RESPONSE USING A GWM STRIP
APPROACH
February 26, 2015
Preface
The thesis in front of you has been written as the main deliverable of the graduation research for the title of master of science in maritime technology at the department of Ship Hydromechanics and Structures of the TU Delft under supervision of Miros law Kaminski and Pieter Maljaars. This research has been carried out as intern at Bureau Veritas (BV) both in Rotterdam under supervision of Martijn Nieuwenhuijs and at the research department at Neuilly-sur-Seine under supervision of ˇSime Malenica and J´erˆome de Lauzon.
52 Laplacian operator
ωe Excitation frequency
ωn Natural frequency ∂η
∂t Local time derivative of free surface ∂φ
∂n description ∂φ
∂t Local time derivative of potential Dφ
Dt Substantioal derivative of the potential
φ(x, y) Velocity potential of the fluid ρ Density per unit volume ξr Rigid body rotations
ξt Rigid body translations
d Distance
F Force
g(x) Sampling function K Stiffness matrix
m Mass
nsection Normal vector of slam strip
n Normal vector of the body surface
p Pressure
p0 Atmoshperical pressure
TEB Transformation matrix from earth to body
rb Point in body coordinates
rp Location of point p
td Period of pulse load
Tn Natural period
Trise Rise time
U Flow velocity
V Velocity of the body surface
wi0 Weighing function for discrete sampling points w(x) Weighing function
xGP Gauss point coordinates
xP oS Coordinates of pressure point on slam strip
x0i Discrete position x(t) Position
Abbreviations
b.c. Boundary condition BEM Boundary Element MethodCPU Central Processing Unit DAF Dynamic Amplification Factor FEA Finite Element Analysis FEM Finite Element Method
GP Gauss Point
GWM Generalized Wagner Method LI Linear interpolation method MLM Modified Logvinovich Method
NN Nearest Neighbor interpolation method inf Infinite
1 Introduction 7 1.1 Context . . . 7 1.2 Research approach . . . 8 1.3 Thesis outline . . . 9 2 Abstract 11 3 Theory 12 3.1 Hydro-structural models . . . 12 3.2 Slamming model . . . 14 3.3 Structural model . . . 20 4 Model Overview 26 4.1 Scope of project . . . 26 4.2 Inputs . . . 28
4.3 Analysis of slamming sections . . . 33
4.4 Obtaining outputs . . . 35
4.5 Verification . . . 39
5 Case 49 5.1 Case description . . . 49
5.2 Case results . . . 52
5.3 Using the response for design purposes . . . 74
6 Conclusions 78
CONTENTS CONTENTS
Introduction
The goal of this thesis is to use a strip approach based on the Generalized Wagner Method (GWM) to set up a method to find the local ship response due to slamming.
The problem is that efficient and validated slamming calculations are 2 dimensional, and the structural response is 3 dimensional. Two different methods to interpolate pressures found on slamming strips to the structural mesh are presented, a nearest neighbor (NN) approach and a linear interpolation (LI) approach.
The main requirements of the model are that the structural response is accurate and reliable. Furthermore the CPU effort should not be excessive, so that slamming calculations can be done on a normal computer.
1.1
Context
The effect of slamming is investigated so this component of the loading and response can be obtained in the design phase, this is done to make sure ships remain intact during operations. Currently, the strength against slamming response is based on heuristic rules, which fulfill their purpose as long as ships stay the same.
With the advent of ultra large container ships, understanding of slamming response is necessary as there is no empirical data on container ships of such a large size. Container ships still tend to increase in size to increase capacity, which reduces shipping costs. However, these types of ships tend to suffer from slamming induced whipping response and local damage.
Illustrating the increase in container capacity is the changing of hands of the title of largest con-tainer ship, in November 2014 the CSCL Globe (19,000 TEU [27]) took the title of the largest containership in the world from MV Maersk McKinney Moller (18,270) [35], which obtained the title of largest container ship in February 2013. Currently the MSC Oscar holds the title of largest container ship with a marginal increase of TEU capacity to 19,224 TEU [7].
Other types of ships that are influenced by slamming are smaller ships that operate in high sea-states, these ships can suffer from high impact loads which change vessel motions and can cause damage.
Besides structural damage, slamming can also be very uncomfortable for people on board. In cargo ships, the crew is usually far away from the bow of the ship, where most violent response occurs. However if slamming occurs on cruise ships, see figure 1.1 for example, this is a problem as these ships operate to provide people with a comfortable journey across the seas.
The purpose of slamming analysis is to find out whether the ship structure is strong enough to resist damage. The information sought is the pressure on the ship structure to calculate the local
1.2. RESEARCH APPROACH CHAPTER 1. INTRODUCTION
Figure 1.1: Cruise ship slamming
response, so the pressure loads must be accurate on a local plating level.
The solution presented in this thesis is a 2D strip model based on the Generalized Wagner Method and interpolates 2D-pressures on a 3D structural finite element model consistently. The benefits of this method are that the 2D strips are relatively CPU efficient and distinguishes slamming loads from other types of wave loading.
The pressure calculation is consistent in the sense that the pressures corresponding to Gauss points on the 3D body are recalculated on the 2D strips.
Creating the model was done using source code of the BV developed program ”Homer”. This tool, which is developed for hydro-structural interaction, will be upgraded with a module based on strip theory applied on the slamming problem.
The procedure followed for slamming calculation is illustrated in figure 1.3 and can be summarized as follows: several strips are taken from a mesh and the relative velocity between the strips and free surface is obtained as slamming inputs. Pressure at sampling points are found during slamming calculations, these pressures are interpolated in time and space to obtain pressures on the 3D mesh. These pressures are used to create inputs for a Finite Element Method (FEM)
1.2
Research approach
The main research question is: how can local slamming response be predicted accurately on struc-tural details using a GWM impact strip approach?
To find an answer to this question a methodology is set up to calculate pressures due to slamming in detail. Loads are calculated using a strip approach, the pressure is recalculated on the 2D strips for each Gauss point and the loads on the ship are interpolated to apply loads on a finite element model and making sure the slamming loads are synchronized.
The relevant parameters for input are the geometry and motions of the ship and the free surface. The relevant output velocities are the local stress and deformation which are used to assess whether the ship is strong enough. Parameters which are control the 3D pressures applied are the number of slamming strips, section inclination, time step size, number of Gauss points per element and interpolation method.
The input parameters are dependent variables, the goal is to find the slamming response for a ship with certain motions in a specific sea state. When the solution is independent of analysis parameters (e.g. number of GPs or sections) an accurate model is set-up since the output stress and deformation only depend on the input parameters and not the model itself.
Figure 1.2: Research field, the grey area represents the scope of the research, culminating in local quasi-static slamming response.
Since all loads are calculated quasi-statically and the actual impact is a transient process, dynamic amplification factors due to the effects of an impulsive loading is treated as well. Using frequency content of the loading and dynamic amplification factors will give an insight in what the magnitude of the response will be.
1.3
Thesis outline
This thesis is divided into chapters containing information on the background theory, strip slam-ming model and a case study on a general cargo ship.
In the theory chapter hydro-elastic models are treated and the position of slamming within these models. 2D slamming methods are treated with the main focus on Generalised Wagner Method (GWM) which was used to calculate the loads on slamming strips. Finally the structural model loading, response and evaluation is treated.
Chapter model overview consists of the scope of the work performed, which is also shown in figure 1.3, inputs for slamming strips, the subsequent GWM analysis, how to use the results to get Finite Element (FE) nodal loads and finally verifications are performed on simple cases and on a ship model. The ship model is compared with a model towing test to show that slamming pressure trends are realistic in shape, not in magnitude. Constraints and assumptions made in the model are treated as well.
The case study focuses on a model of a general cargo ship on which a parameter convergence study is analyzed in detail. The parameters and results give an insight in how the strip slamming model works. The effect of a more rapid convergence when linear interpolation instead of nearest neighbor interpolation is used is shown, as well as the effect of varying the number of Gauss points, slamming strips, time step size and slamming strip inclination.
1.3. THESIS OUTLINE CHAPTER 1. INTRODUCTION Time out Slam time tn Interpolation of 2D pressures ݑ
Relative velocity of slam strips 2D slamming calculation
3D pressures and FEM analysis
Abstract
The goal of this thesis is to obtain local response due to slamming loads. The main problem is that there is no satisfactory 3D slamming model, so a strip approach is applied.
Slamming loads are included in hydro-structure interaction models when fully non-linear wave loads are applied. The slamming model used is the generalized Wagner method (GWM), which is based on wedge entry in water and uses potential flow assumptions. The structural analysis is done by using finite element analysis and an impulse response spectrum based on blast loads. The calculation procedure uses a 3D ship geometry and its motions to obtain 2D strips with their relevant velocities. These 2D strips are analyzed using GWM and pressures are calculated on the 2D strips for each element on the 3D hull. To use the pressures obtained, a temporal and spatial interpolation is applied. The resulting loads are analyzed using a quasi-static finite element model and checking whether the modes are excited near their natural frequencies.
From a convergence study on a general cargo ship, the structural loads converge with increasing the number of slamming strips and decreasing time step size, this means that loads can be obtained accurately. The slamming loads are very local, a linear spatial interpolation method proved to converge faster than nearest neighbor interpolation. The number of Gauss points on the hull proved to be not important, since the mesh is already relatively fine. The inclination angle proved to lead to an increase in velocity, but when more realistic ship motions are analyzed the sensitivity to inclination angle proved to be not so big.
Quasi-static structural analysis is used to find response of the structure. Using a triangular im-pulse response spectrum and Fourier analysis on the loads showed a significantly large static load component to justify the quasi-static analysis.
Chapter 3
Theory
The main goal of slamming analysis is to obtain the design loads and response for a given ship and operational profile. This is used to determine the structural strength of the ship.
Hydro-structural interaction means the coupling between hydrodynamics and structural models to obtain correct loads on the ship structure, this information is used to assess whether the structural strength is sufficient.
This chapter first treats hydro-structural models in general, to explain the context of why and when slamming analysis is done. The second section treats slamming models in general and explains why and how generalised Wagner method (GWM) is used for slamming analysis. The third section treats the structural model and how loads are applied and interpreted.
3.1
Hydro-structural models
The complexity of a hydro-structural analysis depends on the modeling of the hydromechanical loads and the structural response. A complete overview is given in [20]. The structural model can be either quasi-static, so that the ship is modeled as a rigid body, or the structural model can be dynamic, which means deformations have an influence on hydromechanics. In order of increasing complexity, the hydromechanical loads are either based on linear wave pressures, weakly non-linear wave pressures or fully non-linear wave pressure, which includes impulsive wave loads. An overview of hydro-structural categories is given in table 3.1.
The hydromechanical loads can be found efficiently using a boundary element method (BEM) [20] in case of linear and weakly non-linear wave loads. Linear wave loads take classical linear diffraction loads into account. The weakly nonlinear loading is based on the Cummins equation [8] to take non-linear wave loading into account. The fully non-linear approach also takes slamming into account, this is achieved by adding a strip theory based slamming model.
When elastic deformations of the ship structure are taken into account in the loads on a ship, this is known as a hydro-elastic ship model [3]. So when structural dynamics are taken into account during hydromechanical analysis, this implies hydro-elasticity.
Hydro-structural analysis
Hydro model Structural model Linear wave loads Quasi-static response Non-linear non-impulsive wave loads Transient response
Fully non-linear wave loads
Linear wave loading on a quasi-static structure is the most basic type of hydro-structure analysis. The analysis is usually performed in the frequency domain for efficiency. A big advantage is that the structural and hydromechanical calculations can be performed separately.
Hydro-elasticity also implies coupling from the structural to the hydromechanical model and vice versa. There exists several terms for this coupling [2] and [25], for example, use the term two-way coupling for a hydro-elastic model and one-way coupling for a rigid ship model whereas [20] and [29] refer to hydro-elastic models as coupled and rigid models as uncoupled.
The difference in terminology originates from the approach used for the structural analysis, which can be either a FEA or modal analysis. Modal analysis can be incorporated into hydromechanical code, where the term coupling means that global elastic modes are taken into account along with rigid body modes, whereas FEA is usually performed in a separate code so there is at least coupling from the hydromechanical model to the structural model necessary and hydro-elasticity is achieved when there is two-way coupling.
Whether elastic deformations become important depends on structural natural frequencies and wave excitation frequency. To include global deformations in hydromechanics usually the modal approach is used [29]. Local analysis usually remains uncoupled, since the CPU effort is very high when performing a dynamic detailed FEM calculation and use the deformations in each hydro analysis time step.
Promising developments of incorporating local deformations in hydromechanical analysis are made in several fields but are not yet suitable for practical purposes, some sources are mentioned here. Efforts to couple a FEM with a BEM [16] require too much CPU effort for practical purposes. Other advancements include a totally different solver using smoothed particle hydrodynamics [24] in which both the structure and the fluid are modeled as interacting particles without the need for a interface between the fluid and the structure, however this method is also far too CPU intensive for practical purposes.
3.1.1
Hydro-Structure interaction
In this section the difference between coupled and decoupled slamming calculations are treated. When the fluid and structure start interacting this means that the pressure on the ship hull is dependent on the motions of that same hull. This can be categorized in global and local hydro-elasticity.
When hydro-elasticity is included the impact energy results in both elastic and rigid motions. Compared to a rigid body, this means the rigid body decelerations are less severe leading to a lower impact force [2].
3.1.2
Coupled and decoupled slamming calculations
A coupled seakeeping slamming simulation method has been developed by Tuitman taking global hydro elasticity into account[30, 31]. This method uses a panel method for the seakeeping calcu-lations, generalized coordinates of rigid mode shapes in combination with a number of bending mode shapes to obtain the structural response and for slamming a seperate module is used when an impact occurs.
The advantage of decoupled calculations is that slamming calculations can be done as post-processing of the seakeeping calculations. This means the slamming loads are effectively calculated with imposed velocities.
Major advantages of the decoupled method are that the free surface does not need to be coupled between calculation methods and that calculation effort can be improved by selecting the most significant slamming events encountered for structural analysis.
3.2. SLAMMING MODEL CHAPTER 3. THEORY
means that there is no certainty on how often this type of slamming takes place, therefore a reliable statistical analysis of how often a certain type of response occurs is not possible to construct from decoupled slamming calculations.
The main advantage with the coupled method is that slamming loads are obtained more realistic, since the ship motions are calculated using both impulsive and non-impulsive wave loads. This means that a reliable statistical analysis of the structural response can be made.
The main disadvantage of a coupled method is that calculation time is long and not only loads and velocities need to be coupled, but also the free surface and structural deformations. This leads to a very complicated program as well.
A big problem in validating any slamming response model is that there is little validation data of the coupled method and data specifically for slamming loads and response is very hard to obtain since loads measured are due to both seakeeping and slamming.
If the calculation methods can be improved, it might be possible to use coupled seakeeping-slamming methods to obtain more accurate design loads in the design stage for the Ultimate Limit State (ULS). The elastic and rigid (global) ship motions are found by using a coupled method. The local response is found by imposing ship motions on a detailed mesh after ship motions have been found.
3.2
Slamming model
The hydromechanical model focuses on the slamming model, since the non-impulsive wave loads are not treated in this analysis. Since slamming is characterized by inertial loads, so according to momentum theory, potential flow is a good method to solve the slamming problem.
An overview of many used definitions is shown in figure 3.1. A distinction is made in the undis-turbed free surface and the additional ’uprise’ induced by the jet flow. The contact point is shown taking the uprise into account, which is consistent with (generalized) Wagner method. The Von Karman method places the contact point at the undisturbed free surface.
The deadrise angle is an important parameter for the magnitude of slamming pressures, as the flatter the impacting body is the bigger the initial pressures.
Slamming overview since this is an important part in the load calculation GWM since it is used as slamming calculation method Other methods are briefly treated for sake of completeness
3.2.1
Potential Flow
In hydromechanics, the fluid flow can be modeled very efficiently as a potential function, which is a continuous function that satisfies the conservation of mass and boundary conditions at the free and body surfaces. This function has the pleasant characteristic that the gradient of the potential at a certain point in space and time equals the flow velocity at that point. Potential theory only takes inertial loads into account, so viscous forces are neglected. This is a valid simplification when wave loads are dominant. For further information on potential flow see [28] or [18].
As stated above, the assumption is made that a velocity potential which satisfies mass continuity exists. Other assumptions are that the flow is:
• Incompressible • Irrotational • Inviscid
For slamming calculations a Boundary Element Method (BEM) is used. Mathematically speaking, the BEM solves an integral equation throughout a domain by fitting an integral at boundary
Deadrise angle α
Undisturbed free surface Uprise
Impacting body
Contact point
Definitions used in 2D slamming
z y
Jet flow
Figure 3.1: Overview of definitions used to describe 2D slamming
elements, where boundary conditions are defined. In the case of potential flow, a potential function is defined throughout the fluid domain.
This potential function is found solving several boundary conditions at the boundary elements. The Green’s equation is used to fit the potential using the boundary elements. Once the integral equation has been set up, it can be used to calculate the solution at any point in the domain. The governing equation is based on conservation of mass and is also known as a Laplacian function: 52(φ) = 0
The boundary conditions (b.c.) used are either kinematic or dynamic. Kinematic b.c.’s define flow motions and velocity at the boundary whereas dynamic b.c.’s define force or pressure at boundaries. The kinematic body boundary condition is that the ship hull is watertight. Mathematically speak-ing, this means the potential gradient normal to the ship hull is equal to the hull ship speed in that direction.
Dynamic free surface b.c. is that the total pressure on the free surface equals the atmospheric pressure. This leads to the dynamic pressure and gravity head canceling each other out. Kinematic free surface b.c.: a particle on the free surface will follow that surface.
A property of the BEM which is used a lot to obtain structural loads: the pressure potential can be recalculated anywhere since the potential throughout the entire fluid domain is continuous. This property can be exploited in seakeeping and slamming analysis to recalculate pressures at any point [21]:
φ(xs) =
R
Sσ(xh)G((xh); (xs))dS
3.2.2
Slamming overview
The problem of water impact on ship structures can be classified into several types of impact [2]. The type of water impact treated here is bottom and bow flare slamming. An overview of slamming research is given in [15]. The focus is on the generalised Wagner method for 2D slamming and strip methods to go from 2D slamming to 3D loads.
3.2. SLAMMING MODEL CHAPTER 3. THEORY
points and computation of pressure. The fluid surface uprise and intersection points are difficult to find since there is no exact solution for finding the uprise on an arbitrary body. Computation of pressures can be solved when the potential is known, singularities in the potential pose problems however.
Fundamental work has been done by Von Karman [32] and Wagner [33] who investigated the impacts of seaplane floats. Von Karman derived a formulation for the impact of seaplane floats during landing by modeling the problem as a falling triangular wedge, taking added mass into account and assuming water is incompressible. Momentum theory is used to calculate the force of the impact and pressures.
Parameters used for pressure calculation are the body and free surface positions, density of the fluid, velocity, deadrise angle, weight of the body (per unit length) and Mach number (when compressibility is taken into account). Maximum pressure was found by [32] to be at the tip of the wedge at the moment of first contact. An approximation of flat bottom impact pressure was found by taking the compressibility of the fluid into account.
The Von Karman theory [32] made a lot of simplifications, the free surface was assumed to remain undisturbed and the impact was reduced to a flat disk on the still water surface. Wagner improved on this flat-disk theory by taking water uprise along the wedge into account, which is often referred to as pile-up.
Zhao and Faltinsen [36] improved on the Wagner theory by generalizing the shape of the impacting object, this is referred to as generalized Wagner method (GWM). This is done by satisfying the boundary conditions on the exact body surface by using a panel method. They also improved the method by taking flow separation into account e.g. at knuckles.
A comparison of how the body surface is modeled according to the Von Karman model, Wagner model and GWM is displayed in figure 3.2. What can be seen is how the wetted body is described as a flat disk with differing widths by the Von Karman and Wagner models and that the GWM describes the body without flat disk approximation.
There are also other slamming models like the modified Logvinovich model (MLM [19]) which is based on Wagner theory with an expansion of the potential. Another model is the Vorus model, in which non-linear terms are included in the free surface boundary condition.
Limitations and assumptions on slamming
Many physical effects take place at once during slamming, it can be difficult to determine what effects are important and what can be neglected. A list of some of the most important physical effects:
• Impulse load
• Local hydro-elastic vibrations
• Air cushion and compressibility effects • Three dimensional flow effects
• Jet flow seperates from free surface
The impulse load is attributed to the sudden acceleration of fluid close to the body surface according to momentum theory. This impulsive character is one of the key elements to investigate slamming as it is useful information in determining the structural strength.
According to [2] air-cushion effects were found to be significant for flat bottom impacts while according to [10] air cushion effects are much less significant than local deflections. Whereas [15] mentions that air-cushions effect applied loads significantly in inital stages of (near) flat impacts. From a structural point of view, air-cushions can be neglected since bottom slamming structural response is most significant after initial slamming phase. However in case of flat-bottom impacts, air cushions and compressibility of fluid reduce the loads on the body significantly. Therefore, when considering flat-bottom impacts compressibility becomes important.
Flat disk approximation based on immersed width in undisturbed free surface
Flat disk approximation based on immersed width including uprise
Exact body boundary condition Von Karman model:
Situation:
Situation:
Generalized Wagner model: Situation:
Wagner model:
Slamming models visualized
New Section 1 Page 1
Figure 3.2: Slamming models comparison, the difference between Von Karman, Wagner and GWM is in the definition of the contact point and how the body boundary conditions are applied.
3.2. SLAMMING MODEL CHAPTER 3. THEORY
Local hydro-elastic vibrations are not taken into account during slamming calculations since local response is analyzed quasi-statically. However, an amplification factor based on the impulsive time-history can be used in post-processing to still get a good sense of what the local response will be like. A piece of information that gets lost however is how the local vibrations affect the slamming loads.
Global motions can be coupled with slamming impacts [29], [31]. Excluding global motions is a conservative approximation of the response, since slamming causes the ship to decelerate and a decelerating ship will have lower slamming loads. Neglecting the effect motions have on slamming is acceptable for large ships, as their momentum is not significantly altered by slamming. Small vessels which encounter slamming have significant changes in motions and velocities due to slamming. Three dimensional flow effects are not considered significant by [10]. Also it is not possible to take 3D flow effects into account in 2D simulations. Still in a scenario where 3D flow effects are significant excluding them leads to a conservative estimate of loads.
The jet flow is often neglected as it does not contribute significantly to the structural loads. In [36] a fully non-linear model including jet flow is presented along with the GWM, the jet flow did not contribute significantly to slamming loads but cost a lot of CPU effort.
3.2.3
Generalised Wagner Method
The Generalized Wagner Method (GWM) was developed by Zhao and Faltinsen [36]. In this section a presentation by de Lauzon is also used as source of information [9] in which the singularity of the potential near the contact points is dealt with.
The main difference between GWM and Wagner’s original theory is that on the body the exact boundary condition is satisfied instead of using a flat plate approximation. This means that slamming loads can be calculated on arbitrary body shapes.
The assumptions, initial and boundary conditions can be summarized as follows:
• The fluid is modeled as potential flow and unbounded by depth • Pressure is calculated using law of Bernouilli
• A two-dimensional body impacts the water surface
• The fluid surface is initially at rest and the water particles do not leave the fluid surface (jet flow is neglected)
• No air pocket is created between body and free surface
• The origin of the x-axis is located on the undisturbed water surface, y-axis goes through symmetry line of the cross section
• Physical effects that are neglected: water compressibility, local vibrations, air cushion effects, 3D flow effects
Additional assumptions for numerics are that at the initial time step the body has a small draught, since the boundary elements need to be defined somewhere. A numerical problem is that this discontinuity of velocity potential at the moment of initial contact causes an infinite potential according to [15].
Important parameters for GWM are the deadrise angle,
GWM can be simplified by cutting off the jet flow. Neglecting the jet is done to improve calculation time, however it also means that at a knuckle or seperation point (e.g. a bulb on the bow) the flow which should leave tangentially is cut off. This leads to numerical problems in case of inwards inclined sections, so the geometry of the body is enhanced to always be monotonically increasing in height with increasing width. This results in 2D geometry being modified for inclined parts, at the bulb for example. From a physical point of view this simplification is acceptable since the flow would separate at inclined parts [31].
Figure 3.3
The boundary conditions are described mathematically as: Dφ Dt = 1 2( ∂φ ∂x 2 +∂φ ∂y 2 ) on free surface (3.1) ∂φ ∂z = ∂η ∂t on free surface (3.2) ∂φ ∂n = V · n on body (3.3) φ(x, y, t) → 0 f or x, y → ∞ (3.4)
The next step in the calculations is to solve the potential φ for the flow. This is done using Green’s function which uses points on the boundary elements to calculate the potential anywhere in the flow. The mathematics here are quite complicated and are described in [9].
The time step estimation is a bit tricky in GWM. First a contact point is assumed for the next time step, then the time is calculated between time steps i and i + 1. It is assumed that the vertical velocity is constant for free surface at the contact x-coordinate between time steps, see figure 3.3. Pressure is calculated using Bernoulli’s equation equation (3.5). The time derivatives can be calculated since the potential and time step sizes are known. The the potential gradient is not known yet which is needed to calculate the dynamic pressure.
p − p0= −ρ ∂φ ∂t − 1 2ρ| 5 φ| 2 (3.5)
The gradient of the velocity potential on the body surface can be constructed from normal and tangential components. These can be both calculated by using the potential, normal on the body and potentials of and distances to the elements next to the point under consideration. With the potential gradient known, pressure can be calculated.
A schematic representation of the GWM procedure is given in figure 3.4.
In case of an asymmetrical impact, conformal mapping has to be applied to obtain the vertical positions of the contact points on the same level for slamming calculations, this maps the problem in such a way that the uprise is a straight line and roll can be included. Mapping the geometry to another domain is quite tricky, for more information see [9]. Mapping the problem back to the physical domain gives the uprise and (relative) roll back.
Green’s function can not handle the discontinuity on the body at the contact points and diverges to infinity at the contact points. An explanation can be found when looking deeper into the workings of the BEM. On each boundary element a source is placed, this source determines a distribution of the panels influence on other panels, e.g. if a potential is found on one panel what the potential will be on another panel. Near the discontinuity this distribution is forced to an increasingly small surface, leading to amplification of potentials near the discontinuity.
The singularity is dealt with by cutting off the slamming pressures. The Bernouilli equation is used to find pressures. The pressures tend to diverge to −∞ this is dealt with by filtering out all pressures smaller than zero. Pressures smaller than zero mean the fluid is moving so fast, that the
3.3. STRUCTURAL MODEL CHAPTER 3. THEORY
Figure 3.4
pressure becomes negative. This moving fluid comes from a jet flow which starts at the intersection between the free surface and body surface. This jet flow does not contribute significantly to the force on the body.
The uprise is computed iteratively by first assuming a new position of the contact point and calculating the time it takes for the uprise to get there. This is done by integrating the free surface boundary conditions from tn−1to tn and keep varying tn−1until an accurate enough time step is
found.
3.2.4
3D-methods for slamming
3D methods to perform slamming methods are too inaccurate for use in ship design. The big problem in 3D methods is that finding the wetted surface is quite difficult. There have been models developed for 3D Wagner impacts, for example in [11].
To improve accuracy and CPU usage, slamming on 3 dimensional sections are in general calculated using 2D methods on several cross sections in a strip theory fashion. This approach has been used for example by [29], [31] and [30].
In strip methods, several slamming sections are defined and pressures are calculated, these pressures are then interpolated over the section surface to be added to the seakeeping nodal force. Difficulties arise from the time step definition of the slam sections, as the size of a time step differs per time step and per section.
These methods have been combined with seakeeping calculations where the structure was analyzed using modal analysis. This lead to good agreement with model tests.
3.3
Structural model
Structural mechanics are an important part hydro-structural interaction. The structural response to hydrodynamic loads lead to deformations and stresses, evaluation of this response tells whether the ship strength is sufficient.
Response is assessed on a global and local scale. Global response is usually represented by mode shapes obtained from the entire ship structure. Local response is assessed using FEA, to find the deformations and stresses on the level of individual plates.
under-stand structural dynamics and FEA are [26] and [6] respectively.
The hydro-structure interaction will be analyzed according to the following components of a struc-tural analysis:
• Load - Loading is determined by the hydrodynamic-slamming analysis.
• Response - The structural response is found by either modal analysis or FEM calculations. • Evaluation - The ship response can be evaluated after running simulations for a certain time
period. Extreme value distribution can be set up for components of interest which is either global or local structural response.
The material used in the cases analysed is steel, which is assumed to remain in the linear-elastic domain. This means that the constitutive equation which governs strain as a function of applied loads is in the form of Hooke’s law:
σ
= Emod
More advanced constitutive equations for steel include plasticity. But since the objective is to find a structure which can withstand slamming loads without deforming plastically, this type of calculation is out of scope.
Other materials used in the maritime industry with more advanced constitutive equations are composite structures. These are not investigated in this thesis.
3.3.1
Impulse loads
Slamming loads are considered impulsive loads as they occur This means that inertia loads are taken into account in the structural analysis and ship hydro-elasticity is taken into account in the hydrodynamic-slamming analysis.
Slamming response can cause local damage and whipping. To take whipping into account a tran-sient analysis must be done. For ultimate local response a static analysis suffices, since a dynamic amplification factor can be used for dynamic increase in response.
When slamming loads are modeled accurately on a global ship structure, whipping response can be modeled by making a simplification of the problem by modeling the ship hull as a beam held afloat by distributed springs and dampers. Since beam ends are assumed to be free, according to solid wave dynamics the wave displacement is doubled at free ends [22]. The whipping effect is reported to cause containers to fall off at the back [17].
Furthermore, much exposure to whipping increases fatigue damage and can lead to an increased value of the global vertical bending moment, which could cause an exceedence of the ultimate limit strength resulting in a broken ship.
Local damage can be found through static analysis since local damage is caused by maximum loads exceeding maximum strength, due to the impulsive nature of slamming, a factor can be used to take dynamic effects into account [14].
The shape of the impulse load during slamming on plates can be simplified to be triangular with a tail [4] which is displayed in figure 3.5, where in the text it is concluded that the tail can be neglected for structural analysis.
3.3.2
Response and dynamic amplification factors (DAF)
The most important choice to be made in analyzing response is whether a quasi-static or dynamic analysis is used.
3.3. STRUCTURAL MODEL CHAPTER 3. THEORY
Figure 3.5: Idealised slamming pressure impulse on a plate [4]
Quasi-static response is defined here as loads calculated using 6 rigid degrees of freedom for ship motions. So no deformation is taken into account when calculating loads. Typically this is when the response of the ship contains very little deformations.
The difference between a structurally dynamic or quasi-static analysis is whether inertia loads and damping based on the time history are taken into account. Static analysis does not take dynamic effects into account which can make a huge difference in resulting response. When the time history does not matter, the structural response can be done after hydrodynamic analysis is finished. Dynamic response includes inertial loads, which means a transient analysis must be done, since inertial loads come from the accelerations of mass. It is possible to do a transient analysis with loads found using a rigid hydrodynamic model, the structural modes need to include added mass, otherwise significant inertial loads would be lost. A major issue with slamming is that the waterline changes rapidly, thus changing the added mass and damping on each mode shape over time. If dynamic response is coupled with hydrodynamics so a hydro-elastic analysis is performed, the added mass due to vibrations is already into account in hydrodynamics, so the ’dry’ mode shapes with modal added mass can be used for response analysis.
From discrete 1-dof dynamics, one should expect stiffness dominated response at excitations with frequencies far below than natural frequency, damping response to excitations near natural fre-quency and inertia dominated response when excitation frequencies are far above natural frefre-quency. To find from the loads obtained by slamming the structural response, one has to perform a full transient analysis with added mass at each node being calculated at each time step and during post-processing assess where and when maximum response occurs.
Another option is to take a dynamic amplification factor (DAF) into account in the response, which is based on natural frequencies of the structure, frequency of excitation and type of excitation. The principle of DAF is also applied in, for example, jack-up platforms under wave loads [13]. The excitation is similar to blast loads according to [4].
In hydro-structure analysis, the first step is to calculate static deflection and find the rise rime and natural period. The next step is to use the response spectrum of a system which is based on equation section 3.3.2 to find the dynamic amplification factor.
x(t) = 1 mωn Trise Z 0 F (τ )sin(ωn(t − τ ))dτ
The advantage of using quasi-static analysis is that the maximum response can be calculated at a specific time step which can then be amplified using the DAF. This DAF can be based on wetted mode shapes at that specific time step. This still means added mass needs to be calculated but
only at one specific time step so the waterline does not change.
An interesting principle from shock load analysis is to use response spectra to determine what the maximum response will be due to an impulse with a certain time history profile and period.
3.3.3
Evaluation
Evaluating whether the structural response is acceptable requires a definition of what is acceptable. If the response is allowable or not depends on the type of analysis which is performed. Either the ultimate, fatigue, accidental or servicability limit strength.
The most violent global response expected from slamming is whipping, which increases fatigue loads and ultimate loads of the global bending and torsional moments [20]. Whipping has been mentioned as one of the causes that the MSC Napoli suffered a total constructive loss [1]. So the global response should not exceed the hull girder strength.
Another expected failure mode is local damage. This usually results in buckled stiffeners and plates which need replacement. So the local response should not exceed the limit strength investigated.
3.3.4
FEM software used
Response is calculated using NX NASTRAN FEM software during post-processing.
FEM software models a structure as a discrete system with a large number of degrees of freedom and solves the displacements due to applied loads.
Nastran is a FEM solver that contains many different analysis types. An important analysis type is the linear-elastic static analysis. What this means is the material behaves linearly and no inertia loads are taken into account. The problem reduces to:
F = [K]u (3.6)
Where F represents the nodal forces, K the stiffness of the discrete structure and u the nodal displacements.
In FEM analysis it is important to consider the level of detail of the model. For global response a coarse mesh suffices, of which the most simple model is a beam model of the entire ship. The inertia and stiffness of the ship need to be modeled to set up a model with several beam elements. This is typically done to get an impression of the global response
The most detailed mesh is where the construction is modeled using solid elements, this is done typically to find stress concentrations for fatigue analysis, this is also a very computational intensive method.
The type of model used to analyze the local structural response is detailed enough to show the local response of stiffened panels. Good candidates for this type of analysis are models which model individual plates and stiffeners as plate and beam elements respectively. Models in which plate stiffeners are modeled as plate elements or solid models are possible as well, but require more computational effort.
When analyzing local response, it is possible to only analyze the local region, instead of the entire ship, if correct boundary conditions are chosen. This leads to much faster analysis, since the calculation time is shortened and a lot of uninteresting data is not produced.
3.3. STRUCTURAL MODEL CHAPTER 3. THEORY
3.3.5
Gauss Points
Gauss points are used in FEA to allow elements to have arbitrary surfaces (isoparametric formula-tion) [6]. For example, the displacement field within an element is approximated by a continuous interpolation function. This interpolation function can be integrated over the element to obtain the stiffness matrix of an element. Gauss points are sampling points with assigned weights to minimize integration error when the integrand is a generic polynomial.
At integration points loads are sampled and given weights so that they yield the result of the exact integral of a polynomial of order 2n − 1, where n is the number of Gauss points.
Usually GP are defined in the domain [−1, 1] at points xiand the corresponding weighting function
wi0 is based on Gauss-Legendre quadrature [34].
An actual element will have its coordinates mapped so that Gauss Points are defined on the element in the local domain [−1, 1].
Z 1 −1 f (x)dx = Z 1 −1 w(x)g(x)dx ≈ n X i=1 w0igi(x0i) (3.7)
For more details on Gauss quadrature see [6] or [34].
3.3.6
Pressure recalculation at Gauss points
For pressure calculation, several derivatives of the potential need to be known [9], the Laplacian derivative and gradient to be precise. The pressure can be racalculated at any point in the domain, even on the boundary elements [21]. This is essential for a consistent pressure recalculation. This method of slamming recalculation is an improvement on [29] as there were a fixed number of points on the 2D geometry used where pressure are calculated. When using pressure recalculation, spatial interpolation is still necessary to take the pressures from 2D strips and apply them on 3D geometries.
The potential can be recalculated using Green’s third identity using the relation section 3.2.1. This means that the pressure at any Gauss Point can be recalculated on the 2-dimensional slamming section. This can reduces the interpolation effort significantly, as the slamming potential is already obtained for the 2D case.
Using the fact that the potential is continuous section 3.2.1, the potential can be recalculated at each point on the body surface [21]. No additional interpolation errors are introduced in this method other than the geometrical inaccuracy between sections. The accuracy of the potential on the body is determined by the number of collocation points, which can be increased by increasing the number of panels, thus creating a finer slamming mesh.
Interpolation of Gauss Points between two sections, interpolation methods like nearest neighbor or linear interpolation is used. More complicated methods were not investigated since the effect of slamming is highly localized, it is not expected that using more than two sections per GP increases accuracy since the slamming phenomenon is very localizes.
There can be issues in (re)calculating potential on a boundary when the constant-source strength is applied, however this does not play a role in the current model. According to [23] the potential at a point P in the domain can be calculated at a distance from the boundary elements without any problem, the potential on the boundary elements is a bit more tricky since it is a limit with terms going to infinity and zero. Trying to find the gradient of the potential on the boundary is not possible as it is unbounded, when using the constant source method.
potential flow around a corner. The problem with calculating the gradient of the potential on the body is solved by using the body boundary condition in the normal direction: ∂φ/∂n = V n. The gradient in tangential direction is calculated for both the singular part and the regular part. The Laplacian derivative is found by numerically differentiating.
Chapter 4
Model Overview
In this chapter the strip slamming model is described in detail. First the scope of the project is given, then the input, analysis and output phase of the slamming calculations are described with more detail.
The program used/developed is HmSlam, this is a program that uses a body and together with its motions and velocities as inputs, creates 2D geometry, initial position and 2D velocities for slamming calculations. Slamming calculations are performed to find pressure on 2D strips. Using the output pressures from 2D slamming calculations, pressures can be interpolated on the ship mesh. The pressures on the ship mesh are the outputs which can be analyzed using FEM software.
4.1
Scope of project
This section describes the scope of the project and requirements of the model. The goal of the slamming model is to evaluate whether slamming response exceeds maximum allowable stress. So obtaining an accurate and reliable response are the key requirements. To make the method practical to use, CPU use must be reasonable, so calculations should take time in the order of minutes.
A software module using 2D GWM strip theory has been developed. This needs to be integrated into a program which can cut a mesh into sections and interpolate the pressures afterwards. The software used for FEA is Nastran NX. The code has been created using Intel Visual Fortran Composer XE 2013. Microsoft visual studio has been used as environment to develop the code. Fortran is used because of the efficiency of calculations.
The project consists of giving inputs for slamming calculations, performing slamming calculations and analyzing output pressures. An example of the calculation process is given in figure 4.1. The inputs given are the motions, velocities and geometry of slamming sections, sampling points for output pressures and BEM parameters.
The slamming calculations are performed using simplified GWM as described in [36] and in sec-tion 3.2.2. The slamming calculasec-tions are performed for each secsec-tion seperately. There is no coupling of the free surface elevation to the slamming problem.
The output pressures are found for fixed points on the slamming sections and interpolated on the 3D mesh. A form of nearest neighbour interpolation (NNI) and linear interpolation (LI) are used for spatial interpolation. Temporal interpolation is also required as the time steps in slamming calculation are different at each time step. More sophisticated interpolation methods do not increase accuracy since the slamming phenomenon is very local.
A wedge and paraboloid geometry are first used to verify whether the spatial and temporal inter-polation work. An issue which is found is the initial displacement, which can cause peaks in force
Figure 4.1: Description of slamming calculation procedure
and pressures at the first time steps.
4.1.1
Assumptions and constraints
In this section the assumptions and constraints on the calculation process are described.
The inputs are given by imposed motions and velocities, no hydro-elastic coupling is made between hydrodynamic/slamming velocities and impact pressures. This means the model is uncoupled which is a valid assumption as long as global motions are not significantly affected by slamming, for local response this is acceptable as local vibrations will not significantly alter global motions. The simplification of cutting off the jet flow and linearizing the uprise velocity, as mentioned in the slamming theory section, requires that sections need to be monotonically increasing in width with increasing height. In the program the strip geometry is modified to fulfill this requirement. Further assumptions made using the simplified GWM are that the shape of the 2D slamming strip does not change. So local hydro elasticity is not taken into account.
4.2. INPUTS CHAPTER 4. MODEL OVERVIEW
project. A direct consequence of this is that a flat free surface is used when calculating slamming impacts which is not coupled. Due to the uncoupled free surface, the uprise is not coupled between slamming sections.
Furthermore, negative pressures are neglected, this is treated in section 4.3.2.
4.2
Inputs
This section described the inputs required to set up slamming strips and to obtain the input data for slamming calculations.
The geometry, velocities and calculation parameters are given as inputs for slamming analysis. The inputs consist of the geometry and velocity of the slamming section, locations of where the pressure is stored and solution parameters.
4.2.1
Geometry input
The slamming strip is a 2D input for the slamming calculations. This 2D input is obtained by projecting a 2D plane on the 3D mesh, the intersection line describes the slamming section geometry.
The 2D plane is defined by a reference point r0and a normal vector n. A point p rp is on the 2D
plane if: n · (rp− r0) = 0
A visual representation of slamming strips on the 3D mesh is given in figure 4.2. Strips with different inclination angles are displayed in red and white. The inclination of the sections is a non-trivial input as it has effect if the impact is not perpendicular to the free surface.
Figure 4.2: The slamsections can be have a ’pitch’ angle as shown in red.
4.2.2
Free surface
The free surface is not yet taken into account, this is necessary to get a full synchronization of slamming calculations and with that a correct representation of the pressures on the 3D-FEM. However, for now coupling between the free surface and slamming calculations is not available as this was out of scope of the project.
Figure 4.3: The original geometry is shown in blue, the modified part is shown in red.
on this in the verification section.
4.2.3
Finding (relative) displacement and velocity
This section describes how the relative displacements are found, which are used for GWM slamming calculations.
The goal is to find relative displacement and velocity for slamming from hydrodynamics program. The velocities that are necessary for the slamming calculations are the vertical, transverse and roll velocities.
Furthermore the slamming sections must stay synchronized, so for each slam section the starting time must be stored and used to synchronize the slamming loads.
Projection vectors are defined by using the normal vector that defines the 2d-cut and its relevant reference point. To get the projection vectors in the first place, use is made of the fact that the pv and nv are orthogonal to each other, so their inner product has to be zero. The y/x ratio remains constant as the projection vector is ’pitching’ down the z-axis. Another constraint is that the vector is normalized, with these constraints the projection vector can be found.
Projection along plane:
n = a, b, c, pv = x, y, z (4.1) n · pv = 0, b a = y x, x 2+ y2+ z2= 1 (4.2) y = bx a , z = p (1 − x2− y2), ax + by + cz = 0 (4.3)
Ship motions of the projection vector are taken into account by using the rotations of the reference point at the keel, which is used to transform the projection vector to its new orientation. An example of how the relative distance is found is shown in figure 4.4.
The subsequent estimation of the relative displacement is done by using the free surface, projection vector and starting point. These are used to generate a line from the starting point along the projection vector until it reaches the free surface. The length of this line is the relative distance between the starting point and the free surface.
Apart from the distance between the body and the free surface, the coordinates of the body must be defined correctly. As mentioned earlier, the 2D slamming section shape does not change, still global hydro elastic motions of 2D sections can be taken into account, when the global rigid and elastic deformations are used to calculate the 2D slamming velocity.
4.2. INPUTS CHAPTER 4. MODEL OVERVIEW
Figure 4.4: Example of finding relative distance from the bottom to the free surfac
The coordinates of the free surface are defined in the earth reference system, the ship deformations are defined in the body reference system. To find a point on the body in the earth reference system rigid and flexible motions are to be taken into account. It is formulated mathematically as: r = ξt+ TEB(ξr)(rb+ rf)
Where rb and rbf are an undeformed point on the ship and its deformation in body coordinates.
T(ξr) is the transformation matrix from the ship to the earth reference system as a function of the
rigid rotations. ξtare the rigid translations of the ship.
The body coordinates are moved using this formula, so the displacement of the flexible and rigid body modes can be used in the velocities of the slamming calculation.
The resulting transverse, vertical and roll velocities of the 2D section are used as input velocities in the slamming calculation.
Furthermore, the initial relative roll is used as input. The free surface is modeled flat during slamming calculation, so the roll is relative to the free surface. Using this method, the relative distance to the free surface at three points is found so it is possible to use this method when the free surface includes waves, as was shown in [29].
4.2.4
Pressure points
This section describes how pressures are sampled to be used for interpolation.
Depending on the spatial interpolation method, one or two pressure points are projected on the slam sections. These pressure points are used during slamming calculation to find the pressure on the slamsection, which is used after slamming calculations to obtain pressure on the 3D-FEM. At the sampling points where the pressure is recalculated during BEM calculation. The advantage of this method is that the accuracy of the output pressures is defined by the hydrodynamic mesh and no additional interpolation on the slamming mesh is needed.
In the code the pressure points are projected on the nearest point on the nearest slam section(s). During GWM calculation, the slam section geometry can be adapted in case of non-monotonically increasing height with increasing width. In that case the pressure point remains on the same height.
The same height is used because when adapting the geometry, it is assumed that the fluid flow separates at inward inclined geometry and so the fluid flow follows the monotonic section more than the original section. The monotonic section has a large gradient, so the pressures will be low or might even be negative. Therefore using the same height of the adapted geometry is the closest
(a) Projection of GP on elements to slamming strips. The white line represents the slamming strip. For clarity, only the two GP closest to the strip are projected.
(b) Correction of GP projections on 2D section, only the coordinate in height remains con-stant. The white line represents the 2D section, the red dots the projection and the red crosses the pressure point on the section
Figure 4.5
thing to getting a correct pressure as long as slamming calculations can not handle non-monotonic sections.
Mesh parameters
This section describes the parameters for the mesh used in the BEM calculations for slamming. Two meshes are used in the slamming calculations: a 2DBEM-mesh of the wetted body and a 2DBEM-mesh of the free surface.
The slamming section boundary element mesh is distributed using a partial cosine spacing on the section geometry. Partial cosine spacing is where a line is divided into segments by projecting a number of points on the curve using a part of a circle.
Using cosine spacing allows to generate a fine mesh near the contact points, where the pressure gradient varies very strongly, and a more coarse mesh near the bottom, where the mesh does not change very strongly. This allows for an efficient way of finding the pressure distribution.
The free surface mesh is exponentially increasing in size. This results in a free surface that is described more accurate near the contact point, since the elements are small there, and far away from the contact point where accuracy of the uprise is less important the calculation is more efficient, since there are less elements there.
4.2. INPUTS CHAPTER 4. MODEL OVERVIEW
Initial displacement
This section describes the effect of the initial displacement on the slamming calculations. The initial displacement is an input parameter which influences the response of the first slamming time steps. See figure 4.6 for a visualization of the initial displacement.
The slamming calculation needs an initial depth to be able to set up a BE mesh. If the initial displacement is chosen in the order of slamming time steps uprise or smaller, the initialization should go smoothly. The bigger the initial displacement, the bigger the error at initialization. According to [15] his error comes from the discontinuity of the velocity potential at the moment of initial contact.
The time trace starts at zero but at a larger displacement, which means that if the initial displace-ment is chosen too big, maximum forces seem to occur earlier than they actually would. Another inaccuracy due to large initial displacement is a pressure peak at initial time steps.
The sensitivity of the initial displacement has been tested by a wedge with width 20 meters, depth 10 meters and a vertical velocity of 10 meters per second. The results are shown in figure 4.7. The total force has a significant error at initial time steps when the initial displacement is 0.1 meter. At an initial displacement of 1 meter no meaningful results are obtained, therefore these results are omitted.
When the initial displacement is chosen to be 0.01 meter or smaller, the total force over time converges quite rapidly.
Another way to quantify if the initial displacement is deep enough is to have a look at the force at the first slamming time step as is done in figure 4.8, note the logarithmic y-axis. Here it can be clearly seen that at 0.001 initial displacement the initial force is converged.
The loads of the different cases show similar trends after initial displacement, where the bigger the initial displacement the earlier maximum force is reached. This is caused by the bigger initial immersion of the body at the first time step, therefore loads are applied earlier on the body. This effect is corrected for in the 3D model, since the trigger to start slamming calculations is based on initial displacement.
Too small time steps have given convergence issues when calculating slamming loads of more arbitrary shapes such as ship sections. Therefore the initial wetting used throughout this project is 0.005 of the section depth. As 0.001 part of the section depth resulted numerical problems with some ship sections.
Figure 4.7: Total force on a wedge with varying initial displacement values relatively to the depth of the wedge.
Figure 4.8: The GWM force is shown as a function of the initial displacement, which is logarith-mically decreasing.
4.3
Analysis of slamming sections
In the analysis phase of the slamming sections, inputs are used to calculate slamming loads. The boundary value problem of slamming is solved and the resulting pressures are used to find
4.3. ANALYSIS OF SLAMMING SECTIONS CHAPTER 4. MODEL OVERVIEW
pressures on the 3D-FEM.
4.3.1
Solving the slamming BVP
The velocity potential is solved each time step using the boundary element method, this means that each time step a boundary value problem is solved.
For the slamming calculations a level free surface is required. So the free surface is mapped using a conformal mapping to get a slamming section with a level free surface. The mathematical details of this mapping process can be found in [9]. The mapped domain is used as a black box, boundary values and boundary conditions are adapted to the mapped geometry using the Jacobian. The details of which are out of scope of this project.
The total potential is decomposed into a singular and regular part. Green’s third identity is used in solving the potential.
The wet body surface is discretized, the potential can be found by using the potential at each boundary element. This results in a system of linear equations which can be solved.
Once the potential is found for a time step, the next time step can be calculated. Time steps are adaptive, because the contact points have to be found for the next time. This is done by choosing a point on the free surface and calculating how much time is necessary for that point to reach the body.
The size of the free surface used in slamming calculations is related to the width of the immersed body.
Time steps for asymmetric bodies are calculated by moving the lowest contact point one time step and moving the free surface along the other side of the body according to its velocity.
4.3.2
Slamming loads
In this section the obtaining of pressures from the (solved) potential is treated. The pressures are calculated by Bernoulli’s equation:
p = −ρ∂φ ∂t − 1 2ρ 5 φ 2= −ρ[Dφ Dt − U 5 φ + 1 2 5 φ 2] (4.4)
The gradient is evaluated in the normal and tangential direction of the body at collocation points. These pressures are integrated over their BE to get the total force on the 2D section.
As mentioned earlier, singularities occur at the contact points. The singularities are due to simpli-fication of the jet at the contact points. These singularities cause the linear gradient term ρU 5 φ to diverge to infinity and the quadratic gradient term −ρ125 φ2to diverge to minus infinity.
The slamming model can handle full immersion of the section. In that case the geometry of the section is extended. Pressures are only calculated (and integrated to obtain force) on the original section.
Negative pressures
Negative pressures occur at several locations: near the contact points, when the geometry is inclined inward or with strong decelerations of the body. These negative pressures occur due to either physical phenomena or simplifications in the model.
In case of negative pressures near the contact points, this is caused by a singularity at the discon-tinuity. The singularity causes the pressure to go to minus infinity. This negative pressure is not
physical, since in reality the uprise is not discontinuous at the ship body, but flows away in a jet. Another case is when the geometry is inclined inwards, this is caused by the gradient of the geometry. Sections containing bulbs are examples where the gradient can become negative. Large decelerations of the body can cause negative pressures if the fluid added mass is still moving. This is a physical cause of negative pressures.
Negative pressures are neglected because they do not cause significant physical structural response. Only the negative pressures caused by the nonphysical singularity near the contact points has an amplitude which causes a significant structural response, however this pressure is nonphysically large and can therefore be neglected.
4.4
Obtaining outputs
The outputs are here defined as outputs from the slamming strip model loads on a FEM mesh from the 2D slamming strips. Before those are obtained a temporal and spatial interpolation is necessary to interpolate the pressures from the slamming calculation to pressures usable in FEM. The results from slamming are pressures over time at fixed points distributed over the slamming sections. These are interpolated in time and subsequently used for spatial interpolation to apply pressures on the Gauss points. So the pressure points are the points where the pressure is known and Gauss points are the point where the pressure has to be applied.
4.4.1
Temporal interpolation
In this subsection the temporal interpolation will be explained. What is important to note in this subsection is that there are different time frames, each with its own time steps.
Time frames
The hydrodynamic time frame has time steps defined by its motions, these time steps can be chosen quite arbitrarily as long as the ship velocity and motions remain correct.
Secondly, the slamming time steps change in size based on the slamming calculations based on the relative velocity and its corresponding uprise. Each slamming section has its own time frame as the geometries and velocities can change per section. Each slamming time frame starts at zero when slamming starts for the slam section.
Finally there are is the structural time and its time step size. This is user defined and gives the output time steps. This time frame is used to synchronize the slamming loads in.
Conserving momentum or pressure accuracy
For the structural loads a single time trace is used. However, since each separate slam section has different time traces in the slamming module since the slamming time step is not constant sized between the slamming sections.
The solution chosen is to interpolate the pressure per Gauss point in time to obtain pressures at the Gauss points. These pressures can be turned into nodal forces / FEM loads used in structural calculations. The question is how to interpolate the pressures at the Gauss points. The choice is to conserve momentum or to describe pressures accurately at each time step.
One option for this is to use a spline interpolation for each section over time. The advantage of this is that the pressures are accurately described at the time steps selected in the structural time frame. The drawback is that momentum is not necessarily conserved. This option gives the height of the pressures accurately, so can be a good method to find the ultimate structural response.
