# Very Flexible Floating Structures (VFFS)

Authors: Sjoerd van Hoof and Oriol Colomés Gené

Published: July 2021

Gridap version: Gridap@0.14

This tutorial shows how a Fluid Structure Interaction (FSI) in a 2D domain is modelled. Potential flow is used to model the fluid and on top a Dynamic Euler-Bernoulli beam is located that serves as the floating structure. 3D model

# Problem description

First of all, let's dive in to the mathematics behind the problem. Potential flow is based on the principle that the velocity field can be described by the spatial derivatives of a scalar function, this is called the potential function. Moreover, the fluid is considered to be incom­pressible. This consideration implies that the divergence of the velocity is equal to zero. The potential function then satisfies the Laplace equation:

$\left\{\begin{array}{l} \nabla \cdot \vec{u}=0 \\ \nabla \phi=\vec{u} \end{array} \Leftrightarrow \nabla \cdot(\nabla \phi)=\Delta \phi=0 \quad\right. \text { in } \quad \Omega$

Where $\Omega$ denotes the 2D domain.

Now it is time to set the boundary conditions of the domain. In this case, three different boundaries need to be applied:

• The bottom boundary

• The free surface boundary

• The structure boundary

## Bottom boundary

The bottom boundary (also called the sea bed boundary) states that the boundary is impermeable and given as:

$\nabla \phi \cdot \vec{n}=0 \quad \text { on } \quad \Gamma_{L} \cup \Gamma_{b t m} \cup \Gamma_{R} \quad \text { bottom b.c. }$

In this model, not only the bottom of the domain ($\Gamma_{btm}$) is impermeable, but also the left ($\Gamma_L$) and right ($\Gamma_R$) hand side of the domain (see figure below). The top boundary, however, is not impermeable, but is the free surface boundary ($\Gamma_{fs}$). The fluid is able to move freely up and down, but mustn't leave the domain. Two conditions need to be applied here:

• The dynamic boundary condition

• The kinematic boundary condition

## Dynamic boundary condition

The dynamic free surface boundary condition states that the pressure above the free surface is constant and equal to zero. Using Bernoulli’s equation in the lin­earised form and assuming that there is no y­-direction (since the model is only 2D), the boundary condition is given as:

$\frac{\delta \phi}{\delta t}+g \eta=0 \quad \text { on } \quad \Gamma_{f s} \quad \text { dynamic free surface b.c. }$

Where $g$ is the gravity constant and $\eta$ is the surface elevation. With this boundary condition the formula of the potential flow can be derived and is equal to:

$\phi=\frac{\xi_{i} g}{\omega} \cdot \frac{\cosh \left(k_{i}(d+z)\right)}{\cosh \left(k_{i} d\right)} \cdot \sin \left(k_{i} x-\omega t\right)$

Where $\xi_i$ , $\omega$ and $k_i$ are the amplitude, the angular frequency and the wave number of the incident wave, respectively and $d$ the water depth.

## Kinematic boundary condition

Finally, the third boundary condition is the kinematic free surface boundary condition which describes that the vertical velocity of the free surface has to be equal to the vertical motion of the flow and is given as follows:

$\frac{\delta \phi}{\delta z}=\frac{\delta \eta}{\delta t} \quad \text { on } \quad \Gamma_{f s} \quad \text { kinematic free surface b.c. }$

## Cauchy-Poisson condition

Using this boundary condition the wave number can be related to the wave frequency. By differenti­ating the dynamic free surface boundary condition and inserting the kinematic free surface boundary condition one obtains the Cauchy-­Poisson condition:

$\frac{\delta \phi}{\delta z}+\frac{1}{g} \frac{\delta^{2} \phi}{\delta t^{2}}=0$

Substituting the potential function into the Cauchy-Poisson equation gives the dispersion relation:

$\omega^{2}=g k_{i} \tanh \left(k_{i} d\right)$

Using these equation, one is able to establish the free surface and the fluid domain can be modelled. However, we are interested in a fluid structure interaction, which means that there should also be a boundary condition at the location of the floating structure ($\Gamma_b$). This is done by imposing the Dynamic Euler-Bernoulli beam on a slice of the free surface.

## Dynamic Euler-Bernoulli beam

The dynamic Euler-Bernoulli beam is a one-dimensional equation that is valid if the loads are purely lateral and the deflections remain small. The general form of the Euler-Bernoulli beam describes a relation of the deflection of the beam ($w$) and an external force ($f(t)$) as function of time:

$E I \frac{\delta^{4} w}{\delta x^{4}}=-\mu \frac{\delta^{2} w}{\delta t^{2}}+f(x)$

Where $EI$ is the bending stiffness and $\mu$ is the mass per unit length.

Regarding the boundary conditions, the kinematic b.c. remains the same as for the free surface. However, the dynamic boundary condition changes to a non-zero solution and is multiplied by the fluid density $\rho_w$:

$-\rho_{w} \frac{\delta \phi}{\delta t}-\rho_{w} g \eta=p$

The pressure $p$ is equal to the external force of the Euler-Bernoulli beam:

$p=\rho_{b} h \frac{\delta^{2} \eta}{\delta t^{2}}+\frac{E I}{B} \frac{\delta^{4} \eta}{\delta x^{4}}$

Where $\mu$ is equal to the product of the structure density $\rho_b$ and the thickness of the structure $h$. The bending stiffness has been divided by the width ($B$) of the structure, as the model is only two-dimensional and the deflection $w$ has been replaced by the surface elevation $\eta$. Rewriting this into one equation results in the altered dynamic boundary condition:

$\rho_{b} h \frac{\delta^{2} \eta}{\delta t^{2}}+\frac{E I}{B} \frac{\delta^{4} \eta}{\delta x^{4}}+\rho_{w} \frac{\delta \phi}{\delta t}+\rho_{w} g \eta=0 \quad \text { on } \quad \Gamma_b$

Rewriting this equation and inserting the same governing kinematic free surface boundary condition results in an altered Cauchy­-Poisson condition with an added factor; three terms in between brackets:

$\left(\frac{\rho_{b} h}{\rho_{w} g} \frac{\delta^{2} \eta}{\delta t^{2}}+\frac{E I}{\rho_{w} g B} \frac{\delta^{4} \eta}{\delta x^{4}}+1\right) g \frac{\delta \phi}{\delta z}=-\frac{\delta^{2} \phi}{\delta t^{2}}$

Assuming that the solution of the free surface elevation is wave:

$\eta = \xi_b \cos{(k_b x - \omega t)}$

With $\xi_b$ and $k_b$ the amplitude and the wave number of the hydroelastic wave, respectively. Its second derivative in time and fourth derivative in space are given as:

$\begin{array}{l} \eta_{t t}=-\omega^{2} \xi_{b} \cos \left(k_{b} x-\omega t\right) \\ \eta_{x x x x}=k_{b}^{4} \xi_{b} \cos \left(k_{b} x-\omega t\right) \end{array}$

Inserting these wave terms and assuming initial conditions ($x = 0, t = 0$) the equation is given as:

$\left(-\frac{\omega^{2} \rho_{b} h}{\rho_{w} g}+\frac{k_{b}^{4} E I}{\rho_{w} g B}+1\right) g \frac{\delta \phi}{\delta z}=-\frac{\delta^{2} \phi}{\delta t^{2}}$

## Damping zone

At the end of the domain the wave energy should be dissipated to prevent wave reflection. This is done by adding a viscosity term to the set of equations. There are several ways to achieve this, but the most effective way is to use a method by Kim Woo Min who changed the kinematic boundary condition by adding two terms which dissipate the wave energy:

$\frac{\delta \eta}{\delta t}-\frac{\delta \phi}{\delta z}+\mu_{1} \eta+\frac{\mu_{2}}{g} \phi=0 \quad \text { kinematic b.c. damping zone}$

Where $\mu_1$ and $\mu_2$ are damping coefficients that follow from the iteratively chosen $\mu_0$, and are dependent on each other to ensure that no dispersion occurs:

$\begin{array}{l} \mu_{1}(x)=\mu_{0}\left[1-\cos \left(\frac{\pi}{2}\left(\frac{x-x_{d}}{L_{d}}\right)\right)\right] \\ \mu_{2}=-\frac{\mu_{1}^{2}}{4} \end{array}$

Where $x_d$ is starting point of the numerical damping zone and $L_d$ is the length of the damping zone.

# Numerical model

As the mathematics behind the model have been shown, it is now time to rewrite the system of equations into the weak formulation and insert them into the numerical model. However, in order to do so, we should first set up the numrical model.

## Wave parameters

Let's start with initiating a simple regular wave with a period of $T = 0.5$ s and a steepness of 1/25 with a water depth of $d = 1$ m. Using the just derived Airy wave theory, the wave parameters can be calculated. The module Roots is used to calculate the root of the function.

using Roots
T = 0.5
steepness = 1/25
d = 1

ω = 2*π / T                 # angular frequency
g = 9.81                    # gravity constant
f(k) = √(g*k*tanh(k*d)) - ω
k = abs(find_zero(f, 0.5))  # wave number
λ = 2*π / k                 # wavelength
ξ = λ * steepness / 2       # wave amplitude

## Material parameters

Next to the wave parameters, the structure also contains several properties, such as a stiffness, a thickness and a density. The fluid density has been set equal to the sea and the VFFS is made of a 0.005 m rubber-like material called Neoprene, which is very flexible. The width of the structure has not been taken into account and it is assumed that the cross-section is a rectangle.

ρ_w = 1025    # fluid density
ρ_b = 116     # structure density
E = 560e3     # Young's modulus
h = 0.005     # thickness
I = h^3 / 12  # moment of inertia
EI = E * I    # bending stiffness

For 3D applications the bending stiffness $EI$ should be replaced by the flexural rigidity $D$ and can be calculated as follows:

$D = \frac{E h^3}{12 (1-\nu^2)}$

Where $\nu$ is the Poisson's ratio.

ν = 0.4
D = E * h^3 / (12 * (1-ν^2) )

However, this tutorial will focus on a 2D domain, so this will be discarded.

## Domain

The domain of the model is defined such that the free surface waves can develop and travel a few wavelengths before they reach the structure. The structure must be longer than the incident wavelength in order to be assumed a VFFS. Finally, the waves will leave the structure and are free surface waves again. Before the waves reach the end of the domain the wave energy should be dissipated to prevent wave reflection at the right hand side of the domain ($\Gamma_R$) The picture below shows the set-up of the model. To define the domain the Gridap module is used and the different zones are specified:

using Gridap
using Gridap.Geometry
using GridapODEs
using GridapODEs.TransientFETools
using GridapODEs.ODETools

Li = 3*λ # length of incident wave zone
Lb = 5*λ # length of VFFS
Lo = 3*λ # length of outgoing wave zone
Ld = 4*λ # length of damping zone

xb₀ = Li        # x-coordinate start VFFS
xb₁ = Li + Lb   # x-coordinate end VFFS
x₂ = xb₁ + Lo   # x-coordinate start damping zone
x₃ = x₂ + Ld    # end of domain

Xmin = 0
Xmax = x₃
Zmin = 0
Zmax = d

domain = (Xmin, Xmax, Zmin, Zmax)

## Damping zone

With these values the formula for the damping zone can be established. It is chosen to use a value of $\mu_0 = 10$:

μ₀ = 10
μ₁(x::VectorValue) = μ₀*(1.0 - cos(π/2*(x-x₂)/Ld)) * (x>x₂)
μ₂(x::VectorValue) = -(μ₁(x)^2)/4 * (x>x₂)

## Resolution

To define the resolution of the domain a partition is given. This is the amount of grid cells that the model contains. The resolution in the x-direction should be high, as we are interested in the exact shape of the wave. For the z-direction, however, we are only interested in the top layer near the free surface. It is assumed that the vertical velocity profile goes to zero when it reaches the bottom, so not much is going on as we proceed towards the bottom of the domain. Therefore, the resolution in x-direction is set to 50 cells per wavelength and for the z-direction only 10 cells are which have been unevenly spaced so that the resolution is fine at the free surface, but becomes coarser as we go down. This is done using the argument map in the function CartesianDiscreteModel. The function simplexify is used to change the mesh to an affine reference map, which is necessary to have the mapping work.

\begin{align*} z_{i}=d-\frac{d}{2^{i}} & \quad \text { for } \quad i < n_{\text {cells}, z} \\ z_{i}=d & \quad \text { for } \quad i=n_{\text {cells}, z} \end{align*}
meshX = 50
meshZ = 10
function f_z(x)
if x == d
return d
end
i = x / (d/meshZ)
return d-d/(2^i)
end
map(x) = VectorValue(x,f_z(x))
Xspan = Int(round((Xmax - Xmin) / λ))
Zspan = Int(round((Zmax - Zmin) / λ))
partition = (Xspan * meshX, Zspan * meshZ)
model_Ω = simplexify(CartesianDiscreteModel(domain,partition, map=map))
Ω = Triangulation(model_Ω)

## Boundaries

As the numerical domain model_Ω has been initiated, we wil now define a discrete model on the boundary corresponding to the free surface and floating structure, model_Γ, and the equivalent triangulation, Γ.

function is_beam(coords)
n = length(coords)
x = (1/n)*sum(coords)
(xb₀ <= x <= xb₁ ) * ( x ≈ d )
end

labels = get_face_labeling(model_Ω)
add_tag_from_tags!(labels_Ω,"surface",[3,4,6])   # assign the label "surface" to the entity 3,4 and 6 (top corners and top side)
model_Γ = BoundaryDiscreteModel(Polytope{1},model_Ω,Γface_to_bgface)
Γ = Triangulation(model_Γ)

We can also identify the entities that belong to the floating structure and assign them to a new label called "beam":

function is_beam_boundary(xs)
is_on_xb₀ = [x≈xb₀ for x in xs] # array of booleans of size the number of points in an element (for points, it will be an array of size 1)
is_on_xb₁ = [x≈xb₁ for x in xs]
element_on_xb₀ = minimum(is_on_xb₀) # Boolean with "true" if at least one entry is true, "false" otherwise.
element_on_xb₁ = minimum(is_on_xb₁)
element_on_xb₀ | element_on_xb₁ # Return "true" if any of the two cases is true
end

labels_Γ = get_face_labeling(model_Γ)
D = num_cell_dims(model_Γ) # spatial dimension of the model_Γ
entity_tag_beam = num_entities(labels_Γ) + 1 # create a new tag for the beam
for d in 0:D # loop over dimensions
grid_Γ_dim_d = Grid(ReferenceFE{d},model_Γ) # construct a grid from the entities of dimension "d" in Γ
coords_grid_Γ_dim_d = get_cell_coordinates(grid_Γ_dim_d) # get the coordinates of entities of dimension "d" of the grid (array of vectors of points). If model_Γ has 10 edges, this will be an array of 10 entries where each entry will contain a vector of 2 points.
beam_mask_in_grid_Γ_dim_d = lazy_map(is_beam,coords_grid_Γ_dim_d) # beam mask with the entities of dimension "d"
beam_boundary_mask_in_grid_Γ_dim_d = lazy_map(is_beam_boundary,coords_grid_Γ_dim_d) # beam boundary mask with the entities of dimension "d"
# Create a list of indices that have value=True in the mask of active entities of dimension "d" in Gamma. Excluding the entities on the beam boundary and on the joint.
# Tag the faces of dimension "d" (all the faces indexed in beam_to_Γ_dim_d)
for face in beam_to_Γ_dim_d
labels_Γ.d_to_dface_to_entity[d+1][face] = entity_tag_beam
end
end

# Add a name to the tag in labels_Γ
add_tag!(labels_Γ,"beam",[entity_tag_beam])

Now we can construct the masks of edges in Ω that will be used to construct the triangulations for the floating structure and the free surface:

Γ_mask_in_Ω_dim_1 = get_face_mask(labels_Ω,"surface",1) # get the mask of edges in Ω that are on Γ
Γ_to_Ω_dim_1 = findall(Γ_mask_in_Ω_dim_1) # Get indices of edges of Ω that are on Γ
Γstr_to_Γ_dim_1 = findall(Γstr_mask_in_Γ_dim_1) # get indices of edges of Γ that are in the beam
Γfs_to_Γ_dim_1 = findall(!,Γstr_mask_in_Γ_dim_1) # get indices of edges of Γ that are in the free surface
Γstr_to_Ω_dim_1 = view(Γ_to_Ω_dim_1,Γstr_to_Γ_dim_1) # get indices of edges of Ω that are in the beam. To create that we "concatenate" two sets (from Γstr to Γ and from Γ to Ω)
Γfs_to_Ω_dim_1 = view(Γ_to_Ω_dim_1,Γfs_to_Γ_dim_1) # Idem for the free surface
# Now we can construct sub-triangulations
Γb = BoundaryTriangulation(model_Ω,Γstr_to_Ω_dim_1)
Γf = BoundaryTriangulation(model_Ω,Γfs_to_Ω_dim_1)

In addition, we construct the set of internal points of the strucutre (skeleton).

Λb_mask_in_Γ_dim_0 = get_face_mask(labels_Γ,"beam",0)
Λb = SkeletonTriangulation(model_Γ,Λb_mask_in_Γ_dim_0)

As well as the other boundaries:

add_tag_from_tags!(labels,"bottom",[1,2,5])
add_tag_from_tags!(labels, "water", )

The inlet of the domain is specified and a vertical velocity profile is imposed which is based on the gradient of the potential function:

$\frac{\delta \phi}{\delta x} =\omega \xi \frac{\cosh k(z+d)}{\sinh k d} \cos (k x-\omega t - \theta)$
Γin = BoundaryTriangulation(model_Ω, tags = ["inlet"])

θ = π/2 # phase shift
function v_inlet(x,t)
return (ω * ξ * ( cosh( k * (x) ) / sinh( k * d ) ) * cos(k * x - ω * t - θ))
end

v_inlet(t::Real) = x -> v_inlet(x,t)

Finally, the quadratures are specified for each boundary:

order = 2
dΩ = Measure(Ω,2*order)
dΓb = Measure(Γb,2*order)
dΓf = Measure(Γf,2*order)
dΓin = Measure(Γin,2*order)

nΛb = get_normal_vector(Λb)
dΛb = Measure(Λb,2*order)

## Finite Element spaces

As the numerical domain has been set and the boundaries have been defined, the test spaces can be constructed. For this type of problem, two spaces will be built; one for the internal domain model_Ω and one for the free surface model_Γ. Both will use lagrangian shape functions and are of order two.

reffe = ReferenceFE(lagrangian,Float64,order)
V_Ω = TestFESpace(
model_Ω,
reffe,
conformity=:H1
)
V_Γ = TestFESpace(
model_Γ,
reffe,
conformity=:H1
)

A TrialFESpace is established using the test spaces and dirichlet boundary conditions. Note that for the "bottom", "water" and "outlet" boundary the value is set to zero using the function u. For the "inlet" boundary, the dirichlet condition is set to a vertical velocity profile which has been specified with the function v_inlet.

u(x,t) = 0
u(t::Real) = x -> u(x,t)
U_Ω = TransientTrialFESpace(V_Ω)
U_Γ = TransientTrialFESpace(V_Γ)

The test spaces and trial spaces are combined in a MultiFieldFESpace indicated by X and Y, respectively.

X = TransientMultiFieldFESpace([U_Ω,U_Γ])
Y = MultiFieldFESpace([V_Ω,V_Γ])

## Numerical time integration

To solve the system in time, a numerical time integration scheme needs to be chosen. As the set of equations consists of a second temporal derivative, the Newmark-beta integration scheme has been chosen, which is widely used in the dynamic response of structures. Generally, the system is explicit, however, by choosing the parameters $\beta = 0.25$ and $\gamma = 0.5$, the system becomes unconditionally stable, which means that the time step can be chosen independently of the grid resolution. In this case, it was found that the results were accurate if the time step was equal to grid resolution in x-direction, resulting in a time step of 1/50 of the wave period.

$\mathbf{M} \ddot{y}+\mathbf{C} \dot{y}+f^{i n t}(y)=f^{e x t}$ $\begin{array}{l} \dot{y}_{n+1}=\dot{y}_{n}+(1-\gamma) \Delta t \ddot{y}_{n}+\gamma \Delta t \ddot{y}_{n+1} \\ y_{n+1}=y_{n}+\Delta t \dot{y}_{n}+\frac{\Delta t^{2}}{2}\left((1-2 \beta) \ddot{y}_{n}+2 \beta \ddot{y}_{n+1}\right) \\ \mathbf{M} \ddot{y}_{n+1}+\mathbf{C} \dot{y}_{n+1}+f^{\text {int }}\left(y_{n+1}\right)=f_{n+1}^{\text {ext }} \end{array}$
γ = 0.5   # Newmark beta factor
β = 0.25  # Newmark beta factor
Δt = T / meshX # time step

## Stabilisation terms

To make sure that the system of equations is coercive, the weak formulation is stabilised by stabilisation terms. Currently, Oriol Colomés Gené is doing research to find the correct set of stabilisation terms to solve this problem. For now, a default value of $αh = 0.5$ is used to display the correct values.

∂ut_∂u = γ/(β*Δt)
∂utt_∂u = 1/(β*Δt^2)
αh = 0.5                                                      # default value
h_m = min((Xmax-Xmin)/partition,(Zmax-Zmin)/partition)
β_f = ∂ut_∂u / (αh*g + ∂ut_∂u)                                # stability parameter fluid
β_b = ∂ut_∂u / (αh*(ρ_b/ρ_w*h*∂utt_∂u + g) + ∂ut_∂u)          # stability parameter structure
γ_m = 1.0e2*order*(order+1)

s((ϕ,η),(w,v)) is a function containing all stabilisation terms that will be included in the weak formulation to ensure a stable system.

s((ϕ,η),(w,v)) = ∫( β_b*EI/ρ_w*( - ( jump(∇(v)⋅nΛb) * mean(Δ(η)) + jump(αh*∇(w)⋅nΛb) * mean(Δ(η)) ) - ( mean(Δ(v)) * jump(∇(η)⋅nΛb) ) + γ_m/h_m * ( jump(∇(v)⋅nΛb) * jump(∇(η)⋅nΛb) + αh*jump(∇(w)⋅nΛb) * jump(∇(ϕ)⋅nΛb) ) ) )dΛb

## Weak formulation

The weak form is the set of equations that will be solved by Gridap. The Newmark-beta scheme distinguishes between three different matrices; the mass matrix $M$ which contains the seond derivative terms, the damping matrix $C$ which contains the first derivative terms, $f_{int}$ and the other terms of the standard form. Finally there is the vector that includes the external loads $f_{ext}$. In Gridap, four functions are used that resemble the mentioned matrices. m((ϕtt,ηtt),(w,v)), c((ϕt,ηt),(w,v)) and a((ϕ,η),(w,v)) form the bi-linear form and b(t,(w,v)) is the vector containing the externeal forces.

m((ϕtt,ηtt),(w,v)) = ∫( ρ_b/ρ_w*h*ηtt*β_b*(v+αh*w) )dΓb

c((ϕt,ηt),(w,v)) =  ∫( β_f*(αh*w + v)*ϕt - w*ηt )dΓf +
∫( β_b*(αh*w + v)*ϕt - w*ηt )dΓb

a((ϕ,η),(w,v)) =  ∫( ∇(ϕ)⋅∇(w) )dΩ +
∫( β_f*(v + αh*w) * g*η - μ₁*η*w - μ₂*ϕ*w/g )dΓf +
∫( β_b*(Δ(v) + αh*Δ(w)) * EI/ρ_w*Δ(η) + β_b*(v + αh*w) * g*η + EI/ρ_w*αh*Δ(w)*Δ(ϕ) )dΓb +
s((ϕ,η),(w,v))

b(t,(w,v)) =  ∫( v_inlet(t) * w )dΓin

The set of equations is combined into one matrix and the numerical solver is set up.

Warning:

Depending on the configuration, the previous formulation might not result in accurate results. Please, check with the authors for the latest version.

## Solver

op = TransientConstantMatrixFEOperator(m,c,a,b,X,Y)
ls = LUSolver()
odes = Newmark(ls,Δt,γ,β)
solver = TransientFESolver(odes)

The initial solution is set to zero.

x₀ = interpolate_everywhere([0.0, 0.0],X(0.0))
v₀ = interpolate_everywhere([0.0, 0.0],X(0.0))
a₀ = interpolate_everywhere([0.0, 0.0],X(0.0))

And all time steps are set, ready to be computed.

t₀ = 0.0
periods = 50
tf = periods * T
sol_t = Gridap.solve(solver,op,(x₀,v₀,a₀),t₀,tf)

The WriteVTK and FileIO modules are loaded to store the results on the computer and a directory is created to store all the files.

using WriteVTK
using FileIO

folderName = "solution"
if !isdir(folderName)
mkdir(folderName)
end

The model consists of three parts; the 2D potential domain and the free surface which can be subdivided into the fluid part and structure part. Each is stored seperately and can be viewed in Paraview.

filePath_Ω = folderName * "/fields_O"
filePath_Γb = folderName * "/fields_Gb"
filePath_Γf = folderName * "/fields_Gf"
pvd_Ω = paraview_collection(filePath_Ω, append=false)
pvd_Γb = paraview_collection(filePath_Γb, append=false)
pvd_Γf = paraview_collection(filePath_Γf, append=false)

Finally, a for loop is created that runs through all the timesteps and stores the potential ϕn and the surface elevation ηn of each time step.

for ((ϕn,ηn),tn) in sol_t
println(tn)

pvd_Ω[tn] = createvtk(Ω,filePath_Ω * "_$tn.vtu",cellfields = ["phi" => ϕn],order=2) pvd_Γb[tn] = createvtk(Γb,filePath_Γb * "_$tn.vtu",cellfields = ["eta" => ηn],nsubcells=10)

## Saving free surface as vector

To save surface elevation as a datastring, a few extra lines need to be added to the for loop. A global variable ηns is created before the for loop. In the loop, for each time step, the surface elevation is stored in a local variable surface and subsequently push!'ed to ηns. The datastring can be stored in a JLD2 file as mentioned in the previous section.

global ηns = []
for ((ϕn,ηn),tn) in sol_t
println(tn)

global cell_values_ηn = get_cell_dof_values(ηn)
surface = []
for i in 1:length(cell_values_ηn)
push!(surface, cell_values_ηn[i])
end
push!(ηns, surface')
ηns = vcat(ηns...)

end

## Calculating energy in the system

To check the amount of energy in the system, the sum of all values can be taken in the for loop as well. Here, we distinguish the energy between potential energy Eₚ_f and kinetic energy Eₖ_f for the free surface. For the structure, two more energy terms are taken into account; the kinetic energy Eₖ_b and the elastic energy Eₚ_b of the structure. In this example the energy is calculated at three different locations; the incident wave zone xp1, the hydroelastic wave zone xp2 and the outgoing wave zone xp3.

global Eₖ_f_p1 = []
global Eₚ_f_p1 = []

global Eₖ_f_p2 = []
global Eₖ_b_p2 = []
global Eₚ_f_p2 = []
global Eₚ_b_p2 = []

global Eₖ_f_p3 = []
global Eₚ_f_p3 = []

global η_0 = interpolate_everywhere(0.0, U_Γ(0.0))

global xp1 = 1 * λ
global xp2 = 3 * λ
global xp3 = 9 * λ

x_p1(x) = (xp1 <= x <= (xp1 + λ))
x_p2(x) = (xp2 <= x <= (xp2 + (λ)))
x_p3(x) = (xp3 <= x <= (xp3 + λ))

for ((ϕn,ηn),tn) in sol_t
println(tn)

push!(Eₖ_f_p1, 0.5 * ρ_w * ∑( ∫(∇(ϕn)⋅∇(ϕn) * x_p1 )dΩ))
push!(Eₚ_f_p1, 0.5 * ρ_w * g * ∑( ∫((ηn*ηn) * x_p1)dΓf ))

push!(Eₖ_f_p2, 0.5 * ρ_w * ∑( ∫(∇(ϕn)⋅∇(ϕn) * x_p2 )dΩ))
push!(Eₚ_b_p2, 0.5 * EI * (∑( ∫(Δ(ηn)⋅Δ(ηn) * x_p2 )dΓb )))
push!(Eₖ_b_p2, 0.5 * ρ_b * h * ∑( ∫((ηₜ*ηₜ) * x_p2 )dΓb ))
push!(Eₚ_f_p2, 0.5 * ρ_w * g * ∑( ∫((ηn*ηn) * x_p2 )dΓb ))

push!(Eₖ_f_p3, 0.5 * ρ_w * ∑( ∫(∇(ϕn)⋅∇(ϕn) * x_p3 )dΩ ))
push!(Eₚ_f_p3, 0.5 * ρ_w * g * ∑( ∫((ηn*ηn) * x_p3 )dΓf ))

η_0 = interpolate_everywhere(ηn, U_Γ(tn))

end