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

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This paper presents a flexible surface grid generation technique for structured grids. The computational domain is sub-divided in regions, where different grid generation techniques may be applied to obtain the desired grid line spacing and/or orthogonality. Examples of application are presented for an elliptical wing and conventional and skewed marine propeller blades.

Engineering

CNMNMFT 2006 1
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

CNMNMFT 2006 2
 A flexible surface grid generation technique for
structural grids that allows the combination of
different grid topologies.
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
Motivation

CNMNMFT 2006 3
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
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.

CNMNMFT 2006 4
Coordinates Transformation
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
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)

CNMNMFT 2006 5
Surface Definition (A)
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
 
 
 
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).

CNMNMFT 2006 6
Transformation between the Dependent
Variables of the Grid Generator (I,J) and the
coordinates of the Surface Definition(s1,s2) (B)
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
 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.

CNMNMFT 2006 7
Two-Dimensional Grid Generation (C)
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
 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.

CNMNMFT 2006 8
Examples of Application
Wing of Elliptical Planform
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
Nearly-
Orthogonal Grid
New Grid
Arrangement

CNMNMFT 2006 9
Examples of Application
Wing of Elliptical Planform
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
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

CNMNMFT 2006 10
Examples of Application
Propeller DTRC P4119
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
Nearly-Orthogonal Grid New Grid Arrangement

CNMNMFT 2006 11
Examples of Application
Propeller DTRC P4119
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
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)

CNMNMFT 2006 12
Examples of Application
Propeller DTRC P4842
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
Conventional Grid Nearly-Orthogonal Grid

CNMNMFT 2006 13
Examples of Application
Propeller DTRC P4842
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006
Nearly-Orthogonal Grid with Control of the Grid Lines Displacement

CNMNMFT 2006 14
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.
FCT-UNL, Monte da Caparica, 8 e 9 Junho
©APMTAC,Portugal, 2006

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