In this paper we examine the errors due to different surface discretizations in the BEM solution of the potential flow past an ellipsoid. Three different approaches are considered: a low order method (LO), a second order normal calculation combined with the low order formulation (HO-normal) and a higher order panel geometry approximation (HOgeom). The potential, velocity and pressure distribution obtained with the different approaches are compared with the analytical solution for a wing-like ellipsoid in the tip region using conventional and orthogonal surface grids. With the LO method, it is shown that the use of orthogonal grids introduces larger errors in the solution near the tip than the use of conventional grids. These errors are significantly reduced already by using a higher order approximation to the element geometry, which improves the surface metrics evaluation.

1. A Study on the Accuracy of Low
and Higher Order BEM in Three-
Dimensional Potential Flows Past
Ellipsoids
J.Baltazar1, J.A.C.Falcão de Campos1 and J.Bosschers2
1Department of Mechanical Engineering,
Instituto Superior Técnico, Portugal
2Maritime Research Institute Netherlands (MARIN),
the Netherlands
5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004

2. Motivations
BEM potential flow calculation has been
extensively used in the analysis of lifting
surfaces and marine propellers
5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Objectives
Analysis and quantification of the numerical
discretization error

3. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
BEM Formulation
Mathematical Model
02
 
 
UV











r
nU
n


if,0

Velocity field:
Laplace Equation:
Boundary Conditions:
Fredholm Integral Equation for Morino Formulation:
   
   
 
















BS
qq
dS
qpRnqpRn
qp
,
1
,
1
2


X Y
Z
U

4. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
BEM Formulation
Numerical Implementation
Low Order Method (LO):
• Bi-linear elements with constant source and dipole distributions.
• Analytical evaluation of the dipole and source influence coefficients
(Morino and Kuo, 1974). Far field multipole expansions were used.
Higher Order Normal Method (HO-normal):
• Bi-quadratic elements defined by nine nodes for the calculation of the
surface metrics at the central point.
• Combination between the LO formulation with a second order normal in
the Neumann boundary condition and tangential velocity.
Higher Order Geometry Method (HO-geom):
• Bi-quadratic elements with constant source and dipole distributions.
• Numerical calculation of the influence coefficients.

5. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Test Cases
•Ellipsoid: with a=1, b=2, c=0.1.
•Conventional grid:
Full cosine stretching is used in spanwise (y) and
chordwise (x) directions.
•Orthogonal grid:
Algebraic procedure constructed by Eça 2002. The
stretching is applied on the surface coordinate lines.
•Discretizations: 16x8, 32x16, 64x32 and 128x64.
1
222


















c
z
b
y
a
x

6. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Results
Approximation of the Surface at the Tip
X
Y
Z
x/a
nx
-0.0015 -0.0010 -0.0005 0.0000 0.0005 0.0010 0.0015
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
hyperboloidal panels
bi-quadratic panels
Exact Surface

7. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Results
Pressure Distribution at Last Panel Strip
x/a
-Cp
0.000 0.005 0.010 0.015 0.020 0.025
0.12
0.14
0.16
0.18
0.20
LO
HO-normal
HO-geom
Analytical
x/a
-Cp
0.0000 0.0005 0.0010 0.0015
-0.05
0.00
0.05
0.10
0.15
0.20
LO
HO-normal
HO-geom
Analytical
Conventional Grid Orthogonal Grid

8. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Results
Error Norms for the Conventional Grid
N
1/2
||e
||
20 40 60 80 100
10
-5
10
-4
10-3
10
-2
L2
: LO
L2
: HO-normal
L2
: HO-geom
L: LO
L
: HO-normal
L
: HO-geom
N
1/2
||eCp
||
20 40 60 80 100
10
-3
10
-2
10-1
10
0
L2
: LO
L2
: HO-normal
L2: HO-geom
L: LO
L
: HO-normal
L
: HO-geom

9. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Results
Error Norms for the Conventional Grid
N
1/2
||e/s1
||
20 40 60 80 100
10
-3
10
-2
10
-1
L2
: LO
L2
: HO-normal
L2
: HO-geom
L
: LO
L: HO-normal
L
: HO-geom
N
1/2
||e/s2
||
20 40 60 80 100
10
-4
10
-3
10-2
10
-1
L2
: LO
L2
: HO-normal
L2: HO-geom
L
: LO
L
: HO-normal
L: HO-geom

10. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Results
Error Norms for the Orthogonal Grid
N
1/2
||e
||
20 40 60 80 100
10
-5
10
-4
10-3
10
-2
L2
: LO
L2
: HO-normal
L2: HO-geom
L: LO
L
: HO-normal
L
: HO-geom
N
1/2
||eCp
||
20 40 60 80 100
10
-3
10
-2
10-1
10
0
L2
: LO
L2: HO-normal
L2: HO-geom
L
: LO
L
: HO-normal
L
: HO-geom

11. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Results
Error Norms for the Orthogonal Grid
N
1/2
||e/s1
||
20 40 60 80 100
10
-3
10
-2
10-1
10
0
L2: LO
L2
: HO-normal
L2
: HO-geom
L
: LO
L
: HO-normal
L: HO-geom
N
1/2
||e/s2
||
20 40 60 80 100
10
-4
10
-3
10-2
10
-1
L2
: LO
L2
: HO-normal
L2
: HO-geom
L
: LO
L
: HO-normal
L
: HO-geom

12. 5th International Conference on
Boundary Element Techniques
Lisbon, Portugal, 21-23 July 2004
Conclusions
• LO geometry approximation leads to large errors in the tip
region for the orthogonal grid due to large discretization error
and large error in the normal in the tip panels.
• The use of a HO geometry approximation for the normal
(HO-normal) already improves the results for the orthogonal
grid significantly, without additional computational burden.
• The use of a complete second order geometry (HO-geom)
introduces an additional but less significant improvement in
the potential flow solution.
•The perturbation velocity along the s2 direction is still not in
the asymptotic convergence region due to the relative large
panel width in s2 direction.