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Timetable (SIPW03)

Ice-structure interaction

Monday 6th November 2017 to Friday 10th November 2017

Monday 6th November 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (INI Director)
10:00 to 11:00 Erland Schulson (Dartmouth College)
Friction of Sea Ice
Friction of sea ice plays a fundamental role in a variety of geophysical and engineering scenarios. Examples include ridging and rafting, ice-induced loads on offshore structures and the physics of brittle compressive failure. In this paper attention is focused on the characteristics of static friction and frictional sliding of ice upon ice at velocities of around 0.1 m s-1 (~10 km/day) and lower, at homologous temperatures greater than around Th =0.85 (>-40 o C). The coefficients of both static and kinetic friction are described and then discussed in terms of the underlying physical processes. At root is creep deformation of asperities. Creep leads to an increase in contact area and thus to an increase in the coefficient of static friction with increasing time under load. Creep leads also to an increase in the coefficient of kinetic friction with increasing velocity at lower speeds. At higher speeds, localized melting of asperities sets in, via frictional heating, and this leads to a decrease in the kinetic coefficient with increasing velocity. Modeling indicates that the velocity that marks the transition scales as $V_t \propto a \Delta T^2$ where $a$ denotes asperity size and $\Delta T$, the difference in temperature between the melting point and the body of ice. Implications for sea ice mechanics are discussed.

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Paul Verlaan
Ice-structure interaction in the Sakhalin-II (Sea of Okhotsk) and Kashagan (NE Caspian) project
In this presentation, it will be described which approach was taken to specify global and local ice loads for the structures in two different oil & gas projects in sub-Artic areas: the Sakhalin II project in the Sea of Okhotsk and the Kashagan project in NE Caspian Sea. In both projects, the approach was a combination of deterministic methods described in the different design codes, probabilistic methods and scale tests in an ice tank.  For the Sakhalin II project, global ice loads were particularly relevant for the design of the two multi-legged platforms (Piltun-B and Lunskoye-A). In contrast, in the NE Caspian, an accurate determination of the global ice loads was mainly relevant for design of the ice protection barriers rather than for design of the islands itself. Instead, much more attention was paid to specifying the risk of ice encroachment onto the islands.  In all cases, it was found that the limited amount of ice data often resulted in a conservative design with still considerable uncertainties in the design loads.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 David Cole
Structure-property relationships for sea ice: Modeling and experimental validation
This talk addresses the constitutive behavior of sea ice, with a focus on the relationships between measurable physical properties and the elastic, anelastic and viscous components of strain. To accommodate attendees with a limited knowledge of sea ice, the presentation includes a brief overview of the microstructure and flaw structure of naturally occurring sea ice. Some attention is paid to the structure and mechanics of columnar and granular freshwater ice for completeness. The components of strain are quantified in terms of crystallographic characteristics (primarily c-axis orientation), dislocation density for the inelastic components, and temperature. The mechanisms of anelastic strain (e.g., time-dependent but recoverable) are associated with basal dislocation glide and grain boundary sliding. Viscous straining is quantified in terms of drag-controlled dislocation glide on the basal planes. It is shown that dislocation density exerts an overwhelming influence on the constitutive behavior of sea ice both at the scale of laboratory experiments (0.1 m) and in-situ experiments ( ≤ 30 m). Recent efforts to account for certain high temperature effects and differences between in-situ vs. in-vitro constitutive behavior of sea ice are described and the associated modifications to the published constitutive model are discussed. An analysis of existing cyclic loading and creep experiments makes it possible to identify the physical basis for the apparent increase in activation energy of inelastic behavior with proximity to the melting point. Additionally, brine drainage from specimens harvested for laboratory experiments is shown to cause a major discrepancy between the in-situ elastic response of warm sea ice vs. that found in laboratory experiments. 
14:30 to 15:30 Andrei Metrikine ; Hayo Hendrikse
Ice-induced vibrations of offshore structures: physics of the process, modelling and remaining challenges
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Aleksey Marchenko (University Centre in Svalbard (UNIS))
Thermo-mechanical loads of sea ice on structures
Thermally induced loads of ice on structures and shorelines occur due to thermal deformations of confined ice. Thermally induced stresses in the ice follow the temperature changes, depend on the coefficient of thermal expansion of ice and are reduced due to the relaxation. Temperature changes in sea ice occur due to conductive heat transfer characterized by specific heat capacity and thermal conductivity of ice and due to the advection of liquid brine depending on sea ice permeability. Coefficient of sea ice thermal expansion depends on the amount of liquid brine trapped in closed packets inside the ice. Proportion between the amounts of liquid brines trapped in closed packets and existing in permeable channels depends on the ice temperature and salinity. Relaxation and creep rheology of sea ice also depends on the temperature and ice structure.

Thermo-mechanical model of saline ice taking into account above described properties was recently formulated by Marchenko and Lishman (2017). The dependence of the coefficient of thermal expansion of saline ice from the temperature was reconstructed from the laboratory experiments. In the present work the model equations are used to estimate and compare the heat fluxes and thermal deformation of sea ice caused by the heat conduction and brine advection. Further the model is used for numerical simulations of ice loads on the cofferdam of coal quay in Kapp Amsterdam, Spitsbergen.

Field observations and records of the loads from sea ice confined inside the cofferdam were performed since 2013 (Marchenko et al, 2013; Wrangborg et al, 2015). Sea ice temperature was measured synchronously with the loads over entire ice thickness using thermistor string frozen into the ice. It was discovered that sea water brine migrates through the confined ice with thickness of 2-3 m under the influence of the water overpressure below the ice caused by semidiurnal tide. Horizontal ice loads on the cofferdam walls are also changed according to the semidiurnal cycle. Thermo-mechanical model of saline ice is used in numerical simulations of the observed phenomena by finite element method realized in Comsol Multiphysics software.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 7th November 2017
09:00 to 10:00 Kaj Riska (Total E&P UK Limited); (NTNU)
Theoretical modelling of ship-ice interaction
Ice action on ships occurs when a ship collides with an ice feature – be that level ice, ice floes or iceberg s. The total force acting on the ship depends on the motion of the ice feature and ship. The motion includes rigid body motions (in 6 DoF for ship and ice), elastic deformations of ship and ice and irreversible deformations (ice crushing and ship hull plastic deformation). In principle all these should be taken into account in modeling the collision and the total force. Several simplifications have been made and the lecture describes some of the modeling that has been carried out in order to determine the total collision force. An interaction exists also for local ice loading i.e. ice pressure. In this case the local deformation of the ship hull interacts with the deformation of ice. One example of this is given also in the lecture.
10:00 to 11:00 Claude Daley
Design and Assessment Methods for Ships in Ice
Design and assessment of ships in ice is a topic that concerns a wide range of people, from owners, to designers, builders, regulators and the insurance industry, with the public, the environment and the economy being impacted by arctic shipping risks. The methods that are being used to design and evaluate ice class ships have evolved over many decades and are currently continuing to change. In this discussion, the focus is on the steel structure of ice-going ships, including structurally important ice loads and the nature of the structural response. 

As we seek to better understand the nature of the interaction between ships and ice, we are increasingly focusing on complex multi-body interactions and non-linear behaviours. In such situations, general solutions are not only not currently available, but may not be achievable at all. While we can assemble specific solutions, we must ask whether a general understanding of complex nonlinear systems is possible, and if so, what mathematics can we use to develop them?

Concern for structural ice loads has, until now, mainly focused on a single situation; the “design condition”. For that condition, the structure behaves linearly or pseudo-linearly. The “design” loads have been largely developed empirically, from a combination of measurement data and loads inferred from past successful practice.

While much of the current industrial and regulatory practice still employs relatively simple models of load and response, new sets of tools and approaches are taking shape and are being applied by the most sophisticated ship owners, builders and operators. These new approaches seek to directly model a growing set of complex issues and scenarios, with consequent improvements in the fidelity of ice loads and structural response. The growing computational power and improving simulation tools are enabling these developments. To illustrate these approaches, some of the author’s own efforts will be described.

One recent and developing technique is called safe speed analysis. This method involves modeling a wide variety of discrete ship-ice interaction cases, modelling the load and overload capacity of the vessel and then combining these results to produce a multi-parameter map of acceptable operations (speed, ice size, etc). The method makes use of available tools and solutions, including “Popov” type collision models (algebraic solutions of two-body collisions), and explicit dynamic finite element models (LS-Dyna) to capture interaction kinematics, contact and a full range of structural responses (ductility, dynamics, stability).

A second, but related technique is a model called GEM, which uses simple event solutions and simple equations of motion to model large scale operations of vessels in ice. GEM, while seeking a reasonable level of accuracy, focuses on high simulation speeds and practical decision support rather than on fidelity and universality.

After presenting these two methods, the presentation raises three mathematical challenges. The first challenge relates to the problem of interacting chaotic systems and wonders whether a calculus for such systems is possible. The second challenge questions the application of probabilistic design to ice class ships. The third challenge relates to the simulation of multi-body systems. Such systems are highly nonlinear and computationally costly to model. Could an asynchronous or quasi-synchronous timestep algorithm yield improvements in overall speed?
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Robert Bridges
Influence of broken ice on marine operations – identification of important processes
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Alexander Korobkin
Hydroelastic waves and their interaction with structures
Co-author: S. Malenica (BV), T. Khabakhpasheva (Lavrentyev Institute of Hydrodynamics)  

Linear problems of hydroelastic wave diffraction by structures with vertical walls are studied for a circular cylinder frozen in ice cover of constant thickness and infinite extent. The water depth is constant. The ice plate is modelled by a thin elastic plate clamped to the surface of the cylinder. The cylinder is mounted at the sea bottom. One-dimensional incident hydroelastic wave of small amplitude propagates towards the cylinder and is diffracted on the cylinder.  Deflection of the ice plate and the bending stresses in it are determined by two methods: (a) using the integral Weber transform in radial direction, (b) using the vertical modes for the fluid of constant depth with the rigid bottom and elastic upper boundary. The solution by the second method is straightforward but we cannot prove that the solution is complete because the properties of the vertical modes are not known.  The solution by the Weber transform is more complicated but this solution is unique. We will show that these two solutions are identical. This result justifies the method of the vertical modes in the hydroelastic wave diffraction problems. For a circular cylinder the vertical-mode solution can be also justified by substitution. Different conditions at the contact line between the cylinder and the ice sheet are considered. The wave diffraction problem for broken ice is also considered. It is shown how the problem can be generalised to non-circular cylinders and interaction of several cylinders in ice.
14:30 to 15:30 Yevgeny Aksenov (National Oceanography Centre, Southampton)
Safer Operations in the Sea Ice-covered Oceans: The Tale of the Two Projects
INI Programme Mathematics of sea ice phenomena: SIPW03 (Ice-structures interactions) Safer operations in ice-covered oceans: The tale of the two projects Yevgeny Aksenov(1), Stefanie Rynders(1), Danny Feltham(2), Lucia Hosekova(2), Robert Marsh(3), Nicolaus Skrilis(3), Laurent Bertino(4) and Tim Williams(4) (1)National Oceanography Centre, UK (2)University of Reading, UK (3)University of Southampton, UK (4)Nansen Environmental and Remote Sensing Center (NERSC), Norway Contact: yka[at][dot]uk Increase in the Arctic offshore operations is driven easing of sea ice conditions and improving accessibility of the shipping routes. We present a review of the two projects, focus on the offshore and navigational safety in the high-latitude oceans. The projects develop analysis and forecasting technologies to provide key information for the maritime operations and marine information services. The projects bring together physical oceanography and the mathematics of fluid structure interaction and address the likely extreme loads on a selection of structures and ships, in a wide range of offshore environments, including pack ice areas and Marginal Ice Zones. The innovative capability develop in the project helps the integrated hindcasts and forecasts of ocean currents, tides and waves. We detail the data requirements and discuss existing and emerging datasets on ice thickness, ice drift and ice fragmentation and data on waves, tides, currents and icebergs). The study makes the assessment of the current and future states of the Arctic shipping, including, trans-Arctic navigation, regional Arctic shipping of commodities and transportation of Arctic natural resources. The approach uses sea ice drift, concentration and thickness fields from high-resolution model future projections in conjunction with the climate scenarios. For the study we acknowledge support from the EU FP7 Project ‘Ships and waves reaching Polar Regions (SWARP)’ and from the NERC UK Innovation Grant 'Safer Operations at Sea - Supported by Operational Simulations (SOS-SOS)'.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Kevin Maki
Nonlinear numerical modeling of impact loads of ship sections and floating ice
Wednesday 8th November 2017
09:00 to 10:00 Andrew Palmer
How ought we to structure research so as to make progress in understanding ice interaction?
10:00 to 11:00 Kenneth Johannessen Eik
Effect of uncertainties in ice loading on design of Arctic installations
Co-author: Pavel Liferov (Statoil)

Statoil is operator in field developments offshore East Coast of Canada and in the Norwegian Barents Sea in waters were sea ice intrudes seldom (CA) or extremely rarely (NO). Both developments have in common that FPSOs will operate in open water but still need to consider ice loads as part of design. Norwegian laws require that the knowledge base is included as part of the risk definition. The consequence of this for Arctic Norwegian projects is that uncertainties in design basis shall be compensated with more robust solutions in order to keep risk at the same level as in non-arctic locations. Assessments done for the Johan Castberg project are presented with focus on possible consequences from using immature ice load models. The need for including ice exposure in load assessments and to relate loads to exceedance probability levels is highlighted.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Devinder Sodhi
Correlation of local ice forces across the width of a structure during ice-structure interaction
The concepts presented in this paper pertain to an ice floe moving against a wide structure, having an aspect ratio greater than 10, and are based on results of small-scale and medium-scale indentation tests as well as published data of measured ice forces on full-scale structures.

Data from indentation tests with tactile sensors at the ice-structure interface reveal that: (a) interactions at low ice speed produce ductile deformation of ice, creating a slowly expanding contact area until it covers 100% of the nominal contact area, and (b) interactions at intermediate and high ice speeds produce brittle failure of ice, resulting in the actual contact area to be less than the nominal contact area.

During interactions at intermediate and high speeds, brittle failure of ice creates uneven surface at the ice-structure interface, creating asperities at the ice front. When these asperities advance towards the structure, they generate contacts at a few isolated spots. During an ice-structure interaction, these contact areas expand under contact pressure with time, causing local ice forces to be generated at the ice-structure interface. The summation of all local forces at an instant in time is the global ice force.

The results of indentation tests with a segmented indentor and tactile sensor reveal a decrease in correlation of local ice forces across the width of a structure with increasing ice speed. This results in a decrease in variance of global effective pressure while the average global effective pressure remains in the range of local average effective pressure. These two effects are well supported by experimental results. The correlation of local ice forces is quantified by a correlation-length parameter, which decreases with increasing ice speed as per experimental data. These concepts of ice crushing process have been incorporated in a theoretical ice-structure-interaction model. 
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Clare College
Thursday 9th November 2017
09:00 to 10:00 Frank Thomas Smith (University College London)
Impacts, ice growth and related modelling
The presentation here by Frank Smith and Davide Bella is on interactions involved in impacts, ice formation and related phenomena. There are many such interactions. This mathematical-modelling research, motivated originally by the observed icing of airborne vehicles and the effects of storms, aims to address some of the following issues: first, distortion of water droplets in air flow, where two-way fluid interactions occur taking account of the small density and viscosity ratios; second, air-water interaction during the impact of a droplet onto a solid surface, including pre impact and post impact behaviour; third, ice growth soon after impact, where the interaction is between the ice shape and the water flow over it; fourth, related problems of dynamic fluid-body interaction concerning ice lumps impacting on a solid surface. More basic study of the ice growth within a single droplet on a solid surface will also be presented.
10:00 to 11:00 Manish Tiwari
Passive, nanoengineered anti-icing: An experimental perspective
In this presentation I will share results from some fundamental experiments focussed on investigating freezing of supercooled cold drops in contact with solid surfaces. A common theme running through the presentation will the need for precise surface nanotexturing and wettability control to control ice formation, ice adhesion and impalement resistance of surfaces subjected to impact by cold liquid objects. I will use experimental results to highlight a rational approach to design icephobic surfaces offering passive anti-icing property. Aspects of surface durability will be touched upon followed by a discussion on results from our preliminary work on impact of water drops on nanoengineered, iceohobic surfaces.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Discussion INI 1
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Rocky Taylor
Ice-induced vibrations in offshore structures: coupled dynamic ice-structure interactions over multiple scales
The design of fixed structures for operations in ice environments presents challenges, particularly in terms of determining appropriate levels of structural strengthening for extreme ice loads and accounting for potential ice-induced vibrations in design. The development of improved models of dynamic ice-structure interactions depends significantly on understanding the physics and mechanics of ice compressive failure to enable the effective modelling of ice loads and associated coupling with structural response. While the compressive failure of ice is highly complex, significant progress has been made in recent years in understanding and modelling salient aspects of ice failure during dynamic ice-structure interactions. In this lecture, emphasis is placed on processes associated with the formation and evolution of high-pressure zones (hpzs) and associated load-limiting mechanisms that occur during dynamic ice crushing failure. Recent advances in ice mechanics are discussed, a long with results of recent medium-scale laboratory tests focused on supporting the development of a probabilistic, multi-scale modelling framework as a basis for integrating advances in fundamental ice physics with full-scale ice loads on rigid and compliant structures. 
14:30 to 15:30 Dmitry Onishchenko
Probabilistic aspects of multi-element systems failure
The problem of the determination of ice loads (or actions in ISO terminology) on offshore structures is essentially more difficult, from the methodological point of view, as compared with other environmental factors, such as wind, waves, currents. An engineer needs much more elaborated probabilistic approach for the ice case to determine the design values of ice loads, which yearly exceedance levels are set usually as 0,01 and lower. Reasons for this are well known: first, floating ice is diverse in shape, ice feature morphology is complicated and a number of ice structure parameters are not available for direct measuring at that, and second, in most cases ice load is a straightforward result of ice failure, a process that due to its physical nature has a very high level of internal uncertainty. In fact, it is reasonable to treat ice failure as a structural failure of a multi-element system, and in conditions when the individual elements have a non-trivial properties. The lecture, first, presents some basic ideas on probability aspects of multi-element systems failure that is closely related to the general subject of the reliability of structures. Then, with the help of a number of simple probabilistic models, the cases of level ice, ridges and icebergs impacts on offshore structures are presented. Certain attention i s given to a discussion on the overall reliability of the procedure of design ice load determination, which is closely connected with the accuracy of the probability distribution functions describing ice parameters and the adequacy of load equations.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Kari Kolari
Modelling Brittle Failure of Ice
The modeling of physics and mechanics of compressive failure is of importance in the ice-structure interaction modelling. Horizontal splitting of ice (termed spalling), has been observed in several ice-structure interaction experiments when drifting ice is failed by crushing in the interaction with vertical structure.

One of the greatest challenges of material failure analysis is the modelling of brittle failure. Materials like natural ice and rock are heterogeneous and crystalline, containing pores and flaws and other weaknesses. When these materials are subjected to compressive loading in the brittle regime, they are known to fail by splitting along the loading direction. Formation, growth and interaction of (micro)cracks due to material inhomogeneities and external force are considered to be the mechanism of brittle failure.

In this presentation, I will review three approaches applied in the modelling of brittle failure of ice: 1) Breakable/cohesive bond model; 2) Continuum damage mechanics (CDM) model, starting from Helmholz free energy; 3) WC-CMD-model, where the damage evolution is based on the observed micro-mechanism termed sliding wing crack mechanism.

The WC-CMD-model predicts axial splitting failure mode of granular ice under uniaxial and biaxial compression, and tensile splitting under tension. It also links grain size and strength of granular ice: the model is able to predict Hall-Petch relationship between grain diameter and strength, both under compression and tension. In addition, the model predicts experimentally observed increase in compressive strength with decrease in temperature.

Related Links
Friday 10th November 2017
09:00 to 10:00 Mark Hopkins
Using DEM to model ice sheets and structures
10:00 to 11:00 Renate van Vliet
Determining global ice loads on offshore structures
To design columns, foundations or mooring systems for offshore structures in areas exposed to ice, accurate estimates of global ice loads on the structure are required. In this presentation a comparison is made between the current practice of design of structures in ice and design of structures in waves and how loading is determined. The use of numerical models could increase the accuracy of ice load and structural response predictions, leading to reduced costs and risk of failure. Current use, gaps and ongoing development of numerical models for ice load prediction on structures are discussed. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Jukka Tuhkuri ; Arttu Polojarvi
Discrete Element Simulation of Ice-Structure Interaction
Sea ice load on a marine structure is caused by the failure process of an ice feature against the structure. This failure process is affected by both, the structure and the ice, thus it is commonly called an ice-structure interaction process. Some ice failure processes comprise of large numbers of discrete failure events, which lead to formation of piles of ice blocks. Such failure process can be effectively studied by using the Discrete Element Method, DEM. As an example, when a continuous sea ice sheet is driven by the wind and current against a marine structure, the ice sheet gradually breaks into discrete blocks, which interact with each other and with the structure, and form a rubble pile against the structure. Our recent DEM simulations show that the force from the pushing ice sheet is transmitted to the structure through the rubble pile with chain-like groups of highly loaded ice blocks, force chains. The existence of force chains highlights the granular behavior of ic e rubble and calls for models that account for individual ice blocks. Other important problems where discrete ice blocks and their interactions have key roles, are the strength of sea ice ridges and rubble fields, and the interaction of ships and structures with ridges and rubble piles.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Agnieszka Herman (University of Gdansk)
DEM modelling of wave-induced floe-floe (and floe-structure) collisions
In the marginal ice zone (MIZ), waves entering from the open ocean often induce collisions between neighboring ice floes. Due to limited observational data, it is not known to what degree wave-induced collisions contribute to the stress within the ice and to the dissipation of wave energy in MIZ. In this paper, wave-induced collisions between ice floes are analyzed with a numerical, discrete-element model. The model simulates surge motion of an ensemble of ice floes on a prescribed wave field. It is shown that the collision pattern depends on ice concentration, wave steepness, floe size relative to wavelength, floe-size distribution, as well as the restitution coefficient and drag between the ocean and the ice. For relatively large ice floes, the results are very sensitive to the formulation of wave forcing and ice–water drag. If the forcing correctly takes into account floe size relative to the wavelength, the model accurately reproduces surge RAOs of single floes, as well as accelerations and forces of floes impacting a structure.
14:30 to 15:00 Izolda Sturova (Lavrentyev Institute of Hydrodynamics)
Generation of wave motion in fluid with inhomogeneous ice cover
The dynamic perturbations occurring in fluid and ice cover as a result of the action ofmechanical external force have been thoroughly studied in the linear treatment for aninfinitely extended homogeneous ice cover, which is modeled by a thin isotropicelastic plate floating on the surface of a fluid of constant depth. In reality, the icecover is not homogeneous, since it can cover not the entire upper boundary of a fluid,but only its part, and also there may be the cracks and the patches of ice-free water init. The effect of such complex boundary conditions on the wave motion is in the initialstage of the study.

We present a review of the solutions for several 2-D and 3-D problems. The timeharmonicproblem describing small oscillations (sway, heave and roll) of a horizontalcylinder is considered for four classes of hydroelastic system: i) a floating semiinfiniteelastic plate; ii) two semi-infinite elastic plates connected by the vertical andflexural rotational springs as a model of a partially frozen crack in an ice sheet; iii) afloating elastic platform of finite length; iv) two semi-infinite elastic plates separatedby a region of open water (polynya). The hydrodynamic load (added mass anddamping coefficients) and the amplitudes of vertical displacements of the free surfaceand elastic plates are calculated as functions of the cylinder oscillation frequency andthe location of the cylinder with respect to the plate edges.

The 3-D problem is considered for time-periodic external pressure and a load uniformly moving along the rectilinear edge of the ice cover. By analogy with the 2-D case, the first two classes of hydroelastic system are considered, as well as the effect of the solid vertical wall while the edge of the ice cover adjacent to the wall can be either clamped or free. In the problem of time-periodic external pressure, it is shown that for semi-infinite elastic plate contacting with free surface of water and for two non-identical semi-infinite plates, divided by crack, the directions of predominantwave propagation are allocated under some angle to the crack. In the case of two identical semi-infinite plates, divided by a crack, the edge wave guide mode is excited.The edge wave is most excited is the case of semi-infinite plate with a free edge near a solid vertical wall.

In the case of moving load, the 3-D pictures of generated waves in fluid and in ice cover are built for various speeds of motion. It is shown that in case of two identical semi-infinite plates, divided by a crack, two edge waveguide modes with different wave numbers exist for supercritical values of speed motion. For the vehicle moving with supercritical speed one mode is extending in front of moving load and other mode behind it. The wave forces acting on moving vehicle are investigated for various speeds of motion.
15:00 to 15:30 Vladislav Miryaha
Discontinuous Galerkin method for numerical simulation of ice flow impact on vertical cylinder offshore structure
Co-author - Prof. Igor B. Petrov

This talk describes an approach to numerical simulation of ice field impact on cylinder vertical offshore structure, as well as review of related complications.
Also it provides information on ice rheology, a continuous mechanics model used, which makes it possible to achieve a balance between accuracy and amount of computational resources needed. The description of the numerical method, and also some features of the simulation and techniques, which allow to overcome a number of difficulties associated with the resource-intensive calculations, are given. Typical destruction patterns of the ice fields and pressure distributions on structures are discussed.
15:30 to 16:00 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons