Comparative analysis of the methods to compute the radiation term in Cummins’ equation
 1.9k Downloads
 8 Citations
Abstract
In the present paper, the methods provided in literature to compute the convolution integral in Cummins’ equation are compared. Direct computation of the convolution integral is revised to avoid truncation errors and to save computational cost. The three methods compared are the direct computation of the convolution integral, the approximation of the integral by a state space and the approximation of the impulse response function by Prony’s coefficients. These methods are used to simulate the movement of the water inside an oscillating water column (OWC) and a decay test in heave of a spar buoy. Cummins’ equation results in a system of ordinary differential equations with all the methods. All systems are computed using the same numerical scheme obtaining a fair comparison of the computational cost involved in each method. The results of the OWC are compared against CFD results and the results of the buoy against laboratory experiments. Results obtained by direct computation of the convolution integral show sensitivity to the time step used to precompute the impulse response function, while using state space or Prony’s approximations are dependent on the set of frequencies required for the identification of their coefficients. State space and Prony’s approximations evaluate the radiation force, including it in the matrix of the system, while direct integration computes it outside of the matrix. This modification in the matrix makes these approximations more sensitive to the data used to evaluate the radiation force.
Keywords
Cummins’ equation Radiation Direct convolution integral State space Prony’s approximation1 Introduction
The movement of a floating structure in the time domain can be modelled by Cummins’ equation (Cummins 1962). This equation takes into account information from the excitation forces produced by waves moving the structure and radiation forces produced by the movement of the structure itself, also known as fluid memory effect. Other external forces, such as those produced by moorings and power take off (PTO) systems, can also be included in the equation, but these forces are outside the scope of the present work. When the movement of the structure is only produced by incident waves, Cummins’ equation is an ordertwo differential equation with a convolution integral in it. The presence of the convolution integral in Cummins’ equation makes the solution complicated in the time domain (Kashiwagi 2004). There are methods in the literature to avoid this problem, which can be divided into two main families: approximation of the radiation term by a state space (Jefferys 1984), and approximation of the impulse response function (irf) by a sum of complex exponentials using Prony’s method (Duclos et al. 2001; Paul 1998; de Prony 1795). A recent review of models used in the study of wave energy converters (WECs) is shown in Folley et al. (2012).
Abbreviations
Abbreviations  

ODE  Ordinary differential equation 
OWC  Oscillating water column 
WEC  Wave energy converter 
irf  Impulse response function 
IRF  Method used to compute the radiation term in Cummins’ equation by direct integration of the convolution integral 
SS  Method used to compute the radiation term in Cummins’ equation by a state space 
Prony  Method used to compute the radiation term in Cummins’ equation by Prony’s coefficients 
RAO  Response amplitude operator 
DOF  Degree of freedom 
BEM  Boundary element method 
TF  Transfer function 
SWL  Still water level 
The second method of approximation of the convolution integral considered in this work is approximation by a state space, a method hereinafter called SS. This method can be argued to be the most popular method to compute the convolution integral in Cummins’ equation. Several approaches have been applied in the literature to identify the matrices of the state space and they can be classified into two main families: frequency domain methods (Jefferys 1984; McCabe et al. 2005) and time domain methods (Kristiansen and Egeland 2003; Kristiansen et al. 2005; Yu and Falnes 1995, 1998). A review of these techniques can be found in Perez and Fossen (2008) and Taghipour et al. (2008). Recently, Perez and Fossen (2011) stated some important aspects that all methods identifying the coefficients applied to marine structures should fulfill. Applications of these methods can be found, for example, in the study of OWCs (Alves 2012; Iturrioz et al. 2014; Kurniawan et al. 2012), the study of one body heave converter involving control strategies (Abraham and Kerrigan 2013; Fusco and Ringwood 2011; Schoen et al. 2011) or the study of more than one body (de Andres et al. 2013; Yu et al. 2010). This method is also used in naval applications, for example in Spyrou and Tigkas (2011). As mentioned before, state spaces have been recently implemented to avoid solving the convolution integral in FAST (Duarte et al. 2013). Another model to study offshore floating vertical axis wind turbines (VAWTs), known as FloVAWT, which uses state space approximation, is presented by Shires et al. (2013).
The third method available in the literature to evaluate the convolution integral of Cummins’ equation is the approximation of the irf by a sum of complex exponentials using Prony’s method. Prony’s method was developed by de Prony (1795), and it was applied to the approximation of the convolution integral by Duclos et al. (2001) and Paul (1998). This method is hereinafter called Prony. This methodology is used in the study of the SEAREV (Babarit and Clément 2006; Babarit et al. 2006) and to study a spar buoy used to extract wave energy (Grilli et al. 2007). More recently, it has also been used with a heaving buoy WEC (Bailey and Bryden 2012), a heaving buoy attached via a tether to a directdrive generator (Crozier et al. 2013) and a submerged wave energy point absorber (Guanche et al. 2013).
This paper presents a comparative analysis of these available methods. Previous works (Kurniawan et al. 2011; Ricci et al. 2008; Taghipour et al. 2008) have presented similar studies where IRF and SS methods are compared. In Taghipour et al. (2008). Simulink is used to compute the problem embedding the solution of the convolution integral as an Sfunction with the trapezoidal integration method programmed in C. As a conclusion, SS approximation is reported to be between 8 and 80 times faster than IRF. In Kurniawan et al. (2011), Matlab is used to calculate the convolution integral and different methods to evaluate the ODEs system generated by the SS method are tested. Kurniawan et al. (2011) conclude that IRF is slow, but the accuracy of the method is guaranteed when time step is small enough. In order to compute the irf, Kurniawan et al. (2011) truncate the infinite integral to a highfrequency value. This approach may yield errors if the truncating high frequency is not properly selected. In this article, this kind of error is avoided since the integral is analytically calculated until infinity (Kashiwagi 2000, 2004). In Ricci et al. (2008), the identification of state space is done in the time domain, while in this article this identification is done in the frequency domain following Perez and Fossen (2011). Furthermore, the resulting ODE system obtained using SS and Prony’s method is computed using a 5th order Runge–Kutta method, while the direct integration of the radiation term is done using backward Euler and Crank–Nicholson methods. So far, the models used to compute the convolution integral have been always compared using different methods to obtain the solution of the resulting system of equations.
The purpose of this article was to summarise and compare the three methods used in the literature. The methods under study are the direct solution of the convolution integral, IRF, approximation of the convolution integral by state spaces, SS, and approximation of the irf by Prony’s method, Prony. In this paper, the results obtained are compared using the same technique to calculate the resulting system of firstorder ODEs.
The programs used to calculate the convolution integral by the three methods were written by the authors. In all three cases, the convolution integral, matrices and the computation of irf for a predefined set of times is done in Python and the results are written in files. Programs, made in Fortran, read these files and compute the ODEs using ODEPACK (Hindmarsh 1983), or VODE (Brown et al. 1989). Also, wave excitation force is computed in Python and saved into a file. This way, in the study of the movements of one WEC under several wave conditions, the only file that needs to be changed is the one containing the wave excitation force information. The matrices and irf information, if needed, would be computed only once for the complete set of wave conditions under study.
The computational cost, shown in tables throughout the paper, only contains the cost of the computation of the ODEs systems. Exceptionally, in the case of the response amplitude operator (RAO), the cost of the whole process is shown. The solution of the system provided by Prony’s method contains complex values, so the complex version of VODE (Brown et al. 1989), ZVODE, is used. VODE is based on ODEPACK. A comparison between ODEPACK and VODE has been performed providing both the same results at very similar computational costs. All computations have been performed in just one core and always in the same computer, a desktop computer. As ODEPACK and VODE use variable time step methods, the same restrictions to the computation of the time step were applied in all cases. Therefore, for the first time, the different techniques used to evaluate the convolution integral can be compared in a fair way in this work.
In this article, different methodologies have been applied to the study the OWC and a spar buoy. A first set of tests compare the results obtained in the study of an OWC, as presented by Iturrioz et al. (2014), with results from numerical models. Results and the computational effort required for each method are compared with data from an inhouse CFD software IH2VOF (Losada et al. 2008) and WADAM (WADAM 2014). The first study case in this paper is the comparison of the results with those of a decay test obtained with IH2VOF. Decay tests are suitable for the purpose of comparing the methods in literature (IRF, SS and Prony) to evaluate the radiation term, as only radiation and hydrostatic forces are considered. The second test, used for the OWC, is to compare the RAO obtained from each method with the one calculated with WADAM. Finally, results are compared with those produced by IH2VOF for irregular waves.
IH2VOF evaluates the 2D Reynolds Averaged Navier–Stokes (RANS) equations (Liu et al. 1999; Hsu et al. 2002), based on the decomposition of the instantaneous velocity and pressure fields and the \(k\varepsilon \) equations for the turbulent kinetic energy (k) and the turbulent dissipation rate (\(\varepsilon \)). The model has been under a continuous development process based on an extensive validation procedure carried out for lowcrested structures (Garcia et al. 2004; Losada et al. 2005; Lara et al. 2006), wave breaking on permeable slopes (Lara et al. 2006), overtopping on rubble mound breakwaters and lowmound breakwaters (Lara et al. 2008; Losada et al. 2008; Guanche et al. 2009), pore pressure damping in rubble mound breakwaters (Guanche et al. 2015) and the study of OWCs (Armesto et al. 2014; Iturrioz et al. 2014). The detailed validations provided by the aforementioned studies have tested the capability of the IH2VOF model to satisfactorily reproduce the wave–structure interactions.
After the validations using the OWC, a second set of comparisons uses a decay test of the spar buoy based on the Hywind offshore wind project, and results from all methods are compared with results from laboratory. A sensitivity analysis is performed for the time step, the memory of the fluid and the frequencies obtained from different runs of WADAM.
This paper is organised in three parts. The first part describes the different numerical approximations considered: IRF, SS and Prony. In the second part, the results yielded by each of the methods are given and discussed. The last part presents the conclusions of the study.
2 Numerical model
The three alternatives used in literature to compute the convolution integral in Cummins’ equation are reviewed in the following sections.
2.1 Solving the convolution integral (IRF method)
Truncating the infinite integral, to a high value of frequency, is common practice in the literature (Kurniawan et al. 2011; Perez and Fossen 2008; Ricci et al. 2008; Yu and Falnes 1995) to compute the irf. The solution adopted in this work (Kashiwagi 2004) avoids the truncation, as the integral of the exponential function is analytically evaluated to infinity.
2.2 Approximation of the convolution integral by a state space (SS method)
Once again, Cummins’ equation is written as a system of firstorder ODEs. In this case, the order of the system is twice the number of DOFs under study plus the sum of the orders of all the state space equations used to approximate the convolution integral of every coupling of DOFs. For example, in the study of a 6 DOFs system, there are 36 convolution integrals to approximate, which may be reduced depending on the symmetry of the body.
2.3 Approximation of the convolution integral by Prony’s estimation (Prony method)
3 Comparison
In order to compare the results yielded by each method and the computational time employed in their solution, two sets of tests are performed. In the first set, the OWC employed in Iturrioz et al. (2014) is used to analyse different cases. The heave of the OWC is studied as the heave movement of the mass of water inside the OWC at still water level (SWL). The first case is a decay test where results are compared with those obtained by IH2VOF (Losada et al. 2008). In the second test, the RAO is produced with each radiation method, and the result compared with the RAO given by the BEM model used. Sensitivity of the results to \(t^{*}\) and the value of \(\Delta t\) used to precompute the irf are presented in this second test. The last case of study employing the OWC is an irregular sea state. This case requires the calibration of linear and nonlinear friction coefficients to compare the results with those obtained by IH2VOF (Iturrioz et al. 2014). In all cases, the computational cost of solving the final system of ODEs obtained with each method is compared.
In the second set, the three methods were used in the study of a heave decay of the SPAR buoy based on the Hywind offshore wind project. The results are compared with data from laboratory experiments.

An interpolation of the results given by the BEM model to 512 equally spaced frequencies between the minimum and the maximum frequencies used by the BEM model (for OWC, \(\omega \,{\in }\, [1.532,12.566], \Delta \omega \,{=}\, 0.0216\)). This set is called ‘Interpolated’.

An interpolation of the results given by the BEM model to 512 equally spaced frequencies between the minimum frequency and 0.6 times the maximum frequency used by the BEM model (for OWC, \(\omega \,{\in }\, [1.532,7.540], \Delta \omega \,{=}\, 0.0118\)). This set is called ‘Truncated high’.

An interpolation of the results given by the BEM model to 512 equally spaced frequencies between 1.5 times the minimum frequency and the maximum frequency used by the BEM model (for OWC, \(\omega \,{\in }\, [2.299,12.566], \Delta \omega \,{=}\, 0.0201\)). This set is called ‘Truncated low’.
3.1 Oscillating water column
In this section, the study carried out for an OWC device is described. All OWC cases were analysed with volume, \(V \,{=}\, 20 \,{\times }\, 30 \,{\times }\, 68\) cm\(^3\), and added mass at infinity frequency \(A_{\infty } \,{=}\, 31.43\) kg (Iturrioz et al. 2014). The computational cost of solving the final system of ODEs obtained with each method and for every test is presented. The solution of all the systems of ODEs is done using the implicit Adams method with relative tolerance of \(10^{7}\) and absolute tolerance of \(10^{9}\) (Brown et al. 1989; Hindmarsh 1983).

Method of resolution using the identification toolbox: 2

Maximum order of the identification: 20

Identification threshold: 0.99

Number of iterations for the identification \( = 50\).
3.1.1 Decay
Natural period, in seconds, obtained by each method and every preprocessing studied to compute the corresponding decay test
SS  IRF  Prony  

Interpolated  1.145  1.15  1.166 
Truncated high  1.143  1.15  1.166 
Truncated low  1.146  1.15  1.166 
Summary of the executed cases for the heave decay of the OWC
Radiation method  Frequencies  \(t^{*}\) (s)  \(\Delta t \) (s)  

OWC  SS  Interpolated  3  0.02 
Truncated high  3  0.02  
Truncated low  3  0.02  
IRF  Interpolated  3  0.02  
10  0.02  
3  0.01  
Truncated high  3  0.02  
Truncated low  3  0.02  
Prony  Interpolated  3  0.02  
Truncated high  3  0.02  
Truncated low  3  0.02 
Integral of the velocity of the free surface along the executed time normalised with the initial position of the OWC, 0.12 m
SS  IRF  Prony  

Interpolated  \(5.61 \times 10^{2}\)  \(7.07\times 10^{2}\)  \(7.63\times 10^{2}\) 
Truncated high  \(2.99 \times 10^{2}\)  \(7.07\times 10^{2}\)  \(7.63\times 10^{2}\) 
Truncated low  \(4.78 \times 10^{2}\)  \(7.07\times 10^{2}\)  \(7.63\times 10^{2}\) 
3.1.2 RAO
A set of 47 frequencies, between 1.53 and 12.56 rad/s, were used in the BEM model. In order to reproduce the RAO, 47 simulations of regular waves were executed, using the frequencies studied in the BEM model.
Computational effort, in seconds, required by each method to compute the corresponding RAO
SS  IRF (\(t^{*} = 10\) s)  IRF (\(t^{*} = 3\) s)  Prony  

CPU (s)  32  55  37  32 
The RAOs have been repeated using the different sets of frequencies described before: ‘Interpolated’, ‘Truncation high’ and ‘Truncation low’. In case of IRF, two temporal discretizations are included \(\Delta t \,{=}\, 0.02\) s and \(\Delta t \,{=}\, 0.01\) s. The results can be seen in Fig. 10, where the differences in the amplitude at the natural period are in the order of 6 % when using the different sets of frequencies for the SS method.
3.1.3 OWC under irregular waves
The inhouse CFD model IH2VOF (Losada et al. 2008) is used to compare the results of the OWC under the action of an irregular sea state. In this case, a peak period of \(T_p \,{=}\, 3.2\) s with significant wave height, \(H_s\,{=}\,0.06\) m, is run for 150 s using \(\Delta t \,{=}\, 0.02\) s. The calibration of the linear and nonlinear friction coefficients produced the values \(k_\mathrm{l} = 147.01\) and \(k_\mathrm{nl} \,{=}\, 0\) (Iturrioz et al. 2014). The computational effort required for each method to compute the resulting system of ODEs is presented in Table 7. SS and Prony require similar computational cost, which is almost half the computational time required by the best case of IRF. The results obtained are compared in Fig. 11, where solutions from all methods are expressed. The results are similar to one another even using \(\Delta t \,{=}\, 0.02\) s.
The movements of the OWC under irregular waves are computed using the superposition principle. Data from the BEM code are obtained for a set of frequencies. A different set of frequencies is obtained from the spectral decomposition of irregular waves, so interpolation is required from the known frequencies used in the BEM model. Small discrepancies can result specially in the interpolated phase as they are \(2\pi \) periodic. The time series presented is \(t \,{\in }\, (50,100)\), so small differences in the interpolated phases could cause small discrepances in the phase, as in Fig. 11. So, this time lag is the same in all three methods, because the lag comes from the wave excitation force, which is the same in all methods.
Summary of the executed cases for the RAOs of the OWC
Radiation method  Frequencies  \(t^{*}\) (s)  \(\Delta t\) (s)  

OWC  SS  Interpolated  3  0.02 
Truncated high  3  0.02  
Truncated low  3  0.02  
IRF  Interpolated  3  0.02  
3  0.01  
Truncated high  3  0.02  
3  0.01  
Truncated low  3  0.02  
3  0.01  
Prony  Interpolated  3  0.02  
Truncated high  3  0.02  
Truncated low  3  0.02 
Computational effort required by each method to compute the resulting system of ODEs for each method in the case of irregular waves
SS  IRF (\(t^{*} = 10\) s)  IRF (\(t^{*} = 3\) s)  Prony  

CPU (s)  0.164  0.484  0.284  0.16 
3.2 SPAR buoy
In this section, the analysis carried out for a SPAR buoy is described. A SPAR buoy based on the Hywind offshore wind project is used in this calibration because of the availability of laboratory results. In this study, three different BEM files have been applied: one using equidistant frequencies between 0.1 and 5 rad/s every 0.1 rad/s called ‘Original’; one using equidistant periods between 1 and 60 s every second called ‘Equal’; and finally, one with variable increments between 1 and 60 s, smaller close to the natural period and larger increments for the rest of periods called ‘Variable’. Infinity added mass has been externally computed to adjust the laboratory peak period in all the cases. Furthermore, linear and nonlinear coefficients are also used to adjust the damping measured at the laboratory, with the same coefficients in all simulations, \(k_\mathrm{l} \,{=}\, 2\,{\times }\, 10^{4}\) and \(k_\mathrm{nl} \,{=}\, 4\,{\times }\, 10^{4}\) (Iturrioz et al. 2014).
Natural period, in seconds, obtained by each method and every BEM file to compute the corresponding decay test of the SPAR buoy
SS  IRF  Prony  

Original  30.464  30.91  30.918 
Equal  30.464  30.91  30.918 
Variable  30.447\(^\mathrm{a}\)  30.91  30.932\(^{\mathrm{a}}\) 
Computational effort, in seconds, required by each method to compute the system of ODEs for the corresponding data from BEM
SS  IRF (\(t^{*} = 30\) s)  Prony  

Original  0.76 (8)  0.17 (2)  0.46 (22)\(^{\mathrm{a}}\) 
Equal  1.97 (8)  0.18 (2)  0.37 (18) 
Variable  3.22 (22)\(^\mathrm{a}\)  0.16 (2)  0.39 (18) 
The data used to interpolate the coefficients of SS and Prony in the study of the OWC are also used for the SPAR buoy. This means that the maximum order employed in these simulations is 20. The total order of the ODEs system obtained by each method is shown in parentheses in the table. It is obvious that the order of IRF is always 2, but there is a variability in the order used in SS and Prony. This variability is reflected in the computational effort required by each method. The changes in the data used from the BEM model have no influence on the matrix of the system in the IRF method. This is explained because the radiation force is calculated in the RHS of the system, and small discrepancies are not reflected in the solution of the system. On the other hand, SS and Prony compute the radiation term modifying the matrix of the ODEs system. Small changes in the matrix coefficients can modify the properties of the matrix. Such changes will be reflected in the computation of the ODEs system. In the presented analysis of the SPAR buoy, the computational cost of IRF remains almost constant, always lower than the computational cost of SS and Prony.
For the case of ‘Equal’, the test cases have been repeated using the 6 DOFs of the platform instead of only heave. The order of the matrix for IRF is obviously 12 and the computational cost required is 2.6 s. For SS, the order is 112 with a computational effort of 1011 s. Prony requires an order of the matrix of 76 and a computational effort of 4.8 s. The final ODEs system generated by every method is computed by the implicit Adams method that uses variable time steps. The use of variable time steps requires less computational steps when the matrix of the system has appropriate eigenvalues. These eigenvalues change as the coefficients of the matrix vary, due to the different results of the approximations in SS and Prony. IRF and Prony’s methods require similar numbers of evaluations, while SS requires between 2 and 8 times more evaluations of the system matrix for the simulated cases. Every execution simulates 1800 s, writing results every 0.1 s.
Summary of the executed cases for the SPAR
Radiation method  BEM  Frequencies  

SPAR  SS  Original  
Equal  Interpolated  
Truncated high  
Truncated low  
Variable  
IRF  Original  
Equal  Interpolated  
Truncated high  
Truncated low  
Variable  
Prony  Original  
Equal  Interpolated  
Truncated high  
Truncated low  
Variable 
Natural period, in seconds, obtained by each method and every preprocessing studied to compute the corresponding decay test of the SPAR buoy
SS  IRF  Prony  

Interpolated  30.464  30.91  30.932 
Truncated high  31.226  30.91  30.914 
Truncated low  30.464  30.91  30.932 
The differences obtained by SS or Prony are tiny, except in the case of ‘Truncated high’ for the SS. This was not unexpected as, convergence was not reached in the approximation of the polynomials in that case. The proposed truncations, of low and high frequencies, seem to have no influence on the solutions obtained by the IRF method. On the other hand, the results provided by SS and Prony have shown some small impact on the preprocessing performed. The computational cost of these sets of comparisons can be seen in Table 12 and they follow the same pattern as in Table 9, IRF being the fastest one.
Computational effort, in seconds, required by each method to compute the corresponding interpolated and truncated data BEM model “equal”
SS  IRF (\(t^{*} = 30\) s)  Prony  

Interpolated  1.97 (8)  0.18 (2)  0.37 (18) 
Truncated high  0.37 (22)\(^\mathrm{a}\)  0.16 (2)  0.36 (18) 
Truncated low  2.06 (8)  0.17 (2)  0.37 (18) 
4 Conclusions
The three methods employed in the literature to evaluate the convolution term of Cummins’ equation have been implemented and compared. The computation of the irf has been extended to infinity, interpolating the data obtained by a BEM and analytically solving the integral to infinity avoiding truncation errors (Kashiwagi 2004). The systems of ODEs resulting in each method are computed using the same numerical algorithm. The solutions obtained by all methods are similar to one another.
In order to save computational time, the irf values are precomputed for a predefined interval with a fixed time step (Kurniawan et al. 2011). The results obtained by IRF have shown sensitivity to the value of the chosen time step. Large time steps yielded errors around the natural period of the structure under study. On the other hand, their results have not shown sensitivity to different BEM files or to the preprocessing performed to BEM data.
SS method has shown two issues. The first one is the sensitivity of the results to the preprocessing of BEM data in both the decay test and RAOs of the OWC. The differences in the results have always been very small. The second issue is that using the same parameters for identification, the method converged for BEMs cases ‘Original’ and ‘Equal’, but failed to converge in the case ‘Variable’. The only thing that changed for different BEM cases is the set of predefined frequencies chosen.
The Prony method has also shown sensitivity in the results when preprocessing BEM data in the case of the spar buoy, but differences in the results are very small.
The change on the data used from the BEM model has no influence on the matrix of the system in the IRF method. Radiation force is evaluated at the righthand side of the system, and small discrepancies are not reflected on the solution of the system, as the matrix characteristics remain the same. SS and Prony calculated the radiation term modifying the matrix and, therefore, the characteristics of the ODEs system. In conclusion, small changes will be reflected in the solution of the system. Hence, both systems show errors appearing in the solution of the radiation term.
The simulations performed by SS and Prony’s approximations have been computationally cheaper than those by IRF in the cases of the OWC. On the other hand, IRF was computationally cheaper than Prony and SS approximations in the case of the SPAR decay. This seems to highlight that SS and Prony are quicker when the identification is carried out with polynomials of low order and IRF is quicker if high order is needed for the identification. When using more DOFs in the study, the order of the system increases much more in SS and Prony than in IRF, which makes IRF quicker.
With the increase of computational capabilities given by actual computers, the authors recommend the use of direct integration method to compute the radiation term in Cummins’ equation, avoiding the uncertainties seen in the identification of the coefficients in the SS and Prony’s methods. However, a mention has to be made to highlight that the solution given by SS or Prony’s methods should be considered in those cases where the access to modern computational capabilities is reduced.
Notes
Acknowledgments
The authors are grateful to the Spanish Ministry of Economy and Competitiviness and in particular to the State Secretariat for the Research, Development and Innovation funding for VAPEOOcean Climate Variability influence over Wave Energy Converters Power Production project (ENE201348716R), within the National Programme for Research Aimed at the Challenges of Society, modality 1, Research Challenge: Research, Development and Innovation.
References
 Abraham E, Kerrigan E (2013) Optimal active control and optimization of a wave energy converter. Sustainable Energy, IEEE Transactions on 4(2):324–332. doi: 10.1109/TSTE.2012.2224392 CrossRefGoogle Scholar
 Alves MAA (2012) Numerical simulation of the dynamics of point absorber wave energy converters using frequency and time domain approaches. PhD thesis, Universidade Técnica de LisboaGoogle Scholar
 de Andres AD, Guanche R, Armesto JA, del Jesus F, Vidal C, Losada IJ (2013) Time domain model for a two body heave converter: model and applications. Ocean Engineering 72:116–123CrossRefGoogle Scholar
 Armesto JA, Guanche R, Iturrioz A, Vidal C, Losada IJ (2014) Identification of statespace coefficients for oscillating water columns using temporal series. Ocean Engineering 79(0):43–49. doi: 10.1016/j.oceaneng.2014.01.013, http://www.sciencedirect.com/science/article/pii/S0029801814000225
 Babarit A, Clément AH (2006) Optimal latching control of a wave energy device in regular and irregular waves. Applied Ocean Research 28(2):77–91CrossRefGoogle Scholar
 Babarit A, Clément AH, Ruer J, Tartivel C (2006) SEAREV: A fully integrated wave energy converter. In: Proceedings of OWEMESGoogle Scholar
 Babarit A, Hals J, Muliawan MJ, Kurniawan A, Moan T, Krokstad J (2012) Numerical benchmarking study of a selection of wave energy converters. Renewable Energy 41:44–63CrossRefGoogle Scholar
 Bailey H, Bryden IG (2012) Influence of a quadratic power takeoff on the behaviour of a selfcontained inertial referenced wave energy converter. Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment 226:15–22Google Scholar
 Brown PN, Byrne GD, Hindmarsh AC (1989) VODE: a variablecoefficient ODE solver. SIAM Journal on Scientific and Statistical Computing 10(5):1038–1051CrossRefMathSciNetzbMATHGoogle Scholar
 Chitrapu AS, Ertekin RC (1995) Timedomain simulation of largeamplitude response of floating platforms. Ocean Engineering 22(4):367–385CrossRefGoogle Scholar
 Crozier R, Bailey H, Mueller M, Spooner E, McKeever P (2013) Analysis, design and testing of a novel directdrive wave energy converter system. Renewable Power Generation, IET 7(5):565–573CrossRefGoogle Scholar
 Cummins WE (1962) The impulse response function and ship motions. Schiffstechnik 9:101–109Google Scholar
 Duarte T, Alves M, Jonkman J, Sarmento A (2013) Statespace realizatio nof the waveradiation force within FAST. In: Proceedings of the 32 International Conference on Ocean, Offshore and Artic EngineeringGoogle Scholar
 Duclos G, Clément AH, Chatry G (2001) Absorption of outgoing waves in a numerical wave tank using a selfadaptive boundary condition. International Journal of Offshore and Polar Engineering 11(3):168–175Google Scholar
 Folley M, Babarit A, Child B, Forehand D, O’Boyle L, Silverthotne K, Spinneken J, Stratigaki V, Troch P (2012) A review of numerical modelling of wave energy converter arrays. In: 31st International Conference on Ocean, Offshore and Arctic Engineering, pp 535–545Google Scholar
 Fusco F, Ringwood J (2011) Suboptimal causal reactive control of wave energy converters using a second order system model. In: 21st International Offshore and Polar Engineering Conference, pp 687–694Google Scholar
 Garcia N, Lara JL, Losada IJ (2004) 2d numerical analysis of nearfield flow at lowcrested permeable breakwaters. Coastal Engineering 51(10):991–1020. doi: 10.1016/j.coastaleng.2004.07.017 CrossRefGoogle Scholar
 Greenhow M (1986) High and lowfrequency asymptotic consequences of the kramerskronig relations. Journal of Engineering Mathematics 20(4):293–306. doi: 10.1007/BF00044607
 Grilli AR, Merrill J, Grilli ST, Spaulding ML (2007) Experimental and numerical study of spar buoymagnet/spring oscillators used as wave energy absorbers. In: Proc. 17th International Offshore and Polar Engineering Conference., pp 486–489Google Scholar
 Guanche R, Losada IJ, Lara JL (2009) Numerical analysis of wave loads for coastal structure stability. Coastal Eng. doi: 10.1016/j.oceaneng.2008.05.006
 Guanche R, Gómez V, Vidal C, Eguinoa I (2013) Numerical analysis and performance optimization of a submerged wave energy point absorber. Coast Eng 59:214–230Google Scholar
 Guanche R, Iturrioz A, Losada IJ (2015) Hybrid modeling of pore pressure damping in rubble mound breakwaters. Coast Eng 99:82–95CrossRefGoogle Scholar
 Hindmarsh AC (1983) ODEPACK, a systematized collection of ODE solvers. In: Stepleman RS, Carver M, Peskin R, Ames WF, Vichnevetsky R (eds) Scientific computing. IMACS transactions on scientific computation, vol 1. NorthHolland, Amsterdam, pp 55–64Google Scholar
 Hsu TJ, Sakakiyama T, Liu PLF (2002) A numerical model for wave motions and turbulence flows in front of a composite breakwater. Coastal Engineering 46(1):25–50. doi: 10.1016/S03783839(02)000455 CrossRefGoogle Scholar
 Iturrioz A, Guanche R, Armesto JA, Alves MA, Vidal C, Losada IJ (2014) Timedomain modeling of a fixed detached oscillating water column towards a floating multichamber device. Ocean Enginnering 76:65–74CrossRefGoogle Scholar
 Jefferys E (1984) Simulation of wave power devices. Applied Ocean Research 6(1):31–39CrossRefMathSciNetGoogle Scholar
 Jonkman J (2007) Dynamics Modelling and Loads Analysis of an Offshore Floatting Wind Turbine. Tech. rep, National Renewable Energy Laboratory, USACrossRefGoogle Scholar
 Karimirad M (2013) Modeling aspects of a floating wind turbine for coupled wavewindinduced dynamic analyses. Renewable Energy 53:299–305CrossRefGoogle Scholar
 Kashiwagi M (2000) A timedomain modeexpansion method for calculating transient elastic responses of a pontoontype vlfs. Journal of Marine Science and Technology 5:89–100CrossRefGoogle Scholar
 Kashiwagi M (2004) Transient responses of a VLFS during landing and takeoff of an airplane. Journal of Marine Science and Technology 9(1):14–23. doi: 10.1007/s0077300301680 CrossRefGoogle Scholar
 de Kat JO (1988) Large amplitude ship motions and capsizing in severe saa conditions. PhD thesis, University of CaliforniaGoogle Scholar
 Kristiansen E, Egeland O (2003) Frequencydependent added mass in models for controller design for wave motion damping. In: Blanke M, Pourzanjani MMA, Vukić ZZ (eds) Manoeuvring and Control of Marine Craft 2003 (MCMC 2003): A Proceedings Volume from the 6th IFAC Conference. ElsevierGoogle Scholar
 Kristiansen E, Hjulstad A, Egeland O (2005) Statespace representation of radiation forces in timedomain vessel models. Ocean Engineering 32(17–18):2195–2216CrossRefGoogle Scholar
 Kurniawan A, Hals J, Moan T (2011) Assesments of timedomain models of wave energy conversion system. In: Proceedings of the Ninth European Wave and Tidal Energy Conference, University of Southampton, SouthamptonGoogle Scholar
 Kurniawan A, Pedersen E, Moan T (2012) Bond graph modelling of a wave energy conversion system with hydraulic power takeoff. Renewable Energy 38(1):234–244CrossRefGoogle Scholar
 Lara JL, Garcia N, Losada IJ (2006) Rans modelling applied to random wave interaction with submerged permeable structures. Coastal Engineering 53(5–6):395–417. doi: 10.1016/j.coastaleng.2005.11.003 CrossRefGoogle Scholar
 Lara JL, Losada IJ, Guanche R (2008) Wave interaction with lowmound breakwaters using a rans model. Ocean Engineering 35(13):1388–1400. doi: 10.1016/j.oceaneng.2008.05.006, http://www.sciencedirect.com/science/article/pii/S0029801808001194
 Liu PLF, Lin PZ, Chang KA, Sakakiyama T (1999) Numerical modelling of wave interaction with porous structures. Journal of Waterways, Port, Coastal, and Ocean Engineering 125(6):322–330CrossRefGoogle Scholar
 Losada IJ, Lara JL, Christensen ED, Garcia N (2005) Modelling of velocity and turbulence fields around and within lowcrested rubblemound breakwaters. Coastal Engineering 52(10–11):887–913. doi: 10.1016/j.coastaleng.2005.09.008 CrossRefGoogle Scholar
 Losada IJ, Lara JL, Guanche R, GonzalezOndina JM (2008) Numerical analysis of wave overtopping of rubble mound breakwaters. Coastal Engineering 55(10):47–62CrossRefGoogle Scholar
 McCabe AP, Bradshaw A, Widden M (2005) A timedomain model of a floating body using transforms. In: Proceedings of the Sixth European Wave and Tidal Energy Conference, University of Strathclyde, GlasgowGoogle Scholar
 Muliawan MJ, Karimirad M, Moan T (2013) Dynamic response and power performance of a combined Spartype floating wind turbine and coaxial floating wave energy converter. Renewable Energy 50:47–57CrossRefGoogle Scholar
 Nolte JD, Ertekin RC (2014) Wave power calculations for a wave energy conversion device connected to a drogue. Journal of Renewable and Sustainable Energy 6(1):013117CrossRefGoogle Scholar
 Paul J (1998) Modelling of General Electromagnetic Material Properties in TLM. PhD thesis, University of NottinghamGoogle Scholar
 Perez T, Fossen TI (2008) Timedomain vs. frequencydomain identification of parametric radiation force models for marime structures at zero speed. Modelling, Identification and Control 29:1–19CrossRefGoogle Scholar
 Perez T, Fossen TI (2009) A Matlab toolbox for parametric identification of radiationforce models of ships and offshore structures. Modelling, Identification and Control 30:1–15CrossRefGoogle Scholar
 Perez T, Fossen TI (2011) Practical aspects of frequencydomain identification of dynamic models of marine structures from hydrodynamic data. Ocean Engineering 38:426–435CrossRefGoogle Scholar
 Philippe M, Babarit A, Ferrant P (2013) Modes of response of an offshore wind turbine with directional wind and waves. Renewable Energy 49:151–155CrossRefGoogle Scholar
 de Prony BGR (1795) Essai éxperimental et analytique: sur les lois de la dilatabilité de fluides élastique et sur celles de la force expansive de la vapeur de l’alkool, à différentes températures. Journal de l’école Polytechnique 1(22):24–76Google Scholar
 Ricci P, Saulnier JB, de O Falcao AF, Pontes MT (2008) Timedomain models and wave energy converters performance assessment. In: 27th international conference on ocean, offshore and Arctic engineeringGoogle Scholar
 Schoen MP, Hals J, Moan T (2011) Wave prediction and robust control of heaving wave energy devices for irregular waves. Energy Conversion, IEEE Transactions on 26(2):627–638. doi: 10.1109/TEC.2010.2101075 CrossRefGoogle Scholar
 Shires A, Collu M, Borg M (2013) FloVAWT: Progress on the development of a coupled model of dynamics for floating offshore vertical axis wind turbines. 23rd International Offshore and Polar Engineering Conference, International Society of Offshore and Polar Engineers (ISOPE). Anchorage, Alaska, USA, pp 345–351Google Scholar
 SIMO (2008) SIMO  Theory Manual version 3.6, rev: 1. MarintekGoogle Scholar
 Spyrou KJ, Tigkas IG (2011) Nonlinear surge dynamics of a ship in astern seas: Continuation analysis of periodic states with hydrodynamic memory. Journal of Ship Research 55(1):19–28Google Scholar
 Taghipour R, Perez T, Moan T (2008) Hybrid frequencytime domain models for dynamic response analysis of marine structures. Ocean Engineering 35:685–705CrossRefGoogle Scholar
 WADAM (2014) SESAM Users manual, WADAM. WWW page, http://www.dnv.com, accessed Dec 2014
 WAMIT (2014) WAMIT Users manual. WWW page, http://www.wamit.com, accessed Dec 2014
 Yu X, Falzarano JM, Su Z (2010) Timedomain simularion of a multybody floating system in waves. Proceedings of the 29 International Conference on Ocean. Offshore and Artic Engineering, ASME, Shanghai, China, pp 797–803Google Scholar
 Yu Z, Falnes J (1995) Statespace modelling of a vertical cylinder in heave. Applied Ocean Research 17(5):265–275CrossRefGoogle Scholar
 Yu Z, Falnes J (1998) Statespace modelling of dynamic systems in ocean engineering. Journal of Hydrodynamics, Serie B 1:1–17Google Scholar