1. A general variational formulation of the hydroelastic problem with application to VLFS over variable bathymetry
G.A. Athanassoulis & K.A. Belibassakis
National Technical University of Athens
School of Naval Architecture & Marine Engineering
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

2. Plan of the presentation
Introduction
Variational formulation
Local-mode series expansion of the wave potential
Coupled-mode system of horizontal differential equations and boundary conditions
Fully nonlinear water-wave problem / linear thin plate theory
Linearised water-wave problem / linear thin plate theory
Second-order water-wave problem / linear thin plate theory
First numerical results
Conclusions
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

3. Introduction The interaction of free-surface gravity waves with
floating deformable bodies,
in water of intermediate depth with a general bathymetry,
is a difficult-to-solve problem, finding important applications.
Very Large Floating Structures (VLFS, megafloats) are examples of structures for which hydroelastic effects are significant.
Because of their large dimensions, O (km),
the variation of the depth below VLFS is usually non-negligible
Hydroelastic analysis of floating bodies
is also the appropriate context for studying
the interaction between waves and ice sheets
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

4. Introduction
Extended literature surveys of recent research studies for VLFS
have been presented by Masashi Kashiwagi (1999/2000) – [56 refs]
in ISOPE 1999 and in the Journal OPE 2000
and by E. Watanabe et al (2004) - [107 refs]
in Engineering Structures 2004
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

5. Introduction The effect of nonlinear waves
In accordance with E. Watanabe et al (2004)
Future directions of VLFS research should include:
The effect of nonlinear waves
Arbitrary shaped platforms for VLFS
VLFS with non flat hulls
Non uniform seabed topography
Developing simplified models for analysis and design
Smart anti-motion control devices
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

6. Introduction The present method, consistently combines
Variational Principles,
local vertical modes and
smooth bottom approximation
to obtain an efficient coupled-mode system of horizontal equations,
fully equivalent with the nonlinear water-wave / thin plate theory
Floating structure
Sea bed
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

7. Mathematical formulation (Notation and Kinematics)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

8. Mathematical formulation (time domain)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

9. Variational formulation (1/4, Introduction)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

10. Variational formulation (2/4, Introduction)
International Workshop VLFS for the future Trondheim October 2004
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11. Variational formulation (3/4, the functional, 3D problem)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

12. Variational formulation (4/4, the functional, 2D problem)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

13. On the vertical structure of the wave potential (Various expansions used in shallow water wave theories)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

14. On the vertical structure of the wave potential (Overview of the expansion used in the present theory)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

15. A complete vertical modal expansion (1/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

16. A complete vertical modal expansion (2/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

17. A complete vertical modal expansion (3/6)
The above result is generic and can be reformulated and specialized for
any non-uniform waveguide.
Any combination of boundary conditions on the nonuniform boundaries
is possible (Dirichlet – the easiest case, Neumann or Robin)
The differential operator governing the phenemenon might be of a
rather general form
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

18. A complete vertical modal expansion (4/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

19. A complete vertical modal expansion (5/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

20. A complete vertical modal expansion (6/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

21. Derivation of the nonlinear coupled-mode system (1/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

22. Derivation of the nonlinear coupled-mode system (2/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

23. Derivation of the nonlinear coupled-mode system (3/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

24. Derivation of the nonlinear coupled-mode system (4/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

25. Derivation of the nonlinear coupled-mode system (5/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

26. Derivation of the nonlinear coupled-mode system (6/6)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

27. The linearised coupled-mode system
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

28. Crucial questions
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21/11/2018

29. Answers
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

30. Dispersion characteristics of the linearised CMS (1/4, arbitrary water depth, no elastic plate)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

31. Dispersion characteristics of the linearised CMS (2/4, arbitrary water depth, no elastic plate)
analytical
Propagating mode only (mode 0)
N=3: modes 0,1,2
N=5: modes -2,0,1,2,3
N=5: modes 0,1,2,3,4
N=3: modes -2,0,1
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

32. Dispersion characteristics of the linearised CMS (3/4, arbitrary water depth, in the presence of elastic plate)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

33. Dispersion characteristics of the linearised CMS (4/4, arbitrary water depth, in the presence of elastic plate)
propagating mode only (mode 0)
analytical
N=3
modes -2,0,1
N=5
modes
-2,0,1,2,3
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

34. The second-order coupled-mode system (1/3)
International Workshop VLFS for the future Trondheim October 2004
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35. The second-order coupled-mode system (2/3)
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

36. NUMERICAL RESULTS (I): Evolution of incident harmonic waves over a smooth shoal (a=250m and b=750m). The period of the incoming waves is T=15.7sec
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

37. NUMERICAL RESULTS (II): Interaction of incident harmonic waves with a floating elastic plate, with parameters L=500m, δ=10^5 m^4, ε=0, lying over a smooth shoal, extending from x=250m to x=750m.
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

38. Conclusions
International Workshop VLFS for the future Trondheim October 2004
21/11/2018

39. Conclusions
International Workshop VLFS for the future Trondheim October 2004
21/11/2018