Brazil OMAE 2001 / OFT-1230 GENERATION AND NUMERICAL SIMULATION OF PREDETERMINED NONLINEAR WAVE SEQUENCES IN RANDOM SEAWAYS

In this paper a new procedure is proposed to generate predetermined nonlinear wave sequences in random seaw ays.These tailored wave trains are simulated numerically in time domain using the nite element method. Modern nonlinear programming methods are applied to t the twodimensional wave eld to global parameters de ned in terms of signi cant wave height and peak period as well as local target parameters like wave height, crest height, and period of individual waves. Firstly , a linear approximation of the desired w avetrain is computed by optimizing an initially random phase spectrum for a given variance spectrum. This initial guess is further improved b y tting the fully nonlinear wave evolution simulated in a numerical wave tank to the target characteristics. The discrete wavelet transform of the wave board motion allows to identify the relevan t signal information e ciently, as only a small number of w avelet coe cients needs to be considered in the tting problem. Results are presented for a steep and high transient w ave within a tailored group of three successive waves embedded in a random seaw ay. INTRODUCTION The design of safe and economic o shore structures and ships is decisively in uenced by local characteristics of the extreme wave environment and related w ave/structure interactions. Extreme transient individual w avesand w ave groups are subject of intensiv e researc h in the marine communit y.A stochastic description based on integral parameters alone is insu cient as local characteristics in the wave 1 pattern are of great importance for deriving appropriate design criteria. Stansberg (1990) empirically investigates extreme w avesin tank-generated irregular w avetrains. Wave superposition based on linear wave theory is used to simulate randomly occurring large w aveswhich are pic kedout for experimental study. Baldock and Swan (1994) present a description of a tw o-dimensional irregular sea state in which a large transient wave is generated by focusing component w aves.The numerical method is based upon a Fourier series expansion in space and time which is validated by laboratory data. T aylor et al. (1995) describe a theory to constrain a random time series for generating a large crest elev ation of given amplitude at a prescribed time. The technique is a linear process with the extreme surface elevation being indistinguishable from a purely random occurrence of that particular crest. Hua and Ekman (1999) present a numerical procedure based on a boundary element method to simulate irregular steep waves. The wave height and particle v elocities at the wave crest are investigated for three severe wave conditions, and compared with linear and second order wave theories. They conclude that second order wave theory is not appropriate for predicting extreme wave crests as well as the horizontal particle velocity in the vicinity of these crests. A method for generating a strongly asymmetric wave in an irregular wave train is presented by Zou and Kim (2000). A wave elevation time series with random phase is generated by a linear procedure. The largest crest elev ation among the zero-downcrossing waves is distorted by a method similar to Funke and Mansard (1982) from which the complex amplitude spectrum is obtained. Haver and Copyright  2001 by ASME Andersen (2000) discuss the rare existence of freak waves. They present few evidences which support the hypothesis of a separate freak wave population which deviate strongly from a Gaussian and a Rayleigh process. GENERATION OF LINEAR WAVE SEQUENCES At rst, the target wave train needs to be de ned by global parameters as water depth and variance spectrum in terms of signi cant wave height and peak period. In addition, local characteristics of the wave pattern and extreme waves are of great interest. They are de ned in terms of wave and crest height as well as period of individual waves of a particular wave sequence. For de ning an appropriate optimization problem, the objective function as well as equality and inequality constraints are derived from target characteristics. Further restrictions on the wave board motion are taken into consideration. In linear wave theory the wave train is regarded as superposition of independent harmonic waves with particular amplitude, frequency and phase. For a given design variance spectrum the phase spectrum is responsible for all local wave characteristics. To generate the desired wave train, an initially random phase spectrum is optimized by the sequential quadratic programming (SQP). This approach is described in detail by Clauss and Steinhagen (2000). In addition, the paper compares the linear with the fully nonlinear wave evolution which is simulated in a numerical wave tank in time domain applying the nite element method. As a result, the integral parameters are well met but local wave characteristics of the nonlinear wave train deviate from target values. GENERATION OF NONLINEAR WAVE SEQUENCES The nonlinear wave train that originates from the rst optimization process is further improved by applying the subplex method developed by Rowan (1990) for unconstrained minimization of noisy objective functions. The domain space of the optimization problem is decomposed into smaller subdomains which are minimized by the popular Nelder and Mead (1965) simplex method. The subplex method is introduced because SQP cannot handle wave instability and breaking since the gradient of the objective function is di cult to determine in this case. Nonlinear free surface e ects are included in the tting procedure since the values of objective function and constraints are determined from the nonlinear simulation in the numerical wave tank. The basic idea is that only a short sequence of the wave board motion needs to be changed to achieve the desired nonlinear characteristics of an individual wave or wave group at target position. The discrete wavelet transform is introduced to utilize the strong compression capabilities of the transformation scheme which allows to reduce signi cantly the free variables of the tting problem. Hence, the 2 subplex method ts the wavelet coe cients in a certain time frame at the lowest decomposition scale which contain most of the wave board motion signal energy. Constraint violations such as an inadmissible wave board motion and wave breaking are handled by assigning a high value to the objective function indicating infeasible points. NUMERICAL SIMULATION The fully nonlinear free surface ow problem is simulated numerically in time domain using potential ow theory. A nite element method developed by Wu and Eatock Taylor (1994) is used to determine the velocity potential which satis es the Laplace equation for Neumann and Dirichlet boundary conditions. Clauss and Steinhagen (1999) used this method for simulating nonlinear transient waves. The computations have been validated by laboratory data con rming the high accuracy of this approach. DISCRETE WAVELET TRANSFORM The discrete wavelet transform (DWT) is based on the concept of multiresolution analysis developed by Mallat (1989a, 1989b) and Meyer (1992) which allows the implementation of e cient decomposition algorithms. A function f(t) can be expressed in terms of integer translations of the scaling function '(t) by