
English: 
Hello welcome to my talk,All about Fluids,
this talk is the fourth part of the talk on boundary element method for wave-
structure interaction, with a focus on establishing the dynamic equation for
the structure motion in waves. In this talk, I will make a comparison and a
transformation between the dynamic equation of frequency domain and of
time domain.
then a simple example of a cylinder
structure is presented, with the analysis result of the boundary element method
including added mass, damping coefficient, wave exciting force,

English: 
Hello welcome to my talk, All about Fluids.
This talk is the fourth part of the talk on boundary element method for wave-
structure interaction, with a focus on establishing the dynamic equation for
the structure motion in waves. In this talk, I will make a comparison and a
transformation between the dynamic equation of frequency domain and of time domain.
then a simple example of a cylinder
structure is presented, with the analysis result of the boundary element method
including added mass, damping coefficient, wave exciting force,

English: 
the response amplitude operator (RAO) as well as the time series of the
heave motions in an irregular wave, based on solving the time domain equation, and the
transform from the frequency domain result.
if we obtained the forces and the moments, based on the radiation and
diffraction potentials, we can then construct the dynamic equation for the
structure motion, based on the Newton's second law of motion for translational
motion: surge, sway, heave and for rotational motion: roll, pitch and
yaw, respectively. if in a component form, we can write

English: 
the response amplitude operator (RAO) as well as the time series of the
heave motions in an irregular wave, based on solving the time domain equation, and the
transform from the frequency domain result.
if we obtained the forces and the moments, based on the radiation and
diffraction potentials, we can then constructed the dynamic equation for the
structure motion, based on the Newton's second law of motion for translational
motion: surge, sway, heave and for rotational motion: roll, pitch and
yaw, respectively. if in a component form, we can write

English: 
the dynamic equation as this, here the force fi is the total hydrostatic and
hydrodynamic forces, including hydrostatic forces F_S,
the radiation forces F_R and the wave exciting forces, F_EX, and the mass matrix
is given as this, here m is the mass of the structure, I_ij is the moment of
inertia of the structure, calculated as this.
and xg, yg and zg are the coordinates of the center of gravity
in the reference coordinates.
in this slide, the first-order hydrostatic and hydrodynamic forces are
summarized. More details can be seen in my talk on

English: 
the dynamic equation as this, here the force fi is the total hydrostatic and
hydrodynamic forces, including hydrostatic forces F_S,
the radiation forces F_R and the wave exciting forces, F_EX, and the mass matrix
is given as this, here m is the mass of the structure, I_ij is the moment of
inertia of the structure, calculated as this.
and xg, yg and zg are the coordinates of the center of gravity
in the reference coordinates.
in this slide, the first-order hydrostatic and hydrodynamic forces are
summarized. More details can be seen in my talk on

English: 
boundary element method for wave- structure interaction, Part 3.
The hydrostatic forces are given as the net hydrostatic forces, due to the structure
motion away from its equilibrium state, as seen this equation 1.
here the subscript i equals to 1 to 6, corresponding to 6 degrees freedom motion
and non-zero hydrostatic restoring forces coefficient, c_ij, given
as this, it should be noted some of the coefficients would vanish if the

English: 
boundary element method for wave- structure interaction, Part 3.
The hydrostatic forces are given as the net hydrostatic forces, due to the structure
motion away from its equilibrium state, as seen this equation 1.
here the subscript i equals to 1 to 6, corresponding to 6 degrees freedom motion
and non-zero hydrostatic restoring forces coefficient, c_ij, given
as this, it should be noted some of the coefficients would vanish if the

English: 
structures have symmetries about its axes, for instance, a vertical cylinder
would be symmetric about x-axis and about y-axis,
the radiation forces are given as this, in the equation 3, in terms of the added mass,
a_ij and the damping coefficient b_ij, due to the wave radiation defined as the
real and the imaginary part of the radiation force component f_ij. Here it
should be noted that in the radiation force the parameter i here is the
imaginary unit while the subscript i indicates the motion mode, equaling 1 to
6 here. the wave excitation is given, based on

English: 
structures have symmetries about its axes, for instance, a vertical cylinder
would be symmetric about x-axis and about y-axis,
the radiation forces are given as this, in the equation 3, in terms of the added mass,
a_ij and the damping coefficient b_ij, due to the wave radiation defined as the
real and the imaginary part of the radiation force component f_ij. Here it
should be noted that in the radiation force the parameter i here is the
imaginary unit while the subscript i indicates the motion mode, equalling (to) 1 to
6 here. the wave excitation is given, based on

English: 
the Haskind relation, in the equations 3. Here we can see that the wave excitation
can be calculated if the radiation potentials are solved, while the
scattering or diffraction potential may not be necessary to be solved, so for the
reason of reducing computation effort
in the frequency domain if we have complex amplitude XI_j of the motion,
then the velocity and acceleration can be expressed as this. This expression can
be understood that the velocity has a phase difference of 90 degrees, with
regard to the motion, with the imaginary unit i here.
while the acceleration has a 180-degree phase difference with regard to the motion.

English: 
the Haskind relation, in the equations 3. Here we can see that the wave excitation
can be calculated if the radiation potentials are solved, while the
scattering or diffraction potential may not be necessary to be solved, so for the
reason of reducing computation effort
in the frequency domain if we have complex amplitude XI_j of the motion,
then the velocity and acceleration can be expressed as this. This expressions can
be understood that the velocity has a phase difference of 90 degrees, with
regard to the motion, with the imaginary unit i here.
while the acceleration has a 180 degree phase difference with regard to the motion.

English: 
if we substitute the hydrostatic and
hydrodynamic forces into the equation of the structure motion, the hydrodynamic
equation of six degrees freedom motion of the structure in frequency domain
can be given as a mass-spring-damper format under the action of
a sinusoidal format, this term represents the mass term; this is the damping
term and this is the restoring term, wave excitation is the forcing term.
and the parameters are described as this.
and this equation can be written in this simple form,
with the capital C_ij, defined as this, and using the boundary element

English: 
if we substitute the hydrostatic and
hydrodynamic forces into the equation of the structure motion, the hydrodynamic
equation of six degrees freedom motion of the structure in frequency domain
can be given as a mass-spring-damper format under the action of
a sinusoidal format, this term represents the mass term; this is the damping
term and this is the restoring term, wave excitation is the forcing term.
and the parameters are described as this.
and this equation can be written in this simple form,
with the capital C_ij, defined as this, and using the boundary element

English: 
method, these important terms can be calculated.
To solve the radiation potential and the diffraction potential if it is needed, we
can use the commercial boundary element method, such as the commercial software
WAMIT, ANSYS AQWA or use the open-source, Nemoh or the in-house BEM codes.
after solving the relevant potentials, we can calculate the forces
acting on the structure and then the dynamic equation for marine structure can

English: 
method, these important terms can be calculated.
To solve the radiation potential and the diffraction potential if it is needed, we
can use the commercial boundary element method, such as the commercial software
WAMIT, ANSYS AQWA or use the open-source, Nemoh or the in-house BEM codes.
after solving the relevant potentials, we can calculate the forces
acting on the structure and then the dynamic equation for marine structure can

English: 
be solved to obtain the motion amplitude XI_j as this,
To represent the result better, we normally use the response amplitude
operator (RAO), calculated as this. here A is the wave amplitude
that means the response amplitude operator is a motion amplitude in a
unit wave. this response amplitude operator is a complex response, which
would depend on the wave frequency Omega, and the wave incident angle BETA
in this slide the establishment of the frequency domain equations in a linear

English: 
be solved to obtain the motion amplitude XI_j as this,
To represent the result better, we normally use the response amplitude
operator (RAO), calculated as this. here A is the wave amplitude
that means the response amplitude operator is a motion amplitude in a
unit wave. this response amplitude operator is a complex response, which
would depend on the wave frequency Omega, and the wave incident angle BETA
in this slide the establishment of the frequency domain equations in a linear

English: 
dynamic system is discussed. In the real world, the physical
parameters are all time-dependent, for instance, under an action of a sinusoidal
forces, the system response will be a sinusoidal response of the same frequency
then the corresponding motion velocity and acceleration can be calculated as this.
Based on the Newton's second law of
motion, the dynamic system for the structure motion can be given as this
in a form of the mass-spring-damper format, here the added mass from the
radiation forces have been taken as a part of the total mass; the damping is

English: 
dynamic system is discussed. In the real world, the physical
parameters are all time-dependent, for instance, under an action of a sinusoidal
forces, the system response will be a sinusoidal response of the same frequency
then the corresponding motion velocity and acceleration can be calculated as this.
Based on the Newton's second law of
motion, the dynamic system for the structure motion can be given as this
in a form of the mass-spring-damper format, here the added mass from the
radiation forces have been taken as a part of the total mass; the damping is

English: 
the radiation damping and the restoring force is from the net
hydrostatic forces and the forcing is the wave excitation.
so substitute the sinusoidal force, motion, velocity and acceleration in
the physical equation in time domain, we have the equation as this
if we cancel out the time factor in this equation, we can obtain the frequency
domain equation as this. this is exactly same as the frequency
domain equation we used previously
we have seen that the frequency domain equation is derived from the time domain

English: 
the radiation damping and the restoring force is from the net
hydrostatic forces and the forcing is the wave excitation.
so substitute the sinusoidal force, motion, velocity and acceleration in
the physical equation in time domain, we have the equation as this
if we cancel out the time factor in this equation, we can obtain the frequency
domain equation as this. this is exactly same as the frequency
domain equation we used previously
we have seen that the frequency domain equation is derived from the time domain

English: 
equation, however,  when we transfer the
frequency domain equation back to time domain equation, we must be careful
in doing so, since  in the frequency domain equation
for the floating structure, the hydrodynamic parameters, a_ij, b_ij, and F_i
are all frequency dependent, you will see the example later in this talk.
for example, can we transfer the frequency domain equation directly into the time
domain equation like this, the answer for this transform is YES and NO. for YES,
it is because this time domain equation is correct when the forcing fi is the

English: 
equation, however, when we transfer the
frequency domain equation back to time domain equation, we must be careful
in doing so, since in the frequency domain equation
for the floating structure, the hydrodynamic parameters, a_ij, b_ij, and F_i
are all frequency dependent, you will see the example later in this talk.
for example, can we transfer the frequency domain equation directly into the time
domain equation like this, the answer for this transform is YES and NO. for YES,
it is because this time domain equation is correct when the forcing fi is the

English: 
sinusoidal for the linear dynamic system,
and thus the whole dynamic system
has a single frequency. Therefore for this specific frequency, the hydrodynamic
parameters: a_ij, b_ij and fi can be correctly obtained.
and NO, it is because we may often need to examine different forcing, for instance,
the non-linear forcing and this is the time domain equation used for.
for a system of multi-frequencies, the hydrodynamic parameters: a_ij, b_ij and fi

English: 
sinusoidal for the linear dynamic system, and thus the whole dynamic system
has a single frequency. Therefore, for this specific frequency, the hydrodynamic
parameters: a_ij, b_ij and fi can be correctly obtained.
and NO, it is because we may often need to examine different forcing, for instance,
the non-linear forcing and this is the time domain equation used for.
for a system of multi-frequencies, the hydrodynamic parameters: a_ij, b_ij and fi

English: 
cannot be given in a correct way. As such we have to use the time domain equation as this
here a_ij is the added mass at
infinite frequency, while the impulse function k_ij is a Fourier transform of
the radiation damping coefficient, given as this
and in this time domain equation, the forcing term, fi, can be any type of
forcing, thus the time-domain equation can be easily changed for
accommodating the non-linear force or any other types of forces
for many practical wave-structure interaction problems, the linear
solutions are generally very good and reliable, that's why the boundary element

English: 
cannot be given in a correct way. As such we have to use the time domain equation as this
here a_ij is the added mass at
infinite frequency, while the impulse function k_ij is a Fourier transform of
the radiation damping coefficient, given as this
and in this time domain equation, the forcing term, fi, can be any type of
forcing, thus the time-domain equation can be easily changed for
accommodating the non-linear force or any other types of forces
for many practical wave-structure interaction problems, the linear
solutions are generally very good and reliable, that's why the boundary element

English: 
method, for instance, WAMIT,ANSYSAQWA, and some other BEM software packages
have been regarded as the industry standard, for using the boundary
element method for wave-structure interaction, here the commercial code
WAMIT is used as an example: step 1, panelise the wetted surface of the
structure into many panels, as seen in the figure
we can specify whether the sources or doublets to be distributed on the surface. in most
cases, sources are distributed while the doublets are only used for some cases, for

English: 
method, for instance, WAMIT, ANSYS AQWA, and some other BEM software packages
have been regarded as the industry standard, for using the boundary
element method for wave-structure interaction, here the commercial code
WAMIT is used as an example: step 1, panelise the wetted surface of the
structure into many panels, as seen in the figure
we can specify whether the sources or doublets to be distributed on the surface. in most
cases, sources are distributed while the doublets are only used for some cases, for

English: 
instance, the thin structure and both sides are wet.
To solve the Laplace equation for different potential, using the relevant
boundary conditions to decide the strengths of the distributed the
singularities: sources and doublets, for achieving that, you need to specify
the wave frequency, wave incident angle, and the water depth.
Step 3, for calculating the forces and moments acting on the structure, you need
to specify the reference point, for instance, the center of the gravity.
Step 4: for solving the structure motion, based on the Newton's
second law of motion and then the responses of the motion RAO, you need to

English: 
instance, the thin structure and both sides are wet
To solve the Laplace equation for different potential, using the relevant
boundary conditions to decide the strengths of the distributed the
singularities: sources and doublets, for achieving that, you need to specify
the wave frequency, wave incident angle, and the water depth.
Step 3,  for calculating the forces and moments acting on the structure, you need
to specify the reference point, for instance, the center of the gravity.
Step 4: for solving the structure motion, based on the Newton's
second law of motion and then the responses of the motion RAO, you need to

English: 
specify the mass matrix, central gravity and the motion modes, for
instance, whether it is fixed or floating
in the next two slides, some results are given for a cylinder under the wave
action, the cylinder has a radius of 3m and a draft 1.5m
see the figure. its displacement is 43.41m^3,
we discretize the wet surface of the cylinder into the small panels, here the
panels are very coarse for the purpose of an illustration. For this specific

English: 
specify the mass matrix, central gravity and the motion modes, for
instance, whether it is fixed or floating
in the next two slides, some results are given for a cylinder under the wave
action, the cylinder has a radius of 3m and a draft 1.5m
see the figure. its displacement is 43.41m^3,
we discretize the wet surface of the cylinder into the small panels,  here the
panels are very coarse for the purpose of an illustration. For this specific

English: 
structure, it has symmetries about x-axis and about y-axis
As such the heave would be independent of other motions,
and this can be seen from the restoring coefficient, c_34,
and c_35, both are 0 since the symmetries about x-axis and y-axis
and the boundary element method would give a result as this, the added mass
you can see in this small frequencies, the added mass changes quite a lot but it
becomes a constant when Omega is very large as we can see the added mass at
the infinite frequency would not be zero. the radiation damping coefficient is given

English: 
structure,  it has symmetries about x-axis and about y-axis
As such the heave would be independent of other motions,
and this can be seen from the restoring coefficient, c_34,
and c_35, both are 0 since the symmetries about x-axis and y-axis
and the boundary element method would give a result as this, the added mass
you can see in this small frequencies, the added mass changes quite a lot but it
becomes a constant when Omega is very large as we can see the added mass at
the infinite frequency would not be zero. the radiation damping coefficient is given

English: 
as this, it is zero at both very high and low frequencies, it reaches the
maximum at a certain frequency, and wave excitation is the mono function, decreasing
with the wave frequency.
here the heave response RAO is given as this, and from the figure,
We can see the resonance of the heave motion happens at a frequency
1.8 radians per second, corresponding to the wave period 3.9s
at the resonance, RAO is about 2.2
so in the short wave at the high frequencies, the heave response is very

English: 
as this, it is zero at both very high and low frequencies, it reaches the
maximum at a certain frequency, and wave excitation is the mono function, decreasing
with the wave frequency.
here the heave response RAO is given as this, and from the figure,
We can see the resonance of the heave motion happens at a frequency
1.8 radians per second, corresponding to the wave period 3.9s
at the resonance, RAO is about 2.2.
so in the short wave at the high frequencies, the heave response is very

English: 
small here, and this can be understood the cylinder might not move
in the very short way. In the long wave, the the response of the heave motion is
a unit, this means in the long wave, the cylinder rides with the wave.
In this plot, a comparison is made for the solution of the heave motion in time
domain in a irregular wave. And the motion is transformed also from
the frequency domain response. so it can be seen this two time series
are identical in the wave of an average period 5 seconds, and the significant wave
height 1 meter.

English: 
small here, and this can be understood the cylinder might not move
in the very short way. In the long wave, the response of the heave motion is
a unit, this means in the long wave, the cylinder rides with the wave.
In this plot, a comparison is made for the solution of the heave motion in time
domain in an irregular wave. And the motion is transformed also from
the frequency domain response. so it can be seen this two time series
are identical in the wave of an average period 5 seconds, and the significant wave
height 1 meter.
