NATIONAL TECHNICAL UNIVERSITY OF ATHENS
SCHOOL OF NAVAL ARCHITECTURE & MARINE ENGINEERING
Wave propagation in inhomogeneous,
layered waveguides based on
modal expansions and hp-FEM
with application to coastal hydroacoustic/hydrodynamic problems
Kostas Belibassakis
collaboration with Prof. G.A. Athanassoulis and
NTUA-SeaWaves group
School of Naval Architecture and Marine Engineering
National Technical University of Athens
Email: kbel@fluid.mech.ntua.gr
Cargèse summer school " Wave propagation in complex media“, 17-28 Aug 2015
contents
•
propagation of underwater acoustic and surface gravity waves
in inhomogeneous regions
•
coupled-mode methods - description & applications
•
effects of inhomogeneities
•
water waves in coastal regions
- wave seabed interaction (impermeable, porous, ...)
- currents (ambient, shear currents, wave-induced, ...)
- structures (fixed, floating, elastic, ...)
•
conclusions – current and future research
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Introduction – motivation
Wave propagation and scattering in an inhomogeneous waveguide
is an interesting mathematical problem finding important applications, as
underwater acoustic propagation and scattering
in shallow water and seismoacoustics (e.g., Boyles 1984, Jensen et al 1994),
atmospheric acoustics
(e.g., Salomons 2001)
Similar problems governed by the Helmholtz/Laplace equation are also encountered
in water wave propagation and interaction with seabed, current, ice,……
(see, e.g., Dingemans 1997, Mei et al 2005)
in variable cross-section EM waveguides (e.g., Katsenelenbaum et al 1995),…
Several methods for treating this, generally non-separable, b.v.p
have been proposed, ranging from
fully numerical, finite element and finite difference methods to
semi-analytical ones, like wavenumber integration,
boundary integral equations and coupled-mode techniques, as well as
various asymptotic models, like ray theory and the adiabatic and
parabolic approximations
see e.g., Jensen et al (1994), Lee & Schultz (1995)…
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Sea-acoustic inhomogeneous waveguide
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Jensen et al, Computational Ocean Acoustics (1994/2011)
Boyles, Acoustic Waveguides (1984) , Frisk Ocean & seabed acoustics (1994)
Brekhovskikh & Godin, Acoustics of Layered Media (1992), ……
Difficulties of the problem
inhomogeneous, layered environment, variable interfaces
high frequency/large domain, … 3D
increased discretization…
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Formulation of the problem
Cartesian coordinates
p continuous field
1
 k2
∇ ⋅  ∇p  +
p=0
ρ
 ρ
k ( x, z ) = ω / c ( x, z )
Boundary Conditions
on z = η ( x )
p =0
∂p / ∂n = 0 on z = − H
Interface Conditions
1 ∂p
1 ∂p
=
on
ρ j ∂n ρ j +1 ∂n
z = −h j ( x), j = 1, 2,..., M − 1
DOMAIN DECOMPOSITION
Complete normal mode expansions in regions of incidence and transmission
∞
(
p =∑ A e
(1)
n=1
K.A. Belibassakis:
(1) ikn(1) x
n
(1) −ikn(1) x
n
+B e
)
(1)
n
Z (z) , x<a,
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
∞
(
p = ∑ An(3)eikn
(3)
n=1
(3)
x
)
Zn(3) (z) , x>b
Cargèse summer school, 17-28 August 2015, Corsica
Separation of variables – eigensolutions
Cartesian coordinates
∇ 2ϕ + k 2ϕ = 0,
ϕ = 0, z = 0,
Cylindrical coordinates
(axially symmetric )
∂ϕ
= 0, z = −h,
∂z
(singlelayer/ hard bottom)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
In the many layer case Z n(1) ( z ) and kn(1) , n = 1, 2,3..., satisfy the following
vertical eigenvalue problem (VEP) in region of incidence D ( )
1
2
d 2 Z n(1)  (1)
+
k
z
− (kn(1) ) 2  Z n(1) = 0 ,
(
)
(
)
2


dz
(1)
dZ n(1) ( z = − H )
=0,
dz
Z ( z = η1 ) = 0,
(1)
n
Im(k)
(1)
Z ( −h j + 0) = Z (− h j − 0), j = 1, 2, M − 1 , and
(1)
n
(1)
n
(1)
…
exp (±iknx)Zn(z)
(1)
1 ∂Z (−h j + 0)
1 Z (− h j − 0)
=
, j = 1, 2, M − 1 ,
ρj
∂z
ρ j +1
∂z
(1)
n
(1)
n
…
Re(k)
where k (1) ( z ) = ω / c (1) ( z ) (and similar for transmission region D ( ) ).
3
From the properties of
the eigenvalues
{( k
( m)
n
Sturm-Liouville problems
(Coddington & Levinson 1955, Tichmarsh 1962),
) ,n = 1,2..} , m=1,3,
2
are discrete, infinite, with continuously
decreasing moduli, and are subdivided into a finite real subset
{k ( ) , n = 1,2,3...N ( )} and an infinite imaginary one {i k ( ) , n = N ( ) + 1,....} ,
m
n
m
p
m
n
m
p
where N (p ) , denotes the number of propagating modes in D ( ) , m = 1,3 .
m
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
m
Cargèse summer school, 17-28 August 2015, Corsica
Formulation of the problem cylindrical coordinates
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
numerical methods
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Coupled-Mode Methods
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
fast-convergent spectral-type model
A fast-convergent spectral model is presented for treating harmonic wave
propagation and scattering problems in stratified, non uniform waveguides,
governed by the Helmholtz equation.
The present method is based on a local mode series expansion, obtained
by utilizing local eigenfunction systems defined through the solution of
eigenvalue problems formulated along the cross section of the waveguide.
Following Athanassoulis & Belibassakis (water wave problem JFM1999),
Hazzard & Liouneville (IMA/JAM2008), Mercier & Maurel (RSPA2013),
Athanassoulis et al (JCA2008), Belibassakis et al (WM2014),
the local mode series are
enhanced by including additional modes accounting for the effects of
inhomogeneous waveguide boundaries and/or interfaces.
The additional modes provide an implicit summation of the slowly
convergent part of the local-mode series, rendering the remaining part to be
fast convergent, increasing the efficiency of the method,
especially in long-range propagation applications.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Transmission problem
p continuous field
1
 k2
∇ ⋅  ∇p  +
p=0
ρ
 ρ
k ( x, z ) = ω / c ( x, z )
Boundary Conditions
p =0
on z = η ( x )
∂p / ∂n = 0 on z = − H
Interface Conditions
1 ∂p
1 ∂p
=
ρ j ∂n ρ j +1 ∂n
on
z = −h j ( x), j = 1, 2,..., M − 1
Complete normal mode expansions in regions of incidence and transmission
∞
(
p =∑ A e
(1)
n=1
K.A. Belibassakis:
(1) ikn(1) x
n
(1) −ikn(1) x
n
+B e
)Z
(1)
n
(z) , x<a,
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
∞
(
p =∑ A e
(3)
n=1
(3) ikn(3) x
n
)Z
(3)
n
(z) , x>b
Cargèse summer school, 17-28 August 2015, Corsica
Variational formulation of the problem
F ( p ,{Bn(1) } ,{ An(3) }) =
z = η1

1 1
1
+ ∫  p − p( ) Bn( )

2
z =− H
δF
ρ −1 ∇p
∫(2)  (
1
2
)
2
− ρ −1k 2 ( p
D
(1)
({ }) dz −
(1)
∂p
Bn
 ⋅

∂x
({ })
2
1

 p − 1 p(3)
∫  2
z =− H
3)
−∑ ∫ (∇⋅ (ρ ∇p) + ρ k p) δ p dxdz +
j =1
−1
−
−1 2
(2)
Dj
z = η3
∫ (
z =− H
p − p( )
3
)
(3)
n
(3)
({ }) dz
(3)
∂p
An
 ⋅

∂x
({ A })
,
M
Energy type
functional
2
The admissible function space consists of
globally continuous and piecewise smooth functions,
with continuous second derivatives
in the interior of each layer, such that
p[x,z=η(x)]=0
( { } { }) = 0
p ( ) , Bn( ) , An(
z = η3
)  dxdz +
z = η1
∫ (p
−p
(1)
)
z =− H
z = η1
3
1
 ∂p
∂p( )
∂p( ) 

δ
dz − ∫ 
−
 δ p dz +

∂x
∂
x
∂
x

z =− H 
∂p( )
δ
dz +
∂x
x =b
∫
x=a
( z =− H )
1
1 ∂p
δ p dx
ρM ∂z
3
M −1
 1 ∂p
 ∂p
∂p( ) 
1 ∂p 



− ∫ 
−
−
δ p dx = 0
δ p dz − ∑ ∫ 



∂
x
∂
x
ρ
∂
N
ρ
∂
N

j =1 z =−h ( x) j

j +1
z =− H 
j
z = η3
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Local-mode series expansion
A spectral-type representation based on local-modes (briefly denoted by SR) of the
( 2)
wave field p ( x,z ) ∈ D is defined by
∞
p ( x,z ) = ∑U n ( x ) Z n ( z; x )
n =1
The family of local vertical basis functions {Z n ( z; x ) , n = 1, 2 ,3...}
In the expansion, are parametrically dependent on x, obtained by
formulating and solving local, vertical Sturm-Liouville problems
in the z-intervals  − H , η ( x )  , at each horizontal position:
2
d 2Zn 
(1) 2 
+
;
−
(
)  Zn = 0
k
z
x
k
(
)
(
)
n
2



dz
Zn (−hj ( x) +0) = Zn (−hj ( x) −0), j =1,2, M −1
Zn (z =η1 ( x)) = 0,
dZn (z = −H)
=0
dz
1 ∂Zn (−hj + 0) 1 Zn (−hj − 0)
=
, j = 1,2, M −1
ρj
∂z
ρ j +1
∂z
,
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Finite element solution of VEP
Assume a partition of [−H , η ( x )] , of the form
−H = z1 < z2 < ..... < zN +1 = η ( x) , with
N ∈ ℕ and N > M .
Partition is such that interface positions coincide with a node.
We introduce the sequence of F.E. sub-spaces
a ( w, u) = ∫
η
ρ
−H
b( w, u ) = ∫
−1
η
−H
η
dw du
dz − ∫ k 2ρ −1wudz
−H
dz dz
ρ −1wudz
u h ∈ H 1 (−H , η ( x)) : u h


≡ Pℓ (z) and u (η ( x)) = 0 




[ zi , zi+1 ]
V ≜





i = 1, 2,..., N , ℓ ∈ ℕ, x ∈ [a,b]



h
The discrete variational formulation of the local VEP
Find
(λ h , u h ) ∈ ℝ ×V h such that
h
h
a( wh , u h ) = λ h b( wh , u h ) , ∀w ∈ V .
In the following example we assume piecewise linear
approximation for the FEM solution, i.e. l = 1
N
u = ∑ c j N j ( z) ,
h
j =1
where N j ∈ V . the discrete variational formulation
finally becomes an eigenvalue matrix equation
h
A u = λ Bu
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
(a) Exact and computed eigenvalues
for different numbers of elements.
(b) Convergence of the computed
5th and 10th eigenvalue.
Cargèse summer school, 17-28 August 2015, Corsica
2-layer acoustic waveguide
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Finite element solution of VEP
First 5 vertical eigenfunctions
FEM solution for N=160,p=3
K.A. Belibassakis:
Eigenvalue distribution - convergence of the FEM solution
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
convergence plot
Convergence plot of the 5,10,15 eigenfunction in the H1-norm, as calculated by the present FEM
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Enhancement of the series – additional modes
The finite truncation of the local-mode series
1 ∂Zn (−hj + 0) 1 Zn (−hj − 0)
=
, j = 1,2, M −1
ρj
∂z
ρ j +1
∂z
1 ∂p
1 ∂p
=
ρ j ∂n ρ j +1 ∂n
is incompatible with the sloping interface conditions,
whenever
dh j ( x ) dx ≠ 0 ,
j = 1,2,...M −1 ,
rendering the above series to converge slowly (in an
and the coefficients
L2 -sense),
−2
U n to decay slowly like O ( n )
To remedy this inconsistency, an additional mode associated
with each interface is introduced, denoted by
z
U j ( x ) Z j ( z; x ) , j = − M + 2,.., −1,0
n
These modes are called the sloping-interface modes.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
z = hj ( x)
ρj
ρ j +1
Cargèse summer school, 17-28 August 2015, Corsica
Enhanced local mode representation
Thus, we obtain the following enhanced local-mode series
n=0
∑
p ( x, z ) =
n =− M + 2
∞
U n ( x ) Z n ( z; x)
+ ∑ U n ( x ) Z n ( z; x )
n =1
The vertical structure of the sloping-interface modes, for every horizontal
position is any globally continuous and piecewise smooth function
with support in each layer, satisfying the following condition(s):
1 ∂Z n
ρ j ∂z
−
z = − h +j
1 ∂Z n
ρ j +1 ∂z
= 1, j = 1,2,..., M − 1,
z =− h −j
n =1− j
.
Moreover, the function Z0 ( z;x) should satisfy the homogeneous Dirichlet
condition at
z = η ( x) .
Consequently, the M-1 terms U n ( x) Z n ( z; x) , n = −M + 2,.., −1,0 , are
additional degrees of freedom in the bounded subdomain,
permitting the consistent satisfaction of all
interface conditions.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Coupled-Mode System (CMS)
Substituting the enhanced local mode representation in the variational principle the
following coupled-mode system (CMS) of second-order ordinary differential
equations, with respect to the mode amplitudes U n ( x1 ) , n = − M + 2,....0,1, 2,3...
∞
∑
n =− M + 2
amn ( x )
d 2U n ( x )
dx
2
+ bmn ( x )
dU n ( x )
dx
+ cmn ( x )U n ( x ) = 0,
m = − M + 2,....0,1,2,3...
The x-dependent coefficients amn ,bmn ,cmn are defined in terms of Z n ( z; x ) and are
M −1
∂Z n
1
1  dh j

bmn = 2
, Z m + ∑ −
Z n (−h j ) Z m (−h j )

∂x
j =1 
 ρ j ρ j +1  dx
amn = Z n , Z m ,
M −1
∂2Zn ∂2Zn
2
 Z + dh j Z
cmn =
k
Z
,
Z
+
+
+
∑
n
m
n
 n z
∂x 2
∂z 2
dx
j =1 
where
f , g := ∫
K.A. Belibassakis:
η
−H
ρ −1 f ( z ) g ( z ) dz
Zn
u
 1 ∂Z
n
= 
 ρ j ∂u
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM

 Z m (−h j )
x

z =− h+j
1 ∂Z n
−
ρ j +1 ∂u



z =− h−j  ,

j = 1, 2,..., M − 1
.
Cargèse summer school, 17-28 August 2015, Corsica
Numerical results & discussion
H = 100m

75 m , x < a




  x − 300 



 − 0.5 , a ≤ x ≤ b
h1 ( x ) = 50 − 25 tanh 3π 



 400 






25 m ,
x>b



0 , x<a



η ( x) = 
 S o sin( k s x ), a ≤ x ≤ b



0 , x>b


ρ1 = 1g / cm3 , c1 = 1500m / sec
ρ2 = 1.5 g / cm3 , c2 = 1700m / sec
Acoustic pressure (real part).
Excitation by the 1st mode.
Frequency 20 Hz.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Acoustic pressure (modulus).
Excitation by the 1st mode.
Frequency 20 Hz.
Cargèse summer school, 17-28 August 2015, Corsica
CMS - convergence
O(n-2)
O ( n −2 )
O(n-4)
The important effect of the additional modes is to significantly increase the rate of
decay of Z n − Fourier coefficients of the acoustic wave potential (modal amplitudes):
U n ( x ) ≤ C ( x ) n −4 ,
n → ∞ , ∀x ∈ [ a, b] .
The bound C ( x ) is a continuous function and, thus, the previous estimate is global
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
FEM solution of the CMS – error estimate
For the approximation of the solution of CMS for u(x)={Un(x)} by FEM,
a = x1 < x2 < ..... < xN +1 = b .
we assume a partition of [ a,b] of the form
Let Pl ( z) be local polynomial of degree ℓ .
We now set
{
V h ≐ u h ∈ V : uhj
}
≡ Pℓ (x) , i = 1, 2,..., N , j = 1, 2,...., N m . Obviously
[ xi , xi+1 ]
V h ⊂V .
Assuming sufficient regularity of the exact solution, we obtain the standard
Hilbert space error estimate (see Belibassakis et al WM2014, Sec.5)
u − uh
V
2
In addition, an  L (a,b)
≤ Ch ℓ u
 H ℓ+1 (a ,b ) Nm


,
for some positive constant C .
Nm
error estimate is possible
u − uh
H
≤ Chℓ+1 u
 H ℓ+1 (a ,b ) Nm


.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
FEM solution of CMS – error estimate
Results for p=1,2,3 for N=50-300 elements.
The estimated rates of the error decay are calculated to be
1.974, 3.124, 3.995, in compatibility with theoretical predictions.
The dashed lines are used to illustrate p-convergence behavior.
As expected these dashed curves present negative curvature.
K.A. Belibassakis:
Results for p=1,2,3 for N=50-300 elements in the H1-norm.
Rates for p= 2, 3 are calculated to be 2.28, 3.02.
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Numerical results & discussion
Acoustic pressure (real part & modulus).
Excitation by the 1st mode.
Frequency 50 Hz.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Acoustic pressure (real part & modulus).
3layer WG - Excitation by the 1st mode.
Frequency 20 Hz.
Cargèse summer school, 17-28 August 2015, Corsica
Cylindrical coordinates - Point source
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
2layer acoustic waveguide – enhanced local mode series
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
2layer acoustic waveguide – enhanced local mode series
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
sloping interface mode
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Cylindrical coordinates - Point source
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Standard vs enhanced modal series
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Modal series convergence and accuracy of eCMS
Athanassoulis et al (JCA2008)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
acoustic and surface gravity waves
1/μ=g/ω2
acousticwaves
surfacegravity
waves
ϕ−
1 ∂ϕ
= 0, z = 0, µ = ω 2 / g
µ ∂z
acousticgravity
waves
boundary/interface conditions
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Acoustic-gravity waves
Generation / sources
non-linear interactions of pairs of nearly
opposing gravity waves having nearly
equal frequencies. In this case, the lower
frequency part of the spectrum,
frequencies lower than 2 Hz is caused by
the nonlinearity of the hydrodynamic
equations; e.g.,Ardhuin et al (JGR2011),
Ardhuin & Herbers (JFM2013)
Seismic activity
Tsunami generation & propagation
together with acoustic signal
(Stiassnie, J. Eng. Math 2010, Kadri &
Stiassnie, J. Geophys. Res. 2012)
Dominant effects
Inhomogeneous waveguide
Shoaling
Scattering
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
3
6
Cargèse summer school, 17-28 August 2015, Corsica
Governing Equations
Helmholtz Equation
1
 k2
∇ ⋅  ∇p  +
p = 0,
ρ
 ρ
with
k ( x, z ) = ω / c ( x, z )
Boundary conditions:
∂p / ∂z − µ p = 0, at z = 0
2
with µ = ω / g
Normal-mode representations of the A-G
wave field in the semi-infinite strips
(incidence/transmission regions):
∂p / ∂n = ∂p / ∂z = 0, at z = − H
Interface conditions:
1 ∂p
1 ∂p
=
ρ j ∂n ρ j +1 ∂n
Np
p
(1)
=∑A e
n =1
(1) ikn(1) x
n
∞
p
(3)
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Z ( z ) + ∑ Bn(1) e − ikn x Z n(1) ( z )
= ∑ An(3) eikn
n =1
K.A. Belibassakis:
(
∞
(1)
n
(1)
(1)
n =1
x
)Z
(3)
n
( z)
Cargèse summer school, 17-28 August 2015, Corsica
Vertical Eigenvalue Problem
Np
(1)
Wave potential in D :
p
(1)
=∑A e
n =1
(1) ikn(1) x
n
∞
Z ( z ) + ∑ Bn(1) e − ikn x Z n(1) ( z )
(1)
n
(1)
n =1
(1)
(1)
where Z n ( z ) and kn , n = 1,2,3..., satisfy the vertical Eigenvalue problem in
D (1)
2
d 2 Z n(1)  (1)
(1) 2  (1)
+
k
z
−
(
k
)  Z n = 0, k (1) ( z ) = ω / c (1) ( z )
(
)
(
)
n
2



dz
Boundary conditions:
dZ n(1) ( z = 0)
− µ Z n(1) ( z = 0) = 0,
dz
(1)
dZ n ( z = − H )
=0
dz
Interface conditions:
∞
p
1
(
= ∑ An(3) eikn
n =1
Z n(1) ( − h(j ) + 0) = Z n(1) ( − h (j ) − 0), j = 1, 2, M − 1
1
(3)
(3)
x
)Z
(3)
n
( z)
Similarly Z n(3) ( z ) and kn(3) , n = 1, 2, 3...,
are obtained by the solution of the VEP
in D (3)
()
()
(1)
(1)
1 ∂Z n ( − h j + 0)
1 Z n (− h j − 0)
=
, j = 1, 2, M − 1
ρj
∂z
ρ j +1
∂z
1
K.A. Belibassakis:
1
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
3
8
Cargèse summer school, 17-28 August 2015, Corsica
Eigenvalues
Eigenvalues of the regular Sturm-Liouville problems:
{(
kn(
m)
) , n = 1, 2..}, m = 1,3
2
{k ( ) , n = 1, 2,3...}
m
n
{
kn( m ) , n = 1, 2,3...N (pm )
}
U
{i k
( m)
n
}
, n = N p( ) + 1,....
m
∂p / ∂z − µ p = 0, z = 0
k
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
acoustic
parameter
Cargèse summer school, 17-28 August 2015, Corsica
Matching (transmission) BVP
A transmission boundary value problem in D( 2)
can be defined by means of :
Np
p
(1)
=∑A e
(1) ikn(1) x
n
n =1
∞
p
(3)
(
= ∑ An(3) eikn
n =1
∞
Z ( z ) + ∑ Bn(1) e − ikn x Z n(1) ( z )
( 3)
(1)
n
x
)Z
(1)
n =1
(3)
n
( z)
Matching conditions:
p
p
K.A. Belibassakis:
( x, z ) = p ( x, z ) ,
∂ p ( 2) ∂ p (1)
= (1) ,
( 2)
∂x
∂x
x = a , − H < z < 0,
( x, z ) = p ( x, z ) ,
∂ p ( 2 ) ∂ p ( 3)
=
,
∂x
∂x
x1 = b , − H < z < 0
(2)
(2)
(1)
( 3)
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Variational formulation of the transmission problem
Energy-type functional:
F
( p ,{B },{A })
−
(1)
( 3)
n
n
z =η3
1 ( 3)

p− p
2
z =− H 
∫
−1
∫(2)  ρ ( ∇p
1
=
2
∫ (∇ ⋅ ( ρ
2
D( )
({A })
( 3)
n
({ }) dz
( )
An(
 ∂p

∂x

3
3)
δF
)
∇p ) + ρ k p δ p dxdz +
−1
−ρ k
−1 2
(p )
2
D
Variational Principle :
−
)
2
−1 2
 dxdz + 1

2ρ
x =b
∫ µ p dx +
2
1 x =a
(1)
({ }) dz
(1)
Bn
1

 ∂p
+ ∫  p − p (1) Bn(1) 
2
∂x

z =− H 
z =η1
({ })
( p ; {B },{A }) = 0
z =η1
∫
z =− H
(p
(1)
( 3)
n
n
−p
(1)
)
3
∂p( )
∂p ( )
( 3)
δ
dz + − ∫ p − p δ
dz
∂x
∂
x
z =− H
z =η
1
(
)
3
M −1
 1 ∂p
 ∂p ∂p(3) 
 ∂p ∂p (1) 
1 ∂p 
−
−
δ
p
dz
−
−
− ∫ 
−

 δ p dx = 0


∑
 δ p dz
∫
∫
ρ
ρ
∂
x
∂
x
∂
N
∂
N
∂
∂
x
x
j =1 z =− h j ( x ) 
j
j +1
z =− H 


z =− H 

z =η1
z =η3
where,
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
∂p
∂N
z =− h j ( x )
=
∂p dh j ∂p
+
∂z dx ∂x
Cargèse summer school, 17-28 August 2015, Corsica
Enhanced local mode representation
z=0
Spectral type representation of the wave potential in D (2)
∞
p ( x, z ) = ∑ U n ( x ) Z n ( z; x )
Obtained from the
solution of the local
VEP for each horizontal
position a < x < b
n =1
….
z=-H
x
dh j ( x ) dx ≠ 0, j = 1, 2,...M − 1
! Incompatibility with sloping interface conditions when
Additional M-1 sloping-modes, denoted by U j ( x ) Z j ( z; x ) , j = − M + 2,.., −1,0 are used.
Enhanced local mode representation:
p ( x, z ) =
0
∑
n =− M + 2
∞
U n ( x ) Z n ( z ; x ) + ∑ U n ( x ) Z n ( z; x )
n =1
Sloping interface modes
Propagating & evanescent modes
Sloping vertical modes satisfy in the intervals
1 ∂Z n
ρ j ∂z
K.A. Belibassakis:
[−hM −1 ( x), −hM − 2 ( x)],...[−h1 ( x), 0]
−
z =− h j
1 ∂Z n
ρ j +1 ∂z
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
= 1, j = 1, 2,..., M − 1
z =− h j
Cargèse summer school, 17-28 August 2015, Corsica
FEM solution of local VEP
Define the function space:
H 1 E ( x ) = {u : u ∈ H 1 ( − H ,0), x ∈ [a,b]}
Local Eigenvalue problem in variational form:
1
Find (λ , p ) ∈ × H 0 E ( x ) such that a ( w, p ) = λ b ( w, p )
0
a ( w, p ) = ∫ ρ −1
−H
∀w ∈ H 1 E ( x )
0
dw dp
dz − ∫ k 2 ρ −1wpdz + ρ −1µ [ wp ]z =0
−H
dz dz
0
b( w, p) = − ∫ ρ −1wpdz
−H
Define the finite element
subspaces:
{
V h = u h ∈ H 1 (− H ,0) : u h
[ zi , zi +1 ]
}
≡ Pl (z), i = 1,2,..., N , l ∈ , x ∈ [a,b]
Discrete Local Eigenvalue problem in variational form:
Find (λ , p ) ∈
h
h
h
h
h
h
h
× V h such that a( w , p ) = λ b( w , p ) ∀wh ∈ V h
N
Solution assumes the form:
p = ∑ c j N j ( z ), N j ∈V h
h
j =1
After substitution:
A u = λ Bu
where aij = a ( N i , N j ), bij = b( N i , N j )
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Numerical results VEP (h=1000m, f=2Hz)
•
3
Layer 1 properties : ρ1 = 1g / cm , c1 = 1500 m / s
•
Layer 2 properties: ρ 2 = 1.5 g / cm3 , c2 = 1700 m / s
Boundary layer formation.
The first 5 eigenfunctions for f=2 Hz
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Decreasing μ
The first 5 eigenfunctions for f=0,08 Hz
Interface at 100m and 500m depth below free surface
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Comparison between
the computed
eigenvalues against
the exact solution .
Convergence of the finite element solution
for the 5th, 10th and 15th eigenvalues, p= 1, 2
and 3.
Deviation increases with
increasing eigenvalues.
Enhanced rates of convergence
are obtained by raising the
degree of the piecewise
polynomials.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Coupled Mode system (CMS)
In order to calculate the modal amplitudes, the enhanced modal representation
p ( x, z ) =
0
∑
n =− M + 2
∞
U n ( x ) Z n ( z ; x ) + ∑ U n ( x ) Z n ( z; x )
n =1
and the variation of the unknow field,
δ p ( x, z ) =
∞
∑
n =− M +1
Z n ( z; x) δ U n ( x)
are substituted in the variational equation resulting in the following CMS,
d 2U n ( x )
dU n ( x )
amn ( x )
+ bmn ( x )
+ cmn ( x )U n ( x ) = 0
∑
2
dx
dx
n =− M + 2
∞
System coefficients are given by,
amn = Z n , Z m
M −1 
∂Z n
1
1  dh j
bmn = 2
, Zm + ∑  −
Z (−h j ) Z m (−h j )


∂x
ρ j +1  dx n
j =1  ρ j
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Numerical Results – gravity mode
•
Bottom topography
•
Layer 1 properties :
•
Layer 2 properties:
(silt-clay layer)
   x − 3000 

h1 ( x) = 500 − 450 tanh  2π  
 − 0.5   , a ≤ x ≤ b

   4000 
3
ρ1 = 1g / cm , c1 = 1500 m / s
ρ 2 = 1.5 g / cm 3 ,
a = 2800,
b = 7200 m
c2 = 1700 m / s
Acoustic pressure (real part and modulus). Frequency
0,08Hz. Excitation by 1st mode (gravity mode)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Numerical Results: Acoustic-Gravity Modes
Figure 7: Acoustic pressure (real part and modulus). Frequency 2 Hz. Excitation by 2nd mode (acoustic)
Minimal interaction
between the 1st mode
(gravity) and the rest.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Acoustic mode difficult to
observe at the free surface.
Cargèse summer school, 17-28 August 2015, Corsica
Numerical Results: FEM solution of VEP (f=0.2Hz)
•
Bottom topography
•
Layer 1 properties :
•
Layer 2 properties:
(silt-clay layer)
(a)
   x − 3000 

h1 ( x) = 500 − 450 tanh  2π  
 − 0.5   , a ≤ x ≤ b

   4000 
3
ρ1 = 1g / cm , c1 = 1500 m / s
ρ 2 = 1.5 g / cm 3 ,
a = 2800,
b = 7200 m
c2 = 1700 m / s
(b)
(c)
First 5 eigenmodes for f=0,2Hz and interface positioned at (a) 950 m (b) 500 m (c) 50 m.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Results: Oscillating seabed
∞
p ( x, z ) = 2 ρω 2α 0 ∑
n =1
(1)
n
A
(ω ) = 2 ρω α 0
2
Z n(1) ( z ) Z n(1) ( z0 )
k
(1)
n
Z
Z n(1) ( z0 )
kn(1) Z
(1) 2
n
(1) 2
n
( exp ( ik
(1)
n
a0 = 1m, c = 1000m
x + c ) + exp ( ikn(1) x − c )
)
cos ( kn(1) c ) exp ( ikn(1) a )
Yamamoto (1982), Stiassnie (2010)
Acoustic pressure (real part) excited by bottom oscillating block at 0.08 Hz and 1.5 Hz.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Moving seabed
Amplitude spectrum
(corresponding to free surface elevation)
Bottom motion:
a0
τ
H ( c 2 − x 2 ) H ( t (τ − t ) )
0.08Hz
1.5Hz
(a) Pressure distribution at depth 50m approximated from contributions at frequencies f=0.08Hz
and f=1.5Hz and
(b) Pressure signal at depth 50m at the shallow end of the domain b=7200m.
:
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Propagation of surface gravity waves
in nearshore/coastal regions
problems & applications
distinctive features of coastal waters
Massel, S.R., 1989
Hydrodynamics of
coastal zones,
Elsevier
as waves travel to shallow water, wave-seabed interaction becomes important,
their dynamics become progressively more nonlinear and dissipative
energy is transferred from the peak of the spectrum to higher and lower freqs
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
important wave phenomena
-
reflection, refraction, shoaling
wave interaction with bottom topography (mild/steep)
diffraction
dispersion
……..
-
non linear phenomena
bottom friction, wave breaking and energy dissipation
wave-wave and wave-seabed interaction and harmonic generation
wave induced currents
wave run-up
sediment transport, coastal erosion
wave induced porous flow
………
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
wave theories
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
wave models in variable bathymetry
Mild Slope Eq. [ short waves U<8π2 , Z(z)~cosh[k(z+h)] ] frequency domain (MMS)
[ H/h small ]
2
2
∇
CC
∇
ζ
+
k
(1 +ψ ) CCgζ = 0
(
)
g
C − Cg 
∂ζ
2
+ω 
 ζ − ∇ ( CCg ∇ζ ) = 0,
2
∂t
ψ = ψ kh, ∇h, h,′′αβ
 C 
(
Boussinesq Eq.[ U~8π2
)
H/h ~ (λ/h)-2, Z(z)~ (z+h)2m]
∂ζ
+ ∇ [(h + ζ )u)] = 0,
∂t
∂u
1 ∂
1 ∂
+ (u∇)u + g∇ζ = − h2 ∇(∇u) + h ∇[∇(hu)]
∂t
6 ∂t
2 ∂t
NLSWE [long waves U>8π2 , Z(z)~=constant ]
∂ζ
+ ∇ [(h + ζ )u)] = 0,
∂t
K.A. Belibassakis:
∂u
+ (u ⋅∇)u + g ∇ζ = 0
∂t
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
LSWE [ H/h small ]
∂ 2ζ
− g∇( h∇ζ ) = 0,
2
∂t
Cargèse summer school, 17-28 August 2015, Corsica
Enhanced wave models in general bathymetry
• Assumptions (inviscid fluid, irrotational flow,
linear waves / degree of non-linearity,….)
• Variational formumations
- unconstrained (Luke 1967)
- Hamiltonian
(Petrov 1964, Zakharov 1968,
Craig & Sulem 1993, Craig et al 2009,..)
• Complete representations of the wave field,
F(x, z ; t) = fm(x, t) Zm (z ; x, t)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Consistent coupled-mode system (CMS) in general
bathymetry (Athanassoulis & Belibassakis JFM1999)
- waves of small amplitude
- for simplicity
1D depth function h(x)
monochromatic (normally) incident wave
h ( x ) = h1 , x ≤ a ,
K.A. Belibassakis:
( d ζ / dx <<1 )
h = h2 ( x ) ,
a < x < b,
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
h ( x ) = h3 ,
x≥b
Cargèse summer school, 17-28 August 2015, Corsica
normal-mode representations in the two half strips
left half strip
ϕ
(1)
D( ) :
1
( x, z ) = ( exp ( ik0
)
(
)) ( z ) +∑ C Z ( z ) exp ( k ( x − a )) ,
Z ( ) ( z ) = cosh ( k ( ) ( z + h ) ) / cosh ( k ( ) h ) ,
µ h = k ( ) h tanh ( k ( ) h )
(1)
∞
(1)
x + AR exp −ik0 x Z 0
1
n
where
(1)
1
n
1
n
1
n =1
1
(1)
n
1
n
1
(
and similar for ϕ
3)
(1)
n
(1)
n
1
1
n
1
( x,z; µ ) , ( x,z ) ∈ D(3) .
Reformulation as matching b.v.p. in the middle variable bathymetry subdomain
(
for ϕ
2)
( x,z; µ ) ,
in D (
2)
∇2ϕ ( 2) = 0,
- Laplace
- FSBC and BBC
∂ϕ ( 2 )
2
− µϕ ( ) = 0 ,
∂z
∂ϕ ( 2) dh ∂ϕ ( 2)
+
= 0,
∂z
dx ∂ x
z=0
- matching consitions at the vertical interfaces
ϕ
( 2)
K.A. Belibassakis:
(1)
=ϕ ,
∂ϕ ( 2) ∂ϕ (1)
=
∂x
∂x ,
x=a,
z = −h ( x )
x=a και x=b
ϕ
( 2)
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
( 3)
=ϕ ,
∂ϕ ( 2) ∂ϕ (3)
=
∂x
∂x ,
x=b
Cargèse summer school, 17-28 August 2015, Corsica
constant depth strip(s) – vertical eigenfunctions
z
z=0
d 2Zn ( z )
2
−
k
n Z n ( z ) = 0, − h < z < 0
2
dz
dZ n ( z )
− µ Z n ( z ) = 0,
z=0
dz
dZ n ( z )
z = −h
= 0,
dz
Im(k)
…
h
Zn(z)
Z0(z)
ik2
ik1
Re(k)
k0
µ h = kn h tanh ( kn h )
z=-h
Regular Sturm Liouville problem
{Z ( z ) ,
n = 0,1, 2,....} complete orthonormal system
n
∞
f ( z ) = ∑ f n Z n ( z ) , f n = f ,Z n / Z n ,
2
f ,g =
n =0
K.A. Belibassakis:
L2-basis in −h < z < 0
z =0
∫ f ( z ) g ( z ) dz ,
z =− h
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
local mode expansion in the variable batymetry region
ϕ
( 2)
∞
( x, z ) = ϕ−1 ( x) Z−1 ( z; x)
the first term
the rest terms
Z0 ( z; x) =
n =1
ϕ 0 ( x ) Z 0 ( z; x )
propagating mode
ϕ n ( x ) Z n ( z; x ) , n = 1, 2,K
cosh k0 ( z + h) 
cosh ( k0 h)
+ ϕ0 ( x ) Z0 ( z; x ) + ∑ϕn ( x ) Zn ( z; x )
, Zn ( z; x) =
cos  kn ( z + h)
(
)
cos kn h
evanescent modes
, n = 1,2,K µ h ( x ) = k ( x ) h ( x) tanh k ( x ) h ( x ) 
,


The extra term ϕ−1 ( x ) Z −1 ( z; x ) is a correction term
sloping-bottom mode
enables the consistent satisfaction of the Neumann BC on the sloping bottom
∂ Z −1 ( z = 0 )
− µ Z −1 ( z = 0 ) = 0 , z=0,
∂z
∂ Z −1 ( z = − h ( x ) )
≠0,
z=-h(x),
∂z
A possible form is
K.A. Belibassakis:
 z 3  z 2
Z−1 ( z; x ) = h ( x ) 
 + 

h
x
 ( )   h ( x) 

Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
a≤ x≤b
∂ϕ / ∂z
∂ϕ / ∂n


,

Cargèse summer school, 17-28 August 2015, Corsica
Coupled Mode System
(Athanassoulis & Belibassakis JFM1999, 2DH: Belibassakis et al APOR2001)
Using the local mode expansion
ϕ( 2) ( x, z ) = ∑ϕn ( x) Zn ( z; x)
n
in an
energy type variational principle a 2nd order ODE system is derived (CMS)
generalizing the one in Porter & Staziker(1995) and Massel(1993)
∞
∑ a ( x ) ∇ ϕ ( x ) + b ( x ) ∇ϕ ( x ) + c ( x )ϕ ( x ) = 0,
2
n =−1
mn
n
where the coefficients
mn
n
mn
n
amn ( x ) = Z m ( z; x ) , Z n ( z; x )
m = −1, 0,1,...., x = ( x, y ),
and similar for bmn ( x ) , cmn ( x )
boundary/matching conditions, for example, in 1DH propagation at x=a
(
)
(1)
(1)
(1)
ϕ−1 ( a ) = 0 , ϕ 0′ ( a ) − ik0 ϕ 0 ( a ) = −2 i k0 exp −ik0 a , ϕn′ ( a ) + kn(1) ϕn ( a ) = 0, n = 1, 2,.. ,
and similar at x=b
keep only n=0 term (propagating mode)
a00 ( x ) ∇ 2ϕ0 ( x ) + b00 ( x ) ∇ϕ0 ( x ) + c00 ( x ) ϕ0 ( x ) = 0,
Modified MSE
Massel(1993), Chamberlain & Porter(1995)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Harmonic wave potential for frequency ω=2 rad/s in variable bathymetry
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
rate of decay of modal amplitudes
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Effect of sloping-bottom mode on the field
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Berkhoff elliptic shoal (Belibassakis et al APOR 2001)
H=0.1m
T=1sec
θ=20deg
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
elliptic shoal: wave field on the horizontal plane
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
elliptic shoal: vertical structure of the wave field
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
comparisons against exp data (Berkhoff et al 1982)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
steep topography of Scripps and La Jolla submarine
canyons in Southern California (NCEX site)
very complex terrain
complex
Incident spectrum
=>
huge amount of
calculations!
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
offshore spectrum
JONSWAP frequency spectrum
Hs = 1m
Tpeak = 15sec
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Total wave field on the free surface (real part) Incident wave from W of period T=15s
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Comparison of Hs predictions by CMS and SWAN
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
observed nearshore spectra
Point (a)
Point (b)
a
b
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
comparison with observed nearshore spectra
Offshore spectrum, Hs=1.08, Tp=14.3s
(a)
(b)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
scattering by ambient currents (Belibassakis et al APOR 2011)
rip current over sloping bottom (Chen etal CoEng 2005)
σ = ω − U ⋅k
σ 2 = kn g tanh ( kn h )
Zn =
Max current speed
0.3m/s
,
cosh ( kn h )
amn = Z m , Z n
H/2 = 1m,
T = 16sec
Dir = 0deg
Sloping beach 1:50
(9)
cosh  kn ( z + h ) 
bmn =
2iω
U + 2 ∇Zn , Zn +
g
Zn ( −h) Zm ( −h) ∇h
cmn = .......
(
)
1
amn∇ ϕn + b mn∇ϕ n + cmnϕ n − ∇ ⋅ U ( U ⋅∇ ) ϕn  = 0, m = −1, 0,1,..
∑
g
n =−1
2
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
78
elliptic scatterer over horizontal bottom
Vincent & Briggs (1989)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
elliptic scatterer and current due to breaking
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
transformation of wave spectrum over elliptic shoal
in presence of current (Belibassakis et al OE2014)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
scattering by a rip current in a sloping beach region
directional spectrum of
incident wave system
corresponding to
H s = 0.5 m , TP = 10 s ,
mean wave direction Θ = 270 ° .
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
distribution of Hs: CMS against SWAN predictions
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
scattering by shearing current (Belibassakis, JFM 2007)
generalization of mild shear equation (McKee 1987,1996)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
dissipation due to breaking and bottom friction
∞
∑ {a ( x ) ϕ ′′ ( x ) + b ( x ) ϕ ′ ( x ) + ( c ( x ) +
n =−1
mn
n
mn
n
mn
iγ ( x ) k0 ( x ) δ 0 n ) ϕn ( x )} = 0,
m = −1,0,1,...
where γ ( x ) = γ f ( x ) + γ b ( x ) dissipation coefficient
(Dingemans 1997, Massel 1992...)
Mean-flow equations:
(i) depth-averaged mass
∂ηm ∂
+ ( ( h + ηm )U ) = 0
∂t
∂x
(ii) depth-averaged momentum
∂η  ∂S
∂U
dh ∂
 ∂U
%%
ρ ( h + ηm ) 
ρ uwdz
+U
+ g m  + xx = τ S − τ b +
∫
∂x
∂x  ∂x
dx ∂x z =− h( x )
 ∂t
0
where
τ S , τ b : surface and bottom stresses
η ( x ;t )
2
1
2
%
S xx ( x ) = ∫ ( ρ u + p ) dz − ρ g ( h + ηm ) radiation stresses
2
− h( x)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
set-up and mean flow over steep shoal
(Gourlay 1996, Massel & Gourlay CoEng2000)
Numerical results concerning waveheight , set-up and flow , for a steep bottom profile (100%slope),
treated as open domain (including mean flow), Belibassakis et al OMAE 2007.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
set-up and long wave induced due to the
shoaling of short wavegroups (Baldock 2006)
Bottom profile (left subplot) and spatio-temporal evolution of the
short-wave group
η ( x , t ) , of period 2π / ω ≈ 1s, (middle subplot) and
the induced long wave η s ( x , t ) , of period
K.A. Belibassakis:
2π / ∆ ω ≈
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
5s (right subplot)
Cargèse summer school, 17-28 August 2015, Corsica
set-up and long wave induced due to the
shoaling of short wavegroups (Belibassakis OMAE2011)
(a)
(b)
α1 / α 2
(c)
(d)
Long waves (a) and wave set-up/set-down (b), induced by short-wave groups (c)
propagating over the bottom profile (d).
short-wave bichromatic group (f1/f2=0.9/1.1Hz), with mean period of 5sec, and
equal amplitudes of the two monochromatic components 0.03m.
K.A. Belibassakis:
. based on
Wave propagation in inhomogeneous, layered waveguides
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
wave induced porous flow
Sandy beaches consist of unconsolidated sediment and are permeable.
In this case, the changes of pressure associated with both
the mean and the oscillatory wave flow produce
groundwater flow of sea water within the porous medium.
The mean flow component plays a significant role concerning
water table formation and groundwater flow.
Furthermore, this component percolates through the permeable bottom
and influences the wave forces on structures supported by or extending
into the sea bottom.
The oscillatory component of the wave field contributes to damping of
the waves over a porous beach. Knowledge of both the above components
is important concerning
- the interaction of physical processes,
- biodiversity and productivity of sandy beaches,
- sediment transport and coastal structure stability
(see, e.g., Mei 1983, Massel 2004, 2005)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Formulation of the problem
The marine environment consists of a water layer bounded above by the free
surface and below by the sea bottom, separating water from porous medium,
terminated by a flat, impermeable bottom boundary.
The variable water depth is h and the thickness of permeable layer is hB − h .
Coefficient of permeability (or filtration) is denoted by K f
Porosity of the sandy bottom is denoted by
K.A. Belibassakis:
nK
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
equations in poroelastic seabed
n ∂P
Storage equation ( PK is pore pressure): nK ∇ ( u - u S ) + ∇u S = − )K K ,
E ∂t
Momentum equations in the fluid
in the soil ( ρ S , ∇T soil density and effective stress)
nK2 γ
∂u
= −nK ∇PK −
nK ρ
(u - uS )
∂t
Kf
∂u S
nK2 γ
= ∇T − (1 − nK ) ∇PK +
(1 − nK ) ρ S
(u - uS )
∂t
Kf
(see, e.g., Mei 1983, Sec 13, and Massel 2005)
Assuming that the velocity components of the soil matrix are very small
u S / u << 1 ,
the dynamic equations of fluid motion in the porous medium can be simplified,
∂u  K
 ∂P
u = −  nK K + nK ρ  2 f ,
∂x
∂t  nK γ

∂w  K f
 ∂P
w = −  nK K + nK ρ

∂z
∂t  nK2 γ

where u = ( u , w ) denote the velocity components of the flow in the porous medium
Furthermore, mass conservation, in the form of storage equation, takes the form
∂u ∂w
1 ∂PK
+
=− )
, where γ = ρ g
∂x ∂z
Ew ∂t
K.A. Belibassakis:
)
and Ew apparent bulk modulus of the pore water
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
apparent bulk modulus of pore water
)
Ew−1 = SEw−1 + (1 − S ) / p0
Ew = 1.9 109 Nm −2
(1 − S) degree of saturation by air,
p0 is the absolute pressure.
In shallow water, due to wave breaking and the entrance of gas into the porous medium,
production of gases by the organisms living in the sand, the apparent
)
E
bulk modulus of the pore water w depends on the degree S of saturation by water.
Following Massel (2005), we use here the relationship proposed by Verruijt (1969)
,
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Helmholtz equation for pore pressure
Using simplified equations to eliminate the fluid velocities ( u, w ) we finally obtain
the following equation concerning the pore dynamic pressure
∂ 2 PK ∂ 2 PK ρ ∂ 2 PK
nγ ∂PK
)
)
+
−
−
.
2
2
2
∂x
∂z
Ew ∂ t
Ew K f ∂ t
The above equation considered in the frequency domain
 igH

PK ( x, z , t ) = Re −
pK ( x, z ) ⋅ exp ( −iω t )  ,
 2ω

finally reduces to the Helmholtz equation
∂ 2 pK ∂ 2 pK
2
+
+
ψ
pK = 0 ,
2
2
∂x
∂z
ρω 2
ψ = )
2
Ew
nγω
,
+i )
Ew K f
characterised by a complex-valued wavenumber parameter, and
the imaginary part of ψ indicates the dissipative nature of the examined flow
in the porous medium.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Vertical eigenfunctions
h=2m, n=0.26,
K.A. Belibassakis:
Kf=2.910-4ms-1
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
wave induced porous flow (Belibassakis OE 2013)
p / ρ gA
Pressure distribution in the water and in the porous bottom layers, in the
case of short waves T=5s, H=0.3m and longer waves T=8s H=0.5m
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
wave reflection & attenuation by porous rippled bed
waves T=1.3s and amplitude A=1.5cm propagating over a permeable rippled bed
Reflection coefficient for waves over sinusoidal bathymetry vs. incident wavenumber.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
nonlinear effects
Stokes 2nd-order (Belibassakis&Athanassoulis JFM2002)
Harmonic generation over round corner bar (Rey et al 1992)
shoaling ratio 0.375 , μh1=0.78
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Harmonic generation over trapezoidal bar
(Ohyama et al 1995)
shoaling ratio 0.3 , μh1=1.11
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Propagation of NL waves
(Athanassoulis&Belibassakis OMAE2002, DCDS2007,
Belibassakis & Athanassoulis Coastal Engineering 2011)
The wave potential function Φ ( x ,z;t ) , admits of a uniformly convergent
series expansion of the form:
Φ ( x ,z;t ) = ϕ−2 ( x ,t ) Z −2 ( z;h,η ) + ϕ−1 ( x ,t ) Z −1 ( z;h,η ) + ϕ0 ( x ,t ) Z 0 ( z;h,η ) +
free-surface mode
sloping-bottom mode
+
propagating mode
∞
∑ϕ ( x ,t ) ⋅ Z ( z;h,η )
n
n
n =1
,
( x ,z;t ) ∈ D × I .
evanescent modes
ϕ ( x,t ) =
∑ ϕ ( x ,t) = Φ ( x , z = η , t )
n
(since Z n ( z = η ;η , h ) = 1),
n≥−2
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Series expansion of potential in terms of
local vertical modes
µ 0 h0 + 1
µ 0 h0 + 1
2
Z − 2 ( z ;η , h ) =
( z + h) −
(η + h) + 1,
2 (η + h ) h0
2 h0
µ 0 h0 − 1
1
2
Z − 1 ( z ;η , h ) =
( z + h) + ( z + h) ,
2 h0 (η + h )
h0
µ 0 , h0 auxiliary numerical parameters
cos kn ( z + h ) 
cosh  k 0 ( z + h)
, Z n ( z ;η , h ) =
Z 0 ( z ;η , h) =
, n ≥1.


cosh  k 0 (η + h)
cos kn (η + h ) 
µ 0 − k 0 tanh ( k 0 (η + h ) ) = 0
µ 0 + k n tan ( k n (η + h ) ) = 0
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
new Hamiltonian equations
(Athanassoulis & Papoutselis OMAE2015)
ϕ −2


∂ t η + ∇x η ⋅∇x ϕ − 1 + (∇x η ) 
+ µ 0 ϕ = 0 ,

 h 0
2


ϕ −2
1
1
2
2

∂ t ϕ + gη + (∇x ϕ ) − 1 + (∇x η ) 
+ µ 0 ϕ  = 0 .

2
2
 h 0
(
2
)
(
)
and
∑A
mn
(η , h ) ∆ xϕn + B mn (η , h ) ⋅ ∇ xϕ n + C mn (η , h ) ϕ n = 0,
m ≥ −2
n ≥− 2
ϕ (x) = ϕ− 2(x) +
∑ ϕ ( x ),
n
and
[lateral conditions on ϕ n ]
n ≥ −1
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Nonlinear waves - numerical results (BA,CoEng2011)
Ohyama & Nadaoka (1994),
shoaling ratio 0.3, μh1=0.8
Beji & Battjes (1993, 1994) trapezoidal bar
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Beji & Battjes (1994) trapezoidal bar (H=2cm, T=2s)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Beji & Battjes (1994) trapezoidal bar (H=2cm, T=2s)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Beji & Battjes (1994) trapezoidal bar (H=2cm, T=2s)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
wave – floating body interaction in general bathymetry
Belibassakis (EABE 2008)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
effect of bottom corrugations
K.A. Belibassakis:
ISOPE-2012 The 22nd International Ocean and Polar Engineering Conference
Wave propagation in inhomogeneous, layered
waveguides
based
on Rhodes
Cargèse summer
school,
17-28 August
Corsica
Rodos
Palace
Hotel,
(Rodos),
Greece,
June 2015,
17−22,
2012
modal expansions and hp-FEM
transmission coefficient of floating pontoon (BW)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
stochastic response (Belibassakis et al ISOPE 2012)
Distribution of the significant wave height in the region of incidence
and region of transmission around a pontoon with B/h=0.4, T/h=0.2,
over flat horizontal and sloping bottom.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
3D floating body in general bathymetry
ϕ(r) =
∫ σ ( r ) G ( r | r ) dS ( r ) ,
N
0
N
0
0
r ∈ DN ,
GN = 1/ 4π r − r0 ,
where
∂DN
ϕ(r) =
∫ σ ( r ) G( r | r ) dS ( r ) ,
0
∂DB
K.A. Belibassakis:
0
0
r ∈ D \ DN
I
ϕ ( r ) = ∫ σ ( r0 ) G( r | r0 ) dS ( r0 ) ≈ ∑σ ( r0,i ) G( r | r0,i ) Si ,
UEi
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
i =1
Cargèse summer school, 17-28 August 2015, Corsica
3D Green’s function in general bahymetry
Belibassakis & Athanassoulis (JFM, 2004)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Boundary Element Method
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
3D floating bodies in general bathymetry
Belibassakis (ΕΑΒΕ 2008)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
ship wash (Belibassakis APOR 2001, FAST 2008)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
3D diffraction in the vicinity of openings in
coastal structures (Belibassakis et al APOR2014)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Very Large Floating Structures (VLFS)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
VLFS in variable bathymetry regions
Interaction of waves – ice sheet of general form
(from: www.srcj.or.jp/html/megafloat_en/ecology/eco_index.
Shipbuilding Research Centre of Japan)
K.A. Belibassakis:
(from: http://folk.ntnu.no/sveinulo/at327/at327lectures/loset_wave_ice
NTNU)
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
formulation of the problem
Floating elastic body of length L and breadth B in variable bathymetry region,
modelled as thin plate.
The horizontal plane is decomposed into the plate region (E) and the water region (W)
outside the rectangular shaped floating structure.
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Harmonic-time dependence
The wave potential is expressed in the following form
Φ ( x, y, z;t ) = Re {ϕ ( x, y, z ) exp ( −iωt )}
The complex amplitude of the
free-surface elevation (η) is obtained in terms of the wave potential as follows
i ∂ϕ ( x, y, z = 0 )
η ( x, y ) =
∂z
ω
In the area of the elastic-plate,
the deflection (w) is connected with the wave potential by a similar relation
derived from the kinematical condition at the liquid-solid interface,
i ∂ϕ ( x, y, z = 0 )
w ( x, y ) =
ω
∂z
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Differential formulation
in the water layer
(∇
2
+ ∂ 2z ) ϕ = 0,
−h ( x, y ) < z < 0,
in
( x, y ) ∈W U E
,
On free surface part of the horizontal plane
the wave potential satisfies the linearized free-surface boundary condition
∂ zϕ − µϕ = 0
,
on
( x, y ) ∈W .
z = 0,
2
where µ = ω / g is the frequency parameter.
For points on the plate the wave potential satisfies the corresponding
dynamical equation forced by the water pressure
(
)
∇ 2 d ∇ 2 w + (1 − ε ) w =
iµ
ω
ϕ ( x) ,
on
z = 0,
( x, y ) ∈ E
.
On the sea bottom the wave potential satisfies the no-entrance boundary condition
∂ zϕ + ∇h∇ϕ = 0 ,
Finally, at the plate edges
∂ 3w
∂ 3w
+ ( 2 −ν )
=0
,
∂n3
∂n ∂ 2 s
free of shear force
on
z = −h ( x, y ) .
( x, y ) ∈ ∂W
the following conditions apply
∂ 2w
∂ 2w
+ν
=0
∂ n2
∂ s2
free of moment
where n and s denote the normal and tangential derivatives along
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
∂W ,
respectively.
Cargèse summer school, 17-28 August 2015, Corsica
Modal expansion of the wave potential
The studied problem combines the character of water wave propagation and
scattering in inhomogeneous bathymetric terrains, under additional effects due to
the presence of localized hydroelastic scatterer (E).
µ = ( D κ 4 + 1 − ε ) κ tanh (κ h )
Distribution of the roots of dispersion relation on the complex κ-plane.
This type of problems problem in general bathymetry have been treated
by means of the following local mode series expansion which is used
to represent the wave field in the water region:
ϕ ( x, z ) = ϕ−1 ( x ) Z −1 ( z; x ) +
K.A. Belibassakis:
∞
∑ ϕ ( x ) Z ( z; x )
n =0
n
n
,
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
−h ( x ) < z < 0,
x = ( x, y )
Cargèse summer school, 17-28 August 2015, Corsica
Local vertical modes
the vertical structure of each mode Z n ( z; x ) , n = 0,1, 2,3,K ,
of functions describing, at each horizontal position x, are generated by
∂ 2z Z n
( z ) − κ n2 Z n ( z ) = 0 ,
∂ z Zn
in the vertical interval
( z = −h ) = 0 ,
at the bottom
− h ( x ) < z < 0, ,
z = −h ( x ) ,
α (κ ) ∂ z Z n ( z = 0 ) − µ Z n ( z = 0 ) = 0 , at water-elastic body interface z = 0 ,
where
α (κ ) is a function of κ for hydroelastic waves and simplifies to α=1 in the
case of water waves. The solution the above local VEP (Steklov-type) is
Z n ( z ) = cosh κ n ( z + h )  / cosh (κ n h ) ,
{
n = 0,1, 2,3,K ,
}
where the eigenvalues κn , n = 0,1,2... are obtained as the roots of the
(local at any horizontal position x) dispersion relation
µ h = α (κ ) κ h tanh (κ h ) , with
and
K.A. Belibassakis:
α = 1, for x ∈ W ,
α (κ ) = D κ 4 + 1 − ε , for x ∈ E .
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Enhanced hydroelastic local-mode series
f ( x,z ) = ϕ
( 2)
∞
( x,z ) − ϕ−1 ( x,z ) = ∑ϕn ( x ) Zn ( z;x )
⇒ ϕ
n= 0
( 2)
∞
( x,z ) = ϕ−1 ( x) Z−1 ( z;x ) + ∑ϕn ( x ) Zn ( z;x )
n= 0
This will be called the enhanced local-mode representation
The new vertical mode Z−1 ( z;x ) is taken to satisfy inhomogeneous condition on the seabed:
∂Z−1 ( z = −h ( x ) ; x )
∂z
= 1,
which implies
that
ϕ−1 ( x ) =
2
∂ϕ ( ) ( x, z = −h ( x ) )
∂z
.
This extra sloping-bottom vertical mode Z−1 ( z;x ) is taken also to satisfy the condition
∂5Z−1 ( z = 0;x )
∂Z−1 ( z = 0;x )
D
+
1
−
ε
− µ Z−1 ( z = 0;x ) = 0 .
( )
∂z
∂z5
on the elastic-plate surface.
A specific convenient form of Z−1 ( z;x ) is given by :
K.A. Belibassakis:
Z−1 ( z;x ) = h ( x )
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
 z 3  z 2

 +

h
x
 ( )   h ( x ) 


 ,


Cargèse summer school, 17-28 August 2015, Corsica
Coupled-mode system of equations
The CMS is obtained by means of a variational principle (Belibassakis&Athanassoulis 2005),
using the representation of the local mode series expansion of the wave potential
in the water column below the free surface and below the elastic plate
modelling the large elastic floating body.
This permits us to reformulate the problem w.r.t. the unknown modal amplitudes
ϕn ( x ) , n = −1, 0,1, 2,.....,
x ∈W U E .
for
The present CMS takes the following form,
∞
2
a
x
∇
ϕn ( x ) +bmn ( x ) ∇ϕn +cmn ( x ) ϕn ( x ) = iω w ( x ) ⋅ χ ( Ε) , m=−1,0,1,.... .
(
)
∑
mn
n =−1
where χ ( Ε ) denotes the characteristic function of the plate subdomain E
The above system is supplemented by the following equation providing the
coupling between the wave modes and the elastic plate deflection:
∇ ( d ∇ w ( x ) ) + (1 − ε ) w ( x ) =
2
K.A. Belibassakis:
2
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
iµ
ω
∞
∑ ϕ (x)
n =−1
n
, x∈E
Cargèse summer school, 17-28 August 2015, Corsica
Coefficients of the hydroelastic CMS
The horizontally-dependent coefficients of the CMS are given by
amn ( x ) = Z n , Z m ,
bmn ( x ) = 2 ∇Z n , Z m + ∇ h Z n ( z = −h ) Z m ( z = −h ) ,
cmn ( x ) = ∇2 Z n + ∂ 2z Zn , Z m +
 ∂Z ( z = −h )

+  n
+ ∇h ∇Zn ( z = −h )  Zm ( z = −h )
∂z


where
f,g =
z =0
∫
z =− h ( x )
f ( z ) g ( z ) dz .
After solving the CMS, the wave characteristics can be obtained all over the
domain by means of the calculated wave modes ϕ n ( x ) , n = −1, 0,1, 2, 3,.....
Information concerning distributions of moments and stresses are obtained
through the vertical deflection (w); see Belibassakis &Athanassoulis (2005).
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Belibassakis & Athanassoulis (JFM 2005,
APOR 2006, JEME 2009, HSTAM2013)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
calculated wave field (T=15s)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
smooth underwater shoal-effect of the bottom profile
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Papathanasiou et al RSPA(2014), NHESS(2015)
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
conclusions – future research
The present CMS has been applied quite successfully to treat
wave propagation problems in inhomogeneous environment
and hydroacoustic/hydrodynamic problems and applications
in nearshore and coastal regions.
Current and future research include
- optimization of non-linear numerical solver on the horizontal domain
- multichromatic and multidirectional applications (focusing of waves)
- run up on beaches and structures
- 3D wave - current and wave - structure interaction
.. ..........
- study of dissipation effects, oscillatory boundary layers ….
- dissipation due to wave breaking
- estimation of bottom loads and coupling with sediment transport equations
..........
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
extensions – directions of further study
Although in this work linear problems in 2D domains have been considered,
the present model permits its extension to various directions, as
(i)
3D problems (3D waveguides, directional waves, point sources)
(ii)
interaction problems involving localized inhomogeneities
contained in the waveguide
(iii) non-linear problems in inhomogeneous waveguides
Current work also focuses on the development of p-FEM
for the numerical solution of present CMS in 2DH (3D problems),
in conjunction with grid adaptation techniques, based on
the spatial variability of the coefficients of the system
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
various publications/references
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A&B
B, A & G
B&A
B&A
B&A
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B&A
G, B & A
A, B et al
A&B
B&A
B, G & A
B
B
B et al
B et al
P et al
K.A. Belibassakis:
JFM 1999
APOR 2001
JFM 2002
JFM 2004
JFM 2005
(Linear WW, 2D, time harmonic)
(Linear WW, 3D, time harmonic)
(2nd order WW, 2D, frequency domain)
(Linear WW 3D Green function)
(Linear, 2D WW & float. elastic plate,
time harmonic)
APOR 2006 (2nd order, time domain, floating elast. plate)
JOMAE 2008 (WW spectrum evolution 3D bathymetry)
JCA 2008
(hydroacoustic problem – layered medium)
JEME 2009
(WW thick floating elastic bodies)
CoEng 2011
(WW non-linear, steep bathymetry)
APOR11,OE14 (Water wave – current – seabed interaction)
EABE2008
(BEM floating structures)
OE2012
(wave induced groundwater/porous flow)
APOR2014
(waves - coastal structures )
WM 2014
(FEM hydroacoustic, layered medium)
RSPA14,NHESS15 (SW hydroelastic, time domain)
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica
Thank you for your attention!
K.A. Belibassakis:
Wave propagation in inhomogeneous, layered waveguides based on
modal expansions and hp-FEM
Cargèse summer school, 17-28 August 2015, Corsica