A numerical investigation of time integration schemes applied to the dynamic solution of mooring lines

Abstract

This thesis investigates the use of numerical methods to solve the static and dynamic behaviour of a single mooring line, some part of which may be lying on the seabed. For static analysis both the analytic catenary equations and a 3D numerical formulation are presented. Comparisons of results generated by the two approaches show generally good agreement. The numerical static model is used as the initial equilibrium position for the dynamic solution procedure. The equations of motion are developed in accordance with published theory, but whereas published theory have some aspects neglected, here all possible effects are included in the development of the theory and specific effects are only neglected when this can be justified. The Central Difference, Houbolt, Wilson-θ and Newmark time integration schemes are then used to solve the 2D equations of motion. An important aspect of any numerical scheme is the accuracy and stability of the results generated by its use. Here the numerical stability and accuracy of the different time integration schemes used are discussed. Whereas these characteristics can be examined analytically when the equations being solved are linear, the aspects of the solution obtained from nonlinear formulations are generally unique in each case and must be examined numerically. It is therefore important that these aspects be explicitly examined to gain confidence regarding the general application of the solvers presented. The different schemes are therefore used in an extensive numerical study of the resulting predictions to determine the efficacy of these schemes when applied to the solution of nonlinear mooring dynamics. A 3D formulation of the equations of motion is developed and solved with the Houbolt scheme. This more generally applicable solver is used to generate results of relevance to the design of mooring lines. In particular effort is directed towards the calculation of the dynamic tension amplification factors. These generated solutions can then be used to comment upon the safety factors used and required within the rules of the classification societies. An alternative method of solution which uses the eigensolutions of a matrix form of the equations of motion is also presented. This method is common in small displacement finite element dynamics. However, the large displacement requirement imposed for mooring line dynamics requires some important new aspects in the solution procedure to be considered. These include the updating of the modal matrix and the use of different time steps with the uncoupled equations. Considerations such as these do not appear to have been fully appreciated in the literature associated with mooring line dynamics as solved by modal methods. A section detailing the influence of 'ground effects' is also presented. These include the effects of suction and friction upon the grounded portion of the line and the discretisation problems caused by the lifting and grounding of the node masses. This thesis presents for the first time a detailed comparison of the predictions of mooring line dynamics using different time integration schemes. The comparisons indicate which scheme is to be preferred, highlight the numerical problems which can be expected to affect the quality of the solution, and how best to avoid the noted numerical problems.