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2009 | 19 | 1 | 15-37
Tytuł artykułu

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

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
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.
Rocznik
Tom
19
Numer
1
Strony
15-37
Opis fizyczny
Daty
wydano
2009
otrzymano
2008-02-11
poprawiono
2008-08-07
Twórcy
  • Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UMR CNRS 6623, 16, route de Gray, 25030 Besançon, France
Bibliografia
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Cohen, G.C. (2002). Higher-order Numerical Methods for Transient Wave Equations, Scientific Computation, Springer-Verlag, Berlin.
  • 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.
  • Delfour, M. and Zolesio, J. (2001). Shapes and Geometries - Analysis, Differential Calculus and Optimization, SIAM, Philadelphia, PA.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Henrot, A. and Pierre, M. (2005). Variation et optimisation de formes-Une analyse géométrique, Mathématiques et Applications, Vol. 48, Springer, Berlin.
  • Lions, J. and Magenes, E. (1968). Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris.
  • López-Gómez, J. (1997). On the linear damped wave equation, Journal of Differential Equations 134(1): 26-45.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Osher, S. and Fedkiw, R. (1996). Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge.
  • Osher, S. and Fedkiw, R. (2003). Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, Vol. 153, Springer-Verlag, New York, NY.
  • 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.
  • Sokołowski, J. and Żochowski, A. (1999). On the topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.
  • 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.
  • 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.
Typ dokumentu
Bibliografia
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