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CFD LECTUREAE 4012 NUMERICAL AERODYNAMICS
Lecture 11 : Computat ion al Grids
7/17/2019 PN 5013 Aerodinamika Numerik-Lecture11 Algebraic Grids
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Computational Modelling
Real Words Physics Numerical Simulation
Flow Models
Dynamic
approximation
Spatial
approximation
Steadiness
approximation
Space
discretizationMesh definition
Equation
discretizationDefinition of
Numerical schemes
Mathematical
Model
Discretization
Techniques
Resolution of
discrete systemof Equations.
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Road Map
Computational Grids
Structured Algebraic
PDE
PointSurface
H-GridO-Grid
C-Grid
Single Block
Multi-Blocks
Topology
definition
Geometry
definition
Measurementof Grid Quality
HybridChimera
Grid
Generation
Grid Quality
and Adaptation
Unstructured
DelaunayTriangulation
Advancing Front
Quadtree
Adaption Grid
Dynamic /
Moving Grid
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Geometry definition
Grids Generation
Topology definition
Grid Quality and Adaptation
1
2
3
4
Computational grids
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Grid Topology
H-Grid
O-Grid
C-Grid
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Physical and Computational Domains
Physical Domain : x, y, z
Computational Domain : ξ, η, ζ Physical Domain
Computational Domain
O-Grid Topology
x
y
ξ
η
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Topology for Complex Geometry
O-C-Multiblock Grid Topology
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Grid Types
Structured grid Unstructured grid
Hybrid gridChimera grids
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Techniques for Structured Grid Generation
Algebraic Methods
Define boundaries
Distribute points along boundaries
Use Interpolation
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Techniques for Structured Grid Generation
Transfinite Interpolation
TFI :
•Grids can conform to specified boundaries.
•Grid Spacing is under control
• Easily programmed
• Very computationally efficient.
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Techniques for Structured Grid Generation
Algebraic Methods → Interpolation techniques
Uni-directional interpolation
R(0)
R(1)
ξ 1- ξ
U(ξ)
)1()0()1()()( R R RU
10
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Techniques for Structured Grid Generation
Algebraic Methods → Interpolation techniques
Uni-directional interpolation
R(0)
R(1)
ξ 1- ξ
U(ξ)
R(0)
R(1)
η 1- η
V(η)
)1()0()1()()( R R RU
10
)1()0()1()()( R R RV
10
R(0) R(1)
ζ 1- ζ
W(ζ) )1()0()1()()( R R RW
10
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Techniques for Structured Grid Generation
Algebraic Methods → Interpolation techniques
Uni-directional interpolation
R(0) R(1)
η 1- η
V(η)
)1()0()1()()( R R RV
10
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Techniques for Structured Grid Generation
Algebraic Methods → Interpolation techniques
Bi-directional interpolation
R(0,η) R(1,η)
ξ 1- ξ
R(ξ,η)
R(ξ,0)
R(ξ,1)
η
1- η
)1,()0,()1(),1(),0()1(),(),(),(
R R R RV U R
10 10
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Techniques for Structured Grid Generation
Algebraic Methods → Interpolation techniques
Corner effect of Bi-directional interpolation
R(0,η) R(1,η)
ξ 1- ξ
R(ξ,η)
R(ξ,0)
R(ξ,1)
η
1- η )1,0()1,0()1,0( R R R
1,0
)]1,1()1,0()1()0,1()1()0,0()1()1[(
)1,()0,()1(),1(),0()1(
),(),(),(),(
R R R R
R R R R
UV V U R
10 10
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Techniques for Structured Grid Generation
Algebraic Methods → Interpolation techniques
Multi-directional interpolation
R(0,η,ζ)
R(1,η,ζ)
ξ 1- ξ
R(ξ,η,ζ)
R(ξ,0,ζ)
R(ξ,1,ζ)
η
1- η
),,(
),,(),,(),,(),,(),,(),,(),(
UVW
VW UW UV W V U R
10 10
R(ξ,η,0)
R(ξ,η,1)
10
),,1(),,0()1(),,( R RU
),1,(),0,()1(),,( R RV
)1,,()0,,()1(),,( R RW
),0,1()1(),1,0()1(
),1,1(),0,0()1)(1(),,(
R R
R RUV
)0,,1()1()1,,0()1(
)1,,1()0,,0()1)(1(),,(
R R
R RUW
)0,1,()1()1,0,()1(
)1,1,()0,0,()1)(1(),,(
R R
R RVW
)1,1,0()1()1,0,1()1()0,1,1()1(
)1,1,1()1,0,0()1)(1()0,1,0()1()1(
)0,0,1()1)(1()0,0,0()1)(1)(1(),,(
R R R
R R R
R RUVW
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Techniques for Structured Grid Generation
Algebraic Methods → Interpolation techniques
Determination of the blending functions ξ, η, ζ
)(1
0,max,,00,
,
, i ji ji
ji
ji g g h g P
10
10
10
)(10,max,,00,
,
, i ji ji
ji
ji g g h g P
)(*)(1 ,0max,0,max,, j jii ji ji hh g g P