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1
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Arbitrary high-order finite element schemes and high-order mass lumping

100%
Jund S., Salmon S.
International Journal of Applied Mathematics and Computer Science
|
2007
|
tom 17
|
nr 3
375-393
EN
Computers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elementsof order k and show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up to P5 elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.
2
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Conditions for periodic vibrations in a symmetric n-string

100%
Gauthier C.
Open Mathematics
|
2008
|
tom 6
|
nr 2
287-300
EN
A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.
3
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On the nonlinear stabilization of the wave equation

100%
Guesmia A.
Annales Polonici Mathematici
|
1998
|
tom 68
|
nr 2
191-198
EN
We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.
4
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Optimal internal dissipation of a damped wave equation using a topological approach

88%
Münch A.
International Journal of Applied Mathematics and Computer Science
|
2009
|
tom 19
|
nr 1
15-37
EN
We consider a linear damped wave equation defined on a two-dimensional domain Ω, with a dissipative term localized in a subset ω. We address the shape design problem which consists in optimizing the shape of ω in order to minimize the energy of the system at a given time T . By introducing an adjoint problem, we first obtain explicitly the (shape) derivative of the energy at time T with respect to the variation in ω. Expressed as a boundary integral on ∂ω, this derivative is then used as an advection velocity in a Hamilton-Jacobi equation for shape changes. We use the level-set methodology on a fixed working Eulerian mesh as well as the notion of the topological derivative. We also consider optimization with respect to the value of the damping parameter. The numerical approximation is presented in detail and several numerical experiments are performed which relate the over-damping phenomenon to the well-posedness of the problem.
5
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A necessary and sufficient condition for the existence of an exponential attractor

63%
Pražák D.
Open Mathematics
|
2003
|
tom 1
|
nr 3
411-417
EN
We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.
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