FastFEM: Behaviour of fast ships in waves

Lead Research Organisation: University of Leeds
Department Name: Applied Mathematics


The use of fast (10 to 60m) ships has become popular in maritime operations. They nowadays emerge in search and rescue operations, surveillance, Coast Guard activities, interceptor and security missions, anti drugs and piracy actions, in fast supply to offshore platforms, and for crew transport and wind farm maintenance. The main advantage is their speed, while the challenge is to maintain safe seakeeping at high speeds in mild to heavy seaway, while also keeping these ships affordable, reliable, fuel efficient and comfortable. We propose to develop new simulation tools for the dynamics of fast ships and surrounding waves in breaking seas, and to validate these simulations with advanced and novel towing tank laboratory experiments.

Classical calculations of waves are based on (non)linear potential flow wave theory. Such simulations are quite good provided accurate numerical methods are used, but can never deal with wave breaking due to the potential flow Ansatz. In addition, potential flow wave codes in three dimensions need to become computationally more efficient. This requires further scientific development (work package WP1 at the University of Twente, The Netherlands by Jaap van der Vegt with Onno Bokhove), especially when wave and ship dynamics are fully and nonlinearly coupled, as proposed. Another key issue in WP1 is preservation of the variational formulation of wave dynamics, to ensure numerical stability, within our partially existing discontinuous Galerkin finite element methodology (DGFEM). Our first innovation is to simulate wave breaking around the ships, as well as incoming breaking waves, with a new, advanced single-phase mixture theory for the water-air mixture (in WP2 at the University of Leeds). In contrast to other numerical methods that can deal with wave breaking (such as Volume of Fluid methods and Smoothed Particle Hydrodynamics), our method does not lead to (unwanted) numerical wave damping in areas with smooth waves. Our second innovation is that the new method is by design constructed to include our (variational) potential flow limit with nonlinear ship dynamics. Hence, this new methodology for fast ships in breaking seas does require that the DGFEM for the combined ship and potential flow wave dynamics is fully developed as well (in WP1). Essential is the anticipated validation of both (integrated) simulation tools against advanced and new towing tank experiments. Beyond the numerical validation, these towing tank experiments (in WP3 at Delft University of Technology, The Netherlands by Rene Huijsmans) provide direct and new data to the users in the form of the measured (nonlinear and breaking) wave environment around fast ships, pressures on their hulls as well as onboard accelerations. These laboratory measurements will be directly used by our users: Damen Shipyards, MARIN, Royal Netherlands Navy, Royal Netherlands Rescue Organisation (KNRM), Bureau Veritas, and Lloyds Register, while the proposed simulation tools supporting new hull designs will form a sustained investment.

Planned Impact

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Gagarina E (2016) On variational and symplectic time integrators for Hamiltonian systems in Journal of Computational Physics
Description Maths could help search & rescue ships sail more safely in heavy seas

A unique new computer model built on highly complex mathematics could make it possible to design safer versions of the 'fast ships' widely used in search & rescue, anti-drugs, anti-piracy and many other vital offshore operations.

Travelling at up to 23 to 30 knots, fast ships are especially vulnerable to waves that amplify suddenly due to local weather and sea conditions - extreme funnelling effects, for example, may turn waves a few metres high into dangerous waves tens of metres tall that can destabilise ships, resulting in damage, causing injuries and threatening lives.

Developed at the University of Leeds with Engineering and Physical Sciences Research Council (EPSRC) support, by Dr. Kalogirou and Dr. Ambati with Prof. Bokhove, a new model in this line of modelling produces unprecedentedly accurate animations and simulations that can show exactly how sea waves can affect fast ships. This highlights the importance to have accurate predictions of the pressure forces that these craft are subjected to, and so could aid the design of fast ships better able to withstand the effects of rough seas.

To date, the Leeds team has focused on two things: nonlinear sea waves (without ships) that can produce an anomously high freak or rogue wave, and combined wave and ship motions that can be described mathematically by linear equations (see Notes for Editors). The aim is to extend the model over the next three years to include non-linear waves and so produce a comprehensive simulation tool that can be used extensively by ship designers and maritime engineers. As a by-product a laboratory and mathematical model of new wave-energy device has emerged.

Professor Onno Bokhove, who leads the project, says: "Our initial linear wave-ship simulation work represents a vital breakthrough towards development of a unique tool that can assist the design of ships' hulls that absorb or deflect more of a wave's impact, while still ensuring safe seakeeping, reliability and fuel efficiency."

The (non)linear models have been validated through laboratory experiments of a man-made freak or rogue wave, the so-called soliton splash, using test tanks. A comparison with wave and ship motion, for a ship moored on two anchors, has been set up in a small test tank, also used for public demonstrations. Results from the project are disseminated with a range of organisations including the Maritime Research Institute Netherlands (MARIN). A related European Industry Doctorate project with MARIN on rogue and breaking waves against offshore structures has strengthened Bokhove's research on wave impact against ships (funded by EPSRC) and (fixed) offshore structures (funded by the EU).

Fast ships deliver all kinds of services in fields such as disaster response, the fight against crime, the provision of supplies for oil and gas platforms and the transportation of wind farm maintenance personnel. Each year, however, around 100 such ships worldwide are lost or damaged in heavy seas, with around 2500 casualties in 2013 (

Professor Bokhove says: "Describing mathematically the complex behaviour of waves and their interaction with fast ships and then incorporating all of this into a robust computer model has been very challenging. We are delighted to have provided further proof of how advanced mathematics can have real-world applications that help save money and safeguard lives."
[This is the draft EPSRC press release.]

Technical Remarks
The original proposed mile stones (for 1.5 year of funding):
a) Formulation of a mathematical model for breaking waves in year one.
b) Implementation of linear ship dynamics in linear waves in at the end of year 2.

After 20 months (September 2015) we achieved the following:
a) The theoretical model for breaking waves (item a)
has been derived and there is an internal report.
b) The theoretical model for the mathematics and numerical
modelling of linear wave dynamics with a buoy with one degree of freedom (both in 2D and 3D), and
a ship with 6 degrees of freedom (item b) has been formulated
in an internal report to date.
c) The linearized movement of a buoy only moving in the vertical in 2D and 3D waves has been implemented numerically, both for shallow and potential flow water waves:

The interim results using a buoy in 2D and 3D has a direct application for the novel proof-of-principle wave-energy device Prof. Bokhove and Ir. Zweers have designed and build, see:
as well as the youtube-page given above.

These interim results have been introduced to facilitate the step-by-step development of wave-ship interactions.

The most important mathematical and numerical break-through is the development and derivation of a new symplectic/geometric/ variational time integrators for systems with constraints, in this case the imposition of buoy/ship dynamics in (linear) water waves.

We are currently implementing and testing the theoretical model in b) numerically, aiming for a third journal publication.

As part of acquiring the required expertise (as educational stepping stones for the research assistant dr. Anna Kalogirou)
we have submitted the following papers:
1) O. Bokhove and A. Kalogirou (2015) Variational Water Wave Modelling: from Continuum to Experiment. In: Bridges, T., Groves, M. and Nicholls, D. (eds.) Lectures on the Theory of Water Waves. LMS Lecture Note Series. Cambridge University Press, United Kingdom , 226-260, 2016.
2) A. Kalogirou, E. E. Moulopoulou, and O. Bokhove and . Variational finite element methods for Waves in a Hele-Shaw Tank. Appl. Math. Model., revision submitted, regarding minor revisions, 2016.
3) A. Kalogirou and O. Bokhove (2016) Mathematical and numerical modelling of wave impact on wave-energy buoys. Proceedings of the ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering (OMAE2016), June 19-24, 2016, Busan, South Korea. In press.

Dr. Kalogirou and/or Prof. Bokhove have attended/will attend the following conferences:
i) Water Waves program Isaac Newton Institute of Mathematical Sciences summer 2014.
ii) British Applied Mathematics Colloquium, Cambridge 2015
iii) EGU General Assembly 2015, Vienna, April.
iv) Firedrake/Fenics workshop, IMperial College London, June/July 2015.
v) Scicade conference 2015, Potsdam, September 14-18.
vi) User group meeting with Dutch STW and EPSRC group, Delft Institute of Technology, Nov. 9th 2015.
vii) APS meeting Boston, November 22-24, 2015, submitted abstract.
viii) British Applied Mathematics Colloquium, Oxford 2016.
ix) ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering (OMAE2016), June 19-24, 2016, Busan, South Korea.
Exploitation Route The shipping industry will be interested in the new finite element formulation we have achieved, and in a follow-up proposal we aim to extend this novel mathematical formulation to the non-linear realm, and aim to contact QinetiQ/MOD.
We seek other maritime shipping contacts as well,
and have contact with the Dutch "Maritime Research Institute Netherlands" (MARIN).

The wave energy industry may be interested in our new wave-energy device, and the mathematical modelling we propose based on our interim modelling results of a 3D buoy in linear waves. However, we lack contacts with this industry at the moment.

Finally, there is an EPSRC press release in the making on our research results hitherto.
Sectors Aerospace, Defence and Marine,Energy,Environment,Leisure Activities, including Sports, Recreation and Tourism,Transport
Title Computer Code in Firedrake Finite Element Tool 
Description Variational finite element discretisation of Benney-Luke equations: a reduced water wave model, using Firedrake. "Firedrake is an automated system for the portable solution of partial differential equations using the finite element method (FEM)." 
Type Of Technology Software 
Year Produced 2015 
Impact None to this date; it does however act as stepping stone for further research in my research group. The software tutorial is publicly available.