Isogeometric Finite Element Modelling of Ideal Non-Linear Free-Surface Flows

Abstract

Delft University of Technology has access to a towing tank where a wave maker can generate waves. When generating waves, non-linear start-up effects occur at the start of a wave train which affect the rest of the signal.
The aim of this thesis is to develop a model which can take these effects into account. A numerical modelling approach is adopted, due to the necessity for generating a large space of wave profiles and the need to optimize towards a wave maker input given a wave profile.

A clear assessment of the state-of-the-art revealed a scope of present methods to model non-linear free surface waves. A potential flow assumption proved a clear balance between accuracy versus computational cost. Analysis of this model showed that the state is described by a balance between potential and kinetic energy and should, in principle, be conserved.
For numerical modelling, the Finite Element Method is adopted. The sparse matrices and ease of evaluation of the integrals allow for a better adaptation in present computer architectures. Additionally this method provides more rigorous mathematical tools to demonstrate properties such as stability and convergence. Lastly, the use of Isogeometric Analysis where the solution space is described by splines instead of polynomials could provide an advantage over conventional methods with respect to continuity, convergence and refinement strategies. Subsequently, literature revealed that the potential flow model in conjunction with FEM will result in model that is accurate, stable and fast.

A novel numerical model is presented where the spatial discretization is done using IgA and the resulting semi-discrete Ordinary Differential Equation is integrated in time with a separate method. Analysis shows that this spatial model inhibits the same energy conservation laws as the physical model.

Implementation is done in the DelFI, an in-house fixed domain Navier-Stokes solver with an Open Source back-end named MFEM which utilizes clever FEM abstractions and parallelization. To successfully implement the non-linear problem, first a linear problem is implemented to facilitate computation of variables that are defined on the domain only, in this work the free-surface elevation. This results in introducing an additional problem to compute the elevation on the interior which is dependent on the free-surface, but not vice-versa. This, to ensure the wave problem remains unaffected. Coincidentally, this definition of the free-surface elevation on the interior is used to deform the mesh required to capture non-linear effects in the time-dependent domain.

Results of the linear problem show to agree with literature. Conservation of energy is guaranteed, yet conservation of mass can be attained with sufficient mesh and time resolutions. This affirms the successful implementation of free-surface problems in DelFI. Extension to the non-linear case shows that energy is not conserved, yet analysis shows it should. The same holds for conservation of mass. Still, both quantities can again be contained with sufficient mesh and time resolution. Additionally results demonstrate that implementation of a stabilization scheme is needed. Finally a benchmark case demonstrates that with the current limitations, results agree with others.

This research demonstrated a novel mathematical framework to compute non-linear free surface waves with special emphasis on conservation laws and geometric compliance with the fluid through Isogeometric Analysis. A basis has been laid towards optimization of wave maker signals given a free-surface wave envelope.