Optimal internal dissipation of a damped wave equation using a topological approach

Arnaud Münch

International Journal of Applied Mathematics and Computer Science (2009)

  • Volume: 19, Issue: 1, page 15-37
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Arnaud Münch. "Optimal internal dissipation of a damped wave equation using a topological approach." International Journal of Applied Mathematics and Computer Science 19.1 (2009): 15-37. <http://eudml.org/doc/207917>.

@article{ArnaudMünch2009,
abstract = {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.},
author = {Arnaud Münch},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {shape design; wave equation; level set; topological derivative; numerical viscosity},
language = {eng},
number = {1},
pages = {15-37},
title = {Optimal internal dissipation of a damped wave equation using a topological approach},
url = {http://eudml.org/doc/207917},
volume = {19},
year = {2009},
}

TY - JOUR
AU - Arnaud Münch
TI - Optimal internal dissipation of a damped wave equation using a topological approach
JO - International Journal of Applied Mathematics and Computer Science
PY - 2009
VL - 19
IS - 1
SP - 15
EP - 37
AB - 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.
LA - eng
KW - shape design; wave equation; level set; topological derivative; numerical viscosity
UR - http://eudml.org/doc/207917
ER -

References

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