A Surface Grid Generation Technique for Practical Applications of Boundary Element Methods

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A SURFACE GRID GENERATION TECHNIQUE
FOR PRACTICAL APPLICATIONS OF
BOUNDARY ELEMENT METHODS
J.Baltazar1* and L.Eça1
1MARETEC/IST
Departamento de Engenharia Mecânica,
Instituto Superior Técnico, Lisboa, Portugal
Conferência Nacional de Métodos Numéricos em
Mecânica dos Fluidos e Termodinâmica 2006
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006

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 A flexible surface grid generation technique for
structural grids that allows the combination of
different grid topologies.
Motivation

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Surface Grid Generation
 Introduction of an intermediate coordinate transformation
to make the grid generation process independent of the
surface definition (Baltazar and Eça 2004).
 Combination of different grid topologies by splitting the
computational domain into several regions.

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Coordinates Transformation
x
y
z
s1
s2
1
2
3
4
1
2
3
4


1
2
3
4
s1
s2
1
2
3
4
I
J
Surface
Definition (A)
Aditional Coordinate
Transformation (B)
Two-Dimensional
Grid Generation (C)

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Surface Definition (A)
 
 
 
1 2
1 2
1 2
,
,
,
x x s s
y y s s
z z s s
 




 Analytical surface definition.
 Bi-cubic spline representation of a set of NXNY grid nodes.
 B-splines representation, NURBS (Farin 1990).

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Transformation between the Dependent
Variables of the Grid Generator (I,J) and the
coordinates of the Surface Definition(s1,s2) (B)
 The boundaries of the region to discretize correspond to
the lines I=1, J=1, I=NT, J=NT, with NT arbitrary.
 Definition of (s1,s2) in the (I,J) domain:
• Definition of s1 and s2 at the four boundaries;
• Initial approximation of s1 and s2 in the interior nodes
using linear transfinite interpolation;
• Regularization of the coordinate transformation to
obtain a smooth grid in the physical space;
 Calculation of (s1,s2) by bi-linear interpolation.

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Two-Dimensional Grid Generation (C)
 The third coordinate transformation corresponds to the
generation of a two-dimensional grid between the
coordinates (I,J) and the computational domain (,).
 The computational domain is split into several regions.
Specification of the point distribution on the four boundaries
of each region.
 Algebraic grid generator for each region:
• Linear transfinite interpolation;
• 1-D cubic interpolation, crucial for the connectivity
between the different grids.

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Examples of Application
Wing of Elliptical Planform
Nearly-
Orthogonal Grid
New Grid
Arrangement

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Wing of Elliptical Planform
Parametric Domain (I,J)
I
J
50 100 150 200 250
50
100
150
200
250
New Grid Arrangement
I
J
50 100 150 200 250
50
100
150
200
250
Nearly-Orthogonal Grid

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Propeller DTRC P4119
Nearly-Orthogonal Grid New Grid Arrangement

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I
J
50 100 150 200 250 300
50
100
150
200
250
300
s1
s2
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Parametric Domain (I,J) Parametric Domain (s1,s2)

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Conventional Grid Nearly-Orthogonal Grid

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Nearly-Orthogonal Grid with Control of the Grid Lines Displacement

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Final Remarks
 The present technique is flexible and robust.
 The subdivision of the computational domain in several
regions and a careful control of the slope of the grid
lines at the boundaries of each region allows the
generation of grids with different grid line patterns.
 More sofisticated grid generation methods can be used
in the present technique.

A Surface Grid Generation Technique for Practical Applications of Boundary Element Methods

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