pn 5013 aerodinamika numerik-lecture11 algebraic grids

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

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Moelyadi - 15.04.2014

CFD LECTUREAE 4012 NUMERICAL AERODYNAMICS

Lecture 11 : Computat ion al Grids



Moelyadi - 15.04.2014

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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Transfinite Interpolation

TFI :

•Grids can conform to specified boundaries.

•Grid Spacing is under control

• Easily programmed

• Very computationally efficient.



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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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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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R(0) R(1)

η 1- η

V(η)

)1()0()1()()( R R RV

10



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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



Moelyadi - 15.04.2014



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



Moelyadi - 15.04.2014



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



Moelyadi - 15.04.2014



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

pn 5013 aerodinamika numerik-lecture11 algebraic grids

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