the decay of solutions of the wave equation in some expanding cosmological spacetimes, namely flat Friedmann–Lemaître–Robertson–Walker (FLRW) models and the cosmological region of the Reissner–Nordström–de Sitter (RNdS) solution. By introducing a partial energy and using an iteration scheme, we find that, for initial data with finite higher order energies, the decay rate of the time derivative is faster than previously existing estimates. For models undergoing accelerated expansion, our decay rate appears to be (almost) sharp.

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