the computation of hydrodynamic pressures on a 2d section

8/14/2019 The computation of hydrodynamic pressures on a 2D section

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THE SUMMARY OF Ph. D. THESIS

THE COMPUTING OF GENERAL AND LOCAL LOADS OF THE HULL IN CASE OF VERTICAL PLANE

OSCILLATION, BY NUMERICAL MODELING OF FLUID

Dipl.-eng. Marius Popa

Scientific Coordinator: prof. dr. eng. Liviu Stoicescu cost

Content

Chapter 1. Reason of thesis. Thesis goals

Chapter 2. General and local loading on the hullChapter 3. Basics analysis. Models for hydrodynamic simulation

Chapter 4. The model developed in the thesis. Original software

Chapter 5. The revision of method in comparison with experiments and simulation from bibliography

Chapter 6. The computation of pressure on the studied hull

Chapter 7. The study into the effect on structure of the pressure computed based on the numerical modeling of

fluid

Chapter 8. Personal achievements. Conclusions. Future development

Selective bibliography

General note: The construction of this summary respects the thesis construction.

The notations are similar to those in the thesis.

The numbering of figures is similar to that in thesis.

Chapter 1. Reason of thesis. Thesis goals

All the shipbuilding and shipping-related activities are centered on the safety issue. The administrations signatory of

international conventions, the important classification societies, the insurance companies and even the shipping companies take

into consideration the following: the safety of life on sea, the environment safety, the cargo and ship safety. In order to achieve

this objective it is necessary to identify, with utmost accuracy, the behavior of marine structure with respect to kinematics,

dynamics and strength. In order to achieve these purposes, the pressure distribution on the hull should be computed with utmost

exactitude.

The pressure distribution calculation involves solving the hydrodynamic problem of free- surface fluid flow in the

presence of bodies (fixed or floating).

The reality shows that classic theories are no longer useful for modern ship structures, with their continuous bringing

forth of new, revolutionary concepts. These modern structures involve the construction of long and elastic, optimally computed

hulls, of multi-hull ships, of special end shapes (i.e. twin shafts and rudders aft end), high velocity crafts or large off-shore

structures (platforms, docks or, in the foreseeable future, floating airports). Numerical simulation methods have been used lately

in order to estimate the hull response in such cases. With respect to this, an important classification society (Germanischer Lloyd)

states in one of its periodicals that although the application and advancement of experimental processes in hydrodynamics

remains indispensable, in the foreseeable future, computational fluid dynamics often referred to as the numerical basin will

gain appreciably in significance for the shipyards, owing to the increasing validity and faster availability of the results. Future

assistance measures will therefore concentrate on the field of hydromechanics.

Owing to the finite element method, which was proved to be both efficient and reliable, numerical methods for

structural computing are now very advanced. But, in order to ensure the correct inputs of structural analysis and also for the

computing of real ship kinematics, efficient numerical simulation methods for hydrodynamics should be developed.

The thesis seeks two objectives:

- to present a faithful approach to the free-surface fluid flow with in the presence of a floating body (the hydromechanic

component)

- to carry out a study of the influence on the structure of applying loads with a deep non-linear and dynamic characteristic - a

more faithful approach to reality (the structure component).

Both aspects are of scientific novelty. Thus:

- the fluid is numerically modeled in space-time domain. This is the most faithful-to-reality approach to the modeling ofpotential fluid

- the applying of the pressure, computed by numerical modeling, represents a good base for the optimal design of modern

structures. These pressures are deeply non-linear and time-variable. It is only their average value that can be compared with

the values usually considered at present.

The Hydrodynamic aspect consists in the numerical simulation, based on the fewest possible simplifying hypothesis,

of non-linear interaction between a floating structure and the waves (the simulation of interaction between a fixed structure and

the incoming wave is most facile when the same algorithms, adequately simplified, are used). To that end, the following aspects

have been analyzed:

- numerical generation of a suitable wave;

- numerical modeling of the downstream boundary;

- a homogeneous and robust treatment of the joint area of free and solid surfaces


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- development of a stable algorithm for the time incrementing of solution (time integration of free surface shape and hull

position)

- development of a stable and efficient method for the computing of pressure distribution along the hull.

The Structural aspect consists of a study into the response in time of the structure at dynamic pressure. The study

has the objective to emphasize the loading cycles in areas potentially under or over strengthened and under the condition of loads

resulting from hydrodynamic simulation being significantly different from those estimated by the conventional theories. The

purpose of this study is not only to analyze one specified structure or give red values but, more important, to lay stress on the

significance of a faithful approach.

Chapter 2. General and local loading on the hull

The classic approach to hull strength considers that stresses in the structure can be divided in two types, i.e. general and

local loading. General stresses are the result of general shear forces and bending or torsion moments. These efforts are computed

by the integration over hull length of distributed loads and weights. Local stresses are computed based on locally applied loads

and by taking into account the area characteristics. Global stresses result from the composition of general and local stresses.

This division corresponds to the beam model of hull and reflects the stage of structural analyses at a time when in-

detail computing of large models was not possible. Over the recent years, the development of the finite element method and

hardware support has made it possible to model the entire ship.

Such global analysis involves a high level of precision and requires the most accurate assessment of loads applied on

the structure. Up to present, the computing of hydrodynamic loads has been based on the hydrostatic hypothesis or on a

simplified potential model, namely the strip theory.

In brief, the strip theory computes the ships kinematic response by integration over hull length of the characteristics of

quasi-cylindrical areas. These areas are obtained by dividing the hull with transversal planes. Presumably, these areas could be

studied separately, by translating the real flow around the real section into an ideal analytical flow around a cylinder.

This model involves the acceptance of some hypotheses. The main hypotheses are:- the fluid is ideal (without viscosity), the flow can be described by the potential function ;- the derivatives of this function are small so high order derivatives can be overlooked;

- the fluid domain is unlimited and the amplitude of excitation is constant;

- the excitation is harmonic, with pulsation , and has a small amplitude;- the hull studied are long and sharp

- the ship kinematic response can be divided in two separate groups of distinct displacements: symmetric displacements (yaw,

heave and pitch) and asymmetric displacements (sway, roll and gyration);

- ships response in the form of a response amplitude operator (RAO) depends only on the pulsation of excitation and not

on its amplitude.

The computing of ships response is of great importance, thus, when reality brought forth the nonconformity of strip

theory hypothesis in many new cases, this would engender a search for new methods. The development of hardware and also of

numerical methods has led to the idea of simulation of the involved phenomena.

Chapter 3. Basics analysis. Models for hydrodynamic simulation

The equation that describes the potential flow is the mass conservation equation 0= . The following conditions

are additional:

on the free surface

- kinematics condition in complete form:Dt

Dx

x=

,

Dt

Dy

y=

,

Dt

Dz

z=

(3D case)

- dynamic condition:( )

surfacefree

zgt

_point

2

02

=+

+

on the solid boundaries, the impenetrability condition: nvnun

==

In order to solve the differential equation 0= , the following conditions should be imposed:

- Dirichlet-type conditions, namely values are known. These conditions are mainly imposed for the free

surface. One of the interesting consequences of the above is that on the free surface the waves are generated notnecessarily by the friction between the air and the superficial layer of water, but as an effect of the disturbing of

pressures on a free surface;

- Neumann-type conditions, namely the n values are known. These conditions are imposed for solid

surfaces with known or estimated movement.

If a floating body is also considered, the equation system that describes the phenomena must be completed with the

dynamic equations of equilibrium:

a) horizontal forces equilibrium equation xextxG FFx += && ,

b) vertical forces equilibrium equation zextzG FgFz += && , and

c) moments equilibrium equation extr MMJ += && where

= ships displacement;g = gravitational field acceleration;


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J = the ships mass moment of inertia along its longitudinal axis through weightcenter;

Fxext, Fzext, Mext the exteriors forces acting on the body

Fx, Fz and M the hydrodynamic forces resulted from pressure integration along the bodies

surface :

=uS

uxx dsnpF =uS

uzz dsnpF ( )[ ] uS

G dsnxxpM

u

= x where,

Su= whetted surface = Su(t)

The hydrodynamic analysis is based upon two models: Euler and Lagrange. Eulers model has proven its efficiency in

classical hydrodynamic problems, where the fluid flows in well-known domains. In case of the flow with free surface and/or with

floating bodies, this model is not efficient. The solution for the hydrodynamic study of fluid with moving boundaries is a hybridmethod named Eulero-Lagrangean. According to this method, the fluid particles are studied on their trajectories with the purpose

of computing the time-variable boundaries of domain. The kinematic and dynamic characteristics of fluid and of the body are

computed in well-known points, by the freezing in time of the domain and are time-advanced.

This method can be divided in two essential problems that will be solved successively:

- the kinematic problem consist in solving the continuity equation as a pure space-variables differential

equation (freezing in time of the domain);

- the dynamic problem consist in solving a time-variable system of differential equations (the time

advancement of the solution).

The time advancement of the solution consists in computing the integration by time of:

- the values of potential function on free surface at a new time (by integration of the t term resulted

from the dynamic condition of the free surface);

- the new shape of free surface (by time integration of fluid particles velocities, resulted from the

kinematic condition of the free surface);

- the new positions of floating bodies (by time integration of kinematic parameters resulted from thedynamic equilibrium equations).

The solving of bodys dynamic equilibrium equations involves the computing of pressure distribution on the hull. In

order for this, it is necessary to know the value of t term on a whetted surface. Up to recently, this term was computed based

on the finite difference method. This method has the inconvenient of being most sensitive to grid size and modification and also

to time step. These are the reasons of numerical discontinuities introduced by the method above. It can be demonstrated that the

t term respects the Laplace equation, as does the potential function . Taken into account the above, t can be

computed accordingly, based on the method used for the continuity equation. This idea was announced in [8] as being detailed in

[27] (reference not available). The author of this work has considered and developed it by deriving the impermeability condition.

In order to solve the Laplace equation with t unknown, it is necessary to impose the limit conditions.

On the upward boundary, depending on the excitation type, the Neumann-type condition ( ) knownnt = (for

solid boundary or the analytical wave) or Dirichlet-type condition knownt= (for the analytical wave) were imposed.

As far as the fixed boundaries are concerned, the Neumann condition has a particular form (obtained from the

hypothesis that ( ) 0= t

n at any time, for fixed boundaries, and the time and space are independent variables when the domain is

time-freezed) ( ) ( ) 0== tnnt .

On the free surface, a Dirichlet condition knownt= was imposed, where t values are obtained from the

dynamic condition of free surface.

On the downward boundary, depending on the model, a Neumann-type condition for a fixed surface or a Dirichlet-type

condition knownt= , as obtained from time integration of t term resulted from the Orlanski permeability condition

nct '= , were imposed.

The body was considered to be a moving solid surface and, consequently, a Neumann condition was used.

As far as the hydrodynamic computing is concerned, there exist two models of simulations:

- simulation of real fluid (viscous). Such models are solved mainly by the finite difference method;

- simulation of potential flow. This model is solved by the finite element method (FEM) or by the

boundary element method (BEM).

The models based on finite difference involve significant hardware resources and long running times, but the results are

very accurate. Potential methods require less significant resources and lower running times but the results are influenced by the

ideal fluid approximation right from start.

The viscosity does not significantly influence loads involved in structural computing. Moreover, the complete analysis

of ships behavior requires a complex study of a large number of cases with respect to loading, and the study should be correlated

with various sea states (also defined by a large spectrum of frequencies). The above implies that a complete analysis involves a

huge amount of computation, and, consequently, economical factors, such as the running time and hardware requirements,

become essential.

Over the recent years, this methodology has been of special interest to important classification societies and design

institutes all over the world, confirming the fact that potential methods are optimal for hydrodynamic studies carried out in view

of delivery the inputs for structural analysis.


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From the start, we should make the observation that, irrespective of the method employed, the potential flow is based

upon a sole algorithm. This algorithm solves a kinematic and a dynamic problem for every time-step. What makes this

observation important is that it emphasizes the fact that irrespective of the method employed for the time advancement of

solution, the continuity equation can be solved either based on the finite or on the boundary element method. This means that

whatever difference there exists between the two methods above, it should be found in the kinematic problem.

In all accessed bibliographical reference with respect to FEM, there can be observed that the result of modeling is still

rather different to reality. The past reference used approximation and liniarization with respect to free surface conditions (and

usually t is ignored). The recent reference [50] considers the ship fixed, nevertheless, the free surface is completely

modeled. The authors simulation [76], [77], establishes that FEM is more sensitive to wave raise than BEM. BEM has

successfully modeled the start of wave extreme sharping in the propagation direction. This phenomenon is forerunner of energy

stability loss, namely of overturning, and can not be modeled with FEM. The explanation of the above lies in the fact that FEM

requires elements with rational forms and shapes in order to maintain the numerical stability. From this observation, serious

question has arisen regarding FEMs capacity to model the phenomena in a situation characterized by large displacements of the

body. The boundary resizing, as required by BEM, is more facile than the division of the entire domain, as required by FEM.

Other notable advantage of BEM consists in the reduction with one unit of the geometrical dimension of problem.

Thus, the plane problem becomes a linear one, while the spatial problem becomes a plane one.

The variables attached to the points on the inside domain by FEM are overlooked, since they can be expressed as a

function of boundary points with the same values of variables. BEM also uses a set of variables with physical correspondence,

i.e. the normal-on-boundary velocities. The importance of this detail is worthy of consideration, particularly when the results or

the initial debugging of software require checking.

Nevertheless, during the works he has carried out, the author found an unexpected advantage of FEM, but it is also true

that this can not compensate for the first two advantages of BEM. Thus, the energy conservation principle is the very essence of

FEM, therefore, the numerical dissipation in FEM appears to be far less than in BEM. The waves simulated by FEM maintain

their amplitude approximately constant. The amplitude of waves generated with BEM decrease exponentially and proportionally

with the grid size, the trial function grade and the distance from the excitation source.

In conclusion, the author considers that BEM is the optimum method for the modeling of potential flow of free-surfacefluid, in the presence of a floating body, and, consequently, has employed this method in his thesis.

Chapter 4. The model developed in the thesis. Original software.

In order to model the continuity equation ( 0= ), the thesis has used a boundary element with linear variation along

segment, for both the potential function and its normal-on-boundary derivative n . The equation has the form below:

( ) ( ) ( ) ( ) ( ) ( )

+

++=

++

j

dj

j

cj

j

bjj

ajj xT

nxT

nxTxTxxc

11

For computing the T coefficients, there has been used a local coordinate system, attached to each boundary element.

This system has the origin at the intersection point between the segment and the perpendicular on segment starting from current

point i. Thus, a constant ordinate i and the abscises j and j+1 (corresponding to segment ends points) are obtained. With thiscoordinates, the T coefficients can be computed analytically, without numerical integration (Gauss quadrature).

We should begin by computing the ( )( ) nr ln term. Taking into account that 222 += ir and dn=di we obtain:

( )( ) ( )( )222

22ln

2

1ln

rn

r i

i

ii

i

=

+=+

=

The final formulae are:

+

+

+

=

+++

i

j

i

j

j

j

ji

ji

j

iaj arctgarctg

LLT

1122

21

2

ln2

+

+

=

++

i

j

i

j

j

j

ji

ji

j

ibj arctgarctg

LLT

122

21

2

ln2

+

+

=

+++

i

j

i

j

j

ij

ji

ji

j

icj arctgarctg

LLT

1122

21

22

ln4

( )[ ] ( )[ ]{ }1ln1ln4

1 22221

221 ++

+ ++ jijjij

jL ( ) ( ) 12212 12

21

ln2

ln2

++

++

++

++

jjij

jjji

j

j

LL

+

+

+=

++

i

j

i

j

j

ij

ji

ji

j

idj arctgarctg

LLT

122

21

22

ln*4

( )[ ] ( )[ ]{ }1ln1ln4

1 22221

221 ++

++ jijjij

jL ( ) ( ) jji

j

jji

j

jj

LL

+

+

+ +

+ 222

21

21 ln2

ln2

There can be observed that, when the current point i coincide with j or j+1 nodes, singularities are obtained. In such

cases, despite singularities, the integrals can be computed analytically, if we consider the following:


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a) the integrals containing( )( )

22

ln

+=

i

i

n

rare nulls because i = 0 for every .

b) The integrals containing ln(r) (which for i = 0 became ln (2)/2), are computed based upon the limit

( ) ( )( ) ( )aaxxxdxx aa

lnlnln0

0

== resulting from ( ) 0lnlim0

=

xxx

.

Final expressions in this case are:

0=ajT ; 0=b

jT

For i=0 and j=0

( )[ ]{ } ( ) 12 12

121

21 ln

21ln

41

+++

++ +

+= jjj

jjj

j

cj

LLT ( )[ ]{ }1ln4

1 21

21

= ++ jj

j

dj

LT

For i=0 and j+1=0 the same procedure can be applied.

This modeling is also used for pressures Laplace equation ( ) 0= t . The t terms (implicitly the pressures on

body) and the bodies accelerations are obtained by solving the following equation. Using BEM (as in case of the continuity

equation) the following equation is deduced:

( ) ( ) ( ) ( )( )

( )( )

( )

+

+

+

=

++j

dj

j

cj

j

bj

j

a

j

xTn

txT

n

txT

txjTt

xt

xc11

where the T coefficients are computed in the same manner as for 0= .

From the analysis of the boundary condition for this equation, there can be observed that the ( ) nt term is

difficult to be determined. In order to compute this term, we should start with the impenetrability condition and apply it for the

solid surface of the body nvnun == scalar expressed as uxnx + uznz = vxnx +vznz , where u is the particle velocity and

v is the velocity of solid boundary. By a material derivation

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ndt

duvnv

dt

dnu

dt

dn

dt

dvnv

dt

dn

dt

dunu

dt

d+=+=+ [4.2.1]

The normal-to-surface is a function of instantaneous roll angle ( ( )tr ) and of the current point position (defined by s):

( )( )stnn r ,= . By a material derivation, ( )( ) ( )

dt

ds

ds

nd

dt

d

d

ndn

dt

d r

r

+=

[4.2.2]. The constituent terms are detailed below.

Thus ( ) svudtds = (first term representing the relative velocity while s is the tangent- to-surface unit vector) and

R

s

ds

nd= , where R is the surface radius of curvature in current point. Finally

( )dt

ds

ds

nd = ( )( )

R

ssuv [4.2.3].

The normal-to-boundary unit vector depends on angle (the angle between the tangent to the boundary and the local

horizontal axis function by s parameter) and rolling angle r (the angle between the local horizontal axis and the global

horizontal axis) ( ) ( )( ) ( ) ( )( )( )tstsn rr ++= cos,sin . By derivation with r the following is obtained:

( )( ) ( )( ) ( ) ( )( )( )tttt

d

ndrr

++= sin,cos which is identified with

( ) ( )( ) ( ) ( )( )

++

=

0cossin

100

tsts

kji

nxk

rr

( k is the

normal on study plane unit vector).

=dt

d r and = k so that the expression( )

dt

d

d

nd r

r

= nx [4.2.4].


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Adding [4.2.3] to [4.2.4], the expression [4.2.2] becomes, in its final form,

( )( )

( ) ( )( )

=

R

ssuvnuv

dt

nduv x

With the deduction above, the expression [4.2.1] has the following form:

( ) ( ) ( ) ( )( )

+=

R

ssuvnxuvnv

dt

dnu

dt

d [4.2.5]

Note: for the straight zone, R is infinite so ( ) ( )( ) 0=R

ssuvuv , thus, in this particular case

( ) ( ) ( )( )nuvnvdt

dnu

dt

dx+= .

But( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( )

+

+

+

+

=+

=

z

uu

x

uu

t

u

z

uu

x

uu

t

uuu

t

u

dt

ud zz

zx

zxz

xx

x ,

so( ) ( ) ( ) ( ) ( ) ( ) ( )

zz

zz

xxx

zx

xzz

xx n

z

uu

x

uun

z

uu

x

uun

t

un

t

un

dt

ud

+

+

+

+

+

= [4.2.6]

Because ( )

=

=

txxtu

tx

and also ( )

=

=

tzztu

tz

we obtain

( ) ( )

=

+

=

+

tnn

tzn

txn

t

un

t

uzxz

zx

x . Given the above, the relation [4.2.6] has the form

( ) ( ) ( ) ( ) ( )z

zz

zxx

xz

xx n

z

uu

x

uun

z

uu

x

uu

tnn

dt

ud

+

+

+

+

=

[4.2.7].

By equalizing the expressions [4.2.5] and [4.2.7], the term of interest here, namely ( ) nt is finally deduced:

( )( ) ( )( )

+

+

+=

z

uu

x

uun

z

uu

x

uun

R

ssuvnuvn

dt

vd

tn

zz

zxz

xz

xxxx

Taking into account the basic equality between the potential function derivatives ( xzzx = 22 and

2222zx = ), the two last terms can be expressed as ( )22vn .

For practical use of this expression, we should also detail the ( ) ndtvd term.

For a point on the body, the velocity is expressed as:

rxvv G += , where Gv = the velocity of the bodys weight center,

= the angular velocity of the body,

r= the position of the current point in respect of the bodys weight center.

By material derivation, the velocity it is obtained as below:

( ) ( ) ( )rxxtrxrxvdtvd G +++=&& ,

where: GG av =& = the bodys weight center acceleration vector,

&= the bodys angular acceleration,

vutr = = the relative velocity between the fluid particle and the body.

Unlike the methods usually employed for the computation of ship dynamics, the modulus-of-position vector r is not

constant in time. In respect of this, tr represents the relative velocity while ( ) tv does not represent the acceleration of a

fixed point from the surface. This term represents the variation in time of velocities of points tangentially touched by the same

fluid particle in its movement along the bodys boundary.

The final expression of the ( ) nt term is:

( )( ) ( )( ) ( )

+

+

+++

z

uu

x

uun

z

uu

x

uun

R

uvvunnrxxrxv zz

zxz

xz

xxxG

2

x2 && The computing of last

terms (such as xux or xuz ) is difficult. In order for this, we have noted:

sin= xn respectively cos= xs , where ( ) ( )tt r += andcos= zn and sin= zs

The velocity expressions in the local coordinate system are:

sincos nsx uuu = cossin nsz uuu +=

The derivatives in respect of the global coordinate system are expressed based on thelocal coordinate system by the

formulae:

( ) ( ) ( ) ( ) ( )snx

s

sx

n

nx

+

=

+

=

cossin

( ) ( ) ( ) ( ) ( )snz

s

sz

n

nz

=

+

=

sincos


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Additionally, taking into account that the continuity equation is independent of the coordinates system, there results the

following: nusu ns = . Moreover, because the partial derivatives of potential function do not depend on the derivation

order, we also obtain nusu sn = .

Using the above, the following expressions are obtained:

( )z

u

s

u

s

u

x

u znsx

=

=

cossin2sincos 22

( )x

u

s

u

s

u

z

u zsnx

=

+

=

cossin2sincos 22

As there can be observed, the final goal of this deductions was to obtain expressions composed only by tangential

derivatives and without including normal derivatives (difficult to be computed in nodes). Initially, the author computed this

derivatives based on the finite difference. This solution generates divisions by zero when the ships sides are parallel with one of

the global coordinates axis. To avoid this problem, in a situation close to above (the term closed is defined by an arbitrary

constant) the dangerous derivatives are computed based on the properties generated by the continuity equation and the

independence-to-derivation order. The problem has been seemingly solved, but evident discontinuity appears in force diagrams

(the most sensitive since they represent the primary results of the integration of pressures). The above have determined the author

to try and compute this terms analytically, but he has met serious mathematical difficulties. A more detailed analysis of the goals

and of the disposable tools has emphasized the necessity of establishing a method that is independent of geometric particularities

and uses only the tangential derivatives, which can usually be computed with BEM.

The above explains how we arrived to the a.m. expressions. The s can either be computed based on roughly finite

differences or calculated analytically, if high order boundary elements are used.

The modeling of intersection points between the free surface and the solid surface has some particulars. These points

present discontinuities of boundary tangent, this implying that the derivatives on the right and left side of the point are different.

Thus, for these points, the potential function has the same value, while the normal derivatives of this function have two values.

The potential function value is determined with the free surface condition while one of the normal derivatives results from the

impenetrability condition on solid boundaries. Finally, in these points, only one normal derivative is unknown, this being the

normal velocity of free surface.

The excitation is introduced through the upward boundary. This boundary can model a physically equivalent excitatory

or an analytical wave can by imposed on it.

The model excitatory (fan wing or piston type as used by hydrodynamics research laboratories) is recommended by the

author for full non-linear fluid flow. Thus, from start, the flow is determined based on the uniform set of equations. The mass

error, which is lesser even if the downward boundary is fixed and solid, also evidences the correctness of this method.

Unfortunately, the author has not established a correlation between the movement parameters of excitatory and the amplitude of

generated waves.

The fluid excitation can be produced also by imposing the characteristics of an analytical wave. These characteristics

can be either the potential functions or the velocities obtained from this potential.

In [92], if we impose the Airy linear wave potential function values on the upward boundary, we obtain sawtooth-

type instabilities of free surface. These instabilities are due to the contradiction between the value of potential function from the

Airy expression and computed directly at the point of intersection between the free surface and the upward boundary. The value

determined by time integration of the free surface dynamic condition is different from the value imposed analytically. The

solution found by the references author was to introduce a damping zone where the transition from analytical values tocomputed values would be made. This solution involves some new, supplementary constants, namely the length of damping zone

and the damping coefficient. Generally speaking, the use of constants should be avoided because, usually, their optimal values

are determined by trials. Also, the authors attempts to use this method have failed.

A more faithful approach consists in imposing a velocity field on the upward boundary. This model is the closest one to

reality because the values of potential function in the a.m. intersection points are computed based solely on the dynamics

condition of free surface. Additionally, the tests have led the authors to observe that the a.m. problem does not occur. Also, the

author has ascertained a good correlation between the respective parameters of the analytical wave and the numerically generated

wave (amplitude, wavelength). The latter observation has led to the use of this type of excitation in those cases where a wave

with certain characteristics must be obtained.

The author has modeled the downward boundary in two ways: as a solid fixed surface or as a permeable boundary. The

model based upon a solid fixed boundary is the most realistic one, but has the disadvantage of generating reflected waves that

interfere with the incoming wave, perturbing the ships regular movement. This type of boundary can be successfully used in

correlation with an extended domain or for the study of the body in bounded domains (basins, harbors, and channels). The

permeable boundary is based upon Orlanski condition nct '= , where c is a parameter with the dimensions of a

velocity. In [92] we recommended c=c, where c is the wave velocity. For this situation the reflected wave is not significant. Theauthor has also ascertained this fact.

The mass error evolution establishes the permeability of the boundary. Thus, over a period of time equal to the time

necessary for the wave to cross the domain, the error has a harmonic variation, with a positive average value (the fluid is pushed

inside the domain). Over the next period of time, the error maintains the harmonic aspect, but the average decreases to a negative

value (initially, the fluid is sucked by the permeable boundary) and afterwards tends to stabilize near zero (the stabilizing).


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The solving scheme for fluid flow in the presence of a floating body

Fig. 4.6.1.1 Boundary conditions for 0=

Fig. 4.6.1.1 presents the boundary conditions for the continuity equation.

At time-step ti the following are known:

- the boundary of D domain, namely the shape of free surface, the ships position as defined by Gx and r and the

intersection points between the free surface and other boundaries, as defined by x

- the Neumann-type conditions the normal-on-surface velocities on solid surfaces (i.e. the bottom of domain, the fan wing

and body surface) and on upward boundary

- the Dirichlet-type conditions the potential function values on the free surface and downward boundary

The dynamic problem consist in the computing of the following parameter at time step t i+1 :

- the free surface shape is determined by time integration of the free surface kinematic condition ( ) ( )+

=+

1

0

1

it

i dtDtxDtx where

zjxiDtxD += ;

- ships position by ( )

+ +

=

1 1

0 0

1

i it t

GG dtxtx && and ( )

+ +

=

1 1

0 0

1

i it t

rr dtt &&

- the intersection points between the body and the free surface ( )+

=1

0

it

vux , where u is the velocity of the fluid

particle and v the velocity of the body, both expressed at the intersection point- the intersection points between the free surface and the upward / downward boundaries

- the new fan wing position

- the potential function values on the free surface ( ) ( )+

=+

1

0

1

it

i dtDtDt , where ( ) gzDtD = 22

, as resulted from

the dynamic condition of free surface

Note: is computed in the same manner for every intersection point of the free surface with other boundaries (solidboundaries or boundaries with special condition imposed)

- the potential function values on the downward boundary ( ) ( )+

=+

1

0

1

it

i dttt , where ( )nct = ' (the Orlanski

permeability condition).

In order to compute the Gx , r and v terms, it is necessary to solve the ships movement problem. The ships

movement is easy to know if the instantaneous accelerations were determined. The ships accelerations are obtained based on the

dynamic equilibrium equations, namely extS

G Fdsnpx

u

+= && - the forces equilibrium equation and

( ) extS

r MdsnpxrkJ

u

+

= && - the moments equilibrium equation.


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The pressure p is expressed as ( )( zgtp ++= 22 . In order to compute the t term, we shouldsolve the ( ) 0= t equation.

Fig. 4.6.1.2 Boundary conditions for ( ) 0= t

Fig. 4.6.1.2 presents the boundary conditions for ( ) 0= t equation.

At time ti the following are known:

- the domains boundary

- the Neumann-type conditions - ( ) nt on solid boundaries (i.e. the domain bottom, the fan wing or the

body surface) or on the upward boundary if an analytical wave was imposed

- the Dirichlet-type values - t on free surface (from the dynamic condition on free surface) and on the

downward boundary (from the Orlanski permeability condition)

With the solving of ( ) 0= t system extended with the dynamic equilibrium equations, the body accelerations and also

t values on body surface are obtained.

Original software

The software used for hydrodynamic simulations and for the post-processing of results has been developed by the

author in programming language TURBO PASCAL, release 7.0.

The running times are proportional to the third power of the number of model points. This has led to the attempt by the

author to reduce as much as possible the number of points.

The running effort is substantial for a P.C., especially in the design stage. With the method having been finalized, the

effort appears reasonable.

The software uses the units technique. The main program is DMAIN. This program calls the DTIP, DINIT,

DGRILA, DASM, DGAUSS, DVIT and DFORTE units.

All modules call the DTIP unit, since it contains the type declarations and the global variables.

The DGRILA unit contains procedures that initialize the geometry and on-screen drawing at each time-step.

The DINIT unit contains the procedures which initialize the boundary conditions at each time-step, for 0= and

( ) 0= t equations.

The DASM unit builds the matrices [M] and [Mp] and also computes the c coefficient. This unit calls the DELEM

unit.

The DELEM unit computes the djc

jb

ja

j TTTT ,,, terms that are assembled to form [M] and [Mp].

The DGAUSS unit solves the system of equations based on Gauss elimination method.

The DVIT unit computes the velocities on boundaries; firstly the tangent velocities s and finally the velocities in

the global coordinates system.

The DFORTE unit computes the pressures on the body and integrates them in order to obtain the

hydrodynamics forces.

All the results are post-processed based on the DES_NAV program that was written by the author in the same

programming language.


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Chapter 5. The revision of method in comparison with experiments and simulation from bibliography

In order to verify the method, the results of simulations have been compared with those of similar experiments and

simulations presented in bibliography.

Obviously, the first test consists in waves generation. The waves have been generated both by reproducing the reality

(fan wing in harmonical movement) and by imposing a velocity field derived from the Airy linear wave potential on the upward

boundary. This work presents the case of a wave imposed on the upward boundary, which will also be used to excite the floating

body. The downward boundary is of permeable type.

The author has found a good correspondence to the estimated values. Thus:

- the wave height is near to the theoretical height (taking into account that the attenuation is proportional to the

distance from excitation)- the wave period (computed as difference between successive maximums) coincide with the excitation period (as it

is normal for a forced oscillation)

- the wave length (computed as distance between two maximums on the horizontal axis) is in good concordance

with estimated wave length

- the wave group velocity (computed as ratio of the abscise of a point to the time necessary for the wave to reach it)

corresponds to the theoretical estimation

Fig. 5.1.1 The free surface geometry at t= 10.075 s

Fig. 5.1.2b Variation of free surface elevation versus time at x= 3

From the numerical simulation, which used a fan wing waves generator, it results that the mass error is much less

significant than in the case where a velocity field was imposed on the upward boundary. Moreover, using this type of excitation,


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the author has obtained extreme states of waves, namely shaping and curving in the propagation direction of waves that are

characteristic to the time prior to overturning.

Fig. 5.1.3. The mass error time variation

In order for an optimum presentation of the mass error evolution to be ensured, the duration of wave simulation is of an

18-wave time-period. There can be observed that:

- over the first 3 time-periods, the error has had positive values, which indicates a mass inflow. This inflow is a

consequence of the expression that defines the velocity field;

- over the time between the 3rd and the 7th time-period, the mass error has decreased and became negative. As a

physical phenomena, this represents the moments when the waves reach the permeable downward boundary (the domain

length is of 5). The mass surplus initially introduced starts to be eliminated;- over time between the 7th and the 10th time-period, the mass error tends to reach an equilibrium. The mass flow is

now stabilizing;

- over the period between the 10th and the 18th time-period, the mass flow is stable. The average value is still

positive, but non-significant (1%)

In order to compare the mass error evolution obtained in case of wave generation without a floating body, with that obtained

in a similar case and also in the presence of a body, the relative mass error has been calculated as ratio to the floating body mass

(the mass of the model used in the experiments presented in [92]). The formula used is: (V-Vo)/M*100 [%], where V / Vo are

the instantaneous / initial fluid volumes, the fluid density and M is the body mass.

The next important step to be taken in view of method confirmation is to compare the ships simple free motions (free

roll and heave oscillations) computed by numerical simulation and by experiments (as presented in [92]).

Fig. 5.2.1.2 The ships vertical relative displacement (h/H)


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Fig. 5.2.1.4 The dynamic component of pressure on the ship at the times t=0.460 s and t= 1.060 s

The free heave motion demonstrates a good concordance between the experiment and simulation. The results that we

compared were the time period of oscillation and the logarithmic attenuation factor.

Fig. 5.2.2.2. Time evolution of ships relative roll angle (u/U)

Fig. 5.2.2.4 The dynamic component of pressure on the ship at the times t= 0.52 s and t= 3.76 s

As far as the free rolling motion is concerned, the theoretical estimation is not possible without some extrapolations

from table coefficients. For this reason we have disregarded the comparison with the strip theory estimation. The simulation

results are different from the experimental results. Thus, the experimental roll period (1.83 s) is shorter than the numerical

period (2.16 s), while the experiment shows a more pronounced attenuation than in case of simulation. This tend is also reported


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in [92]. In order to compensate for this difference, the author of the reference [92], interfered numerically in the dynamic

equilibrium equation of rolling, following the classical method. He determined the coefficients particularly involved in that

specific case by making successive comparisons of the numerical results with experimental results. Nevertheless, the author of

this thesis considers that this procedure does not represent an adequate solution. For the future and in order to solve this problem,

the author proposes that a high order element be used. Until then, the results should be considered satisfactory.

As far as both movements are concerned, the non-linearity of the dynamic pressure on the body is emphasized.

Fig. 5.3.1.d Body in free movement at t=3.00 s

Fig. 5.3.1.h Body in free movement at t=6.00 s

Fig. 5.3.1.i Body in free movement at t=9.52 s

The next comparison is to the simulation results presented in [35]. Here, it is presented the complex movement of the

body as a result of the incoming wave. The waves are generated by a fan wing. The comparison emphasizes the difference

between a solid-fixed type downward boundary and a non-reflexive type boundary (as a permeable boundary). In this thesis, the

author has used a solid fix boundary. The bodys evolution over the period of time before the wave reached the downward

boundary proves the correspondence between the methods and the simulation conditions. During the next periods of time,

differences between simulations occur, the most evident being the sway. As far as this thesis is concerned, during these periods of


14/30

time, the ship had the tendency to move backwards, in the direction opposite to wave propagation. Such behavior proves the

existence of a reflected wave generated by a solid-fixed downward boundary. As for the reference, the ship is moving in the wave

direction, proving that the downward boundary is of non-reflexive type. This difference between models is also indicated by the

roll angle evolution. With respect to this, in the reference, the rolling angle is stabilized in time while in this thesis the roll angle

decreases in time, evidencing an opposite direction to the initial one

Fig. 5.3.3. Ships weight center time-space evolution

Fig. 5.3.4 Ships roll angle time variation


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Fig. 5.3.5. The dynamic component of pressure on the ship at the times t=3.56 i 3.88 s

The method has been verified for a large range of frequencies, by comparison of numeric results with the experimental

results presented in [92]. The author of reference [92] tested the model in a Canadian wave flume and in strictly plane flow

conditions. The model was restricted in the sway direction. Such restriction presents certain advantages for the experiment, and

also reproduces the reality since all the ships maintain their route by rudder.

The comparison of the results indicates the existence of an average difference of about 15%. The analysis of

differences finds the explanation for the above in the non-consideration by the reference of parametric rolling and feeding

phenomena.

In [92], the fluid flow was simulated based on a boundary element with constant variation on length, for both the

potential function and its normal-on-boundary derivative ( n, ). The ships accelerations were computed by direct

integration of pressures along the hull and t terms (requested for pressure computation) were determined by finite

difference. All these factors have a linearization effect and lead to a weak modeling capacity of non-linear and transitory

phenomena.

Another characteristic of simulations and experiments in the reference is their reduced duration, i.e. about 10 time

periods.

The feeding phenomena is characteristic of transitory regime. The ships response contains both the forced oscillation

and the ships motion, which is damped in time.

The simulations in this thesis also obtained this phenomenon. The feeding evolution is influenced by the damping

effect. With respect to this, the frequency of feeding is in good correlation with the period estimated theoretically, but the values

are higher and the amplitude is decreased to forced motion values.

The feeding phenomenon explains the difference between simulation and experimental stabilized oscillation.

This phenomenon also appears in case of higher period roll motion.

Fig. 5.4.a Case 1 Adimensionalised ships weight center time motion


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(typical example for heave feeding phenomenon)

Fig. 5.4.c Case 11 Ships roll angle time evolution [degr]

(typical example for roll motion feeding phenomenon)

In most cases, the parametric rolling is triggered after the 10 th time-period. The reference presents simulations and

measurements for a restricted duration. This explains why, the phenomenon above is not evidenced by the reference. This

phenomenon consists in the progressive increasing of roll angle. These big angles are reflected in the non-usual extremes

evidenced by the sway force evolution and by the heave motion.

This parametric roll is different to the classical phenomenon described in bibliography ([15]). In the classical

presentations, the phenomenon occurs as a result of large amplitude pitch motions. In this thesis, the phenomenon occurs as a

result of large amplitude heave motions in beam waves. Similar to the classical situation, the period of waves that trigger the

catastrophic rolling is near half of free roll period.


(typical example for parametric rolling)



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(typical example for stable roll motion)

These phenomena prove that the non-symmetrical motions can not be decoupled from the symmetrical motion. This

conclusion is in contradiction to the strip theory hypothesis.

The reduced duration of the simulation and experiments in [92] is also one of the reasons why this phenomenon does

not appear.

By comparing the relative mass error of cases that include a floating body with that obtained in similar conditions but

without a body, there can be proved that the error is largely due to the upward and downward condition and not to the body

motion or free surface distortion. The mass error has the same evolution in all cases. With respect to quantity, the error amplitude

is proportional to the amplitude of excitation applied on the upward boundary. This observation emphasizes the importance of

boundary conditions.

Fig. 5.4.d Case 6 Relative mass error time evolution [%]

(typical example for relative mass error evolution for comparison with Fig. 5.1.3)

The last comparison is with the results presented in reference [58]. This comparison aims to confirm the accuracy of

pressure computation based on this method.

Reference [58] presents the case of a Millennium-class oil tank, as shown by a study carried out as a result of somedamages being ascertained on the longitudinal side, in the loading lines area (ballast line in one side and full loading line in the

other side). In the reference, the authors consider that these damages are due to the incorrect analysis of dynamic effects based on

ABS methods, particularly with respect to the estimation of unsteady hydrodynamic pressures on hull.

The study consists in testing a towing tank in deep water, in order to confirm the results of SPLASH numerical

simulations.

Fig. 5.5.1. Ships weight center motion [m]


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Fig. 5.5.2. Ships roll angle evolution [degr]

The author of this thesis has made a simulation in conditions identical to those in the reference case (beam wave with a

3.0-m amplitude) and compared the results, which he has found to be in agreement. The results are:

- the amplitude of the relative ship-wave motion in mid-ship section: in the reference this was measured

experimentally at 3.11 m above mark while the numerical simulation measured it at 3.6 m above mark, starting with the 6 th

time period;

- the mean roll angle: 2.7 - experimental, as per the reference, 2.5 - as per this thesis;

- the pressures on windward side, in the cylindrical area, as expressed by Response Amplitude Operators (RAO): at the point situated at 12.0 m above base line: RAO=27.62 kN/m as per the experiment and RAO 25 kN/m asper the numerical simulation

at the point situated at 7.75 m above base line: RAO=27.10 kN/m as per the experiment and RAO 25 kN/m asper the numerical simulation

at the point situated at 3.5 m above base line: RAO=24.93 kN/m as per the experiment and RAO 21.7 kN/m asper the numerical simulation.

Fig. 5.5.4. Pressure evolution at point P13 (12.0 m above B.L.)

Fig. 5.5.5. Pressure evolution at point P10 (7.5 m above B. L.)


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Fig. 5.5.6. Pressure evolution at point P7 (3.5 m above B. L.)

Fig. 5.5.8a Pressure evolution at point F.S.+3.6 m

According to the reference, lower values of pressure RAO - with the same tolerances, are obtained based on SPLASH.

The authors of the reference consider the numerical simulation results to be completely satisfactory. They are of the opinion that

small differences between experiments and simulation are insignificant in comparison with those resulted from classical theories.

As far as this study was concerned, the authors were mainly interested to confirm the real values of pressure in the context of

classical theories, and to find therein an explanation for the damages produced on the side longitudinals. Their explanation lies in

the fact that dynamically computed pressure can be several times different than the static pressure estimated for wave amplitude

being disposed up and down the mark.

The overall conclusion of this verification is that the presented method is both reliable and efficient. Based on a high

order boundary element and on a computation method with low dependency on grid size and time step, transitory and non-linear

effects could be simulated, that are difficult to study even by experiment.

Chapter 6. The computation of pressures on the analyzed hull

In order to study the influence of real pressure calculation on the structure computation, we have chosen a 38000-dwt

oil tank. As far as the loading is concerned, we have considered the case of full loading, with the ship on an approximately even

keel. The ships inertia radius is determined both based on a formula in the bibliography and on the main components of total

inertial moment in midship section (steelwork and cargo), the latter being used for comparison purposes. Both values

demonstrate a high level of coincidence, confirming, thus, the result.

The pressure variation in time is computed at the middle points between the side longitudinals, on windward side The

wave characteristics are the following: time-period: 8.5 s (maximum of energy spectrum for North Atlantic) and amplitude: 2.5

m. Since a parametric rolling was established at advanced times, for the structural computing, we have considered only the first

13.5 time-periods out of the 18 simulated periods (on the trigger of catastrophic rolling, the ship change its course in order to

modify the incoming wave parameters).


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Fig. 7.2.4. Pressure variation at point z= 2.4 m from B.L. (1-st panel)

Fig. 7.2.5. Pressure variation at point z= 7.2 m from B.L. (panel 7)

Fig. 7.2.6. Pressure variation at point z= 12.0 m from B.L. (F.S. level)


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Fig. 7.2.7. Pressure variation at point z= 15.2 m above B.L. (panel 17)

Additionally, we have made simulation of ships motion under waves with a 11-s time-period, and also with several

amplitudes: 1.5 m, 3.0 m and 6.0 m. This simulation proves that the amplification ratio of the motion (the ratio of ships response

to wave height) is highly dependent on the excitation amplitude. The strip theory considers that this ratio is dependent only on

the frequency of excitation. With respect to this, the strip theory can only compute average values of the above, which are useful

for kinematic estimation. The consequence, as far as the structural computing is concerned, is that the pressures determined on

the basis of kinematic response, as computed by the strip theory, are incorrect.

Chapter 7. The study into the effect on structure of the pressure computed based on the numerical modeling of fluid

We should start by emphasizing that the aim of this study is to present a qualitative approach and not a quantitative

one. By this study we have wished to stress the influence of a faithful hydrodynamic computation on the structural stresses. As

presented in [108] (and also indirectly in [58]), the dynamic analysis of ship highly depends on loading and navigation route.

The carrying out of such studies is a most complex work, and the results are very specific to the ship studied, having but a

limited possibility of being extrapolated to certain types of ships. Taking into account the above, the author considers that the

practical way to be followed by the dynamic analyses of ship structures is to establish a reliable method and not to search for a

universal solution.

The structure studied is a double skin of the a.m. oil tank. This structure is studied with the Finite Element Method

program COSMOS. The model is composed of 4 non-linear nodes, plate elements, with membrane and bending properties.

Fig. 7.3.2. View on the structure (top area without side)


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Fig. 6.1. Midship section

In order to compute the structures dynamic response, two methods can be used: the eigenvalues superposition method

or the direct time integration method.

As far as the use of eigenvalues superposition method is concerned, the first step is to compute the natural frequencies

and vibration shapes (eigenmodes). Our try to compute the eigenmodes of such a structure (a thin-skinned tube with longitudinalstrengthening non-fused with the shell) has revealed a response consisting in a mixture of structure and details eigenmodes. This

result is confirmed by the reference [1]. From this experience, the author has concluded that the study of eigenmodes for such a

model should be done either by fusing together the strengthening and the shell or by exciting the structure, with the simulation of

an impulse-type excitation. With last posibility, at least the first natural mode is computed.

Due to the fact that he has not had access to software that uses selectively the raw computed eigenmodes, the author

has used the direct time integration method.

This study has made a comparison between the dynamically computed stresses and the stresses computed based on the

hydrostatic pressure of wave. In this study, the author did not include the longitudinal wave effect, which he considers worthy of

consideration. An extension of simulation in 3D would make the taking into consideration of this effect possible.

The static cases correspond to certain stages of wave: crest, mean and depression. With respect to this, the ship

immersion is up to:

- loading mark line + wave amplitude: the hydrostatic pressure corresponds to a water height H = 12.0+2.5= 14.5 m above

base line (static case 1)- loading mark line: the hydrostatic pressure corresponds to a water height H = 12.0 m above base line (static case 2)

- loading mark line - wave amplitude: the hydrostatic pressure corresponds to a water height H = 12.0-2.5= 9.5 m above base

line (static case 3)

The pressure at a point with z above B.L. is p= g(H - z). On each panel, a constant pressure is applied, which is computedat the middle point of the panel.


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Fig. 7.6.1. x stresses on longitudinal L21

Fig. 7.6.2. x stress time evolution for point P1385.For the first side longitudinal (2800 mm from base line), L21, equivalent stresses have been computed. The stresses

which are significantly different from the rest, are found in the areas below:

- the middle point between the support points to ring (web) frames (P1585),

- at support points to ring frames (P1385).

The maximum of stresses is obtained in points situated at middle distance between support points to first two adjacent totransversal tightness bulkhead webs (P1585).

PointP1385:for static case 1x= -36.4 kN/mm2 and e= 32.0 kN/mm2; for static case 2x = -28.6 kN/mm2 and e=

25.2 kN/mm2 ; for static case 3 x= -20.1 kN/mm2 and e= 18.4 kN/mm2. From the dynamic computation, there results that the

average values of the x extremes are of 27 and 9 kN/mm2 respectively. These values indicate that the use of hydrostaticpressure led to an overloading of structure.

This observation is also valid for point P1585. For static case 1x= +42.2 kN/mm2 and e= 40.2 kN/mm2 ;for static case

2 x= +33.2 kN/mm2 and e= 31.6 kN/mm2 ; for static case 3 x= +24.1 kN/mm2 and e= 23.0 kN/mm2. From the dynamiccomputation, there results that the average values of the x extremes are of +29 and +10.7 kN/mm2 respectively, while those ofthe equivalent stress are of 31 kN/mm2 and 10 kN/mm2.

Fig. 7.6.3. Equivalent stress time variation for point P1585


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Due to the fact that the area is strengthened, the main stresses do not respect the directions of global coordinate system.

For this reason, the equivalent stress is considered to be relevant. The dynamic analysis has shown that the average extremes of

equivalent stress are of about 27 and 9.6 kN/mm2. The overloading effect of hydrostatic pressure has also been observed.

For L32 longitudinal (11600 mm above base line, 400 mm below mark line), the critical points are the same with the

supports to ring frames i.e. pointP1480.

For this point, in static case 1,x= -25.7 kN/mm while being insignificant in the two other static cases (the hydrostatics loads

are very low). The dynamic analysis has shown that the average extremes ofx are of about -28.5 kN/mm and 8.5 kN/mm,proving the fact that the loading cycle resulted from the hydrostatic hypothesis is incorrect.

Fig. 7.6.22. x time evolution for point P1480

The L33 side longitudinal (12400 mm above base line, 400 mm above mark) has been stressed to a low degree based

on the hydrostatic loads (the wave wets this point only during the wave crest). The point 1258 has e= 2.9 kN/mm. Thedynamic loads lead to an equivalent stress with extremes of about 16.3 kN/mm and 4 kN/mm.

Fig. 7.6.23. The equivalent stress time variation for point P1258

The analysis of transversal tightness bulkheads has not revealed any red area.

The ring frames have many critical areas. One of these is the lowest water-cross hole, which represents a stress

concentrator. This situation is corrected by the classification society remarks.

Other points relevant to our study are situated in the support area of ring frame to cross tie. For frame 120, these points

are P1090 and P1089.


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Fig. 7.6.24. The equivalent stress distribution for ring frame 120

Point 1089: static case 1e= 40.1 kN/mm; static case 2e= 29.1 kN/mm and static case 3e= 18.4 kN/mm. The

stress constituent with maximum value is yz. The dynamic analysis has indicated a maximum of equivalent stress of 50 kN/mm(+25% over the hydrostatic). The minimum is of about 7.0 kN/mm. The amplitude of loading cycle is of 22 kN/mm in the static

case and of 43 kN/mm in the dynamic case, which represents an increase of about 100%. Additionally, the form of the

dynamically computed loading cycle is deeply asymmetric.

Fig. 7.6.25. The equivalent stress time variation for point P1089

The comparison between the static and dynamic computations demonstrate that:

- the stresses reflect completely the difference between pressures, as emphasized by the hydrodynamic analyses. The

loading cycles are different as far as the values of extremes and the asymmetrical coefficient are concerned

- the stresses in loading line area are different since the dynamic computation reflects the ship-wave interaction. As

far as this area is concerned, the classic theories impose minimum loadings, but the above is in contradiction to the real

damages on side longitudinals. The dynamic analysis imposes a pulsatory loading cycle

- in supporting areas, the amplitude of stress cycle may increase by 100% over the statically computed cycle.

Chapter 8. Personal achievements. Conclusions. Future development

Personal achievements

The thesis has two goals:

- to develop a faithful approach to the free-surface fluid flow in the presence of a floating body (the hydromechaniccomponent)

- to carry out a study of the influence on structure of applying loads with deeply non-linear and dynamic loads (the

structure component).

The personal achievements of the author refer to these goals.

Thus,

With respect to hydrodynamics, the thesis has:

1) developed a methodology for the simulation, in time-space domain, of free-surface fluid flow. This methodology is based on

the potential flow hypothesis and on the completely non-linear conditions of free surface. This method simulates the waves

as produced prior to a moment in time close to the overturning phenomenon. The results of this simulations are extremely

useful to the study of beach erosion, streams determination or piers effects. Other correlated studies refer to tzunami waves

and to the sloshing phenomenon;


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2) developed the methodology for the simulation of free-surface fluid flow, in the presence of floating body (or fixed) based on

the methodology concerning the free-surface fluid flow. This methodology ensures the computing of body kinematics and

also the determination of hydrodynamic, time-variable pressure distribution on body surface. This method does not depend,

as do the classical ones, on the body shape, fluid depth or small amplitude excitation and motions hypothesis. This method

is applicable to all body shapes, either in bounded or shallower basin conditions or for completely non-linear and large

amplitude excitations. The applicable domains are the following:

- the study of non-linear ship motions, including rare phenomena: i.e. parametric rolling

- the computing of real loading on floating structures

- the hydrodynamics of special bodies (i.e. multi-hulls)

- the study of off-shore structures

- the study of phenomena occurred during anchoring or mooring as a result of body oscillations

- the study of damaged ships stability- the study of ship stability under the influence of inside ship, free-surface moving fluids .

All the above applications of the methodology are of critical importance and have a topical character;

3) pointed out the existence of a real coupling of parametric rolling type, between the heave and the roll. As far as the author

knows, this is a first, classical parametric roll being demonstrated so far only for between the pitch and the roll. This

phenomenon has been constantly observed at a time-period of excitation close to half natural roll period. Despite the fact

that the thesis does not include an analytic approach, the repetition of phenomenon should allow of the conclusion that the

phenomenon is of half-resonance type. This acknowledges that classical theories can not be extrapolated by overlooking the

small amplitude oscillation hypothesis. In the authors opinion, the importance of this conclusion should be enough to

justify all this work.

With respect to structural computation, the analysis demonstrates that the dynamic computation brings more exact

information about the amplitude and form of loading and stress cycles and also regarding the extent of affected areas. This shows

that the dynamics effects must be considered by the structural analysis first of all by a more accurate computing of loads.

Conclusions

Leonardo DaVinci said that there where water starts, the doubt begins. The first step taken in order to eliminate the

incertitude consisted in thorough experimental observations that led to the discovery of rules of hydrodynamics. For a long time,

these rules were expressed by equations with simple form but of a complex essence, so that practical problems were solved based

on a large number of approximations. Nowadays, the numerical methods use a complete set of rules, ensuring the simulation of

hydrodynamic phenomena.

This work has the goal to establish a hydrodynamic computing methodology that simulates the flow of free-surface

fluid in the presence of floating body.

With respect to methodology, there result the following conclusion:

- the best method to be employed by hydromechanical studies of potential flow is the Boundary Element Method.

This method has the advantage of an easy modeling of the geometrical evolution of the time domain. Owing to this

characteristic, extreme states of wave can be simulated. Also, this method decreases with one unit the geometric dimension

of the problem, therefore, the running time and hardware resources can be saved;

- the mobile solid boundary type wave generators reproduce most accurately the reality. However, if it is necessary

to use an excitatory based on the characteristics of theoretical waves, it should be preferable to use the velocity field. The

velocity field does not contradict the free surface conditions.

- the simplified condition for permeable boundary that was used in thesis, offer good results. It has been proven that

the mass error is mainly due to the upward and downward boundary conditions;

- the optimal method for computing the t term is based on the solving of ( ) 0= t equation. This

method offers much flexibility as far as the grid sizing and time stepping are concerned. An additional advantage is that the

same grid as for the continuity equation can be used;

- the Neumann condition on hull, ( ) nt , that was used for solving the ( ) 0= t equation, evidences the

bodies acceleration. This condition must be computed as a function of tangential derivative;

- the methods used for computing t and ( ) nt terms that act on the hull were based on the finite

difference methods and evidenced discontinuities in the simulation of phenomena. This discontinuities were shown clearly

by the time-variation diagrams of forces, i.e. the sawtooth phenomenon;

- the body accelerations were computed by solving an equations system that had been obtained by coupling the

( ) 0= t equation with bodys dynamic equilibrium equations. The solution of the system obtained as per above, has

evidenced the non-linear and transitory effects;

- the time-step used for the time integration based on this method, is long enough to ensure an efficient thesimulation on a P.C..

As far as the hydrodynamic results are concerned, the following conclusions can be drawn:

- the ships kinematics, as computed by time simulation, proves that classical theories provide a global average

kinematic response at the most. Thus, the direct simulation emphasizes the dependence of amplification ratio on the

amplitude of excitation. The classical theories (including the strip theory) do not take this into consideration ;

- the parametric rolling occurs also under the action of transverse waves. This phenomenon is caused by the large

and non-linear time variation of restoring moment as a result of large amplitudes heave motion;

- the amplitude of excitation is essential in those cases where there exist conditions for the occurrence of parametric

rolling. With respect to this, at a same time-period of excitation, the parametric rolling is either triggered or not, depending

on the wave amplitude;


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- the large movements of the ship invalidates the classical hypothesis. With respect to this, the small amplitude

oscillations hypothesis and also the separation of symmetrical movements of the ship from the asymmetrical ones are not

correct;

- the pressure distribution on the hull is largely correlated with the ship-wave relative motion. The pressures, as

resulted from the computation of water heights based on the classical methods, do not include the effect of excitation

amplitude and neither the actual form of pressure distribution in time and space.

The conclusions of hydrodynamic studies prove that the advancement of the structural computing method should be

supported by specific input data. It is known that the resistance of a system is defined by its weakest link. With respect to this, the

development of non-linear, optimal or fatigue computations of structures is futile if the loads are not determined correctly.

The first conclusion resulted from structural computations is that the eigenmodes of thin-skinned structures with distinctlongitudinal strenghtenings, as modeled by finite elements, can not be computed based on the classic method.

The alternative methods proposed are the following:

- the analysis, performed in the classic manner, of a simplified model, obtained by fusing the strenghtenings into the

shell

- the study of the structure response to an impulse-type excitation. Thus there can be estimated at least the first

eigenmodes.

This conclusion was reached as a result of the attempt to use the mode superposition as method for computing the dynamic

response of structures. Since the software that can consider the eigenmodes selectively does not exist, the recommended method

is the direct time integration.

This study acknowledges that the differences between hydrostatically and dynamically computed pressures are reflected in

the extreme values of stresses and in the cycle form. These differences reach dangerous values in the area nearby the load line

due to the fact that the hydrostatic hypotheses are very wrong as far as this area is concerned. Also, at the supporting points,

particularly where the secondary structure it is discharged on the principal one, the amplitude of dynamically computed stress

cycles can increase by 100% over the static estimation. These areas are indicated as critical areas and are carefully manufacturedand monitored.

With respect to the shipbuilding and shipping activities considered globally, the conclusion of this work is that the

hydrodynamic simulation methods have successfully caught up with the numerical methods traditionally applied by structural

analysis. The combination of these methods shall transform the ship design in an extremely exact activity.

From a practical viewpoint, the conclusion is that the method is both efficient and practical. We have proven herein that

such a method can catch deeply non-linear and transitory phenomena that are difficult to be studied experimentally or by

classical theories. The study of extreme or new cases is easier while the total cost is reduced and includes mainly the cost of

initial hardware and the expenses incurred for the development of software. This method is at hand for those interested.

Overall, the conclusion is that the development of hardware and numerical methods make possible a more faithful approach

based on simulations. This is also true for other activities besides shipbuilding and shipping.

Future developments

The most important development is the move towards 3D study. The theoretical ground for 3D studies is the same, but all

the deductions should be adapted to this change. The 3D study opens wide the way to solving many problems characteristic to

shipbuilding science. From the structural viewpoint, this development contributes to a complete computation of the loads acting

on body.

The second development is represented by a thorough study into the generation of excitation of free surface, since the

actual state of the sea must be simulated. With respect to this, the moving boundaries method, which simulates the actual

conditions of wave flume, can be tested. Prior to this, this, the correlation between wing movement parameters and resulted wave

parameters should be determined.

The third development is represented by a thorough study of rolling motion, which should be carried out in order to reduce

the discrepancy between free rolling simulation and experimental results. First step to be taken should be the use of a high order

boundary element (with parabolic variation on segment for the potential function).

The forth development is represented by a combination - with respect to hydroelasticity, of the hydrodynamic study withthe structural one. As far as the above is concerned, the added masses for sink panels vibration should be computed. From a

numerical viewpoint, this problem combines the finite element method used for the computing of structure, in conditions of

pressure deformation, with the boundary element method used for the computing of hydrodynamics pressures.

The most significant development would consist in the simulation of ship hydroelasticity. One of the most important

advantages brought by the above refers to the input data for the study of fatigue mechanism. Some might argue that such a

development would be exaggerated and unrealistic. Probably, the same reaction was generated in the past by the project

mentioned in reference [49], whose objective was the use of a huge floating structure (5 km length, 1 km width and few meters

deep) as airport.

As a final conclusion, we should say that the development of a reliable method for the simulation of free-surface fluid flow,

in the presence or without a floating body, opens wide ways to study. This is of great importance to shipbuilding and shipping

activities since the use of efficient methods would save time, values and human lives.


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- Integrated Fatigue Strengh Analysis

- Hydrodinamic Computation


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1.2. Project Management

2. GL Annual Report 1995

- Research Developemen

the computation of hydrodynamic pressures on a 2d section

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