=Paper=
{{Paper
|id=Vol-1839/MIT2016-p22
|storemode=property
|title= Analysis and design of hybrid pressure vessels
|pdfUrl=https://ceur-ws.org/Vol-1839/MIT2016-p22.pdf
|volume=Vol-1839
|authors=Evgeniya Amelina,Sergey Golushko,Andrey Yurchenko
}}
== Analysis and design of hybrid pressure vessels==
https://ceur-ws.org/Vol-1839/MIT2016-p22.pdf
Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
Analysis and Design of Hybrid Pressure Vessels
Evgeniya Amelina1 , Sergey Golushko1,2 , and Andrey Yurchenko1
1
Institute of Computational Technologies
Siberian Branch of the Russian Academy of Sciences,
Academician M.A. Lavrentjev ave. 6, Novosibirsk, Russia
2
Novosibirsk National Research State University,
Pirogova str. 2, Novosibirsk, Russia
{amelina.evgenia,s.k.golushko,andrey.yurchenko}@gmail.com
http://www.ict.nsc.ru/en
Abstract. The paper presents a computational technology for optimiza-
tion of composite overwrapped pressure vessels (COPV). Mathematical
modeling and numerical optimization were applied to design COPV. The
mathematical models were built using different shell theories and struc-
tural models of composites.
The stress-strain state of the vessels was determined and analyzed based
on three mathematical models. Several solutions of COPV optimization
problem based on different problem statements were obtained. They were
analyzed and verified by substituting of the estimated design parameters
in a direct problem of stress-strain state determination.
The study demonstrated that using of non-constant design parameters,
such as the thickness, the winding angle and the curvature radius of the
composite shell gave the possibility for additional reduction of COPV
mass, while keeping its strength. In addition, acceptability and conve-
nience of using simpler mathematical models for numerical solving the
optimization problems were demonstrated.
Keywords: COPV, mathematical modeling, computational optimiza-
tion, shell theory, structural model of composite material.
1 Introduction
Composite overwrapped pressure vessels (COPV) are used in the rocket and
spacecraft industry due to their high strength and lightweight. Consisting of
a thin, non-structural liner wrapped with a structural fiber composite COPV
are produced to hold the inner pressure of tens and hundreds of atmospheres.
COPV have been one of the most actual and perspective directions of research,
supported especially by NASA [1, 2].
Designing of a highly reliable and efficient COPV requires a technology for
analisys of its deformation behaviour and strength assessment. This technology
should allow one to obtain target COPV parameters through changing vessel’s
geometry, structural and mechanical material parameters while keeping its useful
load.
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Combination of mathematical modeling and numerical optimization makes
it possible to reduce the cost and the duration of identifying the best parameters
for a COPV. However, this approach is characterized by a number of hurdles.
Overcoming these hurdles determine success of an optimum designing of such
structures.
So far, there have been two main approaches in optimization of composite
structures: analytical and numerical ones.
In the first approach the problems are solved basing on their simplified state-
ment, for example using the momentless (membrane) shell theory and the netting
model of composite material (CM) [3–6]. The obtained results may be far from
reality, however they are of value for testing of numerical optimization methods.
Application of the numerical approach in designing, on the other hand, pro-
duces a number of challenges that must be overcome, e.g. lack of reliable methods
for global optimization; nonconvexity and nonlinearity of constraint functions;
ill-conditioned boundary value problems; different scaling of optimization cri-
teria represent just some of the obstacles that prevent reliable optimization of
COPV.
Numerical analysis is usually a computation-intensive process and takes con-
siderable time. One way to solve this problem is approximation of the objective
function using different approaches, such as response surface method [7] and neu-
ral network [8]. Some kinds of numerical analyses use a small number of design
variables, functions and/or corresponding set of their discrete values (analytical
geometry parametrization [9], finite set of feasible winding angles [10]). It leads
to reduction in the number of objective function calculations.
Another way is reasonable simplification of the elasticity problem statement,
for example by using the membrane theory or other shell theories [9,11,12], that
leaves the question of results validity. And this is the approach we have applied
in our study. For validation we have used the Timoshenko [13] and Andreev-
Nemirovskii [14] shell theories, acconting transverse shears with different degrees
of accuracy.
Of course, it should be taken into account that the computed solutions are not
optimum in the strict mathematical sense. However, these solutions could provide
the considerable economy of the weight while keeping the required strength, and,
therefore, they are important from the engineering point of view.
2 The Problem Statement and the Mathematical Models
Let’s consider a multilayer composite pressure vessel at a state of equilibrium
under equidistributed inner pressure. We need to determine the parameters of
structure and CM meeting the following requirements:
𝑉 ≥ 𝑉0 , 𝑃 ≥ 𝑃0 , 𝑀 ≤ 𝑀0 , (1)
where 𝑉 is the volume of the vessel, 𝑃 is inner pressure and 𝑀 is the vessel’s
mass and they are constrained by some preset values 𝑉0 , 𝑃0 , 𝑀0 .
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We define the optimization problems the following way: to find extremum of
one functional from (1) under other constraints.
The structures optimization problem statement includes selection of objective
functional, formulation of constitutive equations and constraints on performance
and design variables.
The mathematical models describing the vessel’s state are based on the fol-
lowing assumptions:
1) the vessel is a multilayer thin-walled structure;
2) the vessel’s layers can have different mechanical characteristics;
3) the reinforced layer’s material is quasi-homogeneous;
4) the vessel’s main loading is high inner pressure, whose alteration happens
rather slowly during operation.
These assumptions allow us to reduce dimension of the corresponding math-
ematical problem and to build the mathematical vessel’s models based on the
different theories of multilayer non-isotropic shells.
Let’s consider the vessel as a shell rigidly compressed on the edges. Taking
into account a symmetry plane in the middle of the vessel, it is enough to cal-
culate and design only it’s half. The type of loading and boundary conditions
allow considering the axisymmetric problem statement.
The half of shell is set by rotation of the generatrix 𝑟 = 𝑟(𝜃) around axis 0𝑧
(fig. 1) where 𝑟 is the current point of the shell radius, 𝜃 is the angle between
the normal to the shell surface and axis 0𝑧 changing within [𝜃0 ; 𝜃1 = 90∘ ]. The
full shell is set by reflection the shell’s half about plane 0𝑥𝑦.
z
z
r1
Meridian Generatix
Winding
angle
y q
0 r2 r,x
j
y
Fig. 1. Shell of rotation geometry
The Kirchhoff—Love shell theory [15] (KLST) and the theories with shear
terms (Timoshenko [13] (TiST) and Andreev-Nemirovskii [14] (ANST)) are used
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to solve the direct calculation problems of multilayer composite vessels, to anal-
yse their behavior and to verify optimization problem solutions. The used coor-
dinate system is (𝜃, 𝜙, 𝜁), where 𝜙 denotes polar angle, 𝜁 – normal to the surface.
The load is equidistributed inner pressure 𝑞 = (0, 0, 𝑞3 ). On the top fixed edge
(𝜃 = 𝜃0 ) all displacement components, angle of normal rotation and additional
shear term (ANST) are equal to zero; on the edge (𝜃 = 90∘ ) we use the symmetry
conditions: transverse force, the first displacement component, angle of normal
rotation and additional shear term (ANST) are equal to zero. One could find
the full systems of equations in the paper [16].
Relations between stresses and strains are defined by the structural models
[17]. The main idea of these models is that CM characteristics are calculated
through matrix and fibers mechanical characteristics, fibers volume content and
winding angles. The stress-strain state of matrix and fibers are evaluated through
stresses and strains of the composite shell. A failure criterion is applied for every
component of CM. Here we use the Mises criterion to determine the first stage
of failure.
The objective function whose minimum is required is the minimum mass:
∫︁ 𝜃1
𝑀 = 2𝜋 𝑟𝑅1 ℎ𝑑𝜃 [𝜌𝑚 (1 − 𝜔𝑟 ) + 𝜌𝑟 𝜔𝑟 ] → min (2)
𝜃0
where 𝜌𝑚 , 𝜌𝑟 are the densities of matrix and reinforcing fibers, 𝜔𝑟 is the volume
content of reinforcement.
We chose the following design functions: the curvature radius 𝑅1 (𝜃) to define
the generatrix; the thickness of the shell ℎ(𝜃); the winding angle 𝜓(𝜃) (fig. 1).
The solution has to satisfy the constraints on the shell’s inner volume:
∫︁ 𝜃1
𝜋 𝑟2 𝑅1 sin 𝜃𝑑𝜃 = 𝑉0 , (3)
𝜃0
and the strength requirement:
max{𝑏𝑠𝑟 , 𝑏𝑠𝑚 } ≤ 1, (4)
where 𝑏𝑠𝑟 , 𝑏𝑠𝑚 are the normalized von Mises stresses in the matrix and fibers [17].
Note that the factor of safety is widely used while solving engineering problems.
It can be considered by correction of the right part of the inequality (4).
We used the following constraints on the design functions:
0 ≤ 𝜓 ≤ 90, ℎ*0 ≤ ℎ ≤ ℎ*1 , 𝑅0* ≤ 𝑅1 ≤ 𝑅1* . (5)
The method of the continuous geodesic winding have been widely used in the
manufacturing of composite shells of revolutions. In this case the winding angles
are defined by the Clairaut’s formula:
𝑟 sin 𝜓(𝑟) = 𝐶, (6)
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
where the constant 𝐶 is defined, as a rule, from the condition at the shell’s
equator. The thickness equation is
𝑅 cos 𝜓𝑅
ℎ(𝑟) = ℎ𝑅 , (7)
𝑟 cos 𝜓(𝑟)
which has the singularity at the edge where the winding angle has to be equal
to 90∘ . The formula (7) is applied into practice at 𝑟 ≥ 𝑟0 + 𝑟𝜔 , where 𝑟𝜔 is equal
to the width of the reinforcement tape. As a result the equation defining the
vessel’s thickness takes the form:
⎧
⎪ 𝑅 cos 𝜓𝑅
⎪
⎨ ℎ𝑅 𝑟𝜔 cos 𝜓(𝑟0 + 𝑟𝜔 ) , 𝑟 ≤ 𝑟0 + 𝑟𝜔 ;
⎪
ℎ(𝑟) = (8)
⎪
⎪ ℎ𝑅 𝑅 cos 𝜓𝑅 ,
⎪
⎩ 𝑟 ≥ 𝑟0 + 𝑟𝜔 .
𝑟 cos 𝜓(𝑟)
We did not consider the problem of fibers slippage. The main goal of the study
was to demonstrate the potentials of using CM in one COPV design approach.
3 Direct Problems. Analysis of the Shell Theories
Estimation of composite vessel stress-strain state using the offered models leads
to the solution of boundary value problems for stiff systems of differential equa-
tions. These problems are ill-conditioned, and their solutions have big gradi-
ents near the edges. Numerical analysis was performed by the spline collocation
and discrete orthogonalization methods, implemented in the COLSYS [18] and
GMDO [19] software. These computing tools have proved to be effective in nu-
merical solving of wide range of composite shell mechanics problems [20].
We investigated the vessel’s deformations by computing of its stress-strain
state based on the different shell theories. The vessel’s shape was a part of a
toroid: 𝑅1 = 2.46 m, 𝜃0 = 0.108∘ , 𝜃1 = 90∘ (the computed half), 𝑟(𝜃0 ) =0.04
m. The CM parameters were: 𝐸𝑚 = 3 · 109 Pa, 𝜈𝑚 = 0.34, 𝐸𝑟 = 300 · 109 Pa,
𝜈𝑟 = 0.3, 𝜔𝑟 = 0.55, 𝑉0 =350 liters where 𝐸𝑚 , 𝐸𝑟 are the Young’s modulus of
the matrix and fibers, 𝜈𝑚 , 𝜈𝑟 — their Poisson’s ratio.
Fig. 2 shows the stress-strain state characteristics of the vessel with the thick-
ness ℎ = 0.6 cm, reinforced in the circumferential direction (𝜓 = 90∘ ) under the
load of 170 atm. On the left are the displacements of the reference surface along
the generatrix 𝑢1 (𝑟) (dashed curves) and the normal displacement of this surface
𝑤(𝑟) (solid curves). On the right are the distribution of normalized von Mises
stresses (nVMS) along the thickness in the matrix 𝑏𝑠𝑚 (𝑟). The solid curves cor-
respond to a slice at the shell edge, the dashed curves — to a slice at 𝜃 = 0.1.
The curves without symbols correspond to KLST simulations, the curves marked
with △ — to those using TiST, and — to ANST.
It’s easy to see that the basic kinematic characteristics coincide both quali-
tatively and quantitatively. Small differences are observed only for the stresses
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
bsm
w, mm
u, mm
q, rad z, mm
Fig. 2. The stress-strain state characteristics of the composite vessel computed using
different shell theories
and deformations near the compressed edge. The maximum results and qual-
itative difference were obtained for ANST. This is due to accounting for the
transverse shears by non-linear distribution in a thickness of a shell. Earlier it
was shown [20] that ANST’s based results were closest to the ones of 3D elastic
theory in most cases.
The winding angle’s influence on the COPV performance was investigated
using parametric analysis.
Dependence of the maximum nVMS in the matrix 𝑏𝑠𝑚 (dashed curves) and
the fibers 𝑏𝑠𝑟 (dash–dotted curves), and the maximum size of the displacement
vector ||𝑣|| (solid curves) are shown in fig. 3. KLST’s results are drawn without
marks, TiST – with symbols △, ANST – with .
bsm, bsr w, mm
||v||,mm bsm, bsr w, mm
||v||,mm
y,o y,o
Fig. 3. The winding angle’s influence on the composite vessel stress-strain state
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
The calculated values are very close in the area of their minima (fig. 3 left
side). The graphs of kinematic function ||𝑣|| coincide qualitatively. Some notice-
able quantitative difference are revealed only for KLST’s results.
The range 𝜓 ∈ (42; 45) corresponds to the zones of mimimum values (fig.
3 right side), which practically coincide (min𝜓 𝑏𝑠𝑚 ≈ 0.65, min𝜓 𝑏𝑠𝑟 ≈ 1.05,
min𝜓 ||𝑣|| ≈ 5 · 10−3 m), as well as the angles, where these values are obtained
(𝜓 ≈ 43.2∘ for 𝑏𝑠𝑚 and 𝑏𝑠𝑟 , 𝜓 ≈ 43.8∘ for ||𝑣||).
It was revealed that the winding angles corresponding to minimum stresses
values were almost insensitive to the thickness variation. The change of ℎ from
0.6 to 1.6 cm corresponded to the angle’s change about 0.2∘ .
Additionally we investigated stress-strain state of the vessel (the thickness
ℎ = 0.6 cm, the winding angles 𝜓 = ±43.2), when nVMS in the matrix and the
fibers were near their minimum (fig. 4). The adopted notation is the same as in
fig. 2.
bsr
bsr
bsm
bsm
q z, mm
Fig. 4. The stress-strain state of the vessel (𝜓 = ±43.2) computed using the three shell
theories
And again the difference is visible only in a very small region near the edge
but now this difference is small enough to be neglected. Moreover the displace-
ment values of the reference surface, the efforts and the moments completely
coincide for all the theories.
All the theories (KLST, TiST, ANST) provided similar estimated character-
istics of stress-strain state. This vessel was characterized not only by essential
decrease of the maximal nVMS in the matrix and fibers, but also by their al-
most uniform distribution along the generatrix. At the same time the values of
bending moments significantly reduced bringing vessel’s stress-strain state close
to momentless.
The performed analysis showed that the optimizing problem can be solved
using rather simple shell theories (KLST, TiST). These theories are characterized
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
by lower computational complexity of corresponding boundary value problem if
compared to ANST. It takes from 10 to 20 times less resources.
One can see that the winding angle as a design parameter gives an oppor-
tunity to increase the vessel’s strength significantly. The difference between the
”best” and ”worst” designs can reach 20 – 35 times comparing their nVMS in
the matrix and fibers. The ”worst” designs have the winding angle close to 90∘ .
In this case are considerable transverse shears near the compressed edge, and
the loading is redistributed to a rather weak matrix while the fibers remain
unloaded.
4 Inverse problems. Optimization of the Vessel
Inverse problems involve not only numerical methods for fast and reliable solv-
ing of direct boundary value problems, but also require numerical optimization
methods for identifying design parameters.
Here we considered conditional optimization problem, including direct con-
straints on design functions and trajectory constraints on the solution imposed
at the end of the interval. The sequential unconstrained optimization is one of the
most widespread approaches to solution of such problems. The main idea of the
method is terminal functional convolution and multiple solutions of one-criteria
problem using different optimization methods [21]. In our study the modified
Lagrange function was used for the convolution.
Hence we sought for solution of a nonconvex problem of finite-dimensional op-
timization [22] by discretization of design functions. The methods implemented
in the OPTCON-A software [23] were used to get the corresponding solution.
In our study several vessels with different type of design parameters were
investigated. The parameters were either functions or constants. Additionally a
design of continuous winding on the geodesic path was considered, where the
only design function was its curvature radius.
Uniform mesh for design functions discretization included 7 points, except for
the geodesic winding design with nonuniform mesh of 17 points. The distance
between points was also the solution of corresponding optimization problem.
Approximation of the design functions was carried out using the 3rd degree
natural splines.
The masses of these vessels are shown in tab. 1. Numbers after ”F” denote
the design parameters-functions, after ”C” – the design parameters-constants
with notations: 1 – 𝜓, 2 – ℎ, 3 – 𝑅1 . ”Geod” denotes the design with continuous
geodesic winding.
We used the mass of C123 design as a basis for further comparisons. It was
about 22 kg (tab. 1).
The considered design with the continuous geodesic winding has been one of
COPV widely used in practice [24]. Its mass was 19 kg while the design F3C12
(the constant winding angles and thickness, the variable geometry) was about
16 kg. The main reason was the circumferential winding near the opening, which
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
Table 1. The different designs comparison
Design Mass, kg 𝑀/𝑀𝐶123 𝜓max − 𝜓min ℎmax /ℎmin 𝑅1max /𝑅1min
F123 16.04 72.5% 4 1.08 1.62
F12C3 16.02 72.4% 12 1.55 1.00
F13C2 16.05 72.5% 2 1.00 1.94
F23C1 17.01 76.9% 0 4.74 3.14
F1C23 20.23 91.4% 15 1.00 1.00
F2C13 16.42 74.2% 0 1.83 1.00
F3C12 16.07 72.6% 0 1.00 1.83
Geod 19.09 86.2% 85 9.95 5.42
C123 22.13 100.0% 0 1.00 1.00
did not allow using all the fiber resources. Therefore it was necessary to increase
the vessel thickness near the holes.
Comparison of the designs with two constants and one function (F1C23,
F2C13, F3C12) shows that the possibility to change the value of parameter along
the radius (parameter’s variability) is the most critical for the shell geometry
(𝑅1 ) and its thickness.
Fig. 5 presents the design parameters and the vessel’s generatrix for several
designs.
Important additional design characteristic is its ”adaptability in manufac-
turing”. For example, the 5 - 10 times difference of thickness along the meridian
would become a serious obstacle for vessels manufacturing. Thus, designs of
nearly minimum mass possessing good properties and satisfying to the given
technological constraints could be of great value, than optimum without them.
Let’s consider the following characteristics of the obtained solutions (tab. 1):
the difference between the maximal and minimum values of angles (𝜓max −𝜓min ),
the maximum thickness ratio (to ℎmax /ℎmin ) and the maximum curvature radius
ratio (𝑅1max /𝑅1min ), which characterizes deviation of the generatrix from a circle
arch.
According to tab. 1 the design with the geodesic continuous winding has
the thickness ratio about 10 and large gradient near the edge. We found the
design F123, where the parameter variance is small – no more than 10% for the
thickness and 4∘ for the winding angle with low gradients. The designs with the
constant thickness (F13C2, F3C12) showed that it was possible to receive the
vessels with weight close to minimum, having on hand only such design functions
as 𝑅1 and 𝜓.
5 Inverse Problem. Verification of the Solutions
We verified the solutions of optimization problem by substituting the obtained
design parameters into the direct problem. Fig. 6 shows key characteristics of
the stress-strain state for the designs F123, F23C1, C123 and Geod.
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
a b
h, m y
q
q
c d
R1, m y, m
q r, m
Fig. 5. The design parameters (a, b, c) and the half vessel’s generatrix (d)
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
a b
M1.10 , N
-3
T1.10 , N/m
-6
q
q
c d
bsr bsm
q q
Fig. 6. The stress-strain state characteristics. (a) – bending moment, (b) – tensile force,
(c) — nVMS for fibers, (d) — nVMS for matrix.
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
It is noteworthy that the stress-strain state of design F123 is almost mo-
mentless, and the fibers are equally stressed. The influence of transverse shear
is minimum.
Let’s substitute this solution in the direct problem. All the three considered
bsr T11, MN/m M11, N
bsm
q q, rad
Fig. 7. The stress-strain state characteristics of the vessel with the optimized design
functions based on the three shell theories
shell theories have yielded close results (fig. 7). The difference is noticeable only
for ANST in narrow zones (less than 1% of all the area of calculation) at the
edges, where non-linear accounting for transverse shear gives difference of about
5%. At the same time the estimated efforts and bending moments are very close
for all the theories, and the bending moments are very small.
Thus, it is possible to use the simplest shell theory to solve such optimization
problem and the estimation of stress-strain state will be close to those obtained
using more complex theories.
6 Conclusions
A technology of COPV optimization has been developed. It makes possible to
obtain high pressure vessel designs that not only meet such requirements as
minimum mass, preset volume and strength, but also possess a number of addi-
tional valuable engineering characteristics including stress-strain state close to
momentless and almost equally stressed fibers.
Non-constant design parameters, such as thickness, winding angles and cur-
vature radius of composite shell give a possibility for additional reduction of
COPV mass while keeping its strength. The obtained design with the variable
design parameters are up to 27% lighter if compared to the best design with the
constant parameters.
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
The optimization problem solutions have been verified by solving the direct
problems with obtained design parameters using the classical shell theory and
theories with shear terms.
Our study has demonstrated acceptability and convenience of using simple
mathematical models based on the Kirchhoff—Love and Timoshenko shell the-
ories for numerical solving of the optimization problems.
Acknowledgments. This study was supported by Integrated program of basic
scientific research of SB RAS No. 24, project II.2 ”Design of computational
technologies for calculation and optimal design of hybrid composite thin-walled
structures” and the scientific project RFBR 15-37-20265.
References
1. Beeson, H.D., Davis D.D., Ross, W.L., Tapphorn, R.M.: Composite Overwrapped
Pressure Vessels. NASA/TP—2002–210769 (2002)
2. Thesken, J.C., Murthy, P.L.N., Phoenix, S.L., Greene, N., Palko, J.L., Eldridge,
J., Sutter, J., Saulsberry, R., Beeson, H.: A Theoretical Investigation of Composite
Overwrapped Pressure Vessel (COPV) Mechanics Applied to NASA Full Scale Tests.
NASA/TM—2009–215684 (2009)
3. Stadler, W., Krishnan V.: Natural structural shapes for shells of revolution in the
membrane theory of shells. Structural Optimization. 1, 19–27 (1989)
4. Banichuk, N.V.: Optimization of axisymmetric membrane shells. Applied Mathe-
matics and Mechanics. V. 1, iss. 4. 578–586 (2007) (in Russian)
5. Obraztsov, I.F., Vasiliev, V.V., Bunakov, V.A.: Optimum reinforcing of composite
shells of rotation. Mashinostroenie, Moscow (1977) (in Russian)
6. Vasiliev, V.V., Krikanov, A.A., Razin, A.F.: New generation of filament-wound com-
posite pressure vessels for commercial applications. Composite structures. 62, 449–
459 (2003).
7. Abbas Vafaeesefat: Optimization of composite pressure vessels with metal liner by
adaptive response surface method. Journal of Mechanical Science and Technology.
25 (11), 2811–2816 (2011).
8. Manolis Papadrakakis, Nikos D. Lagaros: Soft computing methodologies for struc-
tural optimization. Applied Soft Computing. Vol. 3, iss. 3, 283 300 (2003).
9. Cho-Chung Liang, Hung-Wen Chen, Cheng-Huan Wang: Optimum design of dome
contour for filament-wound composite pressure vessels based on a shape factor.
Composite structures. 58, 469–482 (2002).
10. Cheol-Ung Kim, Ji-Ho Kang, Chang-Sun Hong, Chun-Gon Kim: Optimal design of
filament wound structures under internal pressure based on the semi-geodesic path
algorithm. Composite structures. 67, 443–452 (2005).
11. Lei Zu, Sotiris Koussios, Adriaan Beukers: Shape optimization of filament wound
articulated pressure vessels based on non-geodesic trajectories. Composite struc-
tures. 92, 339–346 (2010).
12. Hisao Fukunaga, Masuji Uemura: Optimum design of helically wound composite
pressure vessels. Composite structures. 1, 31–49 (1983).
13. Grigorenko, Ya.M., Vasilenko, A.T.: Static problem of anisotropic inhomogeneous
shells. Nauka, Moscow (1992) (in Russain)
256
Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
14. Andreev, A.N., Nemirovskii, Yu.V.: Multilayer anisotropic shells and plates: bend-
ing, stability, oscillation. Nauka, Novosibirsk (2001) (in Russain)
15. Novozhilov, V.V.: Theory of thin shells. Sudpromgiz, Leningrad (1951) (in Russian)
16. Golushko, S.K.: Direct and inverse problems in the mechanics of composite plates
and shells. Notes on Numerical Fluid Mechanics and Multidisciplinary Design. Vol.
88, 205–227.
17. Golushko, K.S., Golushko, S.K., Yurchenko, A.V:. On modeling of mechanical prop-
erties of fibrous composites. Notes on Numerical Fluid Mechanics and Multidisci-
plinary Design. Vol. 115, 107–120. Springer (2011).
18. Ascher U., Christiansen J., Russel R.D.: Collocation software for boundary value
ODEs. ACM. Trans. on Math. Software. Vol. 7, N. 2, 209–222 (1981)
19. Golushko, S.K., Yurchenko, A.V.: Solution of boundary value problems in me-
chanics of composite plates and shells. Russian Journal of Numerical Analysis and
Mathematical Modelling. Vol. 25, N. 1, 27–55 (2010)
20. Golushko, S.K., Nemirovskii, Yu.V.: Direct and inverse problems of mechanics of
composite plates and shells of revolutions. FIZMATLIT, Moscow (2008) (in Russain)
21. Bertsekas, D.: Constrained optimization and Lagrange multiplier method. Radio
and cvyas, Moscow (1987) (in Russain)
22. Evtushenko, Yu.G.: Methods of solution of extremum problems and their applica-
tion in systems of optimization. Nauka, Moscow (1982) (in Russain)
23. Gornov, A.Yu.: Computational technologies for optimal control solution. Nauka,
Novosibirsk (2009) (in Russain)
24. Lepikhin, A.M., Moskvichev, V.V., Chernyayev, A.P., Pokhabov, Yu. P., Khal-
imanovich, V.I.: Experimental evaluation of strength and tightness of metal–
composite high-pressure vessels. Deformation and rupture of materials. 6, 30–36
(2015) (in Russain)
257