For solid dynamics, the open source multiphysics simulation engine Chrono® is used. Chrono® solves the equations of motion of floating structures and allows the definition of flexible mooring lines using beam theory and FEM. Additionally, mooring statics are solved with a catenary equation solver developed for the purpose of this work which can also be used for initial conditions of the mooring dynamics module. A brief description of these approaches is presented in the remainder of the section.
2.2.1 Rigid body dynamics
The equations of translational and rotational motion for a rigid body can be summarised as follows:
where m is the mass, I* is the 3 x 3 moment of inertia tensor of the rigid body, E/ and Em are the sum of the forces and moments acting on the body, respectively, гь and вi, are the position and rotational vector of the body, respectively, and i'i, denotes acceleration of Г/, (i.e., second-order time derivative). The position vector Г/, corresponds to the absolute position of the centre of mass of the body (surge, sway, heave) in the global frame of reference, and the rotational vector вi, corresponds to the rotation of the body (roll, pitch, yaw) according to the global frame of reference. In case of moored floating structures, forces and moments are typically caused by hydrodynamic forces and the mooring system. For convenience, these equations are often expressed in terms of generalised coordinates or state vector s = (l'b Mb)7 and generalised force vector f = (^ f mj with a 6 x 6 mass matrix M:
In the current implementation, Chrono® is used to discretise Equation (9) in time with a backward Euler scheme. The time step of the discretisation is typically lower than the one used for the fluid dynamics model, due to the cheaper computational cost of an MBD problem when compared to a multi-phase flow FEM problem within the context of a typical FSI simulation.
2.2.2 Mooring statics
Mooring statics allow the geometrical and physical description of mooring lines subject to their own weight (or submerged weight) and at equilibrium. To solve for the layout of the mooring cable, the positions of the anchor and fairlead are known, as well as the position of the seabed (if any) upon which part of the line can rest. The following properties of the mooring line are also given: the unstretched line length L necessary to find the shape of the line, the submerged weight u>o necessary to calculate tensions in the line, and the axial stiffness EA that defines the elasticity of the line (which will affect both the shape and tension in the line). Any catenary line can be analytically represented through the catenary equations:
where s is the distance along the catenary, and a a variable defining the shape of the catenary. The parameter a is found through different means depending on the four possible layouts of the line: fully stretched, fully lifted, partly lifted, and with no horizontal span. The possible configurations are illustrated in Figure 3. When the line is fully stretched (straight between anchor and fairlead) or has no horizontal span (straight between fairlead and seabed), then the line is technically not a catenary and it is straightforward to find a geometrical description of the line and the tension at the fairlead. When the line is partly or fully lifted, the transcendental equations with respect to a (Equation (11)) are typically solved using bisection or Newton-Raphson methods to find the layout of the catenary. For a single catenary mooring line subject to its own weight, the horizontal tension at any point in the lifted line is Th = awo. The vertical tension varies according to the position s within the catenary, and can be found with simple trigonometry as Tv = Th tan (0/) where 0/ is the
Figure 3: Possible configurations of a single catenary line where ra and Г/ are the positions of the anchor and the fairlead, respectively, d and h are the horizontal and vertical distance between the anchor and fairlead, respectively, and a:о and Ls are the horizontal span and lifted line length for the partly lifted line, respectively.
angle formed by the catenary at the point considered. For a fully lifted line, the following transcendental equation is solved:
where Ls is the lifted line length, d the horizontal distance between fairlead and anchor, and h the vertical distance between fairlead and anchor. For a partly lifted line, the following transcendental equation is solved:
where xq is the horizontal span of the line. Technically, a: о is also an unknown so Equation (13) is iteratively solved with updated values of xo until a is found for the right line length. Furthermore, if elasticity and/or multi-segmented lines (i.e., varying properties w and EA) are used, iterations are necessary for both the fully and partly lifted case before converging to a solution. The algorithm developed for solving the procedure above are described in more detail in de Lataillade (2019).
2.2.3 Mooring dynamics
For dynamic simulations, mooring lines are assumed to be elastic cables with isotropic material and circular cross-section. Each line is discretised in ID elements that are governed by flexible beam theory. As isotropic materials have symmetric stress and strain tensors, the constitutive equation linking stresses and deformation is the generalised Hooke’s law for linear elastic material, expressed as:
where a is the 6x1 material stress vector, e is the 6x1 strain tensor and C is the 6x6 stiffness matrix. The dynamic equilibrium equation for linear elasticity that links stresses, deformation and body forces is:
where a is the 3x3 stress tensor, ps is the material density and h is the deformation vector such that, mooring lines are discretised. The deformation of the cross-section of the beams can therefore be assumed negligible (i.e., considered rigid in its own plane). With this assumption, shear stresses become negligible as the positive tensile stress along the longitudinal axis of the line becomes dominant . A local coordinate system £ = (£, t), £) where / is the longitudinal direction and i] and C are the transverse directions, is defined. Assuming that Poisson’s effect is negligible (i.e., v = 0), Equation (14) reduces to:
with the longitudinal strain being:
where кп = and are the curvature of the beam. The axial force Д and
bending moments mv and are expressed as:
Cables are often described in terms of axial stiffness EAo where Aq is the undeformed cross- sectional area and bending stiffness (also known as flexural rigidity) El, where I is the moment of inertia for flexion of the cable (which is the same in the i] and C direction for isotropic materials). For a chain, it is common practice to set the bending stiffness to zero (i.e., I := 0) and to forbid compression stress (i.e., := 0 when < 0) as the mooring line
will deform/buckle before sustaining any compression stress.
In the implementation used here, the beams are defined by the gradient deficient ANCF method provided by Chrono®. According to the formulation proposed by Gerstmayr and Shabana (2006), each beam or cable element is described through the position vector and a single direction gradient for each of its two end nodes. By using the Green strain tensor, the implementation of the ANCF cable elements leads to the following expression for the axial strain:
where Vf is the gradient along the longitudinal axis and x the spatial coordinates along the cable. This formulation does not take into account torsional effects as only a single direction gradient is used. Note that torsional effects are not expected to be of significant importance for the simulations presented in this work. Other formulations can be used to include them if necessary, such as Euler-Bernoulli beams. The numerical solution is calculated through the FEM module of Chrono® using a Backward Euler scheme to solve the system of equations emerging from the mooring line discretisation into segments (Recuero and Negrut, 2016).