Application of the finite segment method to stabilisation of the force in a riser connection with a wellhead
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Abstract
This paper presents a spatial model of riser dynamics formulated using the segment method and its applications. The model has been validated by comparison of the authors’ own results with those obtained from experimental measurements and Abaqus on the basis of forced vibration with large amplitude for the riser submerged in water. The influence of the sea environment is considered. Correctness and numerical effectiveness of the model enable us to formulate and solve the force stabilisation problem. A dynamic optimisation problem is formulated and solved. As a result vertical courses of movement of the upper end of the riser are obtained which compensate the horizontal movement of the base and stabilise the force in the connection of the riser with a wellhead.
Keywords
Dynamics of risers Slender system Finite segment method Optimisation1 Introduction
Deep sea drilling for hydrocarbons requires the use of very sophisticated technical equipment. Thus complex computational methods are used in the design phase of offshore devices. Calculation of statics and dynamics of risers is an important element of this process and is carried out before production. Risers are long elements with bending and longitudinal flexibilities; these risers are loaded with significant forces acting at their ends and large transverse forces caused by sea currents and hydrodynamic forces. Deflections of risers can be very large, and they may undergo elastoplastic deformations [1].
For modelling such structures as risers, the finite element method is used most often [2], which is reflected in many software packages like ANSYS, Abaqus or Riflex. However, many other methods are also used: finite difference method [3], lumpedmass method [4, 5, 6], finite segment method [7, 8, 9, 10], or the similar rigid finite element method [11, 12, 13, 14, 15, 16]. Most of the methods mentioned use Morison equations in order to describe the influence of the sea.
In this paper, we present the authors’ own formulation of the finite segment method for discretisation of a link with bending flexibility. The finite segment method has been used in modelling risers and cables for more than 50 years [7, 10, 17]. Its main idea lies in dividing a slender link into rigid elements connected by spherical joints treated as massless springdamping elements in order to reflect torsional and bending flexibilities, and sometimes also longitudinal flexibility of the discretised links [18]. A similar approach derived from the rigid finite element method [19, 20] is used by Drąg [15, 16].
The finite segment method belongs to a group of methods used for dynamic analysis of slender systems, in which a continuous system is replaced by a system of rigid bodies reflecting inertial features of the system modelled. These rigid bodies are not point masses as in the lumpedmass method, but elements (segments) for which masses and moments of inertia are accurately determined. Using this method, it is not necessary to formulate partial differential equations for dynamics, but the equations of motion are usually derived from the Lagrange equations by means of multibodysystem techniques. The choice of generalised coordinates is an important issue. Drąg in his papers [15, 16] uses absolute coordinates, which are three translational coordinates (describing the position of the chosen point of each rigid element) and up to three rotational coordinates describing the orientation of the rigid element. The translational coordinates are then eliminated by introducing reactions and constraint equations. The advantage of using the absolute coordinates is a pseudodiagonal mass matrix, while the large number of coordinates and the necessity of stabilisation of constraint equations are disadvantages of the method. In the paper, we propose to use joint coordinates. In this case, the vector of generalised coordinates has a minimum number of components; these components are three coordinates of a chosen point of the link discretised and up to a three coordinates describing the orientation of each segment. This means that it is not necessary to introduce reactions and formulate constraint equations. Consequently, there is no need to stabilise the geometrical constraint equations as in the case of absolute coordinates. However, the disadvantage of the method is a full mass matrix. When large deformations are analysed, the mass matrix obtained using both approaches depends on generalised coordinates. In general, the finite segment method with the use of joint coordinates is more numerically effective for a moderate number of segments (less than 100) than when the absolute coordinates are used.
The finite segment method is admittedly not in all respect more efficient than the finite element method applied in commercial software packages, including those for simulation of statics and dynamics of offshore appliances. However, as far as analysis of slender links (cables, risers) is concerned, the finite segment method poses important advantages, such as simple physical interpretation, the ease of describing large deformations of links, and the possibility of special applications, including those presented in the paper.
Analysis of literature concerned with modelling of risers shows that most attention is devoted to analysis of static deflections and bending vibrations both in 2D and 3D analysis. Some research, however, shows that it is necessary to consider torsion. For example, [5, 27] demonstrate how to take into account the bending–torsion coupling effect, especially in the case of seabed interaction. The finite segment formulation presented in this paper, similar to [25, 28, 29, 30], omits the torsional flexibility of the riser model.
Most papers dealing with modelling of risers consider longitudinal flexibility. Consideration of elongation of slender systems is especially necessary when the analysis is concerned with toptensioned risers or catenary lines. The elongation of the riser (500 m long) considered in this paper does not exceed 3 cm. This means that the influence of longitudinal deformations on displacements of the riser is negligible, which is why considerations in this paper are limited to analysis of spatial bending vibrations. Consequently, forces acting at the ends of the riser can be calculated in a different way (to be explained in Sect. 2).
2 Model of a slender link using the finite segment method
The basic idea of the finite segment method lies in the discretisation of a flexible link into rigid elements (segments) reflecting inertial features of the link. The segments are connected by means of joints consisting of massless and nondimensional springdamping elements reflecting stiffness characteristics of the link (Fig. 1).

first rotation about axis \(z^{{\prime }}\) parallel to z of angle \(\psi _i \),

then rotation about axis \({y}^{\prime \prime }\) of angle \(\theta _i \),

finally rotation of angle \(\varphi _i \) about axis \(x_i^{\prime }\) if torsion is considered.
The energy of spring deformation and its derivatives can be calculated in a similar way to that presented by Wittbrodt et al. [20] and Drąg [16].
Angles \({\Delta }\psi _i \) and \({\Delta }\theta _i \) can be calculated in the form presented in [16].
2.1 Influence of the water environment

buoyancy forces,

hydrodynamic forces,

added mass forces.
Parameters of the riser used in the experiment [21]
Parameter  Notation  Value  Measure unit 

Length  L  13.12  m 
External diameter  \(D_\mathrm{out} \)  0.028  m 
Internal diameter  \(D_\mathrm{inn} \)  0.027  m 
Bending rigidity  EI  29.9  \(\hbox {Nm}^{2}\) 
Riser linear density in water  \(\rho _r^L \)  1.47  kg/m 
Water density  \(\rho _\mathrm{w} \)  1000  \(\hbox {kg}/\hbox {m}^{3}\) 
Resistance coefficient  \(D_x^{\prime } \)  0.03  – 
Resistance coefficient  \(D_{yz}^{\prime } \)  0.7  – 
Inertial coefficient  \(c_M \)  1.8  – 
Influence of the number of segments n on natural frequencies for \(T_U =389\hbox {N}\) exerted at the upper end of the riser
Number of segments n  \(\omega _1 \)  \(\omega _2 \)  \(\omega _3 \)  \(\omega _4 \)  \(\omega _5 \)  \(\omega _6 \)  \(\omega _7 \) 

25  0.4029  0.8161  1.2471  1.7042  2.1942  2.7232  3.2964 
50  0.4027  0.8144  1.2418  1.6919  2.1709  2.6841  3.2362 
75  0.4027  0.8141  1.2408  1.6897  2.1666  2.6769  3.2251 
100  0.4027  0.8140  1.2405  1.6889  2.1651  2.6744  3.2212 
125  0.4027  0.8140  1.2403  1.6885  2.1644  2.6732  3.2195 
150  0.4027  0.8140  1.2402  1.6883  2.1640  2.6726  3.2185 
175  0.4027  0.8140  1.2402  1.6882  2.1638  2.6722  3.2179 
200  0.4027  0.8140  1.2402  1.6881  2.1636  2.6720  3.2175 
Influence of the number of segments n on natural frequencies for \(T_U =1925\hbox {N}\) exerted at the upper end of the riser
Number of segments n  \(\omega _1 \)  \(\omega _2 \)  \(\omega _3\)  \(\omega _4 \)  \(\omega _5 \)  \(\omega _6 \)  \(\omega _7 \) 

25  1.0162  2.0393  3.0761  4.1333  5.2175  6.3353  7.4928 
50  1.0157  2.0353  3.0625  4.1013  5.1553  6.2281  7.3233 
75  1.0156  2.0345  3.0600  4.0954  5.1438  6.2083  7.2921 
100  1.0156  2.0343  3.0592  4.0933  5.1398  6.2014  7.2812 
125  1.0155  2.0342  3.0588  4.0924  5.1379  6.1982  7.2761 
150  1.0155  2.0341  3.0585  4.0919  5.1369  6.1965  7.2734 
175  1.0155  2.0341  3.0584  4.0915  5.1363  6.1954  7.2717 
200  1.0155  2.0340  3.0583  4.0913  5.1359  6.1947  7.2706 
Comparison of natural frequencies
Tension  Method  Natural frequency \(\omega \left[ {\frac{rad}{s}} \right] \)  

\(\omega _1 \)  \(\omega _2 \)  \(\omega _3 \)  \(\omega _4 \)  \(\omega _5\)  \(\omega _6 \)  \(\omega _7 \)  
Case 1 \(T_U =389\hbox {N}\)  Experiment [21]  0.405  0.873  1.326  1.804  2.278  2.797  3.389 
VIM [22]  0.402  0.813  1.239  1.687  2.164  2.675  3.224  
calculated SM  0.403  0.814  1.240  1.688  2.164  2.672  3.218  
Case 2 \(T_U=813\hbox {N}\)  Experiment [21]  0.633  1.329  1.954  2.625  3.312  4.028  4.735 
VIM [22]  0.643  1.290  1.948  2.619  3.308  4.020  4.758  
calculated SM  0.635  1.276  1.925  2.588  3.269  3.973  4.702  
Case 3 \(T_U =1185\hbox {N}\)  Experiment [21]  0.782  1.423  2.394  3.203  4.024  4.875  5.704 
VIM [22]  0.795  1.594  2.401  3.219  4.052  4.903  5.777  
calculated SM  0.784  1.571  2.366  3.172  3.993  4.832  5.692  
Case 4 \(T_U =1546\hbox {N}\)  Experiment [21]  0.899  1.815  2.741  3.614  4.561  5.525  6.439 
VIM [22]  0.919  1.842  2.771  3.710  4.662  5.630  6.617  
calculated SM  0.904  1.812  2.726  3.650  4.586  5.539  6.510  
Case 5 \(T_U =1925\hbox {N}\)  Experiment [21]  1.037  2.046  3.084  4.099  5.127  6.222  7.298 
VIM [22]  1.033  2.070  3.112  4.163  5.225  6.302  7.396  
calculated SM  1.016  2.034  3.058  4.091  5.136  6.195  7.271 
Characteristics of the riser model
Parameter  Notation  Value  Measure unit 

Length  L  6.5  m 
External diameter  \(D_\mathrm{out} \)  0.0225  m 
Internal diameter  \(D_\mathrm{inn} \)  0.0127  m 
Young modulus  E  8.847  MPa 
Mass per unit  \(m_\mathrm{L} \)  0.4  kg/m 
Bottom weight in water  W  3.489  N 
Bottom weight length  l  0.06  m 
Bottom weight diameter  \(D_\mathrm{W} \)  0.032  m 
Water density  \(\rho _\mathrm{w} \)  1025  \(\hbox {kg}/\hbox {m}^{3}\) 
Resistance coefficient  \(D_x^{\prime } \)  0.016  – 
Resistance coefficient  \(D_{yz}^{\prime } \)  1.5  – 
Inertial coefficient  \(c_M \)  2  – 
Integrals from (35) are calculated by means of Gauss formulae.
Parameters of the riser
Parameter  Notation  Value  Measure unit 

Length  L  300  m 
External diameter  \(D_\mathrm{out}\)  0.3  m 
Internal diameter  \(D_\mathrm{inn}\)  0.26  m 
Young modulus  E  \(2.07\times 10^{11}\)  \(\hbox {N}/\hbox {m}^{2}\) 
Riser density  \(\rho \)  7850  \(\hbox {kg}/\hbox {m}^{3}\) 
Water density  \(\rho _\mathrm{w}\)  1025  \(\hbox {kg}/\hbox {m}^{3}\) 
Resistance coefficient  \(D^{\prime }_w\)  0.03  – 
Resistance coefficient  \(D^{\prime }_{yz}\)  1.0  – 
Inertial coefficient  \(c_M\)  2.0  – 
3 Validation of the model
3.1 Analysis of frequencies of free vibrations
Analysis of vortexinduced vibrations of risers is essential due to the possible fatigue damage caused by the vibrations. Thus, prediction of natural frequencies of risers is one of the crucial points in the design process of risers. The method presented enables us to calculate frequencies of free vibrations of the riser. In order to validate the model, the natural frequencies obtained using our model are compared with those from the experiment explained in [21], where Chaplin and coauthors describe experimental measurements taken on a model of a riser placed in a water tank and exposed to a stepped current. The data of the riser are presented in Table 1.
The installation of the riser consisted of universal joints at each end of the riser and a tensioning system at the upper end. Measurements of the first seven natural frequencies were taken for five different values of the top tensions.
The results of any discretisation method depend on the number of elements into which the system under consideration is divided. Thus, first the influence of the number of finite segments on the results obtained is examined for two different sets of tension applied at both ends of the riser. The results presented in Tables 2 and 3 show convergence of the results obtained for the segment method and its efficiency.
The comparative analysis is carried out not only with the experimental measurements given in [21] but also with results obtained by means of the variational iteration method (VIM) presented in [22], whose authors use the same experiment. It is assumed that the number of segments used for numerical simulations is \({{n}}={200}\). Table 4 presents comparison of the first seven natural frequencies calculated for five cases of different values of tension. The results calculated using the finite segment method are in very good agreement with both those measured in the experiment and calculated using the variational iteration method.
Analysis of the diagram shows that the error does not exceed 2% for most calculated values. The largest differences are obtained for the first case with the lowest tension, yet still the error is less than 6.5%, and in one case the second natural frequency is calculated with an error of about 10.5%. The distribution of errors is similar to that presented in [22], but it can be seen that in most cases the results from the finite segment method are closer to the experimental measurements than those obtained using the variational iteration method. The analysis presented generally confirms the applicability of the finite segment method for prediction of natural frequencies of rises.
3.2 Regular oscillatory vibrations
In this section, validation of the method is carried out on the basis of experiment used as a reference by many researchers [23, 24, 25]. A laboratory test stand consists of a long thin tube filled with water simulating a riser submerged in water with its upper end attached to a mechanism generating oscillatory motion along the x axis (Fig. 7). Additional weight W is attached at the bottom of the riser in order to keep the model straight in still water.
Cameras installed along the y axis measured x displacements at chosen points. Parameters of the riser are presented in Table 5.
Oscillatory motion exerted on the riser at point \(P_0 =P\) with amplitude 0.1 m is shown in Fig. 8 as \(x_P \). Numerical simulations were carried out for a riser divided into \(n=20\) elements. Comparison of displacements in x direction at points A, C and E (Fig. 7) is presented in Fig. 8. The results of calculations are carried out using the finite segment method are compared with experimental measurements and simulation results presented in [23].
The comparison presented in Fig. 8 demonstrates good compatibility of the results obtained using the method presented in the paper with those from experiment and numerical simulation based on Hamilton’s principle [23].
3.2.1 Irregular forced vibrations
Validation of the model presented has also been carried out by comparing the results obtained from the authors’ own software with those from Abaqus. The comparison is concerned with forced vibrations of the riser both with and without consideration of the sea. The parameters of the riser analysed are presented in Table 6. Displacements of the free end of the riser loaded (Fig. 9a) with two forces changing in time and acting along x and z axes and with constant force \(F_y=\,30\hbox { kN}\) are analysed. Courses of forces changing in time are presented in Fig. 9b.
Calculations have been carried out using the model derived by means of the segment method (SM) dividing the riser into \(n=50\) segments. First order, threedimensional beam elements B31H have been used for discretisation in Abaqus; these elements enable longitudinal and transversal deformations to be taken into account. The same number of elements for both methods has been assumed. Displacements of the end of the riser in directions x, y, and z both without and with consideration of the sea are presented in Figs. 10, 11 and 12, respectively.
Analysis of displacement courses of the free end of the riser in interval \({<}0,100\hbox { s}{>}\) in all directions shows that results obtained by the finite segment method are compatible with those obtained using the finite element method (Abaqus). Enlargements presented in each figure show maximal and minimal displacements, and their values are presented in Table 7.
Maximal and minimal displacements of the end of the riser along x, y, and z axes
Model  Measure  n  

10  20  30  40  50  
Abaqus  \(\max \nolimits _{0\le t\le 100} x \left[ \hbox {m} \right] \)  
Air  18.73  18.76  18.79  18.8  18.8  
Water  7.97  8.00  8.01  8.01  8.01  
\(\max \nolimits _{0\le t\le 100} y \left[ \hbox {m} \right] \)  
Air  \(\) 278.17  \(\) 279.12  \(\) 279.38  \(\) 279.44  \(\) 279.48  
Water  \(\) 295.62  \(\) 295.62  \(\) 295.49  \(\) 295.48  \(\) 295.48  
\(\max \nolimits _{0\le t\le 100} z \left[ \hbox {m} \right] \)  
Air  101.09  103.71  103.14  103.01  102.95  
Water  27.57  27.57  27.84  27.85  27.86  
SM  \(\max \nolimits _{0\le t\le 100} x \left[ \hbox {m} \right] \)  
Air  19.08  18.93  18.90  18.89  18.88  
Water  8.05  8.08  8.08  8.09  8.09  
\(\max \nolimits _{0\le t\le 100} y \left[ \hbox {m}\right] \)  
Air  \(\) 279.74  \(\) 279.64  \(\) 279.59  \(\) 279.56  \(\) 279.55  
Water  \(\) 295.41  \(\) 295.38  \(\) 295.37  \(\) 295.36  \(\) 295.36  
\(\max \nolimits _{0\le t\le 100} z \left[ \hbox {m} \right] \)  
Air  102.16  102.55  102.67  102.72  102.75  
Water  27.87  28  28.03  28.04  28.04  
Abaqus  \(\min \nolimits _{0\le t\le 100} x \left[ \hbox {m} \right] \)  
Air  \(\) 19.92  \(\) 19.70  \(\) 19.65  \(\) 19.63  \(\) 19.62  
Water  \(\) 0.58  \(\) 0.58  \(\) 0.56  \(\) 0.56  \(\) 0.56  
\(\min \nolimits _{0\le t\le 100} y \left[ \hbox {m} \right] \)  
Air  \(\) 300.02  \(\) 300.02  \(\) 300.02  \(\) 300.02  \(\) 300.02  
Water  \(\) 300.01  \(\) 300.01  \(\) 300.01  \(\) 300.01  \(\) 300.01  
\(\min \nolimits _{0\le t\le 100} z \left[ \hbox {m} \right] \)  
Air  \(\) 105.1  \(\) 100.97  \(\) 100.72  \(\) 100.62  \(\) 100.57  
Water  \(\) 0.88  \(\) 0.88  \(\) 0.86  \(\) 0.86  \(\) 0.86  
SM  \(\min \nolimits _{0\le t\le 100} x \left[ \hbox {m} \right] \)  
Air  \(\) 19.59  \(\) 19.62  \(\) 19.62  \(\) 19.62  \(\) 19.62  
Water  \(\) 0.48  \(\) 0.48  \(\) 0.48  \(\) 0.48  \(\) 0.48  
\(\min \nolimits _{0\le t\le 100} y \left[ \hbox {m} \right] \)  
Air  \(\) 300  \(\) 300  \(\) 300  \(\) 300  \(\) 300  
Water  \(\) 300  \(\) 300  \(\) 300  \(\) 300  \(\) 300  
\(\min \nolimits _{0\le t\le 100} z \left[ \hbox {m} \right] \)  
Air  \(\) 100.18  \(\) 100.43  \(\) 100.46  \(\) 100.47  \(\) 100.48  
Water  \(\) 0.73  \(\) 0.74  \(\) 0.74  \(\) 0.74  \(\) 0.74 
Absolute differences \(\varepsilon _b \) for maximal and minimal values of riser displacements calculated using Abaqus and segment method for division of the riser into \(n=50\) elements
\( \max \nolimits _{0\le t\le 100} x \)  \(\max \nolimits _{0\le t\le 100} y \)  \(\max \nolimits _{0\le t\le 100} z \)  \(\min \nolimits _{0\le t\le 100} x \)  \( \min \nolimits _{0\le t\le 100} y \)  \(\min \nolimits _{0\le t\le 100} z\)  

\(\varepsilon _b \left[ \hbox {m} \right] \)  Air  \(\) 0.08  0.07  0.20  0.00  \(\) 0.02  \(\) 0.09 
Water  \(\) 0.08  \(\) 0.12  \(\) 0.18  \(\) 0.08  \(\) 0.01  \(\) 0.12 
The largest difference (20 cm) is related to displacements along the z axis when vibrations in air are considered; however, it has to be noted that displacements in this direction amount to \(100 \hbox { m}\), which means that the relative error is 0.2%. Projections of the trajectory of the end of the riser onto plane xz are also compared, and it can be seen from Fig. 13 that the shape is similar for vibrations both without and with consideration of the sea.
It can be seen that the amplitudes when hydrodynamic damping is considered are almost an eighth of those obtained for vibrations in the air.
4 Stabilisation of the force in the connection with the wellhead
Forces occurring in the connection of a riser with a wellhead can considerably change in heavy seas and strong sea currents. Offshore equipment must be durable and reliable; therefore, it is desirable to ensure that values of these forces are as stable as possible within an accepted range despite the vertical and horizontal motion of the base (platform or vessel) .
The model proposed is numerically very effective and allows dynamic analysis for large displacements to be carried out. For example \(n=25,t \in \langle 0,50\hbox {s}\rangle \), for with integration step \(h=0.01\hbox {s}\), calculation time does not exceed \(4\hbox { s}\). In order to illustrate the possibilities of the model, a dynamic optimisation task is formulated and solved. The problem involves selecting vertical movements of the upper end of the riser (Fig. 14) so that for a given horizontal motion of the base the forces in the connection with the wellhead are stable.
The upper end of the riser (point \(P_0\)) moves in the xz plane, reaching \(\pm \,7\hbox { m}\) and \(\pm \,3\hbox { m}\) alternately and following the trajectory presented in Fig. 15, while displacements and velocities in x and z directions change as in Fig. 16.
Two cases are considered. When simulations start, the upper end of the riser is placed at point \(P_0(7\hbox { m}, 0\hbox { m}, 0\hbox { m})\), while coordinates of its bottom end are \({E}(\,10\hbox { m},\,499.39\hbox { m}, 0\hbox { m})\) in case A, which means that at the beginning the riser is initially slightly bent (Fig. 17a) or \({E}(\,100\hbox { m},\,477.16\hbox { m}, 0\hbox { m})\) in case B when the riser is initially bent (Fig. 17b).
In case B, a rotary spring with stiffness coefficient equal to \(1\times 10^6\hbox { Nm}\) is placed at point \(P_0\) in order to keep the deflection angle of the upper end of the riser within the limits recommended in the literature [32].
This considerable decrease in the force in connection E has been obtained as a result of applying vertical motion at the upper end of the riser in the form presented in Fig. 19.
The results presented are illustrative. Shapes of the riser at different moments of time before and after optimisation are presented in Fig. 20.
Results of calculations for case B are presented in Figs. 21 and 22.
In this case, reduction of the force acting at the bottom end of the riser (point E) is not as great as in case A. The reason is that when the riser is almost vertical (case A) forces acting at point E, when the upper end of the riser move about 14 m in x, z directions in a relatively short time, lead to a large increase in longitudinal forces. In case B, however, when the riser is significantly bent, the longitudinal forces are considerably smaller. It is important to note that as a result of the optimisation procedure for case A (slight bending of the riser) force \(F_E\) is about a half of its initial value, yet still its optimal value is larger than the initial value of force \(F_E\) for case B (large initial bending of the riser) .
5 Conclusions
The paper presents a different formulation of the finite segment method which uses absolute angles in contrast to the classical finite segment method, in which relative angles are used in order to describe the geometry of a slender system. Our method enables us to take into consideration the shortening of the distance between the joints, which means that the length of the slender link measured after deformation along the curvature is constant. The influence of the sea is taken into account. The method is validated by comparing the results obtained with those obtained from experimental measurements and Abaqus. The validation process shows the good correspondence of the results obtained by SM, both in natural frequencies and oscillatory vibrations, with results of measurements and calculations when different methods are applied. In addition, the comparison of results obtained with the method presented and those obtained using professional finite element method software confirms the correctness of the model and its numerical effectiveness. Even when the equations of motion are nonlinear, the integration time for less than 50 segments does not exceed 20 seconds over a time interval of 50 s with an integration step of 0.01s. Good numerical effectiveness of the method allows it to be used in solving the dynamic optimisation problem. In order to solve the optimisation problem, the equations of motion are integrated at each optimisation step (for each combination of parameters describing the objective function). In the application presented, the disturbances caused by horizontal motion of the base are neutralised by vertical motion of the upper end of the riser; this solves the problem of stabilising the force in the connection between the riser and the wellhead. The results of calculations demonstrate that even for unfavourable initial approximation of the optimisation problem (such as assuming that the upper end moves horizontally and slight initial bending) the force in the connection of the riser and wellhead is reduced considerably as a result of optimisation. Such a solution is likely to be feasible by using HCS (heave compensation system). Due to the very short calculation time, the method can be applied when teaching a neural network, which then can be used in control of the system in real time.
References
 1.Szczotka, M.: Pipe laying simulation with an active reel drive. Ocean Eng. 37, 539–548 (2010)CrossRefGoogle Scholar
 2.Wang, P.H., Fung, R.F., Lee, M.J.: Finite element analysis of threedimensional underwater cable with timedependent length. J. Sound Vib. 209, 223–249 (2012)CrossRefGoogle Scholar
 3.Chatjigeorgiou, I.K.: A finite difference formulation for the linear and nonlinear dynamics of 2D catenary risers. Ocean Eng. 35, 616–636 (2008)CrossRefGoogle Scholar
 4.Ghadimi, R.: A simple and efficient algorithm for the static and dynamic analysis of flexible marine risers. Comput. Struct. 29, 541–555 (1988)CrossRefzbMATHGoogle Scholar
 5.Chai, Y.T., Varyani, K.S., Barltrop, N.D.P.: Threedimensional lumped mass formulation of a catenary riser with bending, torsion and irregular seabed interaction effect. Ocean Eng. 29, 1503–1525 (2002)CrossRefGoogle Scholar
 6.Sun, L., Qi, B.: Global analysis of a flexible riser. J. Mar. Sci. Appl. 10, 478–484 (2011)CrossRefGoogle Scholar
 7.Kamman, J.W., Huston, R.L.: Advanced structural application. Modelling of submerged cable dynamics. Comput. Struct. 20, 623–629 (1985)CrossRefGoogle Scholar
 8.Connely, J.D., Huston, R.L.: The dynamics of flexible multibody systems: a finite segment approachI. Theoretical aspects. Comput. Struct. 50, 252–258 (1994)CrossRefGoogle Scholar
 9.Connely, J.D., Huston, R.L.: The dynamics of flexible multibody systems: a finite segment approachII. Example problems. Comput. Struct. 50, 252–258 (1994)CrossRefGoogle Scholar
 10.Kamman, J.W., Huston, R.L.: Modeling of variable length towed and tethered cable structures. J. Guid. Control Dyn. 22, 602–608 (1999)CrossRefGoogle Scholar
 11.AdamiecWójcik, I., Awrejcewicz, J., Brzozowska, L., Drąg, Ł.: Modelling of ropes with consideration of large deformations and friction by means of the rigid finite element method. In: Awrejcewicz, J. (ed.) Springer Proceedings in Mathematics & Statistics, vol. 93, pp. 115–138. Springer, Heidelberg (2014)Google Scholar
 12.AdamiecWójcik, I., Brzozowska, L., Drąg, Ł.: An analysis of dynamics of risers during vessel motion by means of the rigid finite element method. Ocean Eng. 106, 102–114 (2015)CrossRefGoogle Scholar
 13.AdamiecWójcik, I., Awrejcewicz, J., Drąg, Ł., Wojciech, S.: Compensation of top horizontal displacements of a riser. Meccanica 51, 2753–2762 (2016)MathSciNetCrossRefGoogle Scholar
 14.Drąg, Ł.: Model of an artificial neural network for optimization of payload positioning in sea waves. Ocean Eng. 115, 123–134 (2016)CrossRefGoogle Scholar
 15.Drąg, Ł.: Application of dynamic optimisation to the trajectory of a cablesuspended load. Nonlinear Dyn. 84, 1637–1653 (2016)MathSciNetCrossRefGoogle Scholar
 16.Drąg, Ł.: Application of dynamic optimisation to stabilize bending moments and top tension forces in risers. Nonlinear Dyn. 88, 2225–2239 (2017)CrossRefGoogle Scholar
 17.Winget, J.M., Huston, R.L.: Cable dynamics: a finite segment approach. Comput. Struct. 6, 475–480 (1976)CrossRefGoogle Scholar
 18.Xu, X.S., Wang, S.W.: A flexible segment model based dynamics calculation method for free hanging marine risers in reentry. China Ocean Eng. 26, 139–152 (2012)CrossRefGoogle Scholar
 19.Kruszewski, J., Gawroński, W., Wittbrodt, E., Najbar, F., Grabowski, S.: Rigid finite element method (Metoda sztywnych elementów skończonych), in Polish, Arkady, Warszawa (1975)Google Scholar
 20.Wittbrodt, E., AdamiecWójcik, I., Wojciech, S.: Dynamics of Flexible Multibody Systems, Rigid Finite Element Method. Springer, Berlin, Heidelberg (2006)zbMATHGoogle Scholar
 21.Chaplin, J.R., Bearman, P.W., Huera Huarte, F.J., Pattenden, R.J.: Laboratory measurements of vortexinduced vibrations of a vertical tension riser in a stepped current. J. Fluids Struct. 21, 3–24 (2005)CrossRefGoogle Scholar
 22.Chen, Y., Zhang, J., Zhang, H., Li, X., Zhou, J.: Reexamination of natural frequencies of marine risers by variational iteration method. Ocean Eng. 94, 132–139 (2015)CrossRefGoogle Scholar
 23.Senga, H., Koterayama, W.: Development of a calculation method for vortex induced vibration of a long riser oscillating at its upper end. In Engineering Sciences Reports, Kyushu University, pp. 335–341 (2005)Google Scholar
 24.Wang, S.W., Xu, X.S., Yao, B.H., Lian, L.: A Finite difference approximation for dynamic calculation of vertical free hanging slender risers in reentry application. China Ocean Eng. 26, 637–652 (2012)CrossRefGoogle Scholar
 25.Xu, X.S., Yao, B.H., Ren, P.: Dynamics calculation for underwater moving slender bodies based of flexible segment model. Ocean Eng. 57, 111–127 (2013)CrossRefGoogle Scholar
 26.Jensen, G.A., Safstrom, N., Nguyen, T.D., Fossen, T.I.: A nonlinear PDE formulation for offshore vessel pipeline installation. Ocean Eng. 37, 365–377 (2010)CrossRefGoogle Scholar
 27.Chai, Y.T., Varyani, K.S.: An absolute coordinate formulation for threedimensional flexible pipe analysis. Ocean Eng. 33, 23–58 (2006)CrossRefGoogle Scholar
 28.Athisakul, C., Monprapussorn, T., Chucheepsakul, S.: A variational formulation for threedimensional analysis of extensible marine riser transporting fluid. Ocean Eng. 38, 609–620 (2011)CrossRefGoogle Scholar
 29.Chatjigeorgiou, I.K., Mavrakos, S.A.: The 3D nonlinear dynamics of catenary slender structures for marine applications. In: Evans, T. (ed.) Nonlinear Dynamics. InTech, London (2010)Google Scholar
 30.Niedzwecki, J.M., Liagre, P.Y.F.: System identification of distributedparameter marine riser models. Ocean Eng. 30, 1387–1415 (2003)CrossRefGoogle Scholar
 31.Bauchau, O.A.: Flexible Multibody Dynamics. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
 32.Chakrabarti, S.K.: Handbook of Offshore Engineering. Elsevier, New York (2005)Google Scholar
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