A rapid response calculation method for symmetrical floating structures based on state–space model solving in hybrid time-Laplace domain

Highlights

• A rapid response calculation method for symmetrical floating structures is proposed.

• Laplace transform of retardation function is replaced using a pole-residue form.

• State–space model of system is identified from transfer function of floating offshore structure.

• Exciting forces are regarded as an input of state–space model to calculate dynamic response.

• Responses of floating structures match well traditional method and have high computational efficiency.

Abstract

The article focuses on proposing a new hybrid time-Laplace domain response calculation method for symmetrical floating structures by solving Cummins equation, which is depending on the state–space model identified from transfer function. Different from a time or frequency domain method, the proposed approach estimates the state–space model of a floating structure system from the transfer function in the Laplace domain and calculates the response by considering the exciting forces as an input of the state–space model. Implementing complex exponential decomposition to the retardation functions, the retardation functions in the Laplace domain are expressed using a series of poles and the corresponding residues, which avoids the numerical integral of the convolution terms in the Cummins equation and greatly improves the computational efficiency during the process of a dynamic response analysis. Three examples are applied to investigate the validity of the proposed method. The first is a simple single degree of freedom mathematical model excited by an irregular wave. Studies have shown that the response calculated by the proposed method matches well with that of a traditional Newmark-$\beta$ method. Meanwhile, this approach is insensitive to the interval time of the calculation and consumes less calculation time, which means that the proposed method has higher precision and computational efficiency. The last two examples are a spar-type offshore wind turbine and a semi-submersible platform model (SEMI) in SESAM, which extend the proposed method to solve the response estimation problem of marine structures. The results of the spar-type floating structure show that the estimated responses when using the proposed method are in good agreement with the results of the traditional time domain method. By studying the response calculation of SEMI, the following conclusions can be obtained: (1) the estimated responses match well with the traditional time domain method and WASIM code of SESAM, and (2) the calculation time of the new approach is reduced significantly compared with the Newmark-$\beta$ method especially under a long simulation time.

Introduction

Floating offshore structures are the foundation of marine resource and energy development in deep sea locations, such as the exploitation of fossil resources and offshore wind energy capture. Different types of floating offshore platforms have been developed in recent decades, including floating production storage and offloading (FPSO), spar platforms, and semi-submersible platforms. The dynamic response estimation of such platforms is an important part during the design stage. Depending on the calculated response, the shape and size of the platform can be optimized to obtain a better hydrodynamic performance. The motion behavior of the floating offshore structures under wavy conditions can be expressed through a linear model using the Cummins equation (Cummins, 1962), which applies convolution terms of the retardation functions and the motion velocity to reflect the fluid memory effects. Compared with a low-order approximation using constant coefficients to exchange the frequency-dependent added mass and damping, the accuracy of the response computation when using the Cummins equation improves significantly through the introduction of the convolution terms (Taghipour et al., 2008). However, the presence of the convolution terms in Cummins equation makes it inconvenient for the analysis and design of a motion control system (Fossen, 2002, Perez, 2002). In addition, the dynamic response calculation based on the Cummins equation is computationally demanding when solving linear transient or nonlinear problems, and the convolution terms (integral in the time domain) are not convenient to apply in standard simulation packages (Kashiwagi, 2004).

To improve the calculation efficiency, Newman (1977) solved the problem of conducting a dynamic response calculation in the frequency domain when the floating structures are excited by a wave force. Under the assumption that waves can be expressed as a Gaussian process, the first-order potential theory is applied. Before applying this method, it should be assumed that the steepness of the wave is low and that the response caused by the wave exciting force is proportional to the wave amplitude (Faltinsen, 2005). The linear theory can then be used to compute the dynamic response within the frequency domain. Considering that the method is based on linear frequency domain theory, nonlinear effects induced by damping and stiffness or other factors should be avoided to use the linear theory or should be approximated in a linear form. The standard hydrodynamic code of WADAM in the commercial software SESAM was designed using a frequency domain approach to estimate the frequency response of a floating structure, which is implemented through the following three steps: First, the wave spectrum used to describe the sea state of a floating structure location should be selected. Second, the response amplitude operators (RAOs) depending on the potential theory should be calculated. Third, the frequency response of the floating structures depending on the RAOs is obtained by applying the superposition principle (SESAM, 2017). Theoretically, the frequency domain method calculates the frequency response under the condition of a steady process, which means that only a steady-state response can be obtained and not a transient response. Meanwhile, the component of an incident wave acting on a floating structure is defined as a strictly harmonic wave when calculating the RAOs using the frequency method, which ignores the case of a damping harmonic wave. To calculate the time domain response, an inverse Fourier transform should be applied to the obtained frequency response. The limitation of an inverse Fourier transform, such as spectrum leakage, may occur during this process, which should be considered carefully. To address this problem, Liu et al. (2017) calculated the frequency response of floating structures in the Laplace domain, which applies complex exponential decomposition to the retardation functions and uses the calculated poles and corresponding residues in the Laplace transform of the retardation functions, thereby avoiding the restriction of demanding wave forces consisting of purely harmonic components. When considering the nonlinear effects caused by viscous forces and the water entry and exit, a high-order frequency domain approach should be applied to solve the nonlinear dynamic response problem (Bendat, 1998). Nevertheless, high-order frequency domain methods are inconvenient to apply and are computationally inefficient (Taghipour et al., 2008).

Different from a frequency domain method, the time domainmethod used to calculate the response of a floating structure is usually based on a numerical integral to solve the Cummins equation, which can be used to analyze the transient response and avoid the limitation of a Fourier transform applied through a frequency domain method. However, the integral calculation of the convolution terms requires a large amount of computational time and numerous resources when estimating the response of a floating structure, which leads to a low computational efficiency of the response estimation. To improve the analysis efficiency within the time domain, numerous studies have been conducted. Most of these studies have focused on exchanging the convolution terms with constant coefficients or a state–space model, which mainly includes the following three methods (Taghipour et al., 2008): (1) the use of constant coefficients to replace the frequency-dependent added mass and damping, (2) the use of a state–space formulation to exchange the convolution terms, and (3) the use of a state–space model to replace the force-to-motion response function of a floating structure. The first approximation is the simplest possible approach, and uses constant matrices to replace the frequency-dependent added mass and damping values. The low-order approximation method may lead to relatively large errors when analyzing the transient response excited by a single frequency wave or a steady-state response excited by multiple frequency excitations (Holappa and Falzarano, 1999). The second approach replaces the convolution terms with a state–space model estimated from the retardation functions in the Cummins equation, which has been widely discussed owing to its efficiency and accuracy. To the best of the authors’ knowledge, this method was originally proposed by Schmiechen (1973) to analyze the transient response of a ship. Subsequently, the method was applied to calculate the dynamic response of various floating structures. Taghipour et al. (2007) used an alternative state–space model to replace the convolution terms associated with the fluid memory effect, and conducted a hydroelastic time-domain analysis of a flexible marine structure. Duarte et al. (2013) discussed several methods for finding a state–space model for the wave-radiation forces and compared the results with FAST, an offshore wind turbine computer-aided engineering tool. Lu et al. (2019) applied the replacement of the convolution terms using a state–space model to analyze the dynamic response of a semi-submersible platform. Depending on the solution domain, the state–space model estimation method includes a time domain method and a frequency domain method.Nocedal and Wright (1999) described an impulse response curve fitting method to identify the corresponding state–space replacement of the retardation function. This method is a nonlinear optimization problem solved by Gauss–Newton algorithms. Kung (1978) used realization theory to deal with the problem of obtaining a minimal realization. The singular value decomposition (SVD) is applied to the Hankel matrix consisting of the retardation functions, whose results can be used to calculate a minimal realization. A replaced state–space model can then be constructed. Sanathanan and Koerner (1963) applied quasi-linear regression to estimate the transfer function of the retardation function in the frequency domain. The state–space model is constructed depending on the parameters of the transfer function in the form of a rational fraction. The first two methods are time domain methods and the last is a frequency domain method. The third approach to converting the force-to-motion response into a state–space model was originally proposed by Cummins (1962). Inspired by this, Perez and Lande (2006) changed the displacement to velocity using the impulse response from force to velocity, which was used to calculate the state–space model depending on the frequency response function (FRF).

Differing from the method mentioned above, the objective of this study is to estimate the state–space model of floating structures depending on the transfer function in the Laplace domain. The Laplace transfer of the retardation functions is expressed as a sum of the pole-residue form, which is obtained by implementing complex exponential decomposition to the retardation function. The exciting forces are substituted into the estimated state–space model to calculate the response of a floating structure. Depending on the research of Wu and Moan (1996), the nonlinear effects can be considered as “additional” load in the process of dynamic response estimation. Inspired by the study, nonlinear effects can be introduced to the model of the proposed method at the late stage to deal with nonlinear effects. As the advantages of the proposed method, more accurate results can be obtained compared with the step-by-step integral method, and the computational efficiency of the response estimation is greatly improved. Three examples are applied to investigate the performance of the new approach. The first example is a simple single degree of freedom (SDOF) mathematical model, whose retardation function depends on a pure analytical relationship and does not represent a real system. An irregular exciting force synthesized by harmonic waves is simulated to act on the SDOF system. Applying the proposed method and a step-by-step integral method to the calculation of the dynamic response, a comparison of the corresponding results is conducted. The second example is a spar-type offshore wind turbine (Jonkman, 2010) excited by two regular waves and an irregular wave. The results of the response estimation using the proposed approach and the Newmark-$\beta$ method are described in this section. The last example is a semi-submersible (SEMI) platform excited by an irregular wave obeying the JONSWAP spectrum (Hasselmann, 1973). The response comparison and computational efficiency of the simulation are discussed in the example. The final two examples show the potential application of the dynamic response calculation in an ocean engineering project.