One of the important parameters needed to model ship motions in a seaway is the added mass matrix of the hull. Current state-ofthe- art boundary element methods routinely evaluate the 6 x 6 added mass matrices as part of the radiation problem solution. These developments have largely superseded conventional approaches to sectional added mass evaluation using conformal mapping techniques. However, conformal mapping techniques are still attractive in terms of their mathematical explicitness and computational simplicity. The recurrent form of Bieberbach Method of conformal mapping was developed for mapping the exterior of a closed curve i.e. the two-dimensional ship cross section and its mirror image, into the exterior of the circle oscillating vertically at free surface and to compute the added mass coefficients. By incorporating a strip theory approximation the added mass coefficients of a three dimensional structure can be estimated from its two-dimensional section coefficients at different drafts. In this paper we have applied this method to calculate the heave, pitch and heave induced pitch added mass coefficients of a tanker. The applicability of these conformal mapping techniques to floating platforms under consideration is discussed, by comparing the results with state-of-the-art industry standard boundary element methods, AQWA and SESAM

Hem Lata, W.

Thiagarajan, K. P.

University of Queensland eSpace

16th Australasian Fluid Mechanics Conference 
Crown Plaza, Gold Coast, Australia 
2-7 December 2007 
Comparison of added mass coefficients for a floating tanker evaluated by conformal mapping 
and boundary element methods 
 
W. Hem Lata and K P Thiagarajan1 
School of Mechanical Engineering 
The University of Western Australia, Crawley, WA 6009, AUSTRALIA 
 
Abstract 
One of the important parameters needed to model ship motions in 
a seaway is the added mass matrix of the hull. Current state-of-
the-art boundary element methods routinely evaluate the 6 x 6 
added mass matrices as part of the radiation problem solution. 
These developments have largely superseded conventional 
approaches to sectional added mass evaluation using conformal 
mapping techniques. However, conformal mapping techniques 
are still attractive in terms of their mathematical explicitness and 
computational simplicity.  
 
The recurrent form of Bieberbach Method of conformal mapping 
was developed for mapping the exterior of a closed curve i.e. the 
two-dimensional ship cross section and its mirror image, into the 
exterior of the circle oscillating vertically at free surface and to 
compute the added mass coefficients. By incorporating a strip 
theory approximation the added mass coefficients of a three 
dimensional structure can be estimated from its two-dimensional 
section coefficients at different drafts. In this paper we have 
applied this method to calculate the heave, pitch and heave 
induced pitch added mass coefficients of a tanker. The 
applicability of these conformal mapping techniques to floating 
platforms under consideration is discussed, by comparing the 
results with state-of-the-art industry standard boundary element 
methods, AQWA and SESAM. 
 
Introduction 
Added Mass is the pressure force per unit acceleration acting on 
an oscillating floating body, due to the acceleration field set up in 
the surrounding fluid. It is different in different degrees of motion 
and depends upon the geometry of the body. The governing 
equations of motion for a floating rigid body are given by:  
 
( ) ( )ij ij j ij j ij j iM a x b x k x F t+ + + =                    (1)
  
where 
Mij =  Oscillating mass/ moment of inertia 
aij = Added mass induced in i due to unit acceleration in j  
bij = Damping 
kij =  Restoring stiffness  
Fi = Exciting forces in the ith direction 
,
 ,j j jx x x  = the displacement, velocity and acceleration of the 
vessel in the jth direction 
i, j = 1, .. 6, denote the six degrees of freedom surge, sway, 
heave, roll, pitch and yaw respectively as shown in 
Figure 1. 
An added mass coefficient Cij is the ratio of the added mass to the 
mass of the body as given below 
ij
ij
ij
a
C
M
=             (2)1 
                                                          
1
 Corresponding author, krish.thiagarajan@uwa.edu.au 
 
. 
Figure 1.  6 DOF of a rigid body2 
 
Motion prediction of floating platforms is accomplished using 
one of the several boundary element packages available in the 
market. Out of these packages, AQWA [1], and SESAM [2] are 
used by offshore industry worldwide. The common procedure 
adopted in these methods is to discretize the underwater surface 
into discrete elements and solve for the incompressible, 
irrotational velocity potential of the flow around the body. The 
boundary value problem is formulated using Green’s function 
from linear diffraction and radiation theory. One solution 
outcome is the 6x 6 added mass matrix as a function of the 
frequency of oscillation. 
 
There are no known sources of experimental or 3-D 
computational results for radiation forces on systematic series of 
hull forms, which is perhaps not surprising due to the substantial 
effort involved in either case. On the other hand, such data has 
been available for 2-D forms for quite some time. Vugts [3] 
published a comprehensive set of experimental data, along with 
some theoretical results, for 2-D cylinders including semicircles, 
triangles, several ship-like sections, and rectangles at range of 
drafts.  
 
In the ‘60s, Professor Landweber and M. Macagno from Iowa 
Institute of Hydraulic Research attempted a number of added 
mass coefficient calculations methods including two parameter, 
three parameter and conformal mapping developments to 
calculate added mass coefficients of two dimensional forms [4, 5, 
6]. The method of conformal mapping was found comparatively 
better than the other two methods, and provides higher level of 
accuracy as compared to the other two methods [7]. The first two 
methods are based on the assumption that a ship section having 
the same principal geometric characteristics as a member of one  
                                                          
2
 (http://www.answers.com/topic/ship-powering-maneuvering-and-
seakeeping?cat=technology) 
 
1388
 . 
 
Figure 2. Two dimensional strips in heave [6]. 
 
of the particular families, the Lewis forms or the three-
parameters forms, will have the same added mass coefficient as 
that member. This assumption is not exact and no level of 
accuracy can be provided [6]. This method of conformal mapping 
thus has gone largely unnoticed by software developers. 
 
Once coefficients of a two-dimensional section are known, the 
coefficients for the entire ship may be found by strip theory (see 
for e.g. [8] or [9]).  This theory is applicable to slender bodies i.e. 
when the length of the body is much larger compared to the 
lateral dimension as shown in Figure 2. It is based on the 
assumption that the radiated wave lengths are of same order of 
magnitude as beam of the tanker and short as compared to the 
length of the tanker. Strip theory has the limitation of predicting 
transverse motions better than longitudinal motions. 
 
We present here the basics of the theory of Landweber and 
Macagno [6] followed by its implementation into a strip theory 
formulation.  Results for added mass coefficients from this theory 
are compared with boundary element packages for an offshore 
floating tanker vessel. It is to be noted that this method is 
basically applicable to the case of infinite frequency only. 
 
 
Theory 
 
The radiation problem comprises of a body shape oscillating at 
an angular frequency ω.  The boundary condition on the 
underwater surface is basically satisfaction of no flow through 
the surface, and is given in terms of the velocity potential as 
            
y
g
∂
∂
=
φφω 2                                                              (3) 
The limiting condition when frequency ω→∞ is Ф = 0. 
 
The conformal method is based on mapping the exterior of a 
ship-section and its mirror image about x = 0, in z-plane into the 
exterior of the circle in ζ-plane as shown in Figure 2. Here B and 
T are the beam and draft of the floating body respectively.  Using  
 
Figure 3 Mapping the z-plane one-to-one into ζ-plane. 
 
a double-body contour eliminates free surface influence in the 
problem.  The boundary condition is satisfied by supposing that 
the entire shape oscillates as a single shape with instantaneous 
velocity V which gives 
yV
n n
φ∂ ∂
=
∂ ∂
 (4) 
The above equation relates the velocity potential to the shape of 
the contour.  At infinite frequency, the amplitude of oscillation 
diminishes infinitesimally, resulting in a finite velocity.  The 
kinetic energy resident in the fluid due to this oscillation may be 
directly related to the added mass as  
 
2
33
1
2
KE a V dρ φ ψ= = − ∫  (5) 
where Ψ is the stream function.  Solving for KE subject to the 
boundary conditions gives the added mass coefficient as 
   ( )233 121 2V o SC C r bb pi = = + −             (6)  
Where 
CV   = Added Mass Coefficient due to vertical (heave) oscillations 
b    = half beam of the ship section 
ro = Mean radius of the ship section. 
S = Area of the ship section  
b1 = the first coefficient of mapping from z-plane to ζ-plane. 
 
It remains to find the parameters b1, r0 and S. Let  us consider a 
double ship like contour cc’, symmetrical with respect to the x, y 
axes with the x-axis in the free surface as shown in Figure 3. The 
transformation as per the Bieberbach method of conformal 
mapping is given by [6] 
......7
4
5
3
3
21 +++++= ζζζζζ
aaaa
z       (7) 
Where iz re θ=  is the complex coordinate.  Its inverse given by 
......7
4
5
3
3
21 +++++=
z
b
z
b
z
b
z
b
zζ          (8) 
is the transformation of mapping the exterior of the closed curve 
(Ship section and its mirror image) in the z-plane into the exterior 
of the closed circle in ζ-plane (Figure 3). The form of these 
mapping functions as expansions of odd powers of z and ζ with 
real coefficients is required in order to satisfy the condition that 
both the  original section cc’ and its transformation in  ζ-plane be 
symmetrical with respect to their coordinate axes.  Here ai and bi 
are the coefficients of the mapping.  
 
ζ=f(z) 
1389
The Bieberbach method is based on the property that among the 
closed curves obtained from the conformal mapping, by 
transformations of Eq. 8, of the exterior of the given closed curve 
bounding a simply connected region, the circle will bound the 
maximum area. Ritz procedure is applied to find the values of 
finite number of coefficients of mapping i.e. bi’s. in Eq. 8 such 
that they yield maximum area subject to this restriction. 
 
The area bounded by the curve CC’ as shown in Figure 3 is given 
by 
2
' '
2
cc cc
zdz i r d iSθ= =∫ ∫                                                  (9) 
Thus we get the area as  
 
2
'
1
2 cc
S r dθ= ∫                                                      (10) 
 
Let C1 be the closed curve in ζ-plane obtained by mapping CC’ 
one-to-one in z-plane. From Eq. 8, we obtain 
iSd
cc
2
'
=∫ ζζ                                                              (11) 
 Substituting Eq. 8, we get 
dz
z
bk
z
b
zd
n
k
k
k
cc
n
j
j
cc
j 




 −
−








+= ∑∫ ∑∫
==
−
−
1
2
'
1
'
)12(112ζζ  
 (12) 
Substituting values in Eq. 12 from Eqs. 9 and 11, the area, S’ 
bounded by this curve is given by, 
∑∑∑
== =
+−=
n
j
jj
n
j
n
k
kjjk bebbSS
11 1
2' β
        (13) 
Where  







=−
>
−
−
=⇒








−
−−=
∫
∫
∫
−
−
−
pi
pi
θθ
θθ
2
0
2
0
22
'
2
1,2cos.ln
1,2cos
22
12
)12(1
4 12
jdr
jd
r
j
j
j
e
dz
z
zj
z
i
e
j
j
cc jj j
                     (14) 
And  
θθβ
β
pi
d
r
kj
kj
kj
dz
zz
k
zz
ji
kjjk
cc kjjkjk
∫
∫
−+
−−
−
−+
−−
=⇒






−
−
−
−=
2
0
222
' 212212
)22cos(
)1(2
)12)(12(
1212
4
          (15) 
 
Simpson’s Rule of integration may be used for the integration of 
Eq.15.  For Ritz condition that S’ be a maximum Area, the 
derivative of S’ with bj should be zero. 
nj
b
S
j
,.....2,10' ==
∂
∂
                                                       (16) 
Using Eq. 13 we get a set of linear equations with the coefficients 
bk. 
j
n
k
kjk eb =∑
=1
β                                                                      (17)      
This system of linear equation is solved to get b’s, the values of 
coefficients of mapping. 
The radius of the circle, obtained by the mapping is the mean 
radius of the curve cc’ 
∫=
pi
θ
pi
2
02
1 ldro                                                             (15) 
Substituting mean radius, Area, b1 and half beam values in Eq. 6 
gives the vertical added mass coefficient of the two-dimensional 
section. 
 
The procedure included obtaining the series of added-mass 
coefficients at each station. By incorporating strip theory we can 
calculate the added mass coefficients of three-dimensional 
tanker-shaped offshore production platform from the added 
masses of the two dimensional cross sections of the tanker at 
each section as shown in Figure 2. Heave-induced heave added 
mass (A33), pitch-induced pitch added mass (A55), and heave-
induced pitch added mass (A53) or pitch-induced heave added 
mass (A35) are obtained from the two-dimensional heave added 
mass (a33) by the strip theory formulations as below 
∫= dxaA 3333          (16) 
∫−== xdxaAA 333553        (17) 
∫= dxxaA
2
3355         (18) 
 
 
Implementation & validation  
 
The above formulation has been implemented for the calculations 
of added mass coefficients of a Floating Production Storage and 
Offloading (FPSO) tanker with dimensions shown in Table 1. the 
draft was varied to obtain a range of B/2T ratios.  The sectional 
heave added mass thus evaluated was compared against 
published data for ship-shaped sections by Vugts [3] in Figure 4. 
 
The data from conformal mapping shows approximately constant 
value for various B/2T with a slight decrease in the limit of B/2T 
→ 0 i.e. a very deep structure. In contrary Vugts data shows a 
very small value of a33 as B/2T→0. The agreement is seen to be 
 
Table 1. FPSO tanker particulars 
Parameter Value 
Length Over All (LOA) 329 meters 
Length between perpendiculars (Lpp) 318 meters 
Beam (b) 57.24 meters 
Depth (d) 28.2 meters 
Draft (D) variable meters 
Displacement (W) 145800 Tonnes 
 
 
0
0.5
1
1.5
2
2.5
1 1.5 2 2.5B/2T
a
33
/ρ
A
Conformal Mapping
Vugts 1968
 
Figure: 4 Non-dimensional heave added mass (a33/ρA) Vs B/2T 
1390
 reasonable at realistic values of B/2T of around 2 (as shown in 
Figure 4). At very high values of B/2T (shallow draft) the 
conformal mapping shows an increase in the added mass 
coefficient, which is contrary to Vugt’s data. The shallow draft 
limitation of strip theory is well known (for example, Lewis [9]) 
and is being studied further. Landweber and Macagno [4] 
mentioned that the mapping twice gives more accuracy in the 
added mass coefficient values. This aspect will be considered in 
the future work.  
 
Figures 5 and 6 show curves of heave-induced pitch (A53) pitch-
induced pitch added mass (A55) and pitch induced heave (A35) vs. 
B/2T.  Since these are obtained from Eqs. 17 and 18, they follow 
an amplified monotonic trend derived from Figure 4.  
 
The added mass values and corresponding coefficients of the 
entire hull were independently computed using both AQWA and 
SESAM for two different drafts.  As is well known, computations 
with boundary element software becomes quite unreliable at very 
low wave lengths, corresponding to higher frequencies.  For 
realistic offshore structures this limits application of these 
methods to periods less than 5 seconds (frequencies of 0.2 Hz). 
The computational results at the highest frequency are compared 
with the present strip theory results in Tables 2 and 3.  The 
agreement is shown to be very reasonable with maximum errors 
of less than 10% in all cases.   
 
Conclusions 
 
We have presented the theory and implementation of a conformal 
mapping technique and compared with published data as well as 
with industry standard software results.  Following conclusions 
have been drawn.  
• Conformal mapping method of calculating added mass 
coefficients is quicker for calculations at varying drafts 
as compared to the other two industry accepted 
methods. 
• It achieves the level of accuracy in calculating added 
mass of the three dimensional floating tanker. 
Further studies in progress: 
• Analysis of the increase in the 2-D sectional added 
mass coefficient at higher B/2T, which is contrary to 
the Vugts data. 
• Repetitive application of conformal mapping to achieve 
higher level of accuracy  
 
Acknowledgements 
The authors acknowledge the support of Jayanth Munipalli and 
Shan Varghese for providing AQWA and SESAM results. 
 
15
20
25
30
35
1 1.5 2 2.5 3B/2T
A5
3/
(ρA
2)
 
Figure 5. Heave induced pitch added mass (A53/ρA2) vs. B/2T. 
 
 
55
75
95
115
135
155
1 1.5 2 2.5 3B/2T
A5
5 /(
ρ
A2
B)
 
Figure 6. Two-dimensional pitch induced pitch added mass 
(a55/ρA2B) vs. B/2T. 
 
Table 2. Comparison of added mass 
 
At draft 10 m 
Method A33 (kg) A53 (kg – m) A55 (kg – m2) 
Draft = 10 m    
Conformal 
Mapping 
2.57E+08 
-4.70E+10 1.01E+13 
AQWA 2.66E+08 
 
-4.71E+10 
 
9.36E+12 
 
Draft = 10.56 m    
Conformal 
Mapping 
2.82E+08 -5.16E+010 1.11E+013 
AQWA 2.86E+08 -6.07E+10 1.05E+13 
SESAM 2.79E+08 -4.86E+10 8.29E+12 
 
 
Table 3. Comparison of Added Mass Coefficients 
 
Method C33 C53 C55 
Draft = 10 m    
Conformal 
Mapping 
1.85 -3.38E+02 7.27E+04 
AQWA 1.91 -3.39E+02 6.73E+04 
Draft = 10.56 m    
Conformal 
Mapping 
1.96 -3.59E+02 7.73E+04 
AQWA 1.99 
 
-4.23E+02 
 
7.31E+04 
 
SESAM 1.94 -3.38E+02 5.77E+04 
 
 
References 
[1] AQWA User manual, Horsham (UK), Century Dynamics 
Ltd., RH12 2DT, 2004  
[2] www.dnv.com/software 
[3] Vugts J H, The hydrodynamics coefficients for swqying 
heaving and rolling cylinders in a free surface, Int. Shipbldg, 
Prog. 15, 1968.  
[4] Landweber, L. and Macagno, M., Added Mass of Two-
Dimensional Forms Oscillating in a Free Surface, J. Ship 
Research, 1, (3), 1957. 
[5] Landweber, L. and Macagno, M., Added Mass of a Three 
Parameter Family of Two-Dimensional Forms Oscillating in 
a Free Surface, J. Ship Research,  2, (4), 1959. 
[6] Landweber, L. and Macagno, M., Added Mass of Two 
Dimensional Forms by Conformal Mapping, J. Ship 
Research, April 1967. 
[7] Macagno, M., A Comparison of Three Methods For 
Computing Added Mass of Ship Sections, .J Ship Research, 
April 1967. 
[8] Lewandowski, E M, The Dynamics of Marine Craft, 
Advanced Series on Ocean engineering, 22, 2004. 
[9] Lewis, E V, Principles of Naval Architecture, 3, SNAME 
Publ,1988. 
1391


Comparison of added mass coefficients for a floating tanker evaluated by conformal mapping and boundary element methods

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Comparison of added mass coefficients for a floating tanker evaluated by conformal mapping and boundary element methods

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