A general variational formulation of the hydroelastic problem with application to VLFS over variable bathymetry G.A. Athanassoulis & K.A. Belibassakis.

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Presentation transcript:

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 Email: mathan@central.ntua.gr International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

Mathematical formulation (Notation and Kinematics) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Mathematical formulation (time domain) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Variational formulation (1/4, Introduction) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Variational formulation (2/4, Introduction) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Variational formulation (3/4, the functional, 3D problem) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Variational formulation (4/4, the functional, 2D problem) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

On the vertical structure of the wave potential (Various expansions used in shallow water wave theories) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

On the vertical structure of the wave potential (Overview of the expansion used in the present theory) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

A complete vertical modal expansion (1/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

A complete vertical modal expansion (2/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

A complete vertical modal expansion (4/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

A complete vertical modal expansion (5/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

A complete vertical modal expansion (6/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Derivation of the nonlinear coupled-mode system (1/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Derivation of the nonlinear coupled-mode system (2/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Derivation of the nonlinear coupled-mode system (3/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Derivation of the nonlinear coupled-mode system (4/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Derivation of the nonlinear coupled-mode system (5/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Derivation of the nonlinear coupled-mode system (6/6) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

The linearised coupled-mode system International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Crucial questions International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Answers International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Dispersion characteristics of the linearised CMS (1/4, arbitrary water depth, no elastic plate) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

Dispersion characteristics of the linearised CMS (3/4, arbitrary water depth, in the presence of elastic plate) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

The second-order coupled-mode system (1/3) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

The second-order coupled-mode system (2/3) International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

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 28-29 October 2004 21/11/2018

Conclusions International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018

Conclusions International Workshop VLFS for the future Trondheim 28-29 October 2004 21/11/2018