Reducing uncertainty in time domain fatigue analysis of offshore structures using control variates

Highlights

• Method for reducing uncertainty for time domain fatigue analysis with random loads.

• Variance reduction of Monte Carlo simulation (MCS) achieved using control variates.

• Control function constructed via ANN trained from existing MCS data.

• Unlike conventional ANN, proposed method is unbiased and has an error estimate.

• Examples of nonlinear systems show that efficiency is enhanced more than tenfold.

Abstract

This study is concerned with time domain fatigue analysis of offshore structures subjected to random waves. The fatigue damage calculated from a single realization of the stress time history is random, thus the mean damage is typically estimated via Monte Carlo simulation (MCS), by averaging over multiple realizations. This approach is time-consuming, because each realization involves a time domain dynamic analysis, and MCS has a slow convergence rate. Variance reduction techniques can improve the efficiency of MCS, but successful implementation necessitate prior information on the system behavior, which is difficult to acquire for this high-dimensional problem. Herein, a method is developed for reducing the variance of the MCS estimator of the damage, based on a new technique known as auto control variates. The control function is constructed via artificial neural network trained from existing MCS data, thus avoiding the need for prior information or additional simulations. The proposed method has several advantages; it is unbiased, and an error estimate is available. Besides, variance reduction is implemented at the post-processing stage; allowing multiple stress locations to be evaluated from the same dynamic simulation results. The case studies include a nonlinear single-degree-of-freedom system under different scenarios, and a full nonlinear model of a floating system. The proposed method enhances the MCS efficiency for all cases, with speedups ranging from one to two orders of magnitude.

Introduction

Offshore structures are exposed to wave loads that give rise to stress reversals, thus fatigue failure is a key design consideration. A dynamic analysis has to be performed initially, either by frequency domain or time domain analysis. Although frequency domain analysis is fast, it is confined to linear systems. For compliant offshore structures such as marine risers and mooring lines, time domain analysis is necessary due to various nonlinearities such as the drag force and large deflections. After the stress time history is obtained, the stress cycles are extracted via rainflow counting, and the fatigue damage is computed using Miner’s rule with the relevant S-N curve.

In practice, the short-term sea state is usually modeled as the sum of many sinusoidal waves with random amplitudes and phase angles, to account for the irregular and stochastic nature of ocean waves [1]. Consequently, the corresponding fatigue damage is also random. The mean damage can be estimated via Monte Carlo simulation (MCS), by taking the average damage from multiple realizations. This approach is straightforward, but it is time-consuming, because each realization entails a computationally demanding time domain analysis, and numerous realizations are needed due to the slow convergence of MCS.

There are variance reduction techniques for improving the statistical efficiency of MCS; the classical ones are importance sampling, control variates, antithetic variates and stratified sampling [2]. These techniques have the desirable property of being unbiased, but successful implementation necessitate prior information on the system behavior, which is difficult to acquire for complex systems. For the extreme response problem involving the evaluation of small failure probabilities, there are many efficient methods available, including black-box methods that do not require prior information, such as subset simulation [3] and line sampling [4]. In addition, there are fast approximate techniques, in particular the first order reliability method (FORM) [5].

Conversely, for the fatigue problem that is essentially a mean estimation problem, there appears to be a lack of variance reduction methods for improving the efficiency of MCS. One reason is the intricacy of the fatigue problem, as the fatigue damage accounts for the entire stress time series, in contrast to the extreme response problem where only the largest peak is of essence. In addition, the problem is high dimensional due to the random phases, and the complex relationship between the random variables and fatigue damage makes conventional variance reduction techniques difficult to implement. Recently, Low [6] developed an importance sampling approach for efficient rainflow fatigue analysis. Although the approach is efficient and unbiased, it only works for linear systems for which the stress spectrum is available. Jensen [7] observed that in MCS, the mean estimated damage is sensitive to the largest damage samples, and proposed an efficient approximate method that uses FORM to estimate the probability of large values more accurately. However, this method relies on the assumption that the reliability index varies linearly with the damage, and the error cannot be estimated.

The objective of this paper is to develop an efficient method for fatigue analysis based on time domain simulations. The method should be unbiased and provide an error estimate, similar to MCS, and ideally it should not require prior domain information. Metamodeling techniques, in particular artificial neural network (ANN), are widely applied for predicting the dynamic response of offshore structures, to avoid extensive time domain simulations. Yazid et al. [8] adopted a second-order Volterra model to predict the surge motion of a spar. Mohd Zaki et al. [9] estimated the overturning moment of a fixed offshore platform by a finite-memory nonlinear system model. de Pina et al. [10] presented wavelet network models to predict the top tension of mooring lines. Guarize et al. [11] presented an ANN procedure for mooring lines, while de Aguiar et al. [12], Chaves et al. [13], [14], and Cortina et al. [15] used similar techniques for risers. de Pina [16] proposed a Nonlinear AutoRegressive model with eXogenous input (NARX) for the analysis of moorings and risers, while Yetkin and Kim [17] used NARX in conjunction with a Volterra model. Christiansen et al. [18], [19], [20], [21] conducted a series of studies to optimize ANN for applications to slender marine structures.

The abovementioned studies reveal that metamodels are capable of predicting the dynamic response of offshore structures with good speed and reasonable accuracy, and ANN is the most versatile and popular metamodeling method. However, one drawback is that the predicted response has a systematic bias error owing to the limited data available to train these models, and this bias is not easy to quantify since it is embedded in the available data. For fatigue applications, even a small bias error in the predicted stress can be magnified to a large error in the fatigue damage due to the nonlinear relationship between stress and damage. Moreover, the prediction of extreme situations can be compromised if they are outside the range of the training data.

In this work, it is proposed to use a relatively new technique known as auto control variates (ACV) to enhance the efficiency of conventional MCS. The classical control variates technique has many attractive attributes. Variance reduction is achieved in the post-processing stage, without altering the dynamic simulation process. This simplifies the variance reduction, and allows any number of stress locations to be evaluated using the same set of dynamic analyses. Moreover, the estimate remains unbiased, and an error estimate can be obtained from the samples, similar to MCS. A major impediment is that control variates is effective only if one has a control function that is strongly correlated with the response variable. To overcome this limitation, Low [22] proposed an ACV framework to construct the control function based on information extracted from the MCS samples, thus avoiding the need for additional simulations to acquire information regarding the system behavior. Low [22] implemented the ACV scheme for long-term fatigue analysis involving two random variables. Subsequently, Leong et al. [36] showed that ACV works well on six random variables for the prediction of the long-term extreme response.

This study is the first application of ACV on a high-dimensional problem involving the random wave components. To derive the control function, an approximate relationship between the random variables and the response is required, and this complex relationship is modeled using ANN. It is emphasized that ACV is the underpinning concept behind the proposed method, and ANN is only a tool within the method. Unlike traditional ANN, the proposed method enjoys several advantages of MCS, such as being unbiased and having an error estimate.