4. Numerical analysis and design of pressure vessels

Composite overwrapped pressure vessels (COPV) are used in the rocket and spacecraft making industry due to their high strength and lightweight. Consisting of a thin, nonstructural liner wrapped with a structural fiber composite COPV are produced to hold the inner pressure of tens and hundreds atmospheres. COPV have been one of the most actual and perspective directions of research, supported especially by NASA [22, 23].

Designing of a highly reliable and efficient COPV requires a technology for analysis of its deformation behavior 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.

We define the optimization problems the following way: to find extremum of one functional

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25

The structural 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 following assumptions:

These assumptions allow us to reduce dimension of the corresponding mathematical problem and to build the mathematical vessel's models based on the different theories of multilayer

Let us consider the vessel as a shell rigidly compressed on the edge. Taking into account a symmetry plane in the middle of the vessel, it is enough to calculate and design only its one half. The type of loading and boundary conditions allows considering the axisymmetric prob-

The shell is set by rotation of the generatrix r ¼ rð Þ θ around axis 0y (Figure 3) where r is the current point of the shell radius, θ is the angle between the normal to the shell surface and the

The Kirchhoff-Love shell theory [36] (KLST) and the improved Timoshenko [34] (TiST) and Andreev-Nemirovskii [35] (ANST) theories are used to solve the direct calculation problems of multilayer composite vessels, to analyze their behavior and to verify optimization problem

solutions. The full systems of equations were described in the paper [17].

from (Eq. (17)) under other constraints.

nonisotropic shells.

lem statement.

spin axis changing within ½ � θ0; θ<sup>1</sup> .

Figure 3. Shell of rotation geometry.

1. the vessel is a multilayer thin-walled structure;

4. the vessel's main loading is high inner pressure.

3. the reinforced layer's material is quasi-homogeneous;

2. the vessel's layers can have different mechanical characteristics;

Application of combination 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 determines the 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 statement, for example using the momentless (membrane) shell theory and the netting model of composite material (CM) [24–27]. 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, produces a number of challenges that must be overcome, for example, lack of reliable methods for global optimization; nonconvexity and nonlinearity of constraint functions; ill-conditioned boundary value problems; different scaling of optimization criteria represents just some of the obstacles that prevent from reliable optimization of COPV.

Numerical analysis is usually a computation-intensive process and takes considerable time. One way to solve this problem is approximation of the objective function using different approaches, such as response surface method [28] and neural network [29]. Some kinds of numerical analyses use a small number of design variables, functions and/or corresponding set of their discrete values (analytical geometry parametrization [30], finite set of feasible winding angles [31]).

Another way is reasonable simplification of the elasticity problem statement, for example by using the membrane theory or other shell theories [30, 32, 33], that leaves the question of results validity. This is the approach we have applied in our study. For validation, we have used the Timoshenko [34] and Andreev-Nemirovskii [35] shell theories, accounting 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 have high engineering value.

#### 4.1. The problem statement and the mathematical models

Let us 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:

$$V \ge V\_{0\prime} \quad \text{ } P \ge P\_{0\prime} \quad \text{ } M \le M\_{0\prime} \tag{17}$$

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.

We define the optimization problems the following way: to find extremum of one functional from (Eq. (17)) under other constraints.

The structural 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 following assumptions:

1. the vessel is a multilayer thin-walled structure;

target COPV parameters through changing vessel's geometry, structural and mechanical mate-

Application of combination 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

So far, there have been two main approaches in optimization of composite structures: analyt-

In the first approach, the problems are solved basing on their simplified statement, for example using the momentless (membrane) shell theory and the netting model of composite material (CM) [24–27]. The obtained results may be far from reality; however, they are of value for

Application of the numerical approach in designing, on the other hand, produces a number of challenges that must be overcome, for example, lack of reliable methods for global optimization; nonconvexity and nonlinearity of constraint functions; ill-conditioned boundary value problems; different scaling of optimization criteria represents just some of the obstacles that

Numerical analysis is usually a computation-intensive process and takes considerable time. One way to solve this problem is approximation of the objective function using different approaches, such as response surface method [28] and neural network [29]. Some kinds of numerical analyses use a small number of design variables, functions and/or corresponding set of their discrete values (analytical geometry parametrization [30], finite set of feasible winding angles [31]).

Another way is reasonable simplification of the elasticity problem statement, for example by using the membrane theory or other shell theories [30, 32, 33], that leaves the question of results validity. This is the approach we have applied in our study. For validation, we have used the Timoshenko [34] and Andreev-Nemirovskii [35] shell theories, accounting transverse

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 have high engineering value.

Let us 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

where V is the volume of the vessel, P is inner pressure and M is the vessel's mass and they are

V ≥V0, P ≥ P0, M ≤ M0, (17)

rial parameters while keeping its useful load.

testing of numerical optimization methods.

prevent from reliable optimization of COPV.

shears with different degrees of accuracy.

constrained by some preset values V0, P0, M0.

the following requirements:

4.1. The problem statement and the mathematical models

ical and numerical ones.

24 Optimum Composite Structures

determines the success of an optimum designing of such structures.


These assumptions allow us to reduce dimension of the corresponding mathematical problem and to build the mathematical vessel's models based on the different theories of multilayer nonisotropic shells.

Let us consider the vessel as a shell rigidly compressed on the edge. Taking into account a symmetry plane in the middle of the vessel, it is enough to calculate and design only its one half. The type of loading and boundary conditions allows considering the axisymmetric problem statement.

The shell is set by rotation of the generatrix r ¼ rð Þ θ around axis 0y (Figure 3) where r is the current point of the shell radius, θ is the angle between the normal to the shell surface and the spin axis changing within ½ � θ0; θ<sup>1</sup> .

The Kirchhoff-Love shell theory [36] (KLST) and the improved Timoshenko [34] (TiST) and Andreev-Nemirovskii [35] (ANST) theories are used to solve the direct calculation problems of multilayer composite vessels, to analyze their behavior and to verify optimization problem solutions. The full systems of equations were described in the paper [17].

Figure 3. Shell of rotation geometry.

Relations between stresses and strains are described by the structural models [18]. The main idea of these models is that CM parameters are calculated through matrix and fibers mechanical parameters, fibers volume content and winding angles. The stress-strain state of matrix and fibers is 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:

$$M = 2\pi \int\_{\theta\_0}^{\theta\_1} rR\_1 h d\theta [\rho\_m (1 - \omega\_r) + \rho\_r \omega\_r] \to \min,\tag{18}$$

which has the singularity at the edge where the winding angle has to be equal to 90 <sup>∘</sup>

tape. As a result, the equation determining the vessel's thickness takes the form:

hR

8 >><

>>:

hR

h rð Þ¼

demonstrate the potentials of using CM.

4.2. Direct problems: analysis of the shell theories

reinforced in the circumferential direction (<sup>ψ</sup> <sup>¼</sup> <sup>90</sup><sup>∘</sup>

simulations, the curves marked with <sup>Δ</sup>—to those using TiST, and □—to ANST.

(Eq. (23)) is applied into practice at r ≥ r<sup>0</sup> þ rω, where r<sup>ω</sup> is equal to the width of the reinforcement

Rcosψ<sup>R</sup> r<sup>ω</sup> cosψð Þ r<sup>0</sup> þ r<sup>ω</sup>

We did not consider the problem of fibers slippage. The main goal of the study was to

Estimation of composite vessel stress-strain state using offered models leads to the solution of boundary value problems for rigid systems of differential equations. These problems are illconditioned, and their solutions have pronounced character of thin boundary layers. Numerical analysis was performed by the spline collocation and discrete orthogonalization methods, realized in the COLSYS [37] and GMDO [38] software. These computing tools have proved to be effective in numerical solving of wide range of problems of composite shell mechanics [1]. We investigated the vessel's deformations by computing its stress-strain state based on the different shell theories. The vessel's shape was a part of a toroid: <sup>R</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>:46m, <sup>θ</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>:108<sup>∘</sup>

<sup>θ</sup><sup>1</sup> <sup>¼</sup> <sup>90</sup><sup>∘</sup> (the computed half), <sup>r</sup>ð Þ¼ <sup>θ</sup><sup>0</sup> <sup>0</sup>:04 m. The carbon composite parameters were: Em <sup>¼</sup> <sup>3</sup> � <sup>10</sup><sup>9</sup> Pa, <sup>ν</sup><sup>m</sup> <sup>¼</sup> <sup>0</sup>:34, Er <sup>¼</sup> <sup>300</sup> � 109 Pa, <sup>ν</sup><sup>r</sup> <sup>¼</sup> <sup>0</sup>:3, <sup>ω</sup><sup>r</sup> <sup>¼</sup> <sup>0</sup>:55, <sup>V</sup><sup>0</sup> <sup>¼</sup> 350 liters where Em, Er

Figure 4 shows the stress-strain state characteristics of the vessel with the thickness h ¼ 0:6 cm,

Figure 4. The stress-strain state characteristics of the composite vessel computed using different shell theories. Longitudinal displacement u—dashed curves; deflection w—solid curves. The curves without symbols correspond to KLST

are the Young's modulus of the matrix and fibers, νm, νr—their Poisson's ratio.

Rcos ψ<sup>R</sup> r cosψð Þr , r ≤ r<sup>0</sup> þ rω;

Mathematical Modeling and Numerical Optimization of Composite Structures

, r ≥ r<sup>0</sup> þ rω:

. The formula

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(24)

27

,

) under the load of 170 atm. On the left, the

where rm, r<sup>r</sup> are the densities of matrix and reinforcing fibers, ω<sup>r</sup> is the volume content of reinforcement.

We chose the following design functions: the curvature radius R1ð Þ θ to define the generatrix; the thickness of the shell hð Þ θ ; the reinforcement angle ψ θð Þ (Figure 3).

The solution has to satisfy the constraints on the shell's inner volume:

$$
\pi \int\_{\theta\_0}^{\theta\_1} r^2 R\_1 \sin \theta d\theta = V\_0 \tag{19}
$$

and the strength requirement:

$$\max\{b\mathbf{s}\_r, b\mathbf{s}\_m\} \le 1,\tag{20}$$

where bsr, bsm are the normalized von Mises stresses in the matrix and fibers [1]. Note that the factor of safety is widely used while solving engineering problems. It can be considered by correction of the right-hand side of the inequality (Eq.(20)).

We used the following constraints on the design functions:

$$0 \le \psi \le 90, \quad h\_0^\* \le h \le h\_1^\*, \quad R\_0^\* \le R\_1 \le R\_1^\*. \tag{21}$$

The method of the continuous geodesic winding has been widely used in the manufacturing of composite shells of revolutions. In this case the winding angles are defined by the Clairaut's formula:

$$r\sin\psi(r) = \mathbb{C},\tag{22}$$

where C—the constant is defined, as a rule, from the condition at the equator of the shell. The thickness equation is

$$h(r) = h\_R \frac{R \cos \psi\_R}{r \cos \psi(r)},\tag{23}$$

which has the singularity at the edge where the winding angle has to be equal to 90 <sup>∘</sup> . The formula (Eq. (23)) is applied into practice at r ≥ r<sup>0</sup> þ rω, where r<sup>ω</sup> is equal to the width of the reinforcement tape. As a result, the equation determining the vessel's thickness takes the form:

$$h(r) = \begin{cases} h\_R \frac{R \cos \psi\_R}{r\_\omega \cos \psi (r\_0 + r\_\omega)}, & r \le r\_0 + r\_\omega; \\ h\_R \frac{R \cos \psi\_R}{r \cos \psi (r)}, & r \ge r\_0 + r\_\omega. \end{cases} \tag{24}$$

We did not consider the problem of fibers slippage. The main goal of the study was to demonstrate the potentials of using CM.

#### 4.2. Direct problems: analysis of the shell theories

Relations between stresses and strains are described by the structural models [18]. The main idea of these models is that CM parameters are calculated through matrix and fibers mechanical parameters, fibers volume content and winding angles. The stress-strain state of matrix and fibers is 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

where rm, r<sup>r</sup> are the densities of matrix and reinforcing fibers, ω<sup>r</sup> is the volume content of

We chose the following design functions: the curvature radius R1ð Þ θ to define the generatrix;

where bsr, bsm are the normalized von Mises stresses in the matrix and fibers [1]. Note that the factor of safety is widely used while solving engineering problems. It can be considered by

<sup>0</sup> ≤ h ≤ h<sup>∗</sup>

The method of the continuous geodesic winding has been widely used in the manufacturing of composite shells of revolutions. In this case the winding angles are defined by the Clairaut's

where C—the constant is defined, as a rule, from the condition at the equator of the shell. The

Rcos ψ<sup>R</sup>

h rð Þ¼ hR

1, R<sup>∗</sup>

<sup>0</sup> ≤R<sup>1</sup> ≤ R<sup>∗</sup>

rR1hdθ rmð Þþ 1 � ω<sup>r</sup> rrω<sup>r</sup> ½ � ! min, (18)

R<sup>1</sup> sin θdθ ¼ V<sup>0</sup> (19)

max bsr f g ; bsm ≤ 1, (20)

r sinψð Þ¼ r C, (22)

<sup>r</sup> cosψð Þ<sup>r</sup> , (23)

<sup>1</sup>: (21)

The objective function whose minimum is required is the minimum mass:

ð<sup>θ</sup><sup>1</sup> θ0

the thickness of the shell hð Þ θ ; the reinforcement angle ψ θð Þ (Figure 3). The solution has to satisfy the constraints on the shell's inner volume:

> π ð<sup>θ</sup><sup>1</sup> θ0 r 2

0 ≤ψ≤ 90, h<sup>∗</sup>

correction of the right-hand side of the inequality (Eq.(20)). We used the following constraints on the design functions:

M ¼ 2π

stage of failure.

26 Optimum Composite Structures

reinforcement.

formula:

thickness equation is

and the strength requirement:

Estimation of composite vessel stress-strain state using offered models leads to the solution of boundary value problems for rigid systems of differential equations. These problems are illconditioned, and their solutions have pronounced character of thin boundary layers. Numerical analysis was performed by the spline collocation and discrete orthogonalization methods, realized in the COLSYS [37] and GMDO [38] software. These computing tools have proved to be effective in numerical solving of wide range of problems of composite shell mechanics [1].

We investigated the vessel's deformations by computing its stress-strain state based on the different shell theories. The vessel's shape was a part of a toroid: <sup>R</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>:46m, <sup>θ</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>:108<sup>∘</sup> , <sup>θ</sup><sup>1</sup> <sup>¼</sup> <sup>90</sup><sup>∘</sup> (the computed half), <sup>r</sup>ð Þ¼ <sup>θ</sup><sup>0</sup> <sup>0</sup>:04 m. The carbon composite parameters were: Em <sup>¼</sup> <sup>3</sup> � <sup>10</sup><sup>9</sup> Pa, <sup>ν</sup><sup>m</sup> <sup>¼</sup> <sup>0</sup>:34, Er <sup>¼</sup> <sup>300</sup> � 109 Pa, <sup>ν</sup><sup>r</sup> <sup>¼</sup> <sup>0</sup>:3, <sup>ω</sup><sup>r</sup> <sup>¼</sup> <sup>0</sup>:55, <sup>V</sup><sup>0</sup> <sup>¼</sup> 350 liters where Em, Er are the Young's modulus of the matrix and fibers, νm, νr—their Poisson's ratio.

Figure 4 shows the stress-strain state characteristics of the vessel with the thickness h ¼ 0:6 cm, reinforced in the circumferential direction (<sup>ψ</sup> <sup>¼</sup> <sup>90</sup><sup>∘</sup> ) under the load of 170 atm. On the left, the

Figure 4. The stress-strain state characteristics of the composite vessel computed using different shell theories. Longitudinal displacement u—dashed curves; deflection w—solid curves. The curves without symbols correspond to KLST simulations, the curves marked with <sup>Δ</sup>—to those using TiST, and □—to ANST.

displacements of the reference surface along the generatrix u1ð Þr (dashed curves) and the normal displacement of these surfaces w rð Þ (solid curves) are shown. On the right is the distribution of normalized von Mises stress (nVMS) along the thickness in the matrix bsmð Þr . The solid curves correspond to a slice at the shell edge, the dashed curves — to a slice at θ ¼ 0:1.

It is easy to see that the basic kinematic characteristics coincide both qualitatively and quantitatively. Small differences are observed only for the stresses and deformations near the compressed edge. The maximum results and qualitative difference were obtained for ANST. This is due to accounting for the transverse shears by nonlinear distribution in a thickness of a shell. Earlier it was shown [1] that ANST's-based results were the 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 k v ! <sup>k</sup> (solid curves) are shown in Figure 5.

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 displacement values of the reference surface,

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All the theories (KLST, TiST, ANST) provided similar estimated characteristics of stress-strain state. This vessel was characterized not only by essential decrease of the maximal nVMS in the matrix and the fibers, but also by their uniform distribution along the generatrix. At the same time, the values of bending moments significantly reduced bringing vessel's stress-strain state

The performed analysis showed that the optimization problem can be solved using rather simple shell theories (KLST, TiST). These theories are characterized by lower computational complexity of corresponding boundary value problem if compared to ANST. It takes from 10

One can see that the winding angle as a design parameter gives an opportunity 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<sup>∘</sup> . 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.

Inverse problems involve not only numerical methods for fast and reliable solving of direct boundary value problems, but also require numerical optimization methods for finding design

Here we considered conditional optimization problem, including direct constraints 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-criterion problem using different optimization methods [39]. In our study, the

the efforts and the moments completely coincide for all the theories.

Figure 6. The stress-strain state of the vessel (ψ ¼ �43:2), computed using the three shell theories.

close to momentless.

to 20 times less resources.

parameters.

4.3. Inverse problems: optimization of the vessel

modified Lagrange function was used for the convolution.

The calculated values are very close in the area of their minima (Figure 5 left side). The graphs of kinematic function kvk coincide qualitatively. Some noticeable quantitative differences are revealed only for KLST's results.

The range ψ∈ ð Þ 42; 45 corresponds to the zones of minimum values (Figure 5 right side), which practically coincide (minψbsm <sup>≈</sup> <sup>0</sup>:65, min<sup>ψ</sup> bsr <sup>≈</sup> <sup>1</sup>:05, minψkv<sup>k</sup> <sup>≈</sup> <sup>5</sup> � <sup>10</sup>�<sup>3</sup> m), as well as the angles, where these values are obtained (ψ<sup>≈</sup> <sup>43</sup>:2<sup>∘</sup> for bsm and bsr, <sup>ψ</sup><sup>≈</sup> <sup>43</sup>:8<sup>∘</sup> for <sup>k</sup>vk).

It was revealed that the winding angles of 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<sup>∘</sup> .

Additionally, we investigated stress–strain state of the vessel (the thickness h ¼ 0:6 cm, the winding angles at ψ ¼ �43:2), when nVMS in the matrix and the fibers were near their minimum (Figure 6). The adopted notation is the same as in Figure 4.

Figure 5. The winding angle's influence on the composite vessel stress-strain state. KLST's results are drawn without marks, TiST — with symbols <sup>Δ</sup>, ANST — with □.

Mathematical Modeling and Numerical Optimization of Composite Structures http://dx.doi.org/10.5772/intechopen.78259 29

Figure 6. The stress-strain state of the vessel (ψ ¼ �43:2), computed using the three shell theories.

displacements of the reference surface along the generatrix u1ð Þr (dashed curves) and the normal displacement of these surfaces w rð Þ (solid curves) are shown. On the right is the distribution of normalized von Mises stress (nVMS) along the thickness in the matrix bsmð Þr . The solid curves

It is easy to see that the basic kinematic characteristics coincide both qualitatively and quantitatively. Small differences are observed only for the stresses and deformations near the compressed edge. The maximum results and qualitative difference were obtained for ANST. This is due to accounting for the transverse shears by nonlinear distribution in a thickness of a shell. Earlier it was shown [1] that ANST's-based results were the closest to the ones of 3D elastic

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

The calculated values are very close in the area of their minima (Figure 5 left side). The graphs of kinematic function kvk coincide qualitatively. Some noticeable quantitative differences are

The range ψ∈ ð Þ 42; 45 corresponds to the zones of minimum values (Figure 5 right side), which practically coincide (minψbsm <sup>≈</sup> <sup>0</sup>:65, min<sup>ψ</sup> bsr <sup>≈</sup> <sup>1</sup>:05, minψkv<sup>k</sup> <sup>≈</sup> <sup>5</sup> � <sup>10</sup>�<sup>3</sup> m), as well as the angles, where these values are obtained (ψ<sup>≈</sup> <sup>43</sup>:2<sup>∘</sup> for bsm and bsr, <sup>ψ</sup><sup>≈</sup> <sup>43</sup>:8<sup>∘</sup> for <sup>k</sup>vk).

It was revealed that the winding angles of 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

Additionally, we investigated stress–strain state of the vessel (the thickness h ¼ 0:6 cm, the winding angles at ψ ¼ �43:2), when nVMS in the matrix and the fibers were near their

Figure 5. The winding angle's influence on the composite vessel stress-strain state. KLST's results are drawn without

minimum (Figure 6). The adopted notation is the same as in Figure 4.

! <sup>k</sup> (solid curves)

correspond to a slice at the shell edge, the dashed curves — to a slice at θ ¼ 0:1.

bsr (dash-dotted curves), and the maximum size of the displacement vector k v

theory in most cases.

28 Optimum Composite Structures

are shown in Figure 5.

about 0:2<sup>∘</sup>

.

revealed only for KLST's results.

marks, TiST — with symbols <sup>Δ</sup>, ANST — with □.

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 displacement 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 characteristics of stress-strain state. This vessel was characterized not only by essential decrease of the maximal nVMS in the matrix and the fibers, but also by their 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 optimization problem can be solved using rather simple shell theories (KLST, TiST). These theories are characterized 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 opportunity 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<sup>∘</sup> . 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.

#### 4.3. Inverse problems: optimization of the vessel

Inverse problems involve not only numerical methods for fast and reliable solving of direct boundary value problems, but also require numerical optimization methods for finding design parameters.

Here we considered conditional optimization problem, including direct constraints 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-criterion problem using different optimization methods [39]. In our study, the modified Lagrange function was used for the convolution.

have been built. A satisfactory match with the results of mechanical tests has been obtained. The study has proved that the nonlinear properties of polymer matrices and carbon fibers

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31

• The technology of optimization of COPV 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 additional valuable engineering characteristics including stress-strain state close to momentless and almost equally stressed fibers.

• Nonconstant design parameters, such as thickness, winding angles and curvature radius of composite shell give the possibility for additional reduction of COPV mass while keeping its strength. The solutions of the optimization problem have been verified by solving the direct problems with obtained design parameters using the classical and improved shell theories.

• The study has demonstrated acceptability and convenience of using simple mathematical models based on Kirchhoff—Love and Timoshenko shell theories for numerical solving

[1] Golushko S, Nemirovskii Yu. Direct and inverse problems of mechanics of composite

[2] Boyce M, Arruda E. An experimental and analytical investigation of the large strain compressive and tensile response of glassy polymers. Polymer Engineering & Science.

[3] Lagace P. Nonlinear stress-strain behavior of graphite/epoxy laminates. AIAA Journal.

The study was supported by project 18-13-00392 of Russian Science Foundation.

\*Address all correspondence to: s.k.golushko@gmail.com

2 Institute of Computational Technologies SB RAS, Novosibirsk, Russia

plates and shells. Moscow: Fizmatlit; 2008. 432 p. (in Russian)

2004;30(20):1288-1298. DOI: 10.1002/pen.760302005

1985;23(10):1583-1589. DOI: 10.2514/3.9127

1 Novosibirsk State University, Novosibirsk, Russia

optimization problems.

Acknowledgements

Author details

Sergey Golushko1,2\*

References

should be taken into account when calculating and designing real structures.

Figure 7. The stress-strain state characteristics of the vessel with the optimized design functions based on the three shell theories. Longitudinal force T11—dashed curves; bending moment M11—solid curves.

Hence we sought for solution of a nonconvex problem of finite-dimensional optimization [40] by discretization of design functions. The methods realized in the OPTCON-A software [41] were used to get the corresponding solution.

The considered design with the continuous geodesic winding has been one of COPV widely used in practice [42, 43].

Important additional design characteristic is its "adaptability in manufacturing." 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.

According to Amelina et al. [44], the design with the geodesic continuous winding has the thickness ratio about 10 and large gradient near the edge.

We verified the solutions of optimization problem by substituting the obtained design parameters into the direct problem. In [44] shown that all three theories yielded close results (Figure 7). The difference is noticeable only for ANST in narrow zones (less than 1% of all 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.
