Grid Generation and Applications

Transcription

1 Department of Mathematics Vianey Villamizar Math 511 Grid Generation and Applications

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3 Villamizar Vianey Colloquium 2004 BYU Department of Mathematics Transformation of a Simply Connected Domain η Prototypical Antenna. N1 B C E E D T = ( ξ,η) B C D y = y ( ξ,η) A D z 1 A D F 1 Ng N 2 ξ F

4 Department of Mathematics MAA Intermountain Section Meeting 2007 Definition of the Transformation T Elliptic Grid Generators Different Approaches (a) Winslow (No Control Functions) α ξξ 2β ξη + γ ηη = 0, αy ξξ 2βy ξη + γy ηη = 0, defined for all (ξ, η) D α, β, γ are scale metric factors of T, defined by α = 2 η + yη, 2 β = ξ η + y ξ y η, γ = 2 ξ + yξ, 2 and J = ξ y η η y ξ is the jacobian of the jacobian matri, J, of T.

5 η N1 B' C' E' E D' T B C D = ( ξ,η) y = y ( ξ,η) A D z 1 A' D' F' 1 Ng N 2 ξ F

6 Department of Mathematics MAA Intermountain Section Meeting 2007 Definition of the Transformation T Elliptic Grid Generators Different Approaches (a) Winslow (No Control Functions) α ξξ 2β ξη + γ ηη = 0, αy ξξ 2βy ξη + γy ηη = 0, defined for all (ξ, η) D α, β, γ are scale metric factors of T, defined by α = 2 η + yη, 2 β = ξ η + y ξ y η, γ = 2 ξ + yξ, 2 and J = ξ y η η y ξ is the jacobian of the jacobian matri, J, of T.

7 Department of Mathematics Spring Research Conference 2007 BCGC algorithm. Numerical Solution of the BVP. Point SOR Iteration i,j = 1 2(α + γ) i,j [ α i,j ( i+1,j + i 1,j ) + γ i,j ( i,j+1 + i,j 1 ) β i,j 2 ( i+1,j+1 i+1,j 1 i 1,j+1 + i 1,j 1 ) + α i,j φ i,j ( ξ ) i,j + γ i,j ψ i,j ( η ) i,j ], where ( η ) i,j = ( i,j+1 i,j 1 )/2, (y η ) i,j = (y i,j+1 y i,j 1 )/2, ( ξ ) i,j = ( i+1,j i 1,j )/2, (y ξ ) i,j = (y i+1,j y i 1,j )/2 α i,j = (( η ) i,j ) 2 + ((y η ) i,j ) 2 β i,j = ( ξ ) i,j ( η ) i,j + (y ξ ) i,j (y η ) i,j γ i,j = (( ξ ) i,j ) 2 + ((y ξ ) i,j ) 2 i = 1,..., N 2, j = 1,..., N 1

8 Villamizar Vianey Colloquium 2004 BYU Department of Mathematics Numerical Solution of the BVP. SOR Iteration Analogously, ey i;j is obtained replacing by y in the above formula. Convergence is accelerated by updating the approimations. (k) i;j (k 1) e i;j + (1!) i;j! = (k) i;j y (k 1) ey i;j + (1!) y i;j! = Convergence: Stop Criteria. Comparison of two consecutive iterations: fi fi (k) fi (k 1) i;j fi fi ma ; 2»j»N1 1 2»i»N2 1 i;j fi < Tol (9) fi fi (k) fiy (k 1) i;j y fi fi ma ; 2»j»N1 1 2»i»N2 1 fi < Tol (10) i;j

9 Department of Mathematics MAA Intermountain Section Meeting 2007 Show MATLAB Real Time Grid Generation Winter09---MATLAB-CODES---Grid Project---SimplyConnected

10 More General than Winslow

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13 4 Antenna Winslow Grid with 5 Points Clustering at R inf 3060(20) y

14 4 Antenna FPGC Grid 3060(20) with 5 Points Clustering at R inf y

15 Multiply Connected Domains y

16 Fictitous infinite boundary 0.5 C Ω δ

17 ξ,i N 2 B' C' 4 E' Continuity Conditions C 4 C' D' 1 2 C' T = ( ξ,η) y = y ( ξ,η) y C 3 A F C C 1 2 B E 1 A' C' 3 F' 1 N 1 η,j D Computational Domain Physical Domain

18 Villamizar Vianey Colloquium 2004 BYU Department of Mathematics Boundary-Conforming Grid Curves Obtained from the Numerical Solution of the Above BVP for the Three-Leafed Rose. N 2 ξ B' C' 4 E' C 3 C' D' 1 2 C' T C 4 y y A F C 1 C 2 B E 1 A' C' 3 F' D 1 N 1 η

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21 ξ,i N 2 B' C' 4 E' C 4 C 3 C' D' 1 2 C' T = ( ξ,η) y = y ( ξ,η) y F A C C 1 2 B E 1 A' C' 3 F' 1 N 1 η,j D Computational Domain Physical Domain

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23 ξ,i N 2 B' C' 4 E' C 4 C 3 C' D' 1 2 C' T = ( ξ,η) y = y ( ξ,η) y F A C C 1 2 B E 1 A' C' 3 F' 1 N 1 η,j D Computational Domain Physical Domain

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31 Department of Mathematics Spring Research Conference 2007 Numerical Simulation: Fall2006--FinalProject-- MATLABCODES--FORTMATL--ScattWins/ScattBCGC

32 Department of Mathematics Spring Research Conference 2007 Generation of Boundary Conforming Coordinates. Multiply Connected Domain with Several Holes η,j N 2 20 Annular Cut BC BC D' T = ( ξ,η) y = y ( ξ,η) y B 2 B N 1 Computational Domain ξ,i D η-curve Physical Domain ξ-curve

33 Department of Mathematics Spring Research Conference y 0 y Figure 2: Block B 1 BCGC grid (left), and block B 2 BCGC grid (right).

34 Department of Mathematics Spring Research Conference 2007 BCGC Algorithm and the Smoothing Process (cont.) The blocks are joined. The interface is smoothed - The original domain is put back together. In general, the interface at AC is not smooth. - A region containing holes 2 (astroid) is cut out of the domain. It creates a new simply connected region S 1. The region S 1 contains the common interface of the previous blocks. - The BCGC algorithm is applied to this simply connected region. As a consequence, the grid is smoothed along the interface of blocks B 1 and B 2.

35 Department of Mathematics Spring Research Conference y 0 y Figure 2: Block B 1 BCGC grid (left), and block B 2 BCGC grid (right).

36 Department of Mathematics Spring Research Conference 2007 BCGC Algorithm and the Smoothing Process (cont.) The blocks are joined. The interface is smoothed - The original domain is put back together. In general, the interface at AC is not smooth. - A region containing holes 2 (astroid) is cut out of the domain. It creates a new simply connected region S 1. The region S 1 contains the common interface of the previous blocks. - The BCGC algorithm is applied to this simply connected region. As a consequence, the grid is smoothed along the interface of blocks B 1 and B 2.

37 Department of Mathematics Spring Research Conference y 0 y Figure 3: Non-smooth grid (left). Simply Connected Region (right).

38 Department of Mathematics Spring Research Conference Figure 4: Final Smooth BCGC Grid.

39 Department of Mathematics Spring Research Conference Show MATLAB real time grid generation for two holes: Conferences--PresAcoustics--Two-Holes S. Acosta, V. Villamizar, Grid Generation with Grid Line Control for Regions with Multiple Complely Shaped Holes, In Proceedings MASCOT06-IMACS/ISGG Workshop, R. M. Spitaleri (Ed.), Rome, Italy. Accepted for publication on February 2007.

40 Department of Mathematics Spring Research Conference 2007 Initial Boundary Value Problem: Annular Membrane W tt = c2 J 2 (αw ξξ 2βW ξη + γw ηη ) + c2 J 3 (α y ξξ 2β y ξη + γ y ηη ) ( η W ξ ξ W η ), + c2 J 3 (α ξξ 2β ξη + γ ηη )(y ξ W η y η W ξ ), W(ξ, 1, t) = 0, W(ξ, n 2, t) = 0, t > 0, W(ξ, η, 0) = f(ξ, η)), W t (ξ, η, 0) = g(ξ, η), (ξ, η) D, Initial condition: f(ξ(, y), η(, y)) = 0, if r = 2 + y 2 17; sin( (20 r)), 17 if 17 < r 20.

41 Department of Mathematics Spring Research Conference 2007 Time-Dependent Numerical Scheme t (i, j) t = t n+1 (i-1, j+1) (i, j+1) (i+1, j+1) (i-1, j) (i, j) (i+1, j) t = t n η (i-1, j-1) (i, j-1) (i+1, j-1) (i, j) t = t n-1 ξ

42 Department of Mathematics Spring Research Conference 2007 Numerical Simulation: Show SRC Movie