Singularity Phenomena in Ray Propagation. in Anisotropic Media

Transcription

1 Singularity Phenomena in Ray Propagation in Anisotropic Media A. A. Melikyan Institute for Problems in Mechanics, Moscow N. D. Botkin V. L. Turova Center of Advanced European Studies and Research, Bonn PhysCon2005, St.Petersburg, August 24-26, 2005

2 caesar Research areas Multidisciplinary research Smart materials combined center employing about with nanotechnologies 200 researchers Ergonomics: cooperation between humans and machines 1999 foundation of caesar; Coupling of biological and 2003 moving in the new building electronic systems

3 Development of an acoustic wave sensor for biological and medical applications (microbalance) Motivation Operation principles 1. Excitation of acoustic shear waves due to piezolectric properties of a substrate Quartz crystal Y-cut 2. Selective binding of the biomolecules from a contacting liqiud due to aptamers 3. Mass estimation through the measurement of the phase shift in the electric signal Biomolecules Aptamers

4 Methods of modelling Direct modelling with finite elements (recources consuming because of short wave lengths) Dispersion relations (dependence of the wave propagation velocity on the frequency) for multi-layered anisotropic structures Description of wave fronts using Hamilton-Jacobi equations

5 Waves description using optimality principles 1. Characteristic surfaces Wave surface Velocity surface n n wave front V e n V e 0 V V e energy velocity n wave vector 0 V e. n = V - phase velocity

6 Slowness surface (inverse through the origin to the velocity surface) m = n / V V e m 0 n The energy velocity is normal to the slowness surface at all points.

7 2. Fermat s principle Or, using parametrization Thus, feasible rays are solutions of the Euler equation Applicable if V e is a well-defined function.

8 3. Pontryagin s maximum principle p adjoint vector, wave surface Applicable if the wave surface does not have swallow tails.

9 Complicated structure of wave surfaces Slowness surface Wave surface Example of a wave surface for a crystal with cubic symmetry θ 1 a c 3 2 b θ θ a c 0 Variation of the curvature of the slowness surface can yield multiple values for the energy velocity (swallow tails in the wave surface).

10 Description of wave propagation using Hamilton-Jacobi equations Elasticity equations for anisotropic inhomogeneous media WKB-approximation ε << 1 is a small parameter

11 Eikonal equation is equivalent to the following three equations solve the eigenvalue problem where with, which gives three phase velocities (and three polarizations) for the propagation direction n. Using the transformation yields H.-J. equation

12 If the wave surface has swallow tails, the Hamiltonian is non-convex in the impulse variable which motivates to use differential games instead of classical optimality principles

13 Usage of differential games The first player minimizes and the second player maximizes the time of attaining a given terminal set M. Value function T(x) satisfies the Hamilton-Jacobi equation in all points x where T(x) is differentiable. In other points, this equation holds in a viscosity sense. If we find a differential game with the dynamics that provides the equality then level sets of T in the wave propagation problem. yield wave fronts

14 Differential games with simple dynamics are appropriate P and Q are symmetric about the origin convex sets Hamilton-Jacobi equation can be rewritten as or Comparison with the eikonal equation yields the condition on P and Q :

15 Application to the propagation of surface acoustic waves Multi-layered structure x 3 Velocity contour for shear surface acoustic waves x 2 Fluid Protein layer Gold layer SiO 2 - guiding layer Quartz crystal x 1 c α x 2 x 1 N.D. Botkin et al. Numerical computation of dispersion relations in multi-layered anisotropic structures. In: Proceedings of Nanotech, Boston, 2004 x 2 Slowness surface x 1

16 1. Approximation of the phase velocity contour Choice of P Choice of Q The result P Q

17 2. Numerical solution to the differential game M u minimizes and v maximizes the time of attaining the terminal set M Backward step-by-step computation of the attainability set on interval [0,θ] with a step V.S.Patsko, V.L.Turova Numerical solution of two-dimensional differential games, Preprint, Institute of Math. & Mech., Ekaterinburg, 1995.

18 3. Investigation of singular lines A.A. Melikyan. Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games, Boston: Birkhauser, 1998.

19 Wave fronts and singular lines Behavior of optimal trajectories

20 Acoustic waves in anisotropic crystals obey differential games!