FEM Hydroelastic Models with Application to the Nonlinear Response of Large Floating Bodies in Shallow Wave Conditions

A higher order finite element scheme is presented for the study of the transient hydroelastic response of a floating, thin, nonlinear strip in shallow wave conditions. First, nonlinear effects are introduced only in the elasticity model, where large deflections and non-negligible normal stress variation in the lateral direction are assumed. The nonlinear beam is initially coupled with the linearized and subsequently with the full nonlinear Shallow Water equations, introducing nonlinearity in both the hydrodynamics and the elasticity model. The effects of the incorporated nonlinear effects are assessed through a numerical example featuring an elevation pulse of increasing steepness

Αριθμητική προσομοίωση της υδροελαστικής αλληλεπίδρασης κυμάτων tsunami με Μεγάλες Πλωτές Κατασκευές και στρώματα πάγου.

113 σ.Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Υπολογιστική Μηχανική”Οι συνέπειες τηςκλιματικής αλλαγής τόσοστο περιβάλλον, όσο και στην ανθρώπινη δραστηριότητα αποτελεί αντικείμενοεντατικής έρευνας.Με την αύξησητηςθερμοκρασίας, το ενδιαφέρον στρέφεται στιςΠολικές ζώνες και στην ευπαθή ισσορροπία τους.Ως ρυθμιστέςτου παγκόσμιου κλίματος, οι Αρκτικέςζώνες επηρεάζουν την θερμοκρασία και την ωκεάνια κυκλοφορία.Τηνίδια στιγμή, ο καταγεγραμμένος ‘κατακερματισμός’των στρωμάτων πάγου (ice shelves) ή παγετώνων στηνΑνταρκτική και η σημαντική μείωση του θαλάσσιου πάγου στην Αρκτική, φαίνονται να επηρεάζουνάμεσα τιςεμπορικές δραστηριότητες και να επιβεβαιώνουν την αρχή μιας σειράςκλιματικών διαταραχών. Η κυματική διέγερση τωνστρωμάτων πάγου έχει συνδεθείμε τα φαινομενα απόσχισηςσωμάτων πάγων και τοσχηματισμό ροών (ice floes). Η παλιρροιακή δράση και η συνεχής καταπόνηση των κυμάτων, σε συνδιασμό με τις εγγενείς ατέλειες του πάγου οδηγεί σε καμπτική αστοχία και τηντελική απόσχισητμημάτων πάγου. Οιμεγάλεςοριζόντιεςδιαστάσεις, σεσχέσημετο πάχοςτωνστρωμάτων πάγου, καθιστάτιςελαστικές παραμορφώσεις κυρίαρχες των κινήσεων στερεού σώματος. Επομένως, η μελέτητης απόκρισηςστρωματών πάγων υπο κυματική καταπόνιση εμπίπτει στη περιοχή της υδροελαστικότητας. Ανθρώπινες κατασκευές που μοιράζονται τα ίδια χαρακτηριστικάμε τα στρώματα πάγου, όπως οι Μεγάλες Πλώτες Κατασκευές (Very Large Floating Structures, VLFS) αποτελούνεφαρμογές της υδροελαστικότητας.Συνεπώς, η υδροελαστική ανάλυση πλωτών στρωμάτων υπο κυματική καταπόνισηείναι κοινόέδαφος για εφαρμογές τοσοστηγεωφυσικήόσο και στημηχανική κλίμακα. Στην παρούσα εργασία εξετάζεται η απόκριση πλωτώνστρωμάτων ύπο την καταπόνηση μακρών κυμάτων.Για την ανάλυση, επιδιώκεται η μονοδιάστατη σύζευξητουμοντέλουλεπτής, ελαστικής δοκού Euler Bernoulli και τωνγραμμικοποιημένων εξισώσεωνρηχώνυδάτων.Ορίζονται δύοξεχωριστά προβλήματα.Το πρώτο πρόβλημα αφοράμια πλωτή πλάκα μεένα πακτωμέμο άκρο ενώ το δεύτερο εξετάζει μια ελεύθερη πλωτή πλάκα. Στησυνέχεια παρουσιάζεται η ανάλυση ευστάθειας της ισχυρήςδιατύπωσης τωνδύο προβλήματων και μελετάται η αρχήτηςδιατήρησης τηςενέργειας για κάθε σύστημα.Στησυνέχεια, για την επίλυση γίνεται χρήση της μεθόδου των πεπερασμένων στοιχείων.Ειδικά πεπερασμένα στοιχεία κατασκευάζονται μεδιαφορους πολυωνυμικούς βαθμούς, για την προσεγγιση της λύσηςστις περιοχέςτηςυδροελαστικής σύζευξης.Επιπλέον παρατίθεται και η εκτίμησησφάλματος για τηνημι-διακριτή μορφήστοχώρο.Στησυνέχεια η λύσητωνπεπερασμένων στοιχείων συγκρίνεται με διαφορετική μέθοδο (Sturova 2009) που βασίζεται στην ανάπτυξη της ανύψωσης ελεύθερης επιφάνειας σειδιοσυναρτήσεις τηςδοκού in vacuo. Τέλος, παρουσιάζονται δύο παραδείγματα για τοκάθεένα απο τα προβλήματα καθώς και το παράδειγμα τουστρώματος Sulzberger.Επίσης, δίνονται αποτελέσματα που παρουσιάστηκαν στο European Geosciences Union Assembly 2014, μεσυμμετοχήτηςσυγγραφέα.Η ανάλυση επιβεβαιώνειτην επίδραση του πάχουςτης πλάκας στηδιασπορά τουυδροελαστικού κύματος καθώς και στην κατανομήτωνροπών και τεμνουσών.Η απόσταση της μέγιστης ροπής απο το ελευθερο άκρο φαίνεται να εξαρταται απο το πάχος αλλά όχι απο τομήκοςκύματος της καταπόνησης.Ice caps act as climate controllers, regulating temperature, ocean circulation and affecting global weather patterns. The disintegration of ice shelves and sea ice in recent years has gathered scientific attention as the stability of ice formations is being re-evaluated. The detrimental interaction between ice sheets and long waves has been recently advocated, showing the need for a thorough investigation of the phenomenon. Simultaneous technological advancements in marine engineering provide a different motivation for the study of the transient response of Very Large Floating Structures (VLFS) under long wave excitation. In the present thesis, the elastic Euler Bernoulli beam model and the shallow water equations are coupled in order to derive a 1-D hydroelastic system. Two specific problems are defined, the one of a floating cantilever, a fixed-edge plate, able to simulate an ice shelf or moored VLFS, and one of a freely floating plate approximating the configuration of an ice floe or a pontoon VLFS. Stability estimates of the variational form of the governing equations and the energy conservation principle are studied for both problems. The finite element method is employed for the solution of the problems in question. Special hydroelastic elements incorporating various polynomial degrees are developed in order to cater for the coupling in the hydroelasticity dominated regions of the problem. In addition, error estimates for the semi-discrete form are derived. The finite element solution is compared against the eigenfunction expansion method of Sturova (2009). Finally, two examples are explored for each of the problems along with a geophysical case study based on the Sulzberger Ice Shelf calving event in 2011. Additionally, results presented as collaboration, by the author, at the European Geosciences General Assembly 2014 are given. The presence of the fixed boundary and its effect on the bending moment and shear force distributions are explored. Thickness variations are shown to have an effect on shear force distribution while the distance from the free edge of a cantilever plate where the maximum bending moment appears is relatively insensitive of the incoming tsunami wavelength.Αγγελική Ε. Καρπεράκ

Higher-order fem for nonlinear hydroelastic analysis of a floating elastic strip in shallow-water conditions

The hydroelastic response of a thin, nonlinear, elastic strip floating in shalow-water environment is studied by means of a special higher order finite element scheme. Considering non-negligible stress variation in lateral direction, the nonlinear beam model, developed by Gao, is used for the simulation of large flexural displacement. Full hydroelastic coupling between the floating strip and incident waves is assumed. The derived set of equations is intended to serve as a simplified model for tsunami impact on Very Large Floating Structures (VLFS) or ice floes. The proposed finite element method incorporates Hermite polynomials of fifth degree for the approximation of the beam deflection/upper surface elevation in the hydroelastic coupling region and 5-node Lagrange finite elements for the simulation of the velocity potential in the water region. The resulting second order ordinary differential equation system is converted into a first order one and integrated with respect to time with the Crank-Nicolson method. Two distinct cases of long wave forcing, namely an elevation pulse and an N-wave pulse, are considered. Comparisons against the respective results of the standard, linear Euler-Bernoulli floating beam model are performed and the effect of large displacement in the beam response is studied

Propagation of acoustic-gravity waves in inhomogeneous ocean environment based on modal expansions and HP-FEM

A coupled mode model is presented for the propagation of acoustic-gravity waves in layered ocean waveguides. The analysis extends previous work for acoustic waves in inhomogeneous environment. The coupled mode system is derived by means of a variational principle in conjunction with local mode series expansion, obtained by utilizing eigenfunction systems defined in the vertical section. These are obtained through the solution of vertical eigenvalue problems formulated along the waveguide. A crucial factor is the inclusion of additional modes accounting for the effects of spatialy varying boundaries and interfaces. This enhancement provides an implicit summation for the slowly convergent part of the localmode series, rendering the series rapidly convergent, increasing substantialy the efficiency of the method. Particular aspects of the method include high order Lagrange Finite Element Methods for the solution of local vertical eigenvalue problems in the case of multilayered waveguides, and Gauss-type quadrature for the computation of the coupled-mode system coefficients. The above aspects make the present method quite efficient for long range propagation in extended waveguides, such as the ones found in geophysical applications, e.g. ocean basins, as only few modes are needed for the accurate representation of the wave field

FEM Hydroelastic Models with Application to the Nonlinear Response of Large Floating Bodies in Shallow Wave Conditions

A higher order finite element scheme is presented for the study of the transient hydroelastic response of a floating, thin, nonlinear strip in shallow wave conditions. First, nonlinear effects are introduced only in the elasticity model, where large deflections and non-negligible normal stress variation in the lateral direction are assumed. The nonlinear beam is initially coupled with the linearized and subsequently with the full nonlinear Shallow Water equations, introducing nonlinearity in both the hydrodynamics and the elasticity model. The effects of the incorporated nonlinear effects are assessed through a numerical example featuring an elevation pulse of increasing steepness

Hydroelastic analysis of Very Large Floating Structures in variable bathymetry regions by multi-modal expansions and FEM

A novel frequency domain numerical method for Very Large Floating Structure (VLFS) hydroelasticity is developed. The problem is formulated in the 2D ocean waveguide, featuring a realistic seabed bathymetry and the presence of inhomogeneous, elastic plates of varying thickness and negligible draft. An in vacuo modal expansion for the elastic body deflection, modelled as a structural plate, is employed to decouple the hydrodynamics from structural mechanics. The inhomogeneous plate is considered to undergo cylindrical bending, while depending on the structure slenderness and excitation wavelength the classical thin plate theory and Mindlin's model, accounting for first order shear deformation effects are implemented. A weighted residual approach is employed to cast the formulated problems into a mixed weak form for which dimensionality reduction is sought. This is achieved by an enhanced vertical representation for the wave potential, able to accurately account for abrupt bathymetric changes, following Athanassoulis and Belibassakis (1999). The reduced two-field, weak problem is solved by means of the Finite Element Method (FEM). Finally, a series of comparisons are carried out against published results for a range of configurations

A non-linear BEM-FEM coupled scheme for the performance of flexible flapping-foil thrusters

Recent studies indicate that nature-inspired thrusters based on flexible oscillating foils show enhanced propulsive performance. However, understanding the underlying physics of the fluid-structure interaction (FSI) is essential to improve the eciency of existing devices and pave the way for novel energy-ecient marine thrusters. In the present work, we investigate the eect of chord-wise flexibility on the propulsive performance of flapping-foil thrusters. For this purpose, a numerical method has been developed to simulate the time-dependent structural response of the flexible foil that undergoes prescribed large general motions. The fluid flow model is based on potential theory, whereas the elastic response of the foil is approximated by means of the classical Kirchhoff-Love theory for thin plates under cylindrical bending. The fully coupled FSI problem is treated numerically with a non-linear BEM-FEM scheme. The validity of the proposed scheme is established through comparisons against existing works. The performance of the flapping-foil thrusters over a range of design parameters, including flexural rigidity, Strouhal number, heaving and pitching amplitudes is also studied. The results show a propulsive eciency enhancement of up to 6% for such systems with moderate loss in thrust, compared to rigid foils. Finally, the present model after enhancement could serve as a useful tool in the design, assessment and control of flexible biomimetic flapping-foil thrusters

Higher-order fem for nonlinear hydroelastic analysis of a floating elastic strip in shallow-water conditions

The hydroelastic response of a thin, nonlinear, elastic strip floating in shalow-water environment is studied by means of a special higher order finite element scheme. Considering non-negligible stress variation in lateral direction, the nonlinear beam model, developed by Gao, is used for the simulation of large flexural displacement. Full hydroelastic coupling between the floating strip and incident waves is assumed. The derived set of equations is intended to serve as a simplified model for tsunami impact on Very Large Floating Structures (VLFS) or ice floes. The proposed finite element method incorporates Hermite polynomials of fifth degree for the approximation of the beam deflection/upper surface elevation in the hydroelastic coupling region and 5-node Lagrange finite elements for the simulation of the velocity potential in the water region. The resulting second order ordinary differential equation system is converted into a first order one and integrated with respect to time with the Crank-Nicolson method. Two distinct cases of long wave forcing, namely an elevation pulse and an N-wave pulse, are considered. Comparisons against the respective results of the standard, linear Euler-Bernoulli floating beam model are performed and the effect of large displacement in the beam response is studied

Propagation of acoustic-gravity waves in inhomogeneous ocean environment based on modal expansions and HP-FEM

A coupled mode model is presented for the propagation of acoustic-gravity waves in layered ocean waveguides. The analysis extends previous work for acoustic waves in inhomogeneous environment. The coupled mode system is derived by means of a variational principle in conjunction with local mode series expansion, obtained by utilizing eigenfunction systems defined in the vertical section. These are obtained through the solution of vertical eigenvalue problems formulated along the waveguide. A crucial factor is the inclusion of additional modes accounting for the effects of spatialy varying boundaries and interfaces. This enhancement provides an implicit summation for the slowly convergent part of the localmode series, rendering the series rapidly convergent, increasing substantialy the efficiency of the method. Particular aspects of the method include high order Lagrange Finite Element Methods for the solution of local vertical eigenvalue problems in the case of multilayered waveguides, and Gauss-type quadrature for the computation of the coupled-mode system coefficients. The above aspects make the present method quite efficient for long range propagation in extended waveguides, such as the ones found in geophysical applications, e.g. ocean basins, as only few modes are needed for the accurate representation of the wave field