Arvid Naess

Efficient path integration methods for nonlinear dynamic systems

Abstract New approaches for numerical implementation of the path integration (PI) method are described. In essence the PI method is a stepwise calculation of the joint probability density function (PDF) of a set of state space variables describing a white noise excited nonlinear dynamic system. The basic idea behind the proposed procedure is to apply a splines interpolation method to the logarithm of the calculated PDF to obtain an accurate representation of the PDF over the whole domain and not only at the chosen grid points. This exploits the fact that the logarithm of the PDF shows a more polynomial behaviour than the PDF itself, and therefore is better adapted to a splines representation. It is demonstrated that the proposed techniques may lead to significantly improved performance in calculating the response statistics of large classes of nonlinear oscillators excited by white or coloured noise when compared to other available implementations of the PI method. An advantage of the new approaches is that they allow time-variant dynamic systems to be analysed without significant increase in computer time. Numerical results for both 2D and 3D problems are presented.

Response statistics of nonlinear, compliant offshore structures by the path integral solution method

Abstract The paper investigates the applicability of the path integral solution method for calculating the response statistics of nonlinear dynamic systems whose equations of motion can be modelled by the use of Ito stochastic differential equations. The state vector process associated with such a model is generally a diffusion process, and the probability density function of the process thus satisfies a Fokker-Planck-Kolmogorov equation. It is shown in the paper that the path integral solution technique combined with an appropriate numerical scheme constitutes a powerful method for solving the Fokker-Planck-Kolmogorov equation with natural boundary conditions. The method distinguishes itself by high accuracy and numerical robustness even at very low probability levels. These features are highlighted by application to example studies of nonlinear, compliant offshore structures.

Stochastic Dynamics of Marine Structures

1. Preliminaries 2. Dynamics of single-degree-of-freedom linear systems 3. Dynamics of multi-degree-of-freedom linear systems 4. The finite element method 5. Stochastic processes 6. The variance spectrum 7. Random environmental processes 8. Environmental loads 9. The response spectrum 10. Response statistics 11. Statistics for nonlinear problems 12. Short-term and long-term extremes 13. Dynamic load effects for design checks 14. The equations of motion 15. Numerical solution techniques 16. Monte Carlo methods and extreme value prediction.

On the distribution of crest to trough wave heights

Abstract In the present paper we derive the probability distribution function of crest to trough wave heights in a narrow-band, Gaussian stochastic process. It is shown that the distribution function is a one-parameter Rayleigh distribution where the parameter is expressed in terms of the correlation function of the given process. Comparison based on correlation values obtained via sea wave spectra indicate that the derived distribution function agrees well with observed data.

The statistical distribution of second-order slowly-varying forces and motions

A statistical analysis of slow-drift forces and motions is presented. By extending an approach originally due to Kac and Siegert, it is shown that an explicit, closed-form solution may be given to the problem of determining the probability density function of the slow-drift process. From a practical point of view its derivation is carried out without simplifying assumptions, thus a general solution is obtained.

Response Statistics of Nonlinear Dynamic Systems by Path Integration

The paper presents a method for calculating the response statistics of nonlinear dynamic systems excited by a white noise or filtered white noise process. The method, which is based on the path integral solution technique, is still under development, but experience so far indicates that it is singularly well suited for numerical calculation of the response statistics of nonlinear systems to which can be associated a Markov vector process whose probability density satisfies a Fokker-Planck-Kolmogorov (FPK) equation. The method is a viable alternative to the direct numerical solution of the FPK equation. A key feature of the method is the possibility of obtaining highly accurate solutions at very low probability levels. Also, there seems to be almost no limitation on the type of nonlinearity that can be accomodated. On the negative side there are clear limitations of the method concerning required computer resources.

Approximate first-passage and extremes of narrow-band Gaussian and non-Gaussian random vibrations

An approximate method for estimating the probability distribution of first-passage times and extreme values of stationary narrow-band random vibrations is presented. The advantage of the method is that explicit, closed from expressions are obtained. The method is applied to the response process of a simple linear oscillator driven by both Gaussian and non-Gaussian random excitations and, by comparison with published simulation results, good agreement is obtained. For the Duffing oscillator, the results of Markov diffusion models are compared with the present method, and the agreement is fairly good.

Statistical analysis of nonlinear, second-order forces and motions of offshore structures short-crested random seas

Abstract A general theory for statistical analysis of nonlinear, second-order forces and motions of compliant offshore structures in short-crested random seas is described. It is shown that the theory is a natural extension of existing theory for the special case of long-crested waves. Similarly as for this special case, the core of the method is the solution of an eigenvalue problem, which is well suited for numerical analysis. All the information needed for the statistical analysis is contained in the obtained eigenvalues and eigenvectors. It is demonstrated that in the case of slowly-varying, second-order forces and motions, the PDF can be given explicitly and is determined completely by the eigenvalues. An analysis of the extreme values is also presented. In contrast to existing theories, in the present paper no restriction is imposed on the bandwidth of either wave energy or directional spread.

Combination of the peaks-over-threshold and bootstrapping methods for extreme value prediction

The paper investigates the application of the peaks-over-threshold (POT) method in combination with the bootstrapping method for estimating extreme values, in particular, long return period characteristic values of environmental loads, e.g. wind loads, on the basis of observed data. Attention is focused on the effect different statistical estimators of the parameters inherent in the POT method have on the predicted characteristic values. The accuracy of the predicted long return period characteristic values provided by the different methods will be evaluated by application to synthetic data where the relevant characteristic values can be calculated. The bootstrapping method is advocated as a practical tool for estimating confidence intervals on the provided point estimates of the long return period values.

Crossing rate statistics of quadratic transformations of Gaussian processes

The main focus of this paper is the development of a numerical procedure for calculating the average crossing rates of a stochastic process that can be expressed as a sum of a linear and a nonlinear, quadratic transformation of a Gaussian process. Such a representation applies for instance to the motion response of a linear structure subjected to wind loading, when the loading model is proportional to the square of a Gaussian wind velocity process. It is also the standard model for expressing the total wave forces or horizontal excursion responses of a moored floating offshore platform in a random sea way. Knowledge of the crossing rate is a key to many important quantities related to response statistics and reliability applications. It is demonstrated how the proposed numerical procedure can be used for calculating the average crossing rate of the type of response processes considered.