pn 5013 aerodinamika numerik-lecture11 algebraic grids

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CFD LECTUREAE 4012 NUMERICAL AERODYNAMICS

Lecture 11 : Computat ion al 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.

7/17/2019 PN 5013 Aerodinamika Numerik-Lecture11 Algebraic Grids

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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