Mérigaud, Alexis
(2018)
A harmonic balance framework for the
numerical simulation of nonlinear wave energy
converter models in random seas.
PhD thesis, National University of Ireland Maynooth.
Abstract
Numerical simulation is essential, to assist in the development of wave energy technology. In particular,
tasks such as power assessment, optimisation and structural design require a large number
of numerical simulations to calculate the wave energy converter (WEC) outputs of interest, over a
variety of wave conditions or physical parameters. Such challenges involve a sound understanding
of the statistical properties of ocean waves, which constitute the forcing inputs to the wave energy
device, and computationally efficient numerical techniques for the speedy calculation of WEC
outputs. This thesis studies the statistical characterisation, and numerical generation, of ocean
waves, and proposes a novel technique for the numerical simulation of nonlinear WEC models.
The theoretical foundations, the range of validity, and the importance of the statistical representation
of ocean waves are first examined. Under relatively mild assumptions, ocean waves can
be best described as a stationary Gaussian process, which is entirely characterised by its spectral
density function (SDF). Various wave superposition techniques are discussed and rigorously
compared, for the numerical generation of Gaussian wave elevation time series from a given SDF.
In particular, the harmonic random amplitude (HRA) approach can simulate the target statistical
properties with perfect realism. In contrast, the harmonic deterministic amplitude (HDA)
approach is statistically inconsistent (because the generated timeseries are nonGaussian, and
underrepresent the shortterm statistical variability of real ocean waves), but can be advantageous
in the context of WEC simulations since, if it can be verified that HDA results are unbiased, the
HDA method requires a smaller number of random realisations than the HRA method, to obtain
accurate WEC power estimates.
When either HDA or HRA are used for the generation of wave inputs, the forcing terms of
WEC mathematical models are periodic. Relying on a Fourier representation of the system inputs
and variables, the harmonic balance (HB) method, which is a special case of spectral methods, is a
suitable mathematical technique to numerically calculate the steadystate response of a nonlinear
system, under a periodic input. The applicability of the method to WEC simulation is demonstrated
for those WEC models which are described by means of a nonlinear integrodifferential
equation. In the proposed simulation framework, the WEC output, in a given sea state, is assessed
by means of many, relatively short, simulations, each of which is efficiently solved using the HB
method.
A range of four case studies is considered, comprising a flaptype WEC, a spherical heaving
pointabsorber, an array of four cylindrical heaving pointabsorbers, and a pitching device. For
each case, it is shown how the HB settings (simulation duration and cutoff frequency) can be
calibrated. The accuracy of the HB method is assessed through a comparison with a secondorder
RungeKutta (RK2) timedomain integration scheme, with various time steps. RK2 results converge
to the HB solution, as the RK2 time step tends to zero. Furthermore, in a Matlab implementation,
the HB method is between one and three orders of magnitude faster than the RK2 method,
depending on the RK2 time step, and on the method chosen for the calculation of the radiation
memory terms in RK2 simulations. The HB formalism also provides an interesting framework,
for studying the sensitivity of the WEC dynamics to system parameter variations, which can be
utilised within a gradientbased parametric optimisation algorithm. An example of WEC gradientbased
parametric optimisation, carried out within the HB framework, is provided.
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