Second-order monopile wave loads at linear cost

A method to compute the second-order free surface elevation, depth integrated force and mud line moment for a slender circular vertical cylinder is presented. The method is valid for unidirectional irregular waves and includes inertia loads and viscous loads.

We first derive the linear transfer functions for free surface elevation, depth-integrated force and moment from the complex Fourier amplitudes of the velocity potential. Next, the second-order contributions are expressed through closed form quadratic transfer functions, which are further diagonalized through eigen decomposition. Hereby the second-order contributions can be computed as products of pseudo time series calculated by FFT, with the eigenvectors acting as transfer functions on the linear Fourier ampltitudes.

For a sample 3-h sea state, we find that eight modes are sufficient to achieve an accuracy of 1% for the maximum peak value of force and moment and 1.3% for free surface elevation, relative to the standard deviation of each signal. These results are obtained 2500 faster than with the conventional approach and we demonstrate that the computational effort of the new method scales like O (N log N), Similar to wave loads, where N is the number of frequencies. For the eight mode approximation, the error bound of 1% for loads and 4% for free surface evaluation are found to hold across various values of thenormalized peak wave number from shallow to deep water. The accuracy is adjustable thorugh the number of modes and is found to be independent of the time series length. The methods potential in practical design is discussed.