A Numerically Efficient Technique for the Simulation of Random Wave Forces on Offshore Structures
Authors: Samii K. , Vandiver J.K. 
Published: October 2008

ABSTRACT
A technique is presented for numerical time domain simulation of water particle velocities and accelerations corresponding to a desired wave spectrum. Standard practice is to sum a finite number of sinusoidal components with random phase. The new method uses a simple constant coefficient linear prediction equation to generate time series simulations of water particle kinematics using Gaussian white noise as input.
The velocities and accelerations are propagated throughout the offshore structure using a series of numerical convolutions. These methods produce time series whose spectra are smooth continuous functions, not ones approximated by a finite number of sine waves. The new techniques are much more numerically efficient than summing sinusoids, making long simulations feasible.

INTRODUCTION
Current industry practice in the modelling of dynamic response of offshore structures makes extensive use of finite element techniques. The wave forces are generally computed using the Morison equation. This requires that water particle velocities and accelerations be known over an extensive grid of points corresponding to structural elements. The forces on each structural component are computed and applied at the appropriate nodes of the finite element model. To properly account for non-linear mechanisms such as due to drag forces and the actual position of the free surface, these computations must be done in the time domain
The simulation of a unique event, such as the design wave is usually accomplished by prescribing a deterministic non-linear wave or series of waves. Present practice seems quite adequate as the duration of the event is short and not too much computer time is required. For the purpose of fatigue life estimation, long time histories of random waves must be generated. By present practice the wave induced water particle velocities and accelerations are generated by summing a finite number of sinusoidal time histories with random initial phase angle. The amplitude of each sine wave is selected so that the sum provides an approximation to a desired wave spectrum. The more components used the better the approximation. A typical simulation may use 256 components. To obtain reliable statistics of, for example stresses at critical nodes, may require excessive computer time. The principal source of numerical inefficiency with current technology is in the computation of wave kinematics. Another disadvantage of summing sines is that because each component itself is deterministic, the sum eventually repeats itself. Therefore, very long time histories require a very large number of components.
In this paper a method is described, which simulates wave induced velocities and accelerations in a numerically efficient non-repeating way. Furthermore, the desired wave spectrum is approximated, not at a finite number of discrete frequencies but by a continuous wave spectrum. This is beneficial in the simulation of slowly varying drift forces which depend on difference frequencies between various components of the wave spectrum. Finite numbers of components yield drift forces at only a finite number of frequencies, not a continuum as realized in the real ocean. The new method provides a continuum of sum and difference frequencies.