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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  1. Allaire, G., de Gournay, F., Jouve, F. and Toader, A. (2005). Structural optimization using topological and shape sensitivity analysis via a level-set method, Control and Cybernetics 34(1): 59-80. Zbl1167.49324
  2. Allaire, G., Jouve, F. and Toader, A. (2004). Structural optimization using sensitivity analysis and a level-set method, Journal of Computational Physics 194(1): 363-393. Zbl1136.74368
  3. Banks, H., Ito, K. and Wang, B. (1991). Exponentially stable approximations of weakly damped wave equations, Estimation and Control of Distributed Parameter Systems (Vorau, 1990), International Series of Numerical Mathematics, Vol. 100, Birkhäuser, Basel, pp. 1-33. Zbl0850.93719
  4. Bardos, C., Lebeau, G. and Rauch, J. (1992). Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM Journal on Control and Optimization 30(5): 1024-1065. Zbl0786.93009
  5. Burger, M. and Osher, S. (2005). A survey on level set methods for inverse problems and optimal design, European Journal of Applied Mathematics 16(2): 263-301. Zbl1091.49001
  6. Cagnol, J. and Zolésio, J.-P. (1999). Shape derivative in the wave equation with Dirichlet boundary conditions, Journal of Differential Equations 158(2): 175-210. Zbl0959.35017
  7. Castro, C. and Cox, S.J. (2001). Achieving arbitrarily large decay in the damped wave equation, SIAM Journal on Control and Optimization 39(6): 1748-1755. Zbl0983.35095
  8. Cohen, G.C. (2002). Higher-order Numerical Methods for Transient Wave Equations, Scientific Computation, Springer-Verlag, Berlin. 
  9. Degryse, E. and Mottelet, S. (2005). Shape optimization of piezoelectric sensors or actuators for the control of plates, ESAIM Control, Optimization and Calculus of Variations 11(4): 673-690. Zbl1081.49029
  10. Delfour, M. and Zolesio, J. (2001). Shapes and Geometries - Analysis, Differential Calculus and Optimization, SIAM, Philadelphia, PA. Zbl1002.49029
  11. Fahroo, F. and Ito, K. (1997). Variational formulation of optimal damping designs, Optimization Methods in Partial Differiential Equations (South Hadley, MA, 1996), Contemporary Mathematics, Vol. 209, American Mathematical Society, Providence, RI, pp. 95-114. Zbl0931.49019
  12. Freitas, P. (1999). Optimizing the rate of decay of solutions of the wave equation using genetic algorithms: A counterexample to the constant damping conjecture, SIAM Journal on Control and Optimization 37(2): 376-387. Zbl0999.35010
  13. Fulmanski, P., Laurain, A., Scheid, J.-F. and Sokołowski, J. (2008). Level set method with topological derivatives in shape optimization, International Journal of Computer Mathematics 85(10): 1491-1514. Zbl1155.65055
  14. Glowinski, R., Kinton, W. and Wheeler, M. (1989). A mixed finite element formulation for the boundary controllability of the wave equation, International Journal for Numerical Methods in Engineering 27(3): 623-635. Zbl0711.65084
  15. Hébrard, P. and Henrot, A. (2003). Optimal shape and position of the actuators for the stabilization of a string. Optimization and control of distributed systems Systems and Control Letters 48(3-4): 199-209. 
  16. Hébrard, P. and Henrot, A. (2005). A spillover phenomenon in the optimal location of actuators, SIAM Journal on Control and Optimization 44(1): 349-366. Zbl1083.49002
  17. Henrot, A. and Pierre, M. (2005). Variation et optimisation de formes-Une analyse géométrique, Mathématiques et Applications, Vol. 48, Springer, Berlin. 
  18. Lions, J. and Magenes, E. (1968). Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris. Zbl0165.10801
  19. López-Gómez, J. (1997). On the linear damped wave equation, Journal of Differential Equations 134(1): 26-45. Zbl0959.35109
  20. Maestre, F., Münch, A. and Pedregal, P. (2007). A spatiotemporal design problem for a damped wave equation, SIAM Journal on Applied Mathematics 68(1): 109-132. Zbl1147.35052
  21. Münch, A. (2008). Optimal design of the support of the control for the 2-d wave equation: Numerical investigations, Mathematical Modelling and Numerical Analysis 5(2): 331-351. Zbl1242.49091
  22. Münch, A. and Pazoto, A. (2007). Uniform stabilization of a numerical approximation of the locally damped wave equation, Control, Optimization and Calculus of Variations 13(2): 265-293. Zbl1120.65101
  23. Münch, A., Pedregal, P. and Periago, F. (2006). Optimal design of the damping set for the stabilization of the wave equation, Journal of Differential Equations 231(1): 331-358. Zbl1105.49005
  24. Münch, A., Pedregal, P. and Periago, F. (2009). Optimal internal stabilization of the linear system of elasticity, Archive Rational Mechanical Analysis, (to appear), DOI: 10.1007/s00205-008-0187-4. Zbl1169.74008
  25. Osher, S. and Fedkiw, R. (1996). Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge. 
  26. Osher, S. and Fedkiw, R. (2003). Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, Vol. 153, Springer-Verlag, New York, NY. Zbl1026.76001
  27. Osher, S. and Sethian, J.A. (1988). Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics 79(1): 12-49. Zbl0659.65132
  28. Sokołowski, J. and Żochowski, A. (1999). On the topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272. Zbl0940.49026
  29. Wang, M.Y., Wang, X. and Guo, D. (2003). A level set method for structural topology optimization, Computer Methods in Applied Mechanics and Engineering 192(1-2): 227-246. Zbl1083.74573
  30. Zolésio, J.-P. and Truchi, C. (1988). Shape stabilization of wave equation, Boundary Control and Boundary Variations (Nice, 1986), Lectures Notes in Computer Science, Vol. 100, Springer, Berlin, pp. 372-398. 

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