a 3d potential-based and desingularized high order panel method

Ocean Engineering 28 (2001) 1499–1516

A 3D potential-based and desingularized highorder panel method

Jen-shiang Kouh *, Jyh-bin SuenDepartment of Naval Architecture and Ocean Engineering, National Taiwan University, 1 Roosevelt

Road, Section 4, Taipei, Taiwan

Received 7 February 2000; accepted 19 May 2000

Abstract

In this paper, a novel high order panel method based on doublet distribution and Gaussianquadrature was adopted to deal with the potential flow problem. In the geometry representationwe employed both the exact surface and NURBS surface form to construct the surface panel.These data were calculated directly from the mathematical shape definition. Furthermore, nofixed order of doublet density distribution was assumed on each panel. Not only the numberof panels could be chosen, but also the Gaussian order of each panel. The numerical resultsfor sphere, ellipsoid and Wigley hull demonstrated here indicated that the present method wasadapted to the potential flow problem. Moreover, the NURBS surface geometry representationwas capable of further potential flow optimal calculation. 2001 Elsevier Science Ltd. Allrights reserved.

Keywords: High order panel method; Potential-based panel method; Doublet; NURBS surface

1. Introduction

The present method is a surface panel method based on a body perturbation poten-tial function. In computational hydrodynamics, the surface panel method is a highlyeffective means of computing non-lifting potential flow around arbitrarily three-dimensional bodies such as ship hulls. Its simple theoretical principle and easynumerical process have made it ideal for modeling hydrodynamic problems.

Different numerical schemes concerning the geometrical description of the body

* Corresponding author.E-mail address: [email protected] (J.-s. Kouh).

0029-8018/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S 00 29 -8018( 00 )0 0069-X

1500 J.-s. Kouh, J.-b. Suen / Ocean Engineering 28 (2001) 1499–1516

surface are adopted for the panel methods. Hess and Smith (1964), Webster (1975),Jensen (1988) and Soding (1993) use plane panels, while Johnson (1980) and Kehret al. (1996) use quadratic planes and Hsin (1994) uses hyperboloidal panels. Thesemethods require a procedure by which the original surface is approximated by alarge number of panels. Furthermore, a simple source density distribution is assumedover each panel. Hess and Smith, as well as Jensen and Hsin, assume constant sourcedensity, while Johnson and Kehr assume it to be a linear function. Webster andSoding place the sources not directly on the body surface, but inside the body. Withregard to fulfilling the Neumann boundary condition, all the above schemes use acollocation method except that of Soding.

Since all the above methods use an approximation of the surface geometry, errorsmay occur such as leakage in the plane panels of Hess and Smith. Although quadraticpanels used by Johnson and Kehr are better for representing curved surfaces, theydon’t ensure first derivative continuity at the boundary between panels. In addition,problems arise when a simple source density distribution is assumed on each panel,because a jump in source density across panel boundaries exists also in this case.

Gaussian quadrature to discretize the Fredholm integral equation of the secondkind over the body surface is used to develop the present method. The computationalpoints called Gaussian quadrature points are calculated directly from both a math-ematical surface definition and a Non-Uniform Rational B-Spline (NURBS) surfaceof the body.

The Gaussian quadrature points are also used as collocation points, and the doubletdensity at these points is determined by the Neumann boundary condition. There isno assumption about the doublet density distribution within a panel. The body surfacedepending on its complexity can be dealt with as a single surface, or it can besubdivided into a number of panels. The number of Gaussian quadrature points oneach panel, which corresponds to the order of Gaussian quadrature, can be properlyadjusted in accordance with the panel geometry.

Table 1 is a summary of the relative high order panel methods and their main fea-tures.

2. Governing equation

A submerged and closed body is fixed in a uniform flow with velocity›

V�=(U,0,0). The fluid is assumed to be inviscid and incompressible and the flow isirrotational, in other words, a potential flow. Hence, the flow field is governed by avelocity potential � satisfying the Laplace equation

�2��0 (1)

Applying Green’s theorem to � on the boundary surface Sb at its distribution pointp, for the arbitrary point q in the three-dimensional flow field, we obtain the solutionof Eq. (1) as

1501J.-s. Kouh, J.-b. Suen / Ocean Engineering 28 (2001) 1499–1516

Table 1Chronological list of low order and high order panel methods and their main features

Year High order panel Dim. Singularity Singularity Singularity typemethod integration method distribution

1980 Johnson (1980) 3D Analytical method Quadrilateral Linear source1990 Cabral et al. (1990, 2D Numerical method Degree three (B- B-spline curves

1991) spline curves) source1993 Hsin and Kerwin 2D Analytical method Arbitrary (B-spline Arbitrary (B-spline

(1993) curves) curves) source1994 Ushatov et al. 3D Numerical method Degree three (B- Degree three (B-

(1994) spline surfaces) spline Surfaces)source

1995 Hughes and 3D Numerical method Quadrilateral QuadrilateralBertram (1995) source

1996 Kehr et al. (1996) 3D Analytical method Quadrilateral Linear source1996 Kouh and Ho 3D Analytical method Arbitrary Arbitrary source

(1996)1999 Chuang (1999) 2D Analytical method Arbitrary Arbitrary source

�(q)��Sb

�∂fE(p→

)

∂n→p

�∂fI(p

→)

∂n→799p

�G(p→

,q→

)ds��Sb

[fE(p→

�fI(p→

)]∂G(p

→,q→

)∂np

ds�f� (2)

where G(p→

,q→

) is Green’s function in an unbounded free space defined as G(p→

,q→

)=

�(1/4p)(1/r(p→

,q→

)) and r(p→

,q→

)=|p→

�q→

|. The value |p→

�q→

| denotes the distance

between q→

and p→

, fE(p→

) is the external flow velocity potential and fI(p→

) is the internalflow velocity potential. Fig. 1 shows the notation for potential flow over a closedbody.

In Eq. (2), the term [(∂fE(p→

)/∂n→p)�(∂fI(p→

)/∂n→p)] is a source strength, the term

[fE(p→

)�fI(p→

)] is a doublet strength m(p→

) and f� is the uniform flow velocity potential(Fig. 1).

From Eq. (2), kinematic (Neumann) boundary conditions are imposed on the bodysurface Sb

��·n→�0 (3)

Here, we define the total potential � as the sum of the uniform flow potential f�

and the perturbation potential fE. Eq. (3) can then be written as

∂�

∂n→p

�∂f�

∂n→p

�∂fE

∂n→p

�0 (4)

Since the imaginary velocity potential fI of the “interior flow” can be set with an


Fig. 1. Notation for potential flow over a closed body.

arbitrary value, we choose fI a value of �f� so that the total internal potential �I

for the imaginary interior flow becomes zero:

�I�fI�f��f��f��0 (5)

Based on the theory of Kellogg (1967), the external and internal normal velocity iscontinuous across the body surface boundary (∂fE/∂np=∂fI/∂np), if doublets are usedfor distribution over the surface in the surface normal direction. It exists thereforethe following relation:

∂fE

∂n→p

�∂fI

∂n→p

�∂fE

∂n→p

�∂f�

∂n→p

�∂�

∂n→p

�0 (6)

Thus, Eq. (2) becomes

�(q→

)��Sb

[fE(p→

)�f�]∂G(p

→,q→

)

∂n→p

ds�f� (7)

Denoting doublet strength term

[fE(p→

)�fI(p→

)]

by m(p→

), we derive the total potential:

m(p→

)��(p→

)�fE(p→

)�f�.

If point q→

is located on the outer surface, then the external perturbation potential is

fE(q)�m(q

→)

2��

Sb�q

m(p→

)∂G(p

→,q→

)

∂n→p

ds (8)


while q→

is located on the inner surface, the internal perturbation potential becomes

fI(q)��m(q

→)

2��

Sb�q

m(p→

)∂G(p

→,q→

)

∂n→p

ds (9)

Applying the internal Dirichlet boundary condition (5), Eq. (9) becomes

�m(q

→)

2��

Sb�q

m(p→

)∂G(p

→,q→

)

∂n→799p

ds�f��0 (10)

Eq. (10) is an integral equation for solving the unknown doublet distribution m(p→

).

If p→→q

→occurs, the kernel becomes singular. In our numerical scheme, a technique

proposed by Landweber and Macagno (1969) or Jensen (1988) is adopted to over-come this difficulty.

On the body surface Sb where the normal velocity is continuous, the followingequation is valid according to Gauss flux theorem

��Sb

∂G(p→

,q→

)

∂n→p

ds�12

(11)

Eq. (10) can then be rewritten as

�m(q

→)

2��

Sb

�m(p→

)∂G(p

→,q→

)

∂n→799p

�m(q→

)∂G(p

→,q→

)

∂n→p

�ds�m(q

→)

2�f��0 (12)

Thus we get

m(q→

)��Sb

[m(p→

)�m(q→

)]∂G(p

→,q→

)

∂n→799p

ds�f� (13)

Notice that the integral in Eq. (13) is non-singular. Eq. (13) gives a singularity-freeformula for solving the doublet density on the body surface.Eq. (13) is the basis for the potential-based panel method utilizing the Dirichletboundary condition. The doublet strength can be solved with velocities and pressuredistribution obtained by applying the gradients of velocity potential. In order to solve

Eq. (10), it is most important to find the specific discretization scheme, allowingappropriate calculation of the integral. In the present numerical method, we applythe Gaussian quadrature to carry out integration.


3. Geometry representation

We divide the body surface into panels to solve the boundary value problemnumerically. The exact surface form and NURBS surface base are both employedhere to construct the body surface.

3.1. Analytical surface

We apply the exact surface definition to construct the sphere, ellipsoid and Wigleyhull. The parametric representation of quadric surfaces like sphere and ellipsoid arefrequently useful for convincing computer graphic displays and geometry modeling.The ellipsoid parametric representation is given by

x=a cos q sin f 0�q�2p

y=b sin q sin f 0�f�2p

z=c cos f

(14)

The numerical results we demonstrate later will include an ellipsoid whose axislength ratio is a:b:c=4:1:2, and a sphere whose axis length ratio is a:b:c=1:1:1.

The parametric representation of Wigley hull is described by

p→

(u,v)��−8(1−2v)

3.2(2u−u2)(v−v2)

u−1 � with 0�u,v�1 (15)

3.2. NURBS surface

Spline element can be effectively applied to model body surface. Sclavounos andNakos (1988) initially applied the spline element to express the velocity potentialdistribution on the free surface for their Rankine panel method. The principal reasonwhy we chose the NURBS surface to model geometry is that NURBS offers onecommon mathematical form for the precise representation of standard analytic shape(lines, conics, circle, quadric planes) as well as free-form curves and surfaces. Inaddition the use of NURBS for representing surfaces in CAD/CAM applications isbecoming increasingly widespread. Being able to manipulate the control points aswell as the weights, NURBS provides the flexibility to design a large variety ofshapes. Evaluation is reasonably fast and computation is stable. Furthermore,NURBS has clear geometric interpolations, making it particularly useful for thedesigner to model geometry. NURBS has a powerful toolkit such as knot insertion,


refinement, removal, and degree elevation, etc., which can be employed to design,analyze, and process surfaces.

Let p→

(u,v) be the position vectors along the surface u and v direction as a functionof the parameters u and v, the NURBS surface is given by

p→

(u,v)�X→

W→�

�m�1

i�1

�n�1

j�1

wi,jB→

i,jNi,k(u)Mj,l(v)

�m�1

i�1

�n�1

j�1

wi,jNi,k(u)Mj,l(v)

(16)

where

Ni,k(u)

Ni,1= 0, if xi�u�xi+1

1, otherwise

Ni,k(u)�(u−xi)Ni,k−1(u)

xi+k−1−xi

�(xi+k−u)Ni+1,k−1(u)

xi+k−xi+1

Mj,l(v)

Mj,1= 0, if yj�v�yj+1

1, otherwise

Mj,l(v)�(v−yj)Mj,l−1(v)

yj+l−1−yj

�(yj+l−v)Mj+1,l−1(v)

yj+l−yj+1

where wi,j are the weights, B→

i,j forms a control net, and Ni,k(u) and Mj,l(v) are thenormalized B-spline basis function of degree k and l in the u and v directions, respect-ively, defined over the knot vector x and y.

The NURBS surface segments p→

i,j(u,v) are given by

p→

i,j(u,v), u�[ui,ui+1], i=k, . . ., m+1

v�[vj,vj+1], j=l, . . ., n+1(17)

The numerical scheme for our high order panel method requires the position vec-tors of Gaussian quadrature points and tangent vectors along the u and v directionson the body surface and normal vectors at these points. Further, we take the deriva-tives of the u and v directions of the NURBS surface, and then


∂S→

(u,v)∂u

=X→

uW−X→

Wu

W2 =�m�1

i�1

�n�1

j�1

wi,jB→

i,jNi,k(u)Mj,l(v)

�m�1

i�1

�n�1

j�1

wi,jNi,k(u)Mj,l(v)

−�m�1

i�1

�n�1

j�1

wi,jB→

jNi,k(u)Mj,l(v)·�m�1

i�1

�n�1

j�1

Ni,k(u)Mj,l(v)

��m�1

i�1

�n�1

j�1

wi,jNi,k(u)Mj,l(v)�2

∂S→

(u,v)∂v

=X→

vW−X→

Wv

W2 =�m�1

i�1

�n�1

j�1

wi,jB→

i,jNi,k(u)Mj,l(v)

�m�1

i�1

�n�1

j�1

wi,jNi,k(u)Mj,l(v)

−�m�1

i�1

�n�1

j�1

wi,jB→

ijNi,k(u)Mj,l(v)·�m�1

i�1

�n�1

j�1

Ni,k(u)Mj,l(v)

��m�1

i�1

�n�1

j�1

wi,jNi,k(u)Mj,l(v)�2

for all u, Ni,1(u)�0

if k�2, Ni,2(u)�

Ni,1(u)xi+k−1−xi

�Ni+1,1(u)xi+k−xi+1

Ni,k(u)�

Ni,k−1(u)+(u−xi)Ni,k−1(u)

xi+k−1−xi

�(xi+k−u)N

i+1,k−1(u)−Ni+1,k−1(u)xi+k−xi+1

for all v, Mj,1(v)�0

when l�2, Mj,2(v)�

Mj,1(v)yj+l−1−yj

�Mj+1,1(v)yj+l−yj+1

Mj,1(v)�

Mj,l−1(v)+(v−yj)Mj,l−1(v)

yj+1−1−yj

�(yj+l−v)M

j+1,l−1(v)−Mj+1,l−1(v)yj+l−yj+1

(18)

4. Numerical method based on Gaussian quadrature

In the present numerical method, we apply the Gaussian quadrature to carry outintegration. Let closed surface Sb be represented by Np panels. Each panel is defined

parametrically by p→

i(u,v) with �1�u�1 and �1�v�1. The infinitesimal of Sb canbe represented by


dSbi�|∂p→

(u,v)∂u

∂p

→(u,v)∂v |du dv�fi(u,v)du dv (19)

If we apply Gaussian quadrature, Eq. (13) may be written as

m(q→

j(um,vn))�1

4p�NP

i�1

��Ki

k�1

�Li

l�1

wkwl{m(p→

i(uk,vl))

�m(q→

(um,vn))�(p→

i(uk,vl)−q→

j(um,vn))·n→

p→

i

|p→

i(uk,vl)−q→

j(um,vn)|3fi(uk,vl)�

›

V·q→

j for j (20)

�1, . . ., Np, m�1, . . . , Ki, n�1, . . ., L

where Ki and Lj are the chosen Gaussian orders of integration in the u- and v-directionon the ith panel, wk and wl are the weighting coefficients in the u- and v-direction,

p→

i(uk,vl) and q→

j(um,vn) are position vectors of Gaussian quadrature points on the ith

panel and the jth panel, respectively, n→pi and n→qj are their corresponding unit outwardsurface normal vectors as shown in Fig. 2.

Eq. (20) is a linear equation for the doublet density. After solving the doubletdensity, the velocity at a point q in the flow on the exterior of the body can thenbe calculated from the gradient of the velocity potential. Since the velocity potentialat all Gaussian quadrature points is determined, the velocity at these points can becalculated as the numerical derivative of the potential. Let lu be the arc-length of a

u curve (parameter v=const.) on a panel p→

i(u,v), the potential � can be expressed,

by means of curve fitting, as a function of lu. At the point q→

j(u,v) on the bodysurface, the tangential velocity in the u direction can be calculated by

Fig. 2. Surface panels and related geometrical data.


V→

u,J��∂�

∂lu�

J

p→u

J

|p→u

J|J�1, . . ., NPT (21)

where NPT= SNP

i�1(KiLi), p

→uj , p

→uJ is the tangential vector of a u curve at the Jth Gaussian

quadrature point. In a similar way, the v-direction tangential velocity is

V→

v,J��∂�

∂lv�

J

p→v

J

|p→v

J|J�1, . . ., NPT (22)

Thus, the tangential velocity vector V→

t,J can be written as

V→

t,J�V→

u,J�V→

v,J J�1, . . ., NPT (23)

The pressure coefficients at these Gaussian quadrature points are

Cp,J�1�|V→

t,J|U2 (24)

5. Numerical results

In this section, we demonstrate three computational results from the presentmethod. Different combinations of Gaussian orders are selected for accuracy analysisand examination purposes. There are a sphere, an ellipsoid with a:b:c=4:1:2, and amathematical Wigley hull.

5.1. Sphere

With symmetry to the plane y=0 and z=0, only a quarter of the sphere surface isused in the computation scheme. And further, the quarter portion surface is dealtwith as a single panel. In Figs 3 and 4 we demonstrate the computational resultsassociated with Gaussian orders K×L of (5×5) and (7×7). The results in velocitypotential, tangential velocity and pressure coefficients show a good agreement withthe exact solution. In the NURBS surface representation, the quarter portion surfaceis dealt with as 3×3 panels in the u- and v-direction with Gaussian orders K×L of(3×3). The velocity potential, tangential velocity and pressure coefficients results inFig. 5 also show a good agreement with the exact solution.

5.2. Ellipsoids

Figs 6 and 7 show results for an ellipsoid with axis length ratio of a:b:c=4:1:2,with Gaussian orders K×L of (5×5) and (5×9). Again a quarter of the surface is


Fig. 3. Results for a sphere with Gaussian order (5×5) by both velocity and potential based methods.


Fig. 4. Results for a sphere with Gaussian order (7×7) by both velocity and potential based methods.


Fig. 5. Results for an ellipsoids (4:1:2) with Gaussian order (5*5) by both the velocity and potentialbased methods.


Fig. 6. Results for an ellipsoids (4:1:2) with Gaussian order (5*9) by both the velocity and potentialbased methods.


Fig. 7. Results for a sphere by NURBS surface on 3*3 panel and K*L =3*3 geometry definition by thepotential based method.


treated as a single panel. The results in velocity potential, tangential velocity andpressure coefficients also show a good agreement with the exact solution given byLandweber and Macagno (1969).

5.3. Wigley hull

The computational cases use different Gaussian orders of (3×7) and (3×9), respect-ively. Figure 8 shows the pressure coefficient as a function of the x-coordinate. AndFigure 9 shows the Wigley hull geometry.

6. Conclusion

The present method provides a high flexibility in geometry modeling based onGaussian quadrature. The body surface can be defined in any mathematical formsuch as implicit, explicit or parametric or in NURBS surface definition. The geo-metrical data required for computations are the position vectors of Gaussian quadra-

Fig. 8. Pressure coefficient for Wigley hull by the potential based method.


Fig. 9. Wigley hull geometry.

ture points, tangent vectors along curves on the body surface, and surface normalvectors at these points. These data are calculated directly from the mathematicalshape definition. Furthermore, no fixed order of doublet density distribution isassumed on each panel. Not only the number of panels can be chosen, but also theGaussian order of each panel. This flexibility allows a reasonable relation betweenthe Gaussian order and the geometrical complexity of a panel.

Numerical results for sphere, ellipsoid and Wigley hull demonstrate that thepresent method needs only a relatively small number of Gaussian quadrature pointsto give highly accurate results compared with the exact solution. In our method, theconcept of a NURBS surface property such as control net will be useful for furtherbody geometry optimization problems based on the results from this potential flowcomputation method.

References

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a 3d potential-based and desingularized high order panel method

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