J.S. Hoving
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1
A novel boundary formulation is presented by applying the Boundary Element Method (BEM) to a dynamically loaded medium modelled as a discrete system. The two-dimensional medium is divided into a nonlinear discrete lattice in the near field, and a corresponding linear viscoelastic far field. The resulting boundary formulation is derived from the dynamic reciprocal work theorem and describes the far-field response through a Laplace domain force–displacement relation. The involved dynamic compliance matrix is composed of newly derived expressions for the Green’s functions of a viscoelastic half-plane of particles. It is demonstrated that the presented method yields a perfectly non-reflective boundary in the Laplace domain, without the need for artificial absorbing boundaries. Additionally, this contribution shows the successful time-domain application of the boundary method to a medium that exhibits non-smooth behaviour in the vicinity of a load source. In the time domain, the boundary equations are obtained by numerical application of the inverse Laplace transform, and the non-reflectiveness of the boundary is sensitive to the size of the time step. The presented method provides a consistent boundary approach for discrete lattices, and provides an alternative to continuum-based boundary methods for the dynamic response of solid media.
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A novel boundary formulation is presented by applying the Boundary Element Method (BEM) to a dynamically loaded medium modelled as a discrete system. The two-dimensional medium is divided into a nonlinear discrete lattice in the near field, and a corresponding linear viscoelastic far field. The resulting boundary formulation is derived from the dynamic reciprocal work theorem and describes the far-field response through a Laplace domain force–displacement relation. The involved dynamic compliance matrix is composed of newly derived expressions for the Green’s functions of a viscoelastic half-plane of particles. It is demonstrated that the presented method yields a perfectly non-reflective boundary in the Laplace domain, without the need for artificial absorbing boundaries. Additionally, this contribution shows the successful time-domain application of the boundary method to a medium that exhibits non-smooth behaviour in the vicinity of a load source. In the time domain, the boundary equations are obtained by numerical application of the inverse Laplace transform, and the non-reflectiveness of the boundary is sensitive to the size of the time step. The presented method provides a consistent boundary approach for discrete lattices, and provides an alternative to continuum-based boundary methods for the dynamic response of solid media.
Boundary formulations for discrete lattices
To describe the non-smooth dynamic response of solid media in the time domain
This thesis presents a modelling framework to describe the non-smooth dynamic response of solid media in the time domain under dynamic loading conditions, such as those typically found in soil-structure and ice-structure interaction. The approach divides the medium into two domains: a near-field domain capable of capturing nonlinear phenomena, and a surrounding linear far-field domain. The near field is modelled as a discrete lattice incorporating rheological elements, such as springs, dashpots, and dry-friction elements, for example enabling stick-slip behaviour.
The far-field domain is incorporated through a boundary integral formulation that accounts for the domain's properties solely at the interface with the near field, allowing for accurate wave transmission and minimal reflections at the boundary. Boundary integral equations (BIEs) are derived for both continuous and discrete representations of the far field, including one of the first derivations of BIEs for finite or semi-infinite discrete particle systems.
To address the computational challenges of time-domain simulations involving nonlinearities, a novel mixed time-frequency domain (MTFD) method is introduced. This non-iterative hybrid approach combines the efficiency of frequency-domain methods during periods of linear behaviour while accounting for the changing properties of the lattice over time whenever a nonlinear event occurs.
Results demonstrate the effectiveness of lattice models and discrete-based BIEs in capturing non-smooth dynamics, while highlighting the importance of robust numerical implementation. The proposed framework offers a promising tool for simulating wave propagation in nonlinear media and supports improved analysis and design in civil, geotechnical, and offshore engineering applications.
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The far-field domain is incorporated through a boundary integral formulation that accounts for the domain's properties solely at the interface with the near field, allowing for accurate wave transmission and minimal reflections at the boundary. Boundary integral equations (BIEs) are derived for both continuous and discrete representations of the far field, including one of the first derivations of BIEs for finite or semi-infinite discrete particle systems.
To address the computational challenges of time-domain simulations involving nonlinearities, a novel mixed time-frequency domain (MTFD) method is introduced. This non-iterative hybrid approach combines the efficiency of frequency-domain methods during periods of linear behaviour while accounting for the changing properties of the lattice over time whenever a nonlinear event occurs.
Results demonstrate the effectiveness of lattice models and discrete-based BIEs in capturing non-smooth dynamics, while highlighting the importance of robust numerical implementation. The proposed framework offers a promising tool for simulating wave propagation in nonlinear media and supports improved analysis and design in civil, geotechnical, and offshore engineering applications.
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This thesis presents a modelling framework to describe the non-smooth dynamic response of solid media in the time domain under dynamic loading conditions, such as those typically found in soil-structure and ice-structure interaction. The approach divides the medium into two domains: a near-field domain capable of capturing nonlinear phenomena, and a surrounding linear far-field domain. The near field is modelled as a discrete lattice incorporating rheological elements, such as springs, dashpots, and dry-friction elements, for example enabling stick-slip behaviour.
The far-field domain is incorporated through a boundary integral formulation that accounts for the domain's properties solely at the interface with the near field, allowing for accurate wave transmission and minimal reflections at the boundary. Boundary integral equations (BIEs) are derived for both continuous and discrete representations of the far field, including one of the first derivations of BIEs for finite or semi-infinite discrete particle systems.
To address the computational challenges of time-domain simulations involving nonlinearities, a novel mixed time-frequency domain (MTFD) method is introduced. This non-iterative hybrid approach combines the efficiency of frequency-domain methods during periods of linear behaviour while accounting for the changing properties of the lattice over time whenever a nonlinear event occurs.
Results demonstrate the effectiveness of lattice models and discrete-based BIEs in capturing non-smooth dynamics, while highlighting the importance of robust numerical implementation. The proposed framework offers a promising tool for simulating wave propagation in nonlinear media and supports improved analysis and design in civil, geotechnical, and offshore engineering applications.
The far-field domain is incorporated through a boundary integral formulation that accounts for the domain's properties solely at the interface with the near field, allowing for accurate wave transmission and minimal reflections at the boundary. Boundary integral equations (BIEs) are derived for both continuous and discrete representations of the far field, including one of the first derivations of BIEs for finite or semi-infinite discrete particle systems.
To address the computational challenges of time-domain simulations involving nonlinearities, a novel mixed time-frequency domain (MTFD) method is introduced. This non-iterative hybrid approach combines the efficiency of frequency-domain methods during periods of linear behaviour while accounting for the changing properties of the lattice over time whenever a nonlinear event occurs.
Results demonstrate the effectiveness of lattice models and discrete-based BIEs in capturing non-smooth dynamics, while highlighting the importance of robust numerical implementation. The proposed framework offers a promising tool for simulating wave propagation in nonlinear media and supports improved analysis and design in civil, geotechnical, and offshore engineering applications.
Concrete structures in ice prone environments experience abrasion due to ice-structure interaction, where the abrasion is attributed to friction. The sliding friction between concrete and ice is usually described as Coulomb or dry friction and although the physics of dry friction are believed to be well understood, the estimation of the static and kinetic friction coefficients for ice-concrete interaction remains a challenge. Data available in literature is ambiguous and the dependency of friction coefficients on parameters such as normal pressure and velocity is not clear. To contribute to the existing knowledge about friction coefficients and further investigate the influential factors, ice-concrete friction coefficients were estimated experimentally. With a specially designed set-up, stick-slip tests were performed by placing a cylindrical fresh-water ice sample, connected to a fixed structure by springs on either side, on a rotating slab of low-grade, smooth concrete. During the experiment, normal load (0.7-2.0 kg), spring stiffness (80-273 N/m) and concrete velocity (0.15-0.50 m/s) were varied. Following Coulomb’s laws of friction, the static and kinetic friction coefficients were obtained for the given parameters through displacement measurements. In addition, a simplified numerical stick-slip model was developed and validated based on the experimental data. In this paper, the dependency of the static and kinetic ice-concrete friction coefficients on normal load, spring stiffness and relative velocity is discussed based on test results and a comparison between the experiment and the numerical model is made to further identify the frictional behavior between concrete and ice.
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Concrete structures in ice prone environments experience abrasion due to ice-structure interaction, where the abrasion is attributed to friction. The sliding friction between concrete and ice is usually described as Coulomb or dry friction and although the physics of dry friction are believed to be well understood, the estimation of the static and kinetic friction coefficients for ice-concrete interaction remains a challenge. Data available in literature is ambiguous and the dependency of friction coefficients on parameters such as normal pressure and velocity is not clear. To contribute to the existing knowledge about friction coefficients and further investigate the influential factors, ice-concrete friction coefficients were estimated experimentally. With a specially designed set-up, stick-slip tests were performed by placing a cylindrical fresh-water ice sample, connected to a fixed structure by springs on either side, on a rotating slab of low-grade, smooth concrete. During the experiment, normal load (0.7-2.0 kg), spring stiffness (80-273 N/m) and concrete velocity (0.15-0.50 m/s) were varied. Following Coulomb’s laws of friction, the static and kinetic friction coefficients were obtained for the given parameters through displacement measurements. In addition, a simplified numerical stick-slip model was developed and validated based on the experimental data. In this paper, the dependency of the static and kinetic ice-concrete friction coefficients on normal load, spring stiffness and relative velocity is discussed based on test results and a comparison between the experiment and the numerical model is made to further identify the frictional behavior between concrete and ice.
Ships at sea will encounter deck wetness events that usually are just annoying and rarely lead to damage. Discomfort is experienced by spray, as it is limiting view and hampering deck work. In freezing conditions, deck wetting will lead to icing. This study is a continuation of publications regarding the development of a marine icing model based on spray predictions, taking into account ship shape and the physics of wave run-up leading to spray jets. The modelling of jet development from wave run-up against a wall, representing a ship hull, is investigated by experiments and mathematical modelling using nonlinear wave theory. Run-up jets occur frequently for ships at sea and are responsible for most of the generated spray. Detailed measurements of the run-up jet were obtained from high-speed video registration yielding information on the droplet distribution. The measurements indicate that the thickness of the run-up jet above the wall determines the maximum droplet size, and that this size is significantly larger than generally assumed in icing models. Based on these insights, new computations are performed with the ‘SHIPICE’ marine icing model to demonstrate the applicability and accuracy of the approach.
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Ships at sea will encounter deck wetness events that usually are just annoying and rarely lead to damage. Discomfort is experienced by spray, as it is limiting view and hampering deck work. In freezing conditions, deck wetting will lead to icing. This study is a continuation of publications regarding the development of a marine icing model based on spray predictions, taking into account ship shape and the physics of wave run-up leading to spray jets. The modelling of jet development from wave run-up against a wall, representing a ship hull, is investigated by experiments and mathematical modelling using nonlinear wave theory. Run-up jets occur frequently for ships at sea and are responsible for most of the generated spray. Detailed measurements of the run-up jet were obtained from high-speed video registration yielding information on the droplet distribution. The measurements indicate that the thickness of the run-up jet above the wall determines the maximum droplet size, and that this size is significantly larger than generally assumed in icing models. Based on these insights, new computations are performed with the ‘SHIPICE’ marine icing model to demonstrate the applicability and accuracy of the approach.
The implications of installing and operating an Ocean Thermal Energy Conversion (OTEC) plant on the large temporal and spatial scales of the oceanographic features and the interaction between them are yet unknown. The aim of this research is to describe the natural patterns and variability of the ocean currents around Curaçao, an island in the Caribbean Sea and a potential OTEC location, to be able to assess possible risks and opportunities for the OTEC industry. Ten years of data from the Mercator Ocean Model with a spatial resolution of 1/12 ̊ and temporal resolution of one day was analyzed. In the Caribbean Sea, global ocean gyres and wind patterns force the water from the Lesser Antilles in the east towards the Yucatan Channel in the west in a fast-current jet, with a peak from December to March and a trough in October and November, related to wind stress. Largest surface velocities in the order of 1 m/s are found along the coast of Venezuela, where upwelling enhances surface flow to the west but subsurface countercurrents to the east. The period from April to September is characterized by meandering of the jet and the formation of large (diameter > 200 km) anticyclonic eddies that cause large local surface velocities and contribute to the great variability in the Caribbean Sea. Due to upwelling, no OTEC system should be deployed more than 50 km south of Curaçao to avoid cold surface water decreasing the system’s performance. Hydrodynamic forces and moments on the cold-water pipe due to calculated expected maximum velocities do not cause exceeding of the yield stress.
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The implications of installing and operating an Ocean Thermal Energy Conversion (OTEC) plant on the large temporal and spatial scales of the oceanographic features and the interaction between them are yet unknown. The aim of this research is to describe the natural patterns and variability of the ocean currents around Curaçao, an island in the Caribbean Sea and a potential OTEC location, to be able to assess possible risks and opportunities for the OTEC industry. Ten years of data from the Mercator Ocean Model with a spatial resolution of 1/12 ̊ and temporal resolution of one day was analyzed. In the Caribbean Sea, global ocean gyres and wind patterns force the water from the Lesser Antilles in the east towards the Yucatan Channel in the west in a fast-current jet, with a peak from December to March and a trough in October and November, related to wind stress. Largest surface velocities in the order of 1 m/s are found along the coast of Venezuela, where upwelling enhances surface flow to the west but subsurface countercurrents to the east. The period from April to September is characterized by meandering of the jet and the formation of large (diameter > 200 km) anticyclonic eddies that cause large local surface velocities and contribute to the great variability in the Caribbean Sea. Due to upwelling, no OTEC system should be deployed more than 50 km south of Curaçao to avoid cold surface water decreasing the system’s performance. Hydrodynamic forces and moments on the cold-water pipe due to calculated expected maximum velocities do not cause exceeding of the yield stress.
This contribution considers the wave reflection at the interface between a discrete lattice that describes the nonlinear near-field response to a dynamic load and a matching system that describes the linear far-field response. The near-field system is modelled as a one-dimensional discrete system of particles that is capable of describing nonlinear behaviour by including dry friction. The purpose of coupling this nonlinear near-field system to a linear far-field system is to optimize for computational efficiency as, in the time domain, the linear far-field system can be replaced by a single integral force-displacement relation. In the frequency domain, this relation is commonly known as the dynamic stiffness, or inversely, as the dynamic compliance. For an ideal coupling between the two systems an incident wave should propagate through the interface undisturbed and without reflections. For long incident waves, it would suffice to describe the far-field system by a classical continuum, however for short incident waves, the reflections of the classical continuum are significant. Therefore, the linear far-field system is described as a semi-infinite discrete particle system, also known as a semi-infinite cascade, that matches the discrete nature of the near-field lattice. To assess the quality of the applied coupling, we consider the reflection of an incident wave at the lattice-cascade interface, using both time and frequency domain approaches, and compare it to the reflection at the interface of a system where the far-field system is described by a semi-infinite continuum.
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This contribution considers the wave reflection at the interface between a discrete lattice that describes the nonlinear near-field response to a dynamic load and a matching system that describes the linear far-field response. The near-field system is modelled as a one-dimensional discrete system of particles that is capable of describing nonlinear behaviour by including dry friction. The purpose of coupling this nonlinear near-field system to a linear far-field system is to optimize for computational efficiency as, in the time domain, the linear far-field system can be replaced by a single integral force-displacement relation. In the frequency domain, this relation is commonly known as the dynamic stiffness, or inversely, as the dynamic compliance. For an ideal coupling between the two systems an incident wave should propagate through the interface undisturbed and without reflections. For long incident waves, it would suffice to describe the far-field system by a classical continuum, however for short incident waves, the reflections of the classical continuum are significant. Therefore, the linear far-field system is described as a semi-infinite discrete particle system, also known as a semi-infinite cascade, that matches the discrete nature of the near-field lattice. To assess the quality of the applied coupling, we consider the reflection of an incident wave at the lattice-cascade interface, using both time and frequency domain approaches, and compare it to the reflection at the interface of a system where the far-field system is described by a semi-infinite continuum.
Not only in the Arctic, but also well outside the Arctic, icebergs can be a danger to buried offshore pipelines due to seabed gouging. As it is not economically feasible to bury these pipelines at depths where soil deformations are small, these pipelines are buried at sub-gouge depths where the soil experiences significant plastic deformations. Before considering the sub-gouge soil deformations that are vital to determine optimal pipeline burial depths, a better understanding of dynamic iceberg-soil interaction during seabed gouging is required. In the past, several models have been developed to describe the interaction forces between an iceberg and the soil in front of the iceberg keel based on static soil failure theory or from experimental observations. These models do not fully account for the dynamic behaviour of the iceberg, nor do they consider the corresponding response of the sub-gouge soil. In fact, how the extent of sub-gouge deformations depends on the geotechnical parameters of the seabed is not at all well understood. Therefore, a three-dimensional physics-based model is being developed that describes the transient iceberg-soil interaction (i) based on plasticity theory, (ii) accounting for the dynamic response of the iceberg during gouging and (iii) including the hydrodynamics of the iceberg. This paper presents a two-dimensional version of the iceberg-soil interaction model assuming the iceberg keel to be rigid and plane-strain conditions for the soil. Model predictions are presented and compared for different seabed characteristics.
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Not only in the Arctic, but also well outside the Arctic, icebergs can be a danger to buried offshore pipelines due to seabed gouging. As it is not economically feasible to bury these pipelines at depths where soil deformations are small, these pipelines are buried at sub-gouge depths where the soil experiences significant plastic deformations. Before considering the sub-gouge soil deformations that are vital to determine optimal pipeline burial depths, a better understanding of dynamic iceberg-soil interaction during seabed gouging is required. In the past, several models have been developed to describe the interaction forces between an iceberg and the soil in front of the iceberg keel based on static soil failure theory or from experimental observations. These models do not fully account for the dynamic behaviour of the iceberg, nor do they consider the corresponding response of the sub-gouge soil. In fact, how the extent of sub-gouge deformations depends on the geotechnical parameters of the seabed is not at all well understood. Therefore, a three-dimensional physics-based model is being developed that describes the transient iceberg-soil interaction (i) based on plasticity theory, (ii) accounting for the dynamic response of the iceberg during gouging and (iii) including the hydrodynamics of the iceberg. This paper presents a two-dimensional version of the iceberg-soil interaction model assuming the iceberg keel to be rigid and plane-strain conditions for the soil. Model predictions are presented and compared for different seabed characteristics.
In preparation to the SALTO JIP (Safe Arctic Logistics, Transport & Operations) work was done towards developing an improved model for icing due to sea spray at the bow of a ship. The so-called SHIPICE model may be used in a probabilistic risk-based approach and consists of two main segments:
1. Probabilistic modeling of spray water volumes and the break-up in droplets rising above the freeboard, and
2. Modeling the ballistic and thermodynamic processes of these droplets when falling and freezing to the ship.
The present paper deals with the second phase of the SHIPICE model. The modeling of marine spray shows that spray is a phenomenon with sharply increasing volumes with speed and sea state. Furthermore, the droplet size of spray rising above the freeboard has a distinctive effect on the distribution of the spray volume landing on deck. The thermodynamic freezing model takes into account intermittent wetting and run-off, and depends also on the droplet size distribution due to in-flight cooling of the droplets when falling to the ship. The results from the model seem realistic when computed icing growth is compared with two measured icing events reported in literature. The development of SHIPICE will allow users to dimension pre-cautions to the area and time of operation. Additionally, it will provide an improved basis for regulations and guidelines (e.g. ISO TC 67 SC8). For ship owners, it will help evaluate the susceptibility of their ship to marine spray icing.
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In preparation to the SALTO JIP (Safe Arctic Logistics, Transport & Operations) work was done towards developing an improved model for icing due to sea spray at the bow of a ship. The so-called SHIPICE model may be used in a probabilistic risk-based approach and consists of two main segments:
1. Probabilistic modeling of spray water volumes and the break-up in droplets rising above the freeboard, and
2. Modeling the ballistic and thermodynamic processes of these droplets when falling and freezing to the ship.
The present paper deals with the second phase of the SHIPICE model. The modeling of marine spray shows that spray is a phenomenon with sharply increasing volumes with speed and sea state. Furthermore, the droplet size of spray rising above the freeboard has a distinctive effect on the distribution of the spray volume landing on deck. The thermodynamic freezing model takes into account intermittent wetting and run-off, and depends also on the droplet size distribution due to in-flight cooling of the droplets when falling to the ship. The results from the model seem realistic when computed icing growth is compared with two measured icing events reported in literature. The development of SHIPICE will allow users to dimension pre-cautions to the area and time of operation. Additionally, it will provide an improved basis for regulations and guidelines (e.g. ISO TC 67 SC8). For ship owners, it will help evaluate the susceptibility of their ship to marine spray icing.