Multi-floater-mooring coupled time-domain hydro-elastic analysis in regular and irregular waves

Abstract

In this study, a time-domain hydro-elastic model is developed to solve the floater-connector-mooring coupled system. The frequency-domain model is first devised with the discrete-modulus-based method in which the deformable floating structure is modeled with multiple rigid bodies connected by high-order beam elements. The corresponding multi-body hydrodynamic coefficients and loads are estimated by 3D potential theory with the coupling stiffness matrix based on the theory of the Euler-Bernoulli beam and Saint-Venant torsion. The time-domain hydro-elastic model with mooring dynamics is then subsequently developed with multi-body Cummins equation with high-order rod-FE (Finite Element) mooring elements connected to relevant floaters. The present frequency- and time-domain models are validated through comparisons with published experimental and frequency-domain results for a VLFS (Very Large Floating Structure) without mooring lines. The present method directly solves the hydro-elastic problem with mooring lines without using the conventional modal superposition method, for which wet modes with mooring need to be obtained in a priori through cumbersome iterations. The developed hydro-elasticity model is also very useful when complex inner structure-connection conditions, such as discontinuity by multiple hinged joints, are involved. Such an example is also numerically tested, and reliable results are obtained. Vertical flexible displacements, bending and torsional moments, shear forces, and mooring tensions in regular and irregular wave conditions are systematically analyzed and compared with rigid-body and without-mooring cases.

Introduction

Floating-body hydrodynamics as a rigid body with 6-DOF (degree of freedom) motions has been extensively investigated by many researchers. The extension of the fluid-structure interaction theory to elastic or highly deformable floating bodies has also been made by several researchers. The hydro-elastic theory treats the structure as a deformable solid continuum with its own governing dynamic equation. The elastic deformation of the body changes the body's local displacements, accelerations, buoyancy, and stresses. Consequently, the resulting fluid's forces acting on the flexible body are also changed compared to the rigid-body case. This coupling effect between the fluid and deformable body is the main focus of hydro-elasticity studies.

The hydro-elasticity effect depends highly on the flexibility of the structure [1, 2]. The larger the structure, the harder to make the structure stiff enough to be a rigid body. Such examples include Very Large Floating Structures (VLFSs), i.e., floating offshore airports, floating solar power stations, Ultra Large Crude Carrier (ULCC) supertankers, and floating bridges. They have very large length and width so that hydro-elastic effects can be important in the design process since the hydro-elasticity alters its loading distribution and fatigue life. Some WECs (wave energy converters) [3], [4], [5] are intentionally built to be highly deformable to drive the PTO (power take-off) system continuously. In this regard, the importance of hydro-elastic research cannot be too much underscored.

In general, the effect of structural elasticity for VLFSs can be considered through several methods. The most popular method is the modal superposition method. This method depends on the inclusion of all necessary rigid and elastic modes to calculate the generalized body coordinates and normal directions, which then acts as the boundary condition in the fluid dynamics solver [1]. Several studies incorporating this method include Refs. [1, 6, 7], which incorporated “dry” modes in their frequency-domain simulation, most notably on their radiated wave potentials. On the other hand, Refs. [8], [9], [10] incorporated “wet” modes in their frequency-domain simulation, where the effects of added mass and hydrostatic stiffness are considered in their modal analysis. For the determination of the modal characteristics of structures with complex geometry and boundary conditions, Finite Element Method (FEM) was used to determine the structural modes and natural frequencies (Refs. [7, 8, 11]). This method is also compatible with 3D hydrodynamic simulation tools so that the fluid interaction forces can be calculated in 3D in either frequency [1, 2, 12] or time domain [13], [14], [15]. The water contact effect can still be taken into consideration even though the dry mode is used if the hydrodynamic analysis is done in the time domain [12]. Load mapping by considering the hydro-elasticity in time domain under random wave excitation is shown in Ref. [16]. The paper also shows the importance of including the elastic modes in the radiation-damping-related convolution term. The disadvantage of the modal superposition method is that the wet modes are not determined in a priori, and several iterations are needed to get it. Also, in the presence of many mooring lines, finding wet modes become more complicated.

Another method that is very time consuming is a full 3D FEM structural simulation, which is then fully coupled with 3D hydrodynamic simulation tools [17, 18]. This method has the advantage of not having to calculate any modal characteristics beforehand, the ability to incorporate non-linearity if done in time domain, and the ability to simulate complex structural shape. However, huge computational time is a big burden. A compromise to this method can be found in Refs. [19], [20], [21], [22], which utilized a discrete-module-beam (DMB) method to solve the hydro-elasticity problem. This method treats the deformable body as several partitioned floating modules that move independently with each other while connected by a beam of equivalent stiffness of the actual structure. With a sufficient number of body's partition, the DMB method has the ability to solve the hydro-elasticity problem without solving the structure's modal characteristics. Due to this, the DMB method does not need to do iteration to solve the “wet” modes beforehand. Furthermore, Zhang and Lu [23] showed that this method is extensible for a more complex body where the structural stiffness is transformed into its equivalent end node's stiffness by using FEM. Extensive reviews on many different methods to solve the hydro-elasticity problem can further be found in Ref. [24, 25].

For many deformable bodies, station keeping systems such as moorings or tendons are involved and they are important elements of the dynamics of the whole system. However, the effects of structural elastic deformation on these elements and vice versa are still not well studied. The present paper is specifically aimed to examine such a problem. In this study, the hydro-elasticity of VLFS with mooring is investigated by using the DMB-mooring coupled method both in frequency and time domain. The DMB-mooring coupled method in a combined program is hard to find in the open literature. Here, the fully coupled mooring-fluid-elasticity interaction is solved in time domain, where their non-linear interactions can be fully taken into consideration. Using the Cummins equation, the full effects of hydro-elasticity on the added mass, convolution terms, hydrostatics, and excitation forces are also included. Mooring lines are modeled with high-order FE rod elements, and the effects of the mooring to the vessel's elastic response and vice versa are examined. Furthermore, a more complex structural configuration consisting of two elastic bodies connected by a hinge, such as in Ref. [7], is investigated to further examine the robustness of the developed DMB method without solving the modal characteristics beforehand. In the following, for self-containment, multi-floating-body hydrodynamics theory in both frequency and time domains, DMB method for hydro-elasticity, and mooring dynamics by high-order FE rod elements are briefly summarized.