This paper presents an algorithm for shape optimization of composite pressure vessels head. The shape factor which is defined as the ratio of internal volume to weight of the vessel is used as an objective function. Design constrains consist of the geometrical limitations, winding conditions, and Tsai-Wu failure criterion. The geometry of dome shape is defined by a B-spline rational curve. By altering the weights of control points, depth of dome, and winding angle, the dome shape is changed. The proposed algorithm uses genetic algorithm and finite element analysis to optimize the design parameters. The algorithm is applied on a CNG pressure vessel and the results show that the proposed algorithm can efficiently define the optimal dome shape. This algorithm is general and can be used for general shape optimization

1. Appl Compos Mater (2009) 16:321–330
DOI 10.1007/s10443-009-9097-3
Dome Shape Optimization of Composite Pressure Vessels
Based on Rational B-Spline Curve and Genetic Algorithm
Abbas Vafaeesefat
Received: 14 June 2009 / Accepted: 27 July 2009 / Published online: 12 August 2009
# Springer Science + Business Media B.V. 2009
Abstract This paper presents an algorithm for shape optimization of composite pressure
vessels head. The shape factor which is defined as the ratio of internal volume to weight of
the vessel is used as an objective function. Design constrains consist of the geometrical
limitations, winding conditions, and Tsai-Wu failure criterion. The geometry of dome shape
is defined by a B-spline rational curve. By altering the weights of control points, depth of
dome, and winding angle, the dome shape is changed. The proposed algorithm uses genetic
algorithm and finite element analysis to optimize the design parameters. The algorithm is
applied on a CNG pressure vessel and the results show that the proposed algorithm can
efficiently define the optimal dome shape. This algorithm is general and can be used for
general shape optimization.
Keywords Filament wound vessel . Shape optimization . Genetic algorithm .
Tsai-Wu failure criterion . B-spline rational curve
1 Introduction
Filament-wound composite pressure vessels are widely used in commercial and aerospace
industries such as fuel tanks, portable oxygen storage, and compressed natural gas (CNG)
pressure vessel. They utilize a fabrication technique of filament winding to offer a high
stiffness and strength combined with a low weight and an excellent corrosion resistance.
Pressure vessels consist of a cylindrical drum and dome parts. The relative dimensions
of different sections of the vessel are designed based on the space and weight requirements
and the pressure levels. Since filament-wound composite pressure vessels are usually failed
in their dome parts, the design of the dome parts is the main concern. This is due to the fact
that the dome regions undergo the highest stress levels and are the most critical locations
from the viewpoint of structure failure. The desired parameters for a good head shape are
higher burst pressure and internal volume and lower weight.
A. Vafaeesefat (*)
Mechanical Engineering Department, Imam Hussein University, Tehran, Iran
e-mail: Abbas_v@yahoo.com

2. 322 Appl Compos Mater (2009) 16:321–330
Optimum design of dome contours for composite pressure vessels has been the subject
of many researches. The netting theory and orthotropic analysis methods was used to solve
design problems involving dome shapes [1]. Fukunaga et al. [2] presented an analytical
approach to the optimal design of dome shapes based on the performance factor. However,
the performance factor is rather complex and unpredictable because of the lack of a
maximum burst pressure for the dome structure. DiVita et al. [3] described a mathematical
model for determining non-slip winding paths for composite structures of general shape.
Hojjati et al. [4] used the orthotropic plate theory for dome design of the polymeric
composite pressure vessels. Liang et al. [5] investigated the optimum design of dome
contour for filament wound composite pressure vessels. They used the feasible direction
method for maximizing the shape factor. Vafaeesefat and Khani [6] compared different head
shapes and tried to optimize the composite parameters.
Generally, the shape optimization problem consists of finding out the best profile of a
component that improves its mechanical properties and minimizes some properties, for
example, to minimize the weight of the body or reduce high stress concentrations. In recent
years, researchers have carried out many investigations into shape optimization problems
which are proving useful for different optimization problems. They usually use the nodal
coordinates of the discrete finite element model as design variables [7–9]. These approaches
require a large number of design variables and a large number of constraints, which
complicates.
Generally, the choice of any parametric curve to represent the dome profile will result in
a certain degree of restriction of an optimization problem. The objective of shape
optimization of an engineering component is to search for a feasible solution within a
prescribed tolerance. Therefore, an adequate selection of a geometric representation method
for the dome profile and the minimum number of appropriate design variables is of
fundamental importance in order to achieve an automatic design cycle during the shape
optimization process. In this paper, a general approach for shape optimization of vessel
dome is presented. The shape of the design boundaries is modeled by using B-spline
rational curve. The shape of dome profile is altered by changing the control point weights
and the depth of dome. Meanwhile, the winding angle is also optimized to minimize the
Tsai-Wu criterion in the resulted shape. The proposed algorithm is repeated for different
number of layer to find its effects on the optimal dome shape.
2 Winding Conditions
Finding the possible winding patterns on an arbitrary shape is one of the first
necessaries in order to define composite vessels. Since the accuracy of finite element
analysis is directly influenced by the winding information, there is a need for the
winding pattern to be actually modeled. In this paper, the semi-geodesic path method is
used, in which the slippage tendency between the fiber and the mandrel is considered.
The semi-geodesic path for the general filament wound structures is defined as follows
[10]:
À Á
da l A2 sin2 a À rr00 cos2 a À r0 A2 sin a
¼ ð1Þ
dz r A2 cos a
A ¼ 1 þ r2

3. Appl Compos Mater (2009) 16:321–330 323
where α, z, θ, ρ and λ are the winding angle, the axial, circumferential and radial coordinate
parameters, and the slippage tendency between the fiber and the mandrel, respectively.
Moreover, ρ′ and ρ′′ are the first and second derivatives respectively. By setting the slippage
tendency equal to zero in Eq. (1), the geodesic path equation is obtained:
r sin a ¼ cte ð2Þ
The geodesic path introduces the shortest path between two points on a surface.
Therefore, geodesic fiber path is a special kind of semi-geodesic fiber path for which the
slippage tendency is zero. This kind of winding called isotensoid winding is used to obtain
the winding angle at each point, we have:
r sin a ¼ r0 sin a0 ¼ r1 sin 90 ¼ r2 sin 90 ð3Þ
where α0 and ρ0 are the winding angle and the vessel radius in the cylindrical part,
respectively. Also ρ1 and ρ2 denote left and right dome opening radii, respectively.
Therefore, for geodesic winding, two domes must completely have similar opening radii.
From Eq. (3), the winding angle at every point on the head is obtained by
r0 sin a0
sin a ¼ ð4Þ
r
In the composite vessel, the composite thickness is variable along the vessel head
profile. From the junction to the opening, the circumstance on the head from which the
fibers pass is decreased, while the number of fibers passing these circles is constant. Based
on the above reasons, the composite thickness is increased from the junction to the opening.
Composite thickness at every point on the head relates to the thickness on the cylinder and
is obtained by the following equation:
r0 cos a0
t¼ t0 ð5Þ
r cos a
where t and t0 are the helical layer thickness on the head and cylindrical sections,
respectively. Figure 1 shows variations of composite thickness along the vessel head
profile.
Fig. 1 Thickness variations on 60
the vessel head
50
Thickness (mm)
40
30
20
10
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
R (m)

4. 324 Appl Compos Mater (2009) 16:321–330
3 Finite Element Model
The head shape has certain effects on the internal volume, weight, failure criterion, and
burst pressure of the vessel. The internal volume of the vessel is determined by the internal
volume of the domes and the drum. The outer area of the domes affects the whole weight of
the filament wound structure.
Finite element is used to analyze the three dimensional model of the vessel. Since
the structure is symmetric, only half of vessel is modeled. Half of the vessel is modeled
and the opening edge is constrained in the radial and circumferential directions. The
profile of the dome is defined by 100 numbers of points defined by a rational B-spline
curve. When the weights and the depth of the dome are changed by genetic algorithm,
a new 100 points are generated and therefore, a new dome profile is created.
In this study, a CNG vessel with non-metallic liner is selected. The outside diameter of
the cylindrical section is 330 mm and its length is 650 mm (Fig. 1). The working pressure,
test pressure, and burst pressure based on ISO1439 [11] are 200, 300, and 470 bar,
respectively. In the finite element model, a uniform pressure 470 bar is applied (Fig. 2).
There are several failure criteria to predict failure in the composite materials. In this
work, the 3D Tsai-Wu criterion is utilized which is defined by:
x3 ¼ A þ B ð6Þ
where
ðs y Þ ðs xy Þ ðs yz Þ
2 2 2
ð 2 ð 2
A ¼ À ssf xsÞ f À s f s f À ssf zsÞ f þ f 2 þ f 2 þ
xt xc yt yc zt zc ðs xy Þ ðs yz Þ
ðs xz Þ2 Cxy s x s y Cyz s y s z Cxz s x s z
þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðs xz Þ
f 2 f f f f
s xt s xc s yt s tc f f f f
s yt s yc s zt s zc
f f f f
s xt s xc s xt s zc
and
" # " # " #
1 1 1 1 1 1
B¼ f
þ f
sx þ f
þ f
sy þ f
þ f
sz
s xt s xc s yt s yc s zt s zc
and ξ3, σ, s t f and s c f are the three dimensional Tsai-Wu criterion, the applied stresses, tension
strength and compression strength, respectively. Cxz, Cyz and Cxy are coupling coefficients for
Tsai-Wu theory. Indices x, y and z show the fiber direction, it’s perpendicular direction in and
out of the element plane, respectively. Due to the continuous nature of filament winding
process, the thickness of all layers is usually selected a constant value or an integer factor of it.
In this work, the layer thickness is 1 mm. Mechanical properties of the carbon-epoxy
composite material used in this research are detailed in Table 1. Mass density of the composite
material is 1565 kg/m3.
Fig. 2 Vessel dimensions (mm)

5. Appl Compos Mater (2009) 16:321–330 325
Table 1 Materials properties of composite material. (Strengths are in MPa and elastic moduli are in GPa)
Exx Eyy Ezz Gxy Gxz Gyz Vxy Vxz Vyz
110.3 15.2 8.97 4.9 4.9 3.28 0.25 0.25 0.365
f f f
s xt s xc
f
s yt s yc
f
s zt s zc
f
s xy s yz s xz
1918 1569 247 1245 247 1245 68.9 34.5 34.5
4 Rational B-spline Curve
Over the past thirty years, different curves and surfaces representation forms have been
proposed. Currently, B-spline rational curve are the most popular mathematical forms.
B-spline rational curve offers a unified mathematical form not only for representation of
free-form curves and surfaces, but also for the precise representation of close-form shapes.
A rational B-splines curve C(u) is a vector valued piecewise rational polynomial
function of the form
X
n
CðuÞ ¼ Ri; p ðuÞPi ð7Þ
i¼0
where
w Ni; p ðuÞ
Ri; p ðuÞ ¼ Pi
n
Nj; p ðuÞwj ð8Þ
j¼0
wi ! 0 ði ¼ 0; 1; Á Á Á ; nÞ
The B-spline basis function of degree p is defined recursively as
&
1 if ui u uiþ1
Ni; o ðuÞ ¼
0 otherwise ð9Þ
uiþpþ1 Àu
Ni; p ðuÞ ¼ uiþp Àui Ni; pÀ1 ðuÞ þ uiþpþ1 Àuiþ1 Niþ1; pÀ1 ðuÞ
uÀui
Fig. 3 Maximum dome shape w=[1 2 2 2 1] d=0.18 (m)
variations with different weights 0.18
w=[1 1 1 1 1] d=0.10 (m)
(w) and depth of dome (d)
0.15
0.12
R (m)
0.09
0.06
0.03
0
0 0.03 0.06 0.09 0.12 0.15 0.18
Z (m)

6. 326 Appl Compos Mater (2009) 16:321–330
Fig. 4 Schematic diagram of
Number of layers, Initial control points
shape optimization
weights, winding angle, and depth of
dome
New weights, winding
angle, and depth of dome
Finite element analysis
to create a new dome
profile
Genetic algorithm
Optimum dome shape
where Pi are the control points forming a control polygon, wi are the weights and Ri,p(u)
are the pth degree rational basis functions defined over a non-uniform knot vector u∈[1 0]
with u as non-dimensional curve parameter. Degree p, number of knots m +1 and number
of control points n can be related by:
m¼nþpþ1
Shape modification of B-spline rational curve can be achieved by means of knot
values, control points positions, and weights. The extra degrees of freedom in this
model offers the weights, allow a large variety of shapes to be generated. It seldom
happens that the initial design meets the required specifications, and the process of
modification goes on until the shape of the curve satisfies the requirements. B-spline
rational curve permits manipulation of weights and control points thereby offering ease
of design of both standard analytical shapes and free-form curves. In our approach,
shape modifications are achieved by the simultaneous modification of weights and the
last control point to change the depth of dome.
By altering the weights, the shape of the rational B-spline curve is changed within
the convex hull of control points. The negative weights are not allowed. Five control
points are defined in such a way that their convex hull can cover variety of dome
shapes. Moreover, the control points are defined somehow that the first and second
continuities are permitted between the dome and cylindrical sections (see Fig. 3).
Table 2 Results of the shape optimization for different number of layers
No. No. of layer αo w d(m) Tsai-Wu do(m) K
1 17 27 [1, 1.7158, 2.3826, 2.0493, 1] 0.1675 0.924748 0.0749 0.7144
2 16 26 [1, 1.4130, 1.6698, 1.7233, 1] 0.1704 0.776020 0.0618 0.7100
3 15 25 [1, 1.6565, 2.7475, 2.1980, 1] 0.1578 0.731021 0.0671 0.7544
4 14 24 [1, 1.0868, 1.6791, 0.8146, 1] 0.1568 0.865861 0.0671 0.8026
5 13 19 [1, 1.1414, 2.0703, 2.1087, 1] 0.1643 1.138734 0.0537 −1.9056

7. Appl Compos Mater (2009) 16:321–330 327
Fig. 5 Optimum dome shape 0.16 13 layers
under different number of layer
14 layers
0.14
15 layers
0.12 16 layers
17 layers
0.1
Z (M)
0.08
0.06
0.04
0.02
0
0.05 0.07 0.09 0.11 0.13 0.15 0.17
R (m)
5 Shape Optimization Using Genetic Algorithm
The head shape factor is used to compare the performance of the different head shapes.
Since the internal pressure and the material properties are not changed during the
optimization procedure, the head shape factor is simply defined by:
V
K¼ ð10Þ
W
W and V are, respectively, the dome weight and its internal volume.
The optimization technique used in this work is the genetic algorithms due to its great
versatility, easy implementation and its ability for locating the globally optimal solution.
Genetic algorithms are search algorithms based on the mechanics of natural selection and
natural genetics. They usually converge quickly to the optimum structure with a minimum
effort, having to test only a small fraction of the design space to find out the optimum
solution. They can be applied to problems for which it is not possible to have an analytical
relationship for the objective function.
Fig. 6 Optimum dome shape 0.16
with higher shape factor
0.14
0.12
0.1
Z (m)
0.08
0.06
0.04
0.02
0
0.05 0.07 0.09 0.11 0.13 0.15 0.17
R (m)

8. 328 Appl Compos Mater (2009) 16:321–330
Fig. 7 Comparison of optimum 0.9
dome shape between present Liang et al.
method and Liang et al. [5] 0.8 Present method
0.7
0.6
0.5
Z (m)
0.4
0.3
0.2
0.1
0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
R (m)
The procedure for the optimization of the number of dome shape via genetic algorithm is
applied on the predefined number of layer. Genetic Algorithms are most directly suited to
unconstrained optimization. Application of Genetic Algorithms to constrained optimization
problems is often a difficult effort. Several methods have been proposed for handling
constraints. The most common method in genetic algorithms to handle constraints is to use
exterior penalty functions. Therefore, the problem is formulated as bellow:
Max K ðwi ; d; aÞ À R* maxf 0 ; Tsai Wu À 1g
subject to 0 < wi 3
0:1 d 0:18 ðmÞ
8 a 30 ðdeg reeÞ
The design variables of composite vessel consist of weights of control points wi, the
winding angle α, and depth of dome d. The stacking sequence is alternately under +α
and – α. The proposed shape optimization algorithm is presented in Fig. 4.
The initial population and the maximum number of generations for genetic algorithm are
50 and 100, respectively. The allowable change required to consider two solutions different
is set to 1e-6. If this difference becomes less than this defined value, the program will stop.
The mutation and crossover operators have been selected binary and arithmetic,
respectively.
Table 3 Presents the results of the shape optimization for different winding angles
No. αo No. of layer w d(m) Tsai-Wu K
1 27 14 [1, 2.3309 2.7966 1.8923, 1] 0.15661 0.94224 0.8133
2 24 14 [1, 1.5830 2.3096 1.9630, 1] 0.16722 0.93009 0.8150
3 21 15 [1, 1.3280 2.1492 2.2284, 1] 0.16496 0.90767 0.7544
4 18 15 [1, 1.2215 2.1032 1.2488, 1] 0.17233 0.93143 0.7592
5 15 15 [1, 1.3492 2.5380 1.2011, 1] 0.16583 0.93143 0.7518
6 12 16 [1, 0.6558 2.2960 1.6703, 1] 0.16925 0.98053 0.7024
7 9 15 [1, 1.3750 2.1925 1.4063, 1] 0.17883 0.99792 0.7608

9. Appl Compos Mater (2009) 16:321–330 329
Fig. 8 Optimum dome shape :9
under different winding angles 0.16
:12
(9°–27°) :15
0.14
:18
:21
0.12
:24
0.1 :27
Z (M)
0.08
0.06
0.04
0.02
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
R (m)
6 Results
Table 2 presents the results of the shape optimization for different number of layers.
Among different number of layer listed in Table 2, the optimum conditions with higher
shape factor K and the Tsai-Wu less than one is sample No. 4. Figure 5 shows the shape
of the dome profile for five trials listed in Table 2. The optimum shape dome (trail 4) is
shown in Fig. 6.
Figure 7 shows the comparison of optimum dome shape between present method and
Liang et al. [5]. If it is assumed that in both method the number of layer is selected in such a
way that Tsai-Wu failure criterion falls bellow one, the high shape factor is obtained in the
present method. Since instead of control point positions (two-dimension), the weight of
control points are changed during optimization, the number of design variables to change
the dome profile is decreased by half. Therefore, simple optimization procedure and better
results are expected.
Fig. 9 Optimum dome shape
with constant winding angle (24°) 0.16
0.14
0.12
0.1
Z (M )
0.08
0.06
0.04
0.02
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
R (m)

10. 330 Appl Compos Mater (2009) 16:321–330
To verify the optimum shape, the optimization procedure was repeated for constant
winding angles. Table 3 presents the results of the shape optimization for different winding
angles. The optimum shapes of dome profile for different winding angles are shown in
Fig. 8. This figure shows that the profile of optimum dome shape for different winding
angles are follow almost the same pattern. The best result is trail two with 14 numbers of
layer and winding angle 24o. The result is in the good agreement with the last optimization
shape where the numbers of layer were constant. Figure 9 shows the optimum dome shape
of trial 4.
7 Conclusion
In this paper, a general shape optimization algorithm is introduced which can be utilized in
combination with genetic algorithm and F.E. analysis to optimize dome shape of composite
pressure vessels. During optimization procedure, to change the dome shape, the weights of
control points and the position of one control point are changed. Therefore, the number of
design variables and constraints for changing the dome shape are minimal. This makes the
optimization procedure simpler and thus minimizes the number of F.E. analysis required to
reach the optimum solution. The genetic algorithm is used o reach the global shape
optimization. Comparing to other method, the results show that the better dome shape can
be obtained by the present method. This approach is general and can be used for other
shape optimization applications.
References
1. Lubin, G.: Handbook of composites. Van Nostrand Reinhold, (1982)
2. Fukunaga, H., Uemura, M.: Optimum design of helically wound composite pressure vessels. Compos
Struct 1, 31–49 (1983)
3. DiVita, G., Marchetti, M., Moroni, P., Perugini, P.: Designing complex-shape filament-wound structures.
Compos Manuf 3(1), 53–58 (1992)
4. Hojjati, M., Safavi, A.V., Hoa, S.V.: Design of dome for polymeric composite pressure vessels. Compos
Eng 5(1), 51–9 (1995)
5. Liang, C.C., Chen, H.W., Wang, C.H.: Optimum design of dome contour for filament wound composite
pressure vessels based on a shape factor. Compos Struct 58, 469–482 (2002)
6. Vafaeesefat, A., Khani, A.: Head shape and winding angle optimization of composite pressure vessels
based on a multi-level strategy. Appl Compos Mater 14(5–6), 379–391 (2007)
7. Waldman, W., Heller, M., Chen, G.: Optimal free-form shapes for shoulder fillets in flat plates under
tension and bending. Int J Fatigue 23, 185–90 (2001)
8. Heller, M., Kaye, R., Rose, L.R.: A gradientless finite element procedure for shape optimization. J Strain
Anal Eng Des 34(5), 323–36 (1999)
9. Tran, D., Nguyen, V.: Optimal hole profile in finite plate under uniaxial stress by finite element
simulation of Durelli’s photoelastic stress minimisation method. Finite Elem Anal Des 32, 1–20 (1999)
10. Park, J.S., Hong, C.S., Kim, C.G., Kim, C.U.: Analysis of filament wound composite structures
considering the change of winding angles through the thickness direction. Compos Struct 55, 63–71
(2002)
11. ISO 11439(E): Gas cylinders-high pressure cylinders for the on-board storage on natural gas as a fuel for
automotive vehicles (2000)