Nonlinear Solitary Wave Basis Functions for Long Time Solution of Wave Equation

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1 Nonlinear Solitary Wave Basis Functions for Long Time Solution of Wave Equation John Steinhoff Subhashini Chitta Nonlinear Waves --Theory and Applications Beijing - June, 2008

2 Objective Simulate short pulse over long distance As in shock capturing, conserve integrals such as centroid, energy, etc. in normal direction

3 Conventional Methods Conventional Discretization - Not feasible due to long distance Lagrangian Ray Tracing - Difficulty with diverging waves and interference - Difficulty with incorporation of created waves Free Space Greens Function - Not feasible due to varying medium properties and reflections General Greens Function - Very expensive

4 Unsuitability of Direct Discretization of PDE Short initial pulse Pulse spreads even after short propagation and high order discretization Higher order cannot help - only meaningful if significant number of cells in pulse Not Feasible in 3D Requires too many grid points

5 Approach Modify wave equation to generate localized thin nonlinear Solitary Waves (SW s) SW s serve as Basis Functions - stable co-dimension 1 surfaces that propagate/scatter/reflect accurately and carry properties of pulse Implement discretization scheme that preserves these properties

6 Effects to be Included: Long range propagation ( 10 6 λ) in arbitrary directions Varying index of refraction Multiple reflections from complex features Ability to capture and simulate wave on local fine grid and easily project onto global coarse grid

7 Solitary Waves Propagate over indefinitely long distance with zero numerical expansion (dispersion, diffusion) Propagate with fixed (computational) profile Centroids generate characteristic surfaces

8 Simulation on Eulerian Grid Physical pulse width can be much smaller than computed pulse (as in shock capturing) Physical properties of actual pulse such as width (w), amplitude (A), direction ( k r ), etc. can be propagated on grid nodes carried by computed pulse k r w, A,

9 Simple Example Linear 1-D Wave Equation Propagation of short pulse 2 φ c 2 2 t x = φ

10 Nonlinear Solitary Wave Dynamics Solve Propagation PDE with Relaxation Term: Modified Wave Equation tφ = c xφ + t xf F defines structure of pulse Decouples structure relaxation from propagation dynamics Heat Equation in pulse frame: ( F 0)

11 Requirements for F Homogeneous degree 1 - Dynamics independent of scale of φ Contracting (negative dissipation) at longer wavelengths Expanding (saturating) at shorter wavelengths Nonlinear Waves do not interact ( no phase shift or amplitude exchange) in spite of nonlinearity

12 Solitary Wave Equation (1D Example) 2 φ c 2 2 φ 2 t x t x = + Conserves mass (A), Velocity (V), Width (W) F A = φ dx < x >= xφdx A d < x > V = = & φdx/ A dt W x x x x φdx A const ( ) = = /.

13 Features 2 F 0 2x x F 1 2 x F 2 F0, F1 F 2 F = F0 + F1+ F2 - Anti dissipative at small k - Dissipative at large k - Transfers small k to large k - Linear - Nonlinear Analogous to Cahn Hilliard Equation(1958)

14 Structure Factor (PDE) F = αλφ α 2 φ x Basic Form 2 ( φ ) 2 x φ After relaxation sec h ( ( x ct )) γ = λ φ φ γ 0 ( ) x F = α ψ λψ ψ = φ ψ 2 1, 2 λ, αγ>, 0

15 Discretized Structure Factor Good results with ({ }) F = φ Φ φ Φ - type of non-linear mean Do not get instabilities Taylor expansion results in desired PDE Persists at highly discrete level (2~3 grid cells)

16 Discretized 1-D Wave Equation ( ) a ( F F ) φ 2φ + φ = ν δ φ + δ δ n+ 1 n n n 2 n n 1 j j j j j F = μφ εφ ( 2 ) cδt f f f + f, ν =, a = h Δt h 2 j j+ 1 j j 1 2 n Φ j = φ φ φ n n n j+ 1 j j 1

17 Without Confinement With Confinement ε = 0 ~1.5 μ < ε < ~4μ Fifteenth Pass Periodic Boundary Conditions

18 No Interaction Effects (even though nonlinear) Centroid Motion (Colliding Pulses)

19 Pulse Interaction Γ φ φ + + Γ = c + 2 φ 2 2 φ 2 t x t x F t I F 0 ( + I F = F φ + φ + φ ) x ( + φ ) ( φ ) = F + F + F I

20 Change in φ ( ) I I t c x φ = t xf After interaction (Born Approximation) φ I δa δ I ± = dxφ = Γ ± I 0 = = ± I ± x dxxφ A Γ ± Γ 0 Since I F 0 in far field

21 Multidimensions (2D example) φ = 2φ φ + ν φ φ 1 l Φ= N 1 2 ( μφ εφ) 0 μφ εφ Multidimensional Harmonic Mean of (N) neighboring values φ Ach / [ γ ( z z)] ij 0 ε γ = f μ z = xcosθ + y i ( n n ) + Φ n+ 1 n n n aδ 2 μφ ε i, j i, j i, j i, j n i, j i, j j sinθ

22 Demonstration of Linear Interaction Amplitude Contours grid

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24 Acoustic Pulse Scattering from Small Cylinder INITIAL PULSE

25 3-D Pulse Propagation grid

26 Validation Varying Index of Refraction Comparison with Ray Tracing Normalized Index of refraction Wave Contours vs Ray Markers

27 Wave Propagation in Long Duct Effect of Varying Index of Refraction

28 Focusing Moments conserved through focus 2 2 n δ x = 2ν 2 n 2 2 n δ y = 2ν 2 n 2 n n δ xy = 0

29 Focusing

30 Reflection from Immersed Surface Surface Definition For Approximate B.C. s φ set to zero every time-step inside boundary 2 F x Confinement term ( ) t - eliminates tangential stair-case by smoothing - eliminates spreading at boundary by compressing in normal direction

31 Scattering from Complex Objects View Plane

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33 Fine to Coarse Grid Projection Near Field (fine) Far field (coarse)