Grid Generation and Applications

Transcription

1 Grid Generation and Applications

2 Acoustic Scattering from Arbitrary Shape Obstacles.5 Fictitous infinite boundary.5 C.5.5 Ω δ

3 Boundary Value Problem: Acoustic Scattering (p sc ) tt = c p sc, x D, t >, p sc (x s, t) = p inc (x s, t), x s C i, (p sc ) t + c(p sc ) r, when r, p sc (x, ) = f(x), (p sc ) t (x, ) = g(x), x D. Incident wave: p inc (x, t) = I e ik(x cos δ+y sin δ) e iwt,

4 Vibration of Complexly Shaped Annular Membranes time=.5 5 y axis x axis

5 Generation of Boundary Conforming Coordinates. Multiply Connected Domain: Three-Leafed Rose. ξ,i N B' C' E' C ( x,y ) ij ij ( ξ,η ) i j C' C' D' T x = x ( ξ,η) y = y ( ξ,η) y C 3 F A C C E B A' C' 3 F' N η,j D x Computational Domain Physical Domain

6 Definition of the Transformation T. Elliptic Grid Generators BVP governed by the quasi-linear elliptic system of equations αx ξξ βx ξη + γx ηη = J (Px ξ + Qx η ), αy ξξ βy ξη + γy ηη = J (Py ξ + Qy η ), defined for all (ξ,η) D α, β, γ are scale metric factors of T, defined by α = x η + y η, β = x ξ x η + y ξ y η, γ = x ξ + y ξ, and J = x ξ y η x η y ξ is the jacobian of the jacobian matrix, J, of T.

7 Boundary and Interface Conditions: Standard Approach ξ,i N B' C' E' Continuity Conditions C C' D' C' T x = x ( ξ,η) y = y ( ξ,η) y C 3 A F C C B E A' C' 3 F' N η,j D x Computational Domain Physical Domain

8 Winslow s Grid Generation Algorithm P = and Q = αx ξξ βx ξη + γx ηη = αy ξξ βy ξη + γy ηη =

9 Boundary-Conforming Grid Obtained from Winslow s Algorithm N ξ B' C' E' C 3 C' D' C' T C y y A F C C B E A' C' 3 F' D N η x x

10 Grid Lines Control from Distribution of Nodes on a Branch cut ξ,i N B' C' E' Dirichlet Conditions C C' D' C' T x = x ( ξ,η) y = y ( ξ,η) y C 3 F A C C E B A' C' 3 F' N Computational Domain η,j D x Physical Domain

11 Initial Cluster of Grid Points and Modified Winslow s Grid y y x a) b) x

12 Grid Quality Comparison ADO = (N )(N ) N i= N j= ( 9 θ i,j ) ( θ i,j = arccos β i,j (α i,j γ i,j ) / ). Grid Type Mapped BCGC Table : Comparison of Average Deviation from Orthogonality (ADO) between ANSYS mapped and BCGC grids with control parameters.5 7/.8 3

13 Boundary Value Problem in Curvilinear Coordinates: Annular Membrane

14 W tt = c J (αw ξξ βw ξη + γw ηη ) + c J 3 (α y ξξ β y ξη + γ y ηη ) (x η W ξ x ξ W η ), + c J 3 (α x ξξ β x ξη + γ x ηη )(y ξ W η y η W ξ ), W(ξ,, t) =, W(ξ, n, t) =, t >, W(ξ, η,) = f(ξ, η)), W t (ξ, η,) = g(ξ, η), (ξ, η) D, Initial condition: f(ξ(x, y), η(x, y)) =, if r = x + y 5; sin( 5 ( r)), if 5 < r.

15 Time-Dependent Numerical Scheme t (i, j) t = t n+ (i-, j+) (i, j+) (i+, j+) (i-, j) (i, j) (i+, j) t = t n η (i-, j-) (i, j-) (i+, j-) (i, j) t = t n- ξ

16 Cylindrical Scatterer with Circular Cross-Section. Amplitude of the total pressure field for the circular cross-section using a 8 8 5/.3 3 Winslow vs BCGC grid with t =.

17 y- axis x x- axis y y.8 y-axis x x-axis

18 Radar Cross-Section. Far Field Approximation:. p sc (r, θ) eikr r / ( πk e iπ/ m= B m cos mθ ) + O(/r 3/ ). () Radar Cross-Section: Coefficient of Leading Order Term A (θ) = πk e iπ/ B m cos mθ. () m=

19 .5 Numerical Analytical.5 Numerical Analytical Scattering Cross Section.5 Scattering Cross Section θ 3 3 θ Figure : Comparison of numerical and analytical solution of SCS for the circular cylinder scatterer on WoC and BCGC grids: (a) 8 8 WoC grid (b) 8 8 BCGC (. 5/.3 3) grid

20 Total pressure field from Complexly Shaped Scatterers

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22 y axis x axis x axis 6 8.5

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24 y axis x axis x axis 6 8

25 Radar Cross-Section from Numerical Approximation of Pressure Field where and  (θ) = πk e iπ/ N / m= N / ˆbm = ĉ m /H () m (kr ) ĉ m = N (p sc ) N t i,n N e imθ, l= ˆbm ( i) m e imθ, (3) for m = N /,...N /.

26 (a).5 (b) Scattering Cross Section θ Scattering Cross Section θ Figure : Numerical approximation of SCS using BCGC grids: (a) fourpetal rose (b) seashell

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28 y axis x axis

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30 y axis x axis