TEA: A LINEAR FREQUENCY DOMAIN FINITE ELEMENT MODEL
FOR TIDAL EMBAYMENT ANALYSIS
by
J. J.
Westerink, J. J. Connor, K. D. Stolzenbach,
E. E. Adams and A. M. Baptista
Energy Laboratory Report No.
February 1984
MIT-EL 84-012
TEA: A Linear Frequency Domain Finite Element Model
for Tidal Embayment Analysis
by
J.J. Westerink
J.J. Connor
K.D. Stolzenbach
E.E. Adams
A.M. Baptista
Energy Laboratory
and
R.M. Parsons Laboratory for
Water Resources and Hydrodynamics
Department of Civil Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
02139
Sponsored by
Northeast Utilities Service Company
and
New England Power Service Company
under the
M.I.T. Energy Laboratory Electric Utility Program
and by
The Sea Grant Office of NOAA
U.S. Dept. of Commerce
Energy Laboratory Report No. MIT-EL 84-012
February 1984
ABSTRACT
A frequency domain (harmonic) finite element model is developed
for the numerical prediction of depth average circulation within small
embayments. Such embayments are often characterized by irregular
boundaries and bottom topography and large gradients in velocity.
Previously developed finite element based time domain models require
high eddy viscosity coefficients and small time steps to insure numerical
stability, making application to small bays infeasible. Application of the
harmonic method in conjunction with finite elements overcomes theseproblems. The model TEA, for Tidal Embayment
Analysis, solves the
linearized problem and is the core of a fully nonlinear code presently under
development.
This report discusses in detail both the theory behind TEA and
program usage. Furthermore the versatility of TEA is demonstrated in
several prototype examples.
-i-
ACKNOWLEDGMENTS
This
research constitutes part of an overall program for the
development of more efficient and accurate circulation and dispersion
models for coastal waters and is under the direction of Professor
J.J. Connor, Professor K.D. Stolzenbach and Dr. E.E. Adams.
Support for this research was provided by Northeast Utilities Service
Company and New England Power Service Company through the M.I.T. Energy
Laboratory Electric Utility Program and also by the Sea Grant Office of
NOAA, Department of Commerce, Washington, D.C..
The authors would like to thank Miss Diane Hardy for her extensive
help in the preparation of this report.
-ii-
TABLE OF CONTENTS
Page
ABSTRACT . . .
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ACKNOWLEDGEMENTS. ........
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TABLE OF CONTENTS . ......
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iii
1. INTRODUCTION . .......
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1
2. GOVERNING EQUATIONS
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3
....
3. WEIGHTED RESIDUAL AND FINITE ELEMEN 1T
4. PROGRAM USAGE. . ....... * * *
4.1
4.2
4.3
4.4
4.5
Grid Layout. . ......
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Units. . .........
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Specified Phase Shifts . .
Boundary Conditions. ..
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Output . ........
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METHOD FORMULATIONS
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5.1 Massachusetts Bay. ...
5.2 Niantic Bay. . ......
5.3 Brayton Point. . ....
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REFERENCES. . ..........
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5. APPLICATION .
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Appendix 1: Program TEA . . . . . .
1.1 input format . .....
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1.2 output format. . ....
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1.3 example problem. . ....
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1.4 listing. . ........
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Appendix 2: Program RENUMB . . . . . . . *
2.1 inputs/outputs . . . . . . . . . . .
2.2 listing. . ........
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Appendix 3: Massachusetts Bay Example . . .
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3.1 input for Massbay tidal case . . .
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3.2 input for Massbay steady curren t case.
3.3 output for Massbay tidal case.
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3.4 output for Massbay steady curre nt case
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1.
INTRODUCTION
In the past few decades there has been considerable advancement in
the predictive capabilities for tidally induced circulation in
waterbodies.
Many of the more recent numerical models have applied the
finite element method [11,12].
The principal advantage of finite element
methods over the more traditional finite difference methods is
the greater
versatility allowed in grid discretization, both in terms of fitting the
often highly irregular coastal geometry and in terms of refining the grid
in critical areas.
Most of these models have been time based,
the time
dependence being resolved either by using some type of numerical
integration scheme or by applying a combined space-time finite element
scheme.
However,
these time domain models have been plagued with
requirements for small time steps necessary to insure numerical
stability.
The maximum allowable time step decreases along with
decreasing element size, making application to small bays infeasible
because of the extreme costs involved.
Furthermore, these models have
been criticized for their dependence on high eddy viscosity coefficients,
required for numerical damping of small wave length noise, to achieve
reasonable and smooth solutions [3].
Finally, these time domain codes
have the additional shortcoming of requiring start up periods in order to
allow initial transients to be damped out.
To overcome these inherent difficulties associated with time domain
discretizations,
investigators have lately applied the harmonic method in
conjunction with finite elements [7,8,10].
Because of the periodic nature
of the tidal phenomenon, the harmonic method is an intrinsically more
natural solution procedure and was one of the traditional methods for
analysis before the advent of finite difference and finite element
-1-
methods
There are no time stepping limitations as with this
[2].
procedure a set of quasi-steady (or time independent) equations are
generated.
Furthermore, there is no more need for artificially high eddy
viscosity coefficients.
One difficulty which arises, however, when
implementing this frequency domain method, is that non linear terms can
not be included as a part of the solution but must be generated by
iterative superpositioning of several frequencies.
Present research is aimed at advancing the capacities of this
harmonic method by developing a code specifically geared towards small
scale geometries, where the most formidable problems occur with the time
domain codes now available.
This report discusses a highly efficient linear code which shall be
used as the core of a full non linear iterative program presently under
development.
This linear code has been specifically tailored to fulfill
the characteristics which are desirable when applying this frequency
domain method to small bays where the non linearities are often quite
substantial.
However,
for many applications
this linearized finite
element frequency domain model may be quite sufficient and very useful.
In the following two chapters we shall briefly discuss the theory and
numerical techniques involved in the development of this linear model,
such that the user will have a clearer and more comprehensive
understanding of the potentials, limitations and requirements of the
computer code TEA (Tidal Embayment Analysis) developed.
-2-
2. GOVERNING EQUATIONS
The equations to be solved are the depth averaged forms of the
continuity and Navier-Stokes equations.
With the assumptions of constant
density fluid, hydrostatic pressure distribution, negligible horizontal
momentum dispersion (or eddy viscosity) and constant pressure at the
air-water interface,
these equations are:
It
+ [u(h + r)],x + [v(h + r)]
u
+ uu
,t
v
t
+ uv
,x
,x
+ vu
+ vv
,y
,y
= -g n
= -g n
,x
+ fv +
(2.1)
= 0
ph
h
s _
x
-
)
xr)
1h
s
b
- fu + _I ( ) (.
_ rb)
,y
ph H
y
(2.2a)
(2.2a)
(2.2b)
where u,v are the depth averaged velocities in the x and y coordinate
directions, respectively, n is surface elevation above mean water level, h
is mean water depth, t is
time,
H = h + n is
total depth, g is accelera-
tion due to gravity, o is the density of water, f is the Coriolis
parameter, t
b
finally -
x
s
x
and T
b
s
y
are surface stresses in the x and y directions, and
and T are the bottom stresses in the x and y directions.
y
These equations are readily linearized by neglecting the finite amplitude
components of flux in the continuity equation (i.e., the un and vn terms),
by neglecting the convective acceleration terms in the momentum equations,
h
by approximating H
h
h-
1, and by linearizing bottom friction to yield
the following set of equations:
t + (uh),x + (vh)
=
0
-3-
(2.3)
u
,t
v
+ gn
,t
1
+ gn I
+
,x
s
fv oh (x
1
,y + fu-- oh
(
b,lin
=0
(2.4a)
x
s
y
y
b,lin
rt
y
(2.4b)
0
)
Bottom friction has been linearized as:
b,lin
b,lin
x
= Xu
y
and
= Xv
(2.5)
where
X
cfU = linearized friction coefficient
=
f
DW
8
friction factor
Cf =
=
g2
U gh
Darcy-Weisbach
Chezy
Manning
representative velocity of flow
U =
For the tidal case, when the linearization is done on an equivalent work
over a tidal cycle basis,
"maUx
"e
8
X has the following form
*Cf
(2.6)
where
U
max
-
representative maximum velocity during a tidal cycle
Hence, this fully linear set of equations describe tidal wave propagation
with coriolis, surface wind stress and bottom friction effects.
Their
validity are subject to the relative magnitude of the non-linearities
-4-
being small,
which implies that both the long wave and small amplitude
assumptions are satisfied, hence;
a
h
<1
andL
< 1
where
a = wave amplitude
L = /gh
2t
T = --
T = wave length for shallow water waves
= tidal wave period
The linearity of this set of partial differential equations permits
superpositioning of solutions, hence allowing different frequency tides
(eg.,
M2 ,...)
in addition to any steady state components to be solved
for seperately and then to have the results superimposed.
The boundary conditions associated with our linearized governing
equations are the elevation prescribed and normal flux prescribed
conditions which are respectively expressed as,
I(x,y,t)-=
(2.7a)
(x,y,t) on r
and
Q (x,y,t) - Q*(x,y,t) on r
where
r = elevation prescribed boundary
= flux prescribed boundary
-5-
(2.7b)
Normal flux is expressed as:
(2.8)
Qn = anxx +
any y
where
and y direction
Qy are flux components in the x
Qx'
anx,
any are the direction cosines on the boundary
In order to be consistent with our previous linearization we must define
the flux components as,
Qx
=
uh
(2.9a)
Q
= vh
(2.9b)
Qy
A further modification of our equations which is applied in order to
increase the accuracy of computation,
specifically in order to reduce
round off error, is the splitting of elevation into a base component no,
which is representative of tidal elevation amplitude of the entire bay,
and
a local value ri, which describes the amplitude deviation from the base
value.
Hence the actual elevation, n, is now expressed as:
in(x,y,t) = no(t) +
l1 (x,y,t)
(2.10)
Substituting into our linearized governing equations we have,
n
+ '
o,t
+ (uh)
1,t
,x
+ (vh) y =0
,y
-6-
(2.11)
,t
+
v g+
,t
g - fv - ph
1,x
l,y
+ fu
x
s
1
(
oh
y
b,lin)
x
=
0
(2.12a)
b,lin
y
=
0
(2.12b)
bln)
The boundary conditions become:
- no(t)
nl(x,y,t) = n 1 (x,y,t) = I(x,y,t)
on
(2.13a)
Tr
and
Q (x,y,t) = Q
(x,y,t)
(2.13b)
Q
on
Our equations now have the primary variables nl, u and v, and an
additional loading term n o .
The known loading may be -thought of as a pumping mode which forces the
entire basin surface up and down with the base elevation no.
From this
point on we shall refer to no as the base or pumping mode and the
additional local response, %1, as elevation.
Hence as shown in Eq. 2.10
the actual total response equals the base loading, no, plus the computed
elevation,
j.
The program determines the value of the base loading by
averaging the prescribed total elevation boundary loadings (Eq. 2.7a).
We
note that in order to be effective in improving computational accuracy,
the
difference between no and n1 must be at least several orders of
magnitude.
This is indeed the case for small scale embayments where the
difference is typically quite large (e.g.,
2 or 3 orders of magnitude).
We shall now eliminate the time dependence from our equations by
reducing them to the frequency domain.
The basic assumption in this
reduction is that both forcings on the system and responses of the system
in elevation and velocity are of the same periodicity, u,..
-7-
The assumed
periodic response in velocity and total elevation are expressed as:
=
u(xy,t)
u(x,y) e
^
n(x,y,t)
It
=
iWt
(2.14a)
it
(2.14b)
(2.14c)
n(x,y) e
follows that our base loading and elevation are:
To
=
(t)
^
i=t
o e
(2.15a)
A
?1 (x'Y) e
nl(xyt) =
iwt
(2.15b)
It is noted that all quantities with ^ superscripts are complex spatially
varying amplitudes which include both a magnitude and a phase shift.
In
order for these assumed responses to be valid both boundary and surface
forcings must be periodic with frequency, w.
For the flux and elevation
boundary forcings this is expressed as:
^*
r
Q (x,y,t)
Q
SI
nl (x,y,t)
r_
=
(x,y)
^*
ni (x,y)
it
I
e
(2.16a)
int
p
e
(2.16b)
Similarly wind forcings are assumed periodic such that:
^s
St
x
iwt
(2.17a)
e
^s
iWt
(2.17b)
Sy e
-8-
We note that it
is
the wind forcing and not the wind velocity that is
assumed periodic.
The associated wind stress amplitude factors are then
expressed as:
=
pair CD
U
U
cose
(2.18a)
U10 sine
(2.18b)
A
^s
S
pair CD U
where
U 10
is maximum wind speed amplitude
pair is density of air
CD
is a wind drag coefficient
9
w
is the wind approach angle
Finally the periodicity of the linearized bottom friction concurs with the
form of the velocity response and hence is expressed as:
b,lin
x
=
Xu^ e imt
(2.19a)
b,lin
y
-
v ei t
(2.19b)
As previously mentioned these assumed response amplitudes include both a
magnitude and a phase shift which allows reference times to be handled.
This may be somewhat more easily seen by expanding out these periodic
responses/functions as follows:
-9-
iwt
A
A(x,y,t)
[A(x,y) e
=
Re
=
Re IA(x,y) ei(
]
t+ )
(2.20)
cos(t + )
= IA(x,y)
always implied.
We note that the real part of our complex responses is
an amplitude;
Furthermore,
A
(2.21)
A(x,y)I ei
A(
=
IA(x,y)
contains both information about the magnitude
and about the time
phase shift, 6.
It
is
readily seen that the steady state case is
represented by
Furthermore, we note that for this
specifying the frequency, w, as zero.
case phase shifts are not applicable and must always be specified as 6
=
0.
Finally, we note that for w = 0, our wind forcing equations (Eq. 2.18)
reduce to their customary form.
Substituting in the assumed forcings and responses into our governing
partial differential equations and boundary conditions,
taking appropriate
time derivatives and noting that eiwt terms drop from all equations, the
following set of quasi-steady equations
A
iwn
A
1+
generated:
is
A
A
iwmo + (uh),x + (vh)
=
(2.22)
0
^S
iou + gl
1,x
- fv - 1 (x
h
p
- Xu)
-10-
=
0
(2.23a)
^s
i
A
i1 +
A1
-h
,y
y
-
=v)
0
(2.23b)
with boundary conditions:
Q (x,y)
I1(xy) p
Q
=
Qo(x,y)
(2.24a)
nl(x,y)
(2.24b)
We note that these equations are time independent and hence we have
eliminated any time stepping restrictions in solving the equations.
We now wish to develop some numerical technique to solve these partial
differential equations for an arbitrary geometry.
We shall use the finite
element method to do this and shall solve for the spatial variation of the
amplitudes nl(x,y), u(x,y) and v(x,y).
The magnitudes and phase shifts
having been solved for using the finite element method program developed,
we can readily generate the entire time history using Eqs. 2.14.
Moreover,
if we choose to linearly superimpose solutions we superimpose the time
histories generated to resolve the total time history.
-11-
3.
WEIGHTED RESIDUAL AND FINITE ELEMENT METHOD FORMULATIONS
In order to apply the finite element method we must first develop a
weighted residual formulation.
The basic idea behind the weighted residual
method is to establish a set of equations which minimize the weighted error
which is incurred between the exact solution and our approximation to the
solution.
Before proceeding with the weighted residual method we multiply
through the momentum equations by h in
between
order to induce a certain symmetry
the continuity and momentum equations and hence allow for similar
matrices (for the derivative
terms of both equations)
thereby reducing storage requirements.
iwo + in1 + (uh)
to be generated,
Hence we have:
+ (vh)
(3.1)
0
=
^s
iwhu + ghn
1,x
- fhv -
X - Xu)
p
0
(3.2a)
0
(3.2b)
^s
AA
iwhv + ghn 1
,y
+ fhu -
- y - Xv)
p
=
In order to avoid a singular formulation, we shall in later
manipulations substitute the momentum equations into the continuity
equation.
This dictates that the continuity equation be used to establish
the symmetrical weak weighted residual form.
Formulating the fundamental
weak form in this fashion leads to elevation amplitude prescribed boundary
conditions being essential and the normal flux amplitude prescribed
boundary conditions being natural.
We note that it is
therefore mandatory
to prescribe elevation on at least one point of the boundary.
This is
required in order to ensure uniqueness of the solution, that is a reference
depth must be specified.
-12-
Specifically applying Galerkin's method to establish the fundamental
weak form the error in the continuity equation is weighted by the variation
6
in elevation,
I1, and is integrated over the interior domain.
In
addition, the natural boundary error must be accounted for by weighting it
with 611, and integrating over the natural boundary.
It is required
that the combined integrated and properly weighted interior and natural
boundary errors vanish and the following expression results:
+ (uh) ,x + (vh)
ff{iwn 1 + iwn
Applying Gauss'
168
1dP
+ (f-Q
Q
+ Qn16nld
= 0
(3.3)
theorem in order to eliminate derivatives on the flux terms
and combining with boundary relationships leads to the desired symmetrical
weighted residual weak form:
fffliWrn18
1
+ io 0
1
- uh(n1) ,x -vh(6()
,yId
+ rQ n
1 dr
= 0
(3.4)
PQ
The weighted residual forms for the momentum equations are obtained by
weighting the associated errors with residual velocities and integrating
over the interior domain, resulting in:
As
.(f{iwhu + ghn
fffiwhv + gh
-
,
1
yx
1,y
fhv
--
-
P
Xu)6udQ
-
0
(3.5a)
0
(3.5b)
^s
+ fhu - (-
P
v)ydQ
-13-
The functional continuity requirements imposed upon variables are that
linear
n1 and 611 be continuous over the domain, requiring at least a
A
finite element expansion, and that u,
A
A
v, 6u and 6v be finite over the
domain, requiring only constant expansions.
For the full non linear form
v, bu and 6v also require linear expansions.
of the equations, however, u,
Before proceeding with our numerical scheme we rearrange the weighted
residual equations somewhat such that all non variable load terms such as
the pumping mode, boundary flux and wind stress loadings appear on the
right hand side.
f
ff{i
n
1
With these modifications our equations now appear as:
- uh( 6 i),x
-
vh(5 1 1,
dQ =
(fiwn6n dQ -
Q 8ldP
(3.6)
Q
s
A
fffihu
A
+ ku - fhv + ghn
Ar
A
A
1,x
}6udQ - frJl
p
6UdQ
(3.7a)
6vdQ
(3.7b)
s
ff{iwhv + Xv + fhu + ghn
1,y
}6vdQ = ff
0
To generate a system of algebraic equations from the previous
This involves
integral equations the finite element method is applied.
Qe,
dividing the global domain, Q, into element sub-domains,
and
representing-the variables within each element by a polynominal expansion.
Contributions from all elements are summed and inter-element functional
continuity requirements are taken into account in order to generate a
global system of equations.
In our particular case we select triangular
elements and the lowest possible order polynomial mandated for the full
A
non linear form of the equations.
A
A
All variables, nl, u, v, 8i',
A
5u, and
6v, are therefore represented by linear expansion over each element,
-14-
yielding variables which are defined at the nodes.
Furthermore, linear
element expansions are used for mean water level depths h.
In this manner
elevations, velocities and depths are defined at nodes such that
inter-element fluxes are both continuous and clearly defined.
It may be
easily shown that the application of the finite element method to the final
weighted residual equations yields the following set of global algebraic
matrix equations:
S
(iWM
^lin
A
- P
-Q
:--P
-P
- DU
io M r
= 1
A
(3.8)
AT^
+ M + M )U + g
=P
(3.9)
where
r is the global elevation vector (1 elevation per node)
U is the global velocity vector (2 velocities per node)
Mq, M are global mass matrices
M is the global linearized bottom friction matrix'
M is the coriolis matrix
D is the derivative matrix
iwM -n
is the base elevation vector
P
-
P
is the prescribed boundary flux loading vector
-_Q
P is the wind loading vector
We note that all matrices are efficiently banded, the bandwidth depending
on the nodal numbering scheme used. ,Inorder to reduce computational
effort and storage requirements we use lumped mass, bottom friction and
Coriolis matrices.
The lumping for the symmetric mass and friction
-15-
matrices involves combining all terms on a row onto the diagonal and for
the skew symmetric Coriolis matrix combining rows into off-diagonal
terms.
The lumping procedure is in effect a slight redistribution between the
nodes of mass and certain forcings.
insensitive
We have found that results are quite
to these lumpings.
Combining the matrices commonly multiplying the velocity vector in
the
momentum equation as:
=
M
iWM
+ M
(3.10)
+ M
yields a tri-diagonal matrix which can be very economically inverted when
solving for U.
Hence solving for U using the momentum equation we have
M
U
P
- gD
T^
(3.11)
1
Substituting for U into the continuity equation and rearranging somewhat
yields:
T
-1
(iWM
^lin
D
+ gD M
P
- P
^-1
+ DM
P
(3.12)
The left hand side matrix, which is non-symmetrically banded and complex is
then decomposed using an LU decomposition.
The right hand side is combined
A
into a single loading vector and we solve for nl.
Finally in order to obtain the total elevation
we directly solve for U.
Swe
simply
add
back
i
Then using Eq. 3.11
our
base
compoet,
o.
n we simply add back in our base component, ro.
-16-
4.
PROGRAM USAGE
In this chapter we shall briefly comment on usage of program
TEA.
4.1
The exact required input information is shown in Appendix 1.2.
Grid Layout
The layout and degree of refinement used for the grid depends on the
amount of detail desired and also on the expected flow pattern.
Generally
a finer grid is used in areas of rapidly varying flow, near sharp
boundaries and near regions where the boundary
type changes (i.e.,
elevation prescribed to normal flux prescribed).
from
For tidal forcing cases
grid size should be small enough such that;
61
Al
L
<<
1
4
where A -= representative element length scale
L
= tidal wavelength =- gh T
This requirement stems from the nature of our linear element expansion and
the form of a wave.
Regarding grid shapes an attempt should be made to
keep the interior angles of triangular grid cells larger than 30@.
Once the grid layout has been completed, nodes should be numbered
sequentially.
Node numbering should be done such that the arithmetic
difference between node numbers of connecting nodes is minimized.
This
should be done since the matrix bandwidth and hence the efficiency of the
code is determined by the maximum nodal point difference.
Program RENUMB,
a bandwidth optimization code (Appendix 2), is available to renumber the
-17-
nodes in input files of program TEA.
Node connectivity for each element
We note that the sequential element
must be entered counterclockwise.
numbering doesn't effect program efficiency.
Friction factors estimated according to Eq. 2.6 are entered as
constant for each element.
For zero frequency cases,
it
is
necessary that
either friction or Coriolis be specified as non-zero in order to avoid a
singular formulation.
4.2
Units
The gravitational constant should be consistent with other units
Hence if lengths are entered in meters, frequency in sec-1, then
used.
g must be entered as 9.81 m/sec
2
and output will be meters for elevation
and m/sec for velocity.
The forcing frequency is w = 2n/T where T is the tidal period.
For
steady cases T = * and hence w = 0.
The Coriolis parameter is entered in the same time units as frequency
and is
calculated as:
f = 2we sin4
where we = phase velocity of Earth's rotation
(in
rad/time)
i =- degrees latitude.
Wind velocity should be entered in
being calculated.
the same units as water velocity is
The wind friction factor, CD, may be calculated
according to standard formulae (e.g. see Ref. 14) and typically has a value
around 0.001.
-18-
4.3
Specified Phase Shifts
Phase shifts must be applied in
conditions and other forcings.
order to specify time lags in
boundary
Phase shifts should be carefully and
consistently specified in order to avoid a meaningless solution.
They are
specified in rad/sec and are calculated as:
p
where tp is the phase shift time (in seconds if w in sec-1).
4.4
Boundary Conditions
The boundary is segmented into two boundary types, the elevation
prescribed boundary, 17,
and the normal flux prescribed boundary, PQ.
The elevation prescribed boundary is the essential boundary condition and
must be specified at at least one point on the boundary in order to
establish a reference elevation.
being elevation prescribed.
The entire boundary may be specified as
At each node on r. we must specify an
elevation amplitude and the associated phase shift.
Typically elevation
prescribed boundaries are ocean boundaries where tidal forcing exists.
Also elevation prescribed boundaries are used for a constant surface lake
boundary which is connected to a river or estuary where we would specify
zero elevation.
Essential boundaries are exactly satisfied and hence
elevation at these nodes will coincide exactly with our input boundary
conditions.
The normal flux prescribed boundary is the natural boundary
condition.
Any part of the boundary not specified as essential is
automatically taken to be natural.
Unless otherwise specified, program
TEA will assume that the normal flux boundary has zero normal flux across
it, that is it assumes a land boundary.
-19-
When a non-zero normal flux exists
across a natural boundary segment,
flux or a river discharge,
for either
the case of a known tidal
both the segments across which the flow occurs
The
and the value of the flow at the nodes involved must be specified.
segment end node numbers must be specified in clockwise order for shoreline
boundaries and in counterclockwise fashion for islands.
components of flux per unit length (i.e.,
The x and y
velocity x depth)
is
then
specified at each of the nodes on the non-zero flux boundaries.
The user
can either specify the x and y components of the normal flux vector or the
x and y components of the actual flux vector and TEA will find the required
normal components.
Because specified flux is a natural boundary condition
the computed flux on PQ will be satisfied exactly only in the limit (as
grid refinement is increased).
4.5
Output
Program TEA outputs information onto both tape 6 and tape 8.
Tape 6
output is a formated descriptive listing of both inputs and results.
Tape
8 outputs plotting (grids and results) and/or time history generating
information.
For details on outputs see Appendix 1.4.
When dealing with a set of tidal and steady state components,
TEA must be run separately for each different forcing frequency.
program
The
results for each frequency are then superimposed for each node to generate
time history information as follows:
-20-
Anode(t)
IAnode,w lCos(
1t
+ tnode,)
+
Anode,w2 cos(
1
t +
2b
node,
node w 2
where
t is time at which result is desired
or velocity) at node at
Anode,wl is amplitude (for elevation
frequency wl
$node,wl is
corresponding phase shift
-21-
5.
APPLICATION
Program TEA has been verified by comparing computed results with
exact analytical solutions for both the case of a straight channel with
linear friction [5] and the case of a rectangular basin with a slot [1].
In this chapter, however, we shall focus on the application of program TEA
to case studies of several bays and coastal inlets.
Comparisons to the
full non linear time domain finite element code CAFE (12]
shall be shown
and some of the differences shall be discussed.
5.1
Massachusetts
Bay
example is
The first
Massachusetts Bay.
with CAFE (12].
the calculation of the circulation pattern of
The grid used (Fig. 5.1) is identical to one applied
The circulation is driven both by a tidal fluctuation and
a steady coastal current.
The tidal forcing is simply specified by giving
tidal elevation values at ocean nodes and driving the system at a
frequency corresponding to a period of T = 12 hours.
Tidal elevations
vary linearly between Cape Ann and Cape Cod and no phase shifts are
applied.
The steady coastal current is simulated by imposing a linear
elevation gradient along the ocean boundary and driving the system at zero
frequency.
Since there are no fluxes from any land boundary segment, no
natural boundary condition need be specified in either case.
The
particular boundary values used were selected to be identical to the
boundary forcing values used for CAFE.
Program TEA input for both the
tidal forcing component and the steady current component are shown in
Appendices 3.1 and 3.2 and program TEA outputs are shown in Appendices 3.3
and 3.4.
In order to find the resulting currents,
velocity results for both
the tidal and the steady state components are superimposed at any desired
-22-
MASEXAC
CAPE
ANN
ATLANTIC
OCEAN
BOSTON
CAPE
COD
--
Fig. 5.1
Finite Element Grid of Massachusetts Bay
-23-
time.
The resulting circulation patterns at several stages of the tide
are shown in Figs. 5.2, 5.3 and 5.4.
Predictions from the non-linear CAFE
at T/6 after low tide [12] are shown in Figs 5.5 and can be compared with
corresponding output from TEA (Fig. 5.4).
Note that large values of eddy
viscosity were needed in CAFE for numerical damping and are most likely
unrealistically high.
Non-linearities are probably not very important due
to the large depth of Massachusetts
Bay in most areas.
We note that at
some points along the shoreline TEA allows a certain amount of leakage
onto land.
This is
due to the fact that TEA treats flux boundary
conditions as natural and hence allows some error.
Due to the solution
methodology of CAFE, flux boundary conditions are essential and hence the
no flux boundary conditions are strictly enforced.
However, if it is
important in a particular application to conserve total mass,
this
boundary leakage problem of TEA may be readily resolved by doing some grid
refinement in these boundary areas, especially near sharp corners and high
gradient regions.
-24-
" 0.60 m/sec
I
-w
-
...
-a
-
0-0
.-
'- -
-
-
0
.
go&
/
,
-A
/
/
I
//
,,TI Ill
II
'
Fig.
5.2
T/6 after High Tide
Tidal Circulation Computed by TEA at
-25-
0.60 m/sec
>
i
/
/
N
%i
r
N
I.
_Ol
/6'
%
*
Fig. 5.3
,I
Tidal Circulation Computed by TEA at Low Tide
-26-
> 0.60 m/sec
j f
-
"-
I
/
-
._-
/
-
/
/
/
'J
/•/
/
//
/
\I I I i\" ,
v1I/ Ij
I
,
Fig. 5.4
,
\
\-
Tidal Circulation Computed by TEA at T/6 after Low Tide
-27-
a
i d
4e
Le
v-i
ll
P
i
r
V
r
uvV
/
r
.1
(I
1
n4p
I IV
d
qL
Cd~
C
&
20
L
j
i
40 cm/se
~.~p L
Fig. 5.5
Tidal Circulation Computed by CAFE at T/6 after Low Tide
(from Ref. 12)
-28-
5.2
Niantic Bay
The second case study involves computing the tidally induced
circulation in Niantic Bay,
(Fig. 5.6).
in
the eastern part of Long Island Sound
Specifically we desire to resolve the circulation around the
Millstone point,
the site of the Millstone Nuclear Power Station,
and
therefore the grid used (Fig. 5.7) shows a great deal of resolution in
that vicinity.
Tidal forcing boundary conditions were simulated by
specifying a constant tidal elevation at ocean boundary nodes with a phase
lag which increases along the boundary as shown in Fig. 5.7.
It is
stressed that the user must be very cautious in applying correct boundary
TEA is quite sensitive to incorrect
condition values and phase lags.
boundary conditions and unrealistic solutions and/or boundary problems may
develop due to incorrect boundary specifications.
These types of problems
will appear much more dramatically for TEA than for CAFE since TEA has no
eddy viscosity to dampen unrealistic velocity gradients and furthermore
does not allow boundary fluxes to be forced to zero in the manner that
CAFE does.
The resulting circulation computed by TEA at two tidal stages are
5.8 and 5.9.
Results from CAFE [9] corresponding to Fig.
5.9 are shown in Fig. 5.10.
Details for the flow around Millstone Point
shown in Figs.
at maximum flood are shown in Figs. 5.11 and 5.12. Comparisons of velocity
amplitudes and directions to both CAFE and measurements made at seven
sites shown in Fig. 5.13 are presented in Fig. 5.14.
We note that, in general, velocity amplitude results from TEA compare
better to measured results than those of CAFE, which substantially
underpredicts flows at all points.
We note especially the dramatic
-29-
Of
IgJ'
POINT
TWOTREE ISLAND
SEA SIDE
POINT
.
i *.
: BARTLETT
"
REEF
BLACK POINT
:
.
*
t
*t
*
*
LONG ISLAND SOUND
Fig.
5.6
S
l 0
0
.
SCALE 1: 00,000
C &OS 1212
Niantic Bay and Location of Millstone Nuclear Power Station
(from Ref. 9)
GOSIIEN
POINT
MILLSTONE
*
Ocean Boundary Nodes
TL
tidal lag in minutes
TL= 0
TL = 1.5
TL = 12
Fig. 5.7
Finite Element Grid Discretization of Niantic Bay and Millsone Point
-31-
1.0 m/sec
>
o
!
I
1
1
S'e0
I
.
*
!
°e/
.
F
I
00-
Fig. 5.8
Computed Circulation by TEA at Maximum Flood
-32-
>
1.0 m/sec
|
I
I
I
I
I
\
1
I
1I
1
|
I1
I
I
.*
I
S
N ..
/
. a_
/
/
I
/
A
/
/ /
/
//
~~1
- %-"
/
~
-
-A-%
%
I
/d/-
Fig. 5.9
-f-f 1
Computed Circulation by TEA at Maximum Ebb
-33-
-
.5 m/s
*
I"L
*
S I,
'
*
*
-
--
velocity
scale
.1 m/s
1
a
C
*
4
4
4
"
4
*
#
a
*
*
*
*
Ab
S
A
-I
. -00"
\
*'-"*
/*.:-
0*
p
t
.
__
9,t?,/
, , ,._
"
5.10
.
mob
.- .N . - . . .--.
* • ,,,,
"
.. ', "
on- *
Fig.
\. ,
m.-"
qa-
*
'
_---6
Computed Circulation by CAFE at Maximum Ebb (from Ref.
-34-
-.
9)
Fig. 5.11
MILLSTONE
Detail of Grid around Millstone Point
----
- -----------
:, 1.0
n/See
~I I
"
7.
..
•
4J
4-
--
4a4-
4-
-0.4-
4-
44-
Fig. 5.12
Detail of Flow around Millstone Point Computed by TEA
at Maximum Flood
*
current velocity and
direction measurements
Mijoy Dock
Niantic
CM 8
Bay
Yacht
Club
CM 4
CM
CM 7 l
White's
Point
CM 9
CM 5
CM 3
CM 6
Fig.
5.13
Location of Current Measurement
-37-
Sites in
Niantic Bay (from Ref.
9)
1-
*
Range of Measurements
CAFE
-
TEA
Range of yeasurements
.8
*
.6
CAFE
.4
e
Hours
6
12
6
CM 4
MLW
CM 3
SRange
of
Measurements
-
TEA
-
0
CAFE
-
TEA
Hours
12
Range of Measurements
-
*
.6
CAFE
TEA
.4
U
O
Hours
6
MLW
12
5. 14
a-d
6
Hours
12
CM 6
CM 5
Fig.
MLW
Comparison of Current Measurements and Predictions by
CAFE and TEA at various locations
-38-
.8
.6 -
-
Measurement
*
CAFE
- Measurement
.8 -
CAFE
TEA
-TEA
.4 -
.2
m
.2
-
.4
&
11100
~~
1
6
It
i
---
r
.
w . ,
I
Hours
Hours
MLW
12
=
I
12
CM 8
CM 7
!"- Measurement
*
CAFE
----
6
MLW
TEA
Hours
12
C1 9
Fig.
5.14
e-g
Comparison of Current Measurements and Predictions by
CAFE and TEA at various locations
-39-
improvement of currents around Millstone Point which is shown in the
comparison at point 5.
Even though velocities predicted by TEA at the
measurement point itself are not quite high enough,
velocities
do tend to
increase close to Millstone Point, a behavior not exhibited by CAFE.
Again results for Niantic Bay are justified because, for the most part the
bay is quite deep (8 applicable.
15 meters) and hence the linear solution is
Furthermore velocity amplitudes tend to be higher than those
calculated by CAFE since there is no need for artificially high eddy
viscosity values which tend to dampen and smooth out velocities.
Finally there is a dramatic difference in run times between CAFE and
TEA.
For this grid, a run with CAFE required approximately 20 hours of
C-PU times on a Honeywell level 68/DPS computer.
This excessive amount of
CPU time resulted from the maximum time step for numerical stability of 2
seconds required due to the small grid sizes.
For the same grid, program
TEA required only 2.5 minutes of CPU time running on a VAX 11/780, which
is of comparable speed as the Honeywell.
-40-
5.3
Brayton Point
Our final example deals with predicting the combined circulation due
to both tides and a power plant discharge at Brayton Point in Mount Hope
Bay, Somerset, Massachusetts (Fig. 5.15).
Again the main region of
interest is around the power plant and hence the finite element grid
discretization reflects this with a large amount of refinement in
region as shown in Figs. 5.16 and 5.17.
this
The particular case run with
program TEA examines the upper layer circulation pattern produced by tides
and a three unit power plant operation and is analogous to a case study
previously done with CAFE [6].
To simulate upper layer circulation,
the depths used were the
estimated upper layer depths which ranged between 1.5 and 3.5 meters.
Again the tide and the steady state discharge must be handled by two
separate runs of TEA.
In the first run, the tidal component of
circulation is calculated by prescribing tidal amplitude at the ocean
boundary, as shown in Fig. 5.16, ard forcing the system at a frequency
corresponding to T = 12.4 hours.
The second run simulates steady state
power operation and natural river inflow.
We attempt to simulate a jet by
prescribing flow conditions at the near field - far field jet interface.
Hence we prescribed large normal fluxes representing the outgoing jet and
the in-going re-entrainment fluxes surrounding the jet as shown in Fig.
5.17.
We note the very refined region necessary near the jet due to the
large gradient and rapid turning expected in this region.
Power plant
intake flow .issimulated by specifying a normal flux along the east side
of the discharge peninsula.
Additional withdrawals representing
downwelling to the lower layer are simulated by specifying normal fluxes
through elements along the southeastern edge of the domain.
-41-
Taunton River
5w .neSS
soey6so
t
Cole
K>
I
Point
Spar Island
Tiverton
n0utical
mie
: *.--.."
....
,
Fig. 5.15
.-
Site of Brayton Point Generating Station located in Mount Hope Bay,
Somerset, Massachusetts (from Ref. 6)
-42-
BRAYTON7
Fig. 5.16
Finite Element Grid Discretization of Mount Hope Bay
-43-
BRAYTON7
Fig.
5.17
Detail of Grid at Discharge Region at Brayton Point
-44-
flow is specified as a normal flux into the domain through elements in the
northeast corner.
All normal flux information is provided to TEA by
specifying the end nodes, with a correct orientation (see Chapter 4.4), of
each segment through which normal flow occurs and by specifying the x and
y components of the normal flux at the nodes included in these flux
segments.
Finally, zero elevation amplitude is prescribed at the ocean
boundary nodes and the system is driven at zero frequency.
Superimposed flow circulation patterns at two stages of the tide are
shown in Figs. 5.18 and 5.19.
Details of the flow around Brayton Point,
including the discharge and intake locations, are shown in Figs. 5.20,
5.21 and 5.22.
Fig. 5.23.
A result from CAFE corresponding to Fig 5.22 is shown in
When comparing results from TEA and CAFE we note that the jet
is not as well simulated by TEA.
This is due mainly to the fact that TEA
does not include the non-linear momentum terms needed to simulate jet
physics.
TEA drives the discharge only by elevation gradients which
accounts for the rapid spreading of the jet, best exhibited in Fig. 5.22.
Hence we may conclude that a code which includes non-linearities is
superior for simulating the circulation close to jet discharges.
However
program TEA will simulate discharges quite adequately if one is not
interested in the region near the discharge where the momentum effects are
important.
Furthermore, a full non-linear version of TEA is presently
under development.
-45-
1.0 m/sec
>
-
i \\
P
I~
'
I
I~I~
/'
I
t
'
!
I
I
/
/
I
I
,
// /
//
p1
t tf
i I / ff
/
I
I
Fig. 5.18
Circulation Computed by TEA at Maximum Flood
-46-
>
1.0 m/sec
Fig. 5.19
Circulation Computed by TEA at Maximum Ebb
-47-
0.5 m/sec
Fig. 5.20
Computed by TEA
Detail of Circulation at Brayton Point
at Maximum Flood
-48-
-
Fig. 5.21
0.5 m/sec
Detail of Circulation at Brayton Point Computed by TEA
at Maximum Ebb
-49-
-
0.5 m/see
I
s
I
I
I
'
I
!
/I
I
%
i
Fig. 5.22
TEA
Detail of Circulation at Brayton Point Computed by
at High Slack
-50-
Fig. 5. 23 Detail of Circulation at Brayton Point Computed by CAFE
at High Slack (from Ref. 6)
-51-
References
1.
Briggs, D.A. and O.S. Madsen, 1975. "Analytical Models for One- and
Two-Layer Systems in Rectangular Basins", MIT Parsons Lab. T.R. #172.
2.
Dronkers, J.J., 1964. Tidal Computations in Rivers and Coastal
Waters, North Holland Publ. Co., Amsterdam.
3.
Gray, W.G., 1980. "Do Finite Element Models Simulate Surface Flow",
in F.E. in Water Resources #3, ed. S.Y. Wang et al.
4.
Grotkop, G., 1973. "Finite Element Analysis of Long-Period Water
Waves", Comp. Meth. in Appl. Mech. and Engrg., 2:147-157.
5.
Ippen, A.T. (ed.), 1966.
Hill.
6.
"Coupled Near and Far Field
Kaufman, J.T. and E.E. Adams, 1981.
Thermal Plume Analysis using Finite Element Techniques", M.I.T.
Energy Laboratory Report No. MIT-EL 81-036.
7.
Kawahara, M. and K. Hasegawa, 1978. "Periodic Galerkin Finite
Element Method for Tidal Flow", Intl. J. Num. Methods in Engr.,
12:115-127.
8.
Le Provost, C.,
Estuary and Coastline Hydrodynamics, McGraw
A. Poncet and G. Rougier, 1980.
"Finite Element
Computation of Some Spectral Components", in F.E. in Water Resources
#3, ed. S.Y. Wang et al.
9.
"The Effect of Natural
Ostrowski, P. and K.D. Stolzenbach, 1981.
Water Temperature Variation on the Monitoring and Regulation of
Thermal Discharge Impacts - The Role of Predictive Natural
Temperature Models", M.I.T. Energy Laboratory Report No. MIT-EL
81-035.
10.
Pearson, C.E. and D.F. Winter, 1977. "On the Calculation of Tidal
Currents in Homogeneous Estuaries", J. Phys. Ocean., 7:520-534.
11.
Taylor, C. and J.M. Davis, 1975. "Tidal and Long Wave Propagation A Finite Element Approach", Computers and Fluids, 3:125-148.
12.
Wang, J.D. and J.J. Connor, 1975. "Mathematical Modeling of Near
Coastal Circulation", MIT Parsons Lab T.R. #200.
13.
Westerink, J.J., J.J. Connor and K.D. Stolzenbach, 1983. "Harmonic
Finite Element Model of Tidal Circulation for Small Bays", in
Proceedings of the Conference in Hydraulic Engineering, H.T. Shen,
ed., ASCE, N.Y.
14.
Wu, J., 1969. "Wind Stress and Surface Roughness at Air-Sea
Interface", Journal of Geophysical Research, Vol. 74, No. 2.
-52-
Appendix 1:
-53-
Program TEA
1.1
input format
1.2
output format
1.3
example problem
1.4
listing
Appendix 1.1
-54-
Input for TEA is free formated and information should be read as follows:
card 1:
10 character alphanumeric geometry identification
card 2:
10 character alphanumeric run number identification
card 3:
number of elements; number of nodes
card group 4:
node number; x-coord; y-coord; and nodal depth
card group 5:
element number; element node connectivity given by 3
element node numbers (counterclockwise); element linear
friction factor
card 6:
gravitational constant in consistent units
card 7:
forcing frequency
card 8:
Coriolis factor
card 9:
amplitude of wind speed; wind phase shift (in radians);
wind direction (in degrees); wind drag coefficient
Prescribed normal flux section:
card 10:
number of boundary segments which have non-zero normal
flux prescribed (if this is zero skip inputs 10a, 10b,
10c to card 11 input)
card 10a:
end nodes of boundary segments with non-zero normal flux
prescribed with correct orientation (clockwise for land
boundary and counterclockwise for islands)
card group 10b: total number of different nodes inluded in non-zero normal
flux load segments
card group 10c: node number; modulus of x-direction flux at node; phase
shift for x-direction (in radians); modulus of y-direction
flux; phase shift for y-direction (in radians)
Prescribed elevation section:
card 11:
number of nodes where elevation is prescribed
card group 11a: node number; modulus of prescribed elevation at node;
phase shift at node (in radians)
-55-
Appendix 1.2
-56-
As discussed in Section 4.5 program TEA outputs information onto TAPE
6 and TAPE 8. TAPE 6 information is a formated descriptive output and
consists of three main sections:
(1) Print statements which ask for required input in the order needed by
TEA.
This allows the program to be run in an interactive fashion.
Furthermore these prints may also aid the user in checking to make
sure that the input is in proper order.
Finally error messages
concerning dimensioning of matrices may appear here.
(2) A labeled echo print of all input information.
(3) Results of calculations consisting of elevation amplitude and phase
shifts at nodes and velocities in the x and y direction with
corresponding phase shifts at nodes.
Example outputs are shown in Appendices 3.3 and 3.4.
TAPE 8 output consists of information required for plotting of the
grid and results and/or generation of time histories.
TAPE 8 output is
free-formated and consists of the following information groups:
* 10 character alphanumeric geometry identificaton
* 10 character boundary condition identification
* number of elements, number of nodes
* node number, x-coordinate, y-coordinate
* element number, three corresponding node numbers
-57-
* frequency
* node number, elevation amplitude, elevation phase shift
* node number, x-velocity, x-phase shift, y-velocity, y-phase shift
-58-
Appendix
-59-
1.3.1
Here we discuss the boundary condition input requirements for program
TEA with a case which shows a variety of features.
The grid used is very
coarse and is used only for illustrative purposes.
We consider a tidal inlet with an island and several steady state
discharges.
The geometry and the coarse grid used are shown in Fig. Al.
The grid is shown again in Fig. A2 with element and node numbers
included.
The bay is subject to a 12.4 hour tidal forcing of unit
amplitude and zero phase shift along the ocean boundary.
Furthermore we
note the steady state discharge on the land boundary and the steady intake
on the island.
To simulate this case with TEA we perform two runs with two sets of
boundary conditions.
component.
The first run calculates the tidal circulation
The required boundary condition information consists only of
the specification of ocean boundary node numbers and their corresponding
tidal elevation amplitude and phase shift.
TEA input for this run is
shown in Appendix 1.3.2.
The second run calculates the steady state circulation component
(hence zero frequency) and we must input the end nodes of the boundary
segments through which flow occurs (in clockwise order for the land
segments and in counterclockwise order for island segments) and the x and
y components of the normal flow at the nodes involved.
We note that phase
shifts do not apply to the zero frequency case and must be specified as
zero.
Finally we provide information about the open ocean boundary by
specifying zero amplitude at these nodes, allowing the total flow to
balance.
In this case since there is a net inflow into the bay,
will be an outflow into the ocean.
-60-
TEA input for this run is
there
shown in
EXBAY
Fig.
AlExample
Element
Finocite
Grid
Fig.
Al
Example Finite Element Grid
-61-
EXBAY
33
24
23
21
2M2
30
27
2025
2
19
34
31
28
35
29
32
18
38
34
32
28
43
37
S3
2
41
42
17
84
1
17
to
39
11
12
13
3
44
14
71
to
2
44
2
Fig.
A2
5
4
3
Numbers
Example Finite Element Grid with Node and Element
-62-
Included
Appendix 1.3.3.
We note that for a totally enclosed waterbody we would
only specify one elevation at one boundary node.
In that case then there
would only be flows through specified flux (natural) boundary segments.
Furthermore we would have to specify fluxes such that the total flow in
and out of the boundary balanced in order to satisfy continuity.
Outputs for the two sample calculations are shown in Appendices 1.3.4
and 1.3.5.
Since we have solved two linear problems we may now
superimpose the results as discussed in Section 4.5.
-63-
Appendix
-64-
1.3.2
EXBAY1
BC-01
44, 34
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
7500.
9500.
11500.
13000.
7500.
9800.
12000.
5000.
7500.
10500.
13500.
1000.
4000.
6500.
8800.
11500.
14000.
0.
0.
3000.
2500.
6000.
7800.
10000.
12600.
14000.
11800.
13500.
6000.
8500.
10500.
9000.
11000.
12500.
1
2
2
3
3
8
5
6
6
8
8
13
9
12
18
19
800.
1500.
2300.
3000.
2500.
3800.
4500.
3000.
6000.
7500.
8000.
4000.
7000.
9000.
10000.
10000.
10000.
6500.
10000.
10500.
13000.
12000.
11000.
11400.
11300.
11600.
12300.
13000.
14500.
13500.
13300.
15500.
15000.
14000.
2
6
3
7
4
5
6
10
7
13
9
9
10
13
13
13
5
5
6
6
7
9
9
9
10
12
13
14
14
18
19
20
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
17
13
14
20
0.00100
18
19
20
21
22
23
24
25
26
20
19
20
21
29
29
30
22
23
14
20
22
22
22
30
33
23
24
22
21
21
29
30
32
32
30
30
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
-65-
27
3
28
3
29
2
30
3
31
3
32
2
33
1
34
2
35
2
36
2
37
1
38
1
39
1
40
2
41
2
42
1
43
1
44
1
9.81
0.00014075
0.00010000
0.00, 0.00, 0.00,
0
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00
1.00, 0.00
1,00, 0.00
1.00,
1.00,
0.00
0.00
-66-
Appendix
-67-
1.3.3
EXBAY2
BC-01
44, 34
1
2
3
4
5
6
7
8
9
10
11
7500.
9500.
11500.
13000.
7500.
9800.
12000.
5000.
7500.
10500.
13500.
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
800.
1500.
2300.
3000.
2500.
3800.
4500.
3000.
6000.
7500.
8000.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
1000.
4000.
25.
4000.
6500.
8800.
11500.
14000.
0.
0.
3000.
2500.
6000.
7800.
10000.
12600.
14000.
11800.
13500.
6000.
8500.
10500.
9000.
11000.
12500.
1
2
2
3
3
8
5
6
6
8
8
13
9
12
18
19
13
20
19
20
21
29
29
30
22
23
7000.
9000.
10000.
10000.
10000.
6500.
10000.
10500.
13000.
12000.
11000.
11400.
11300.
11600.
12300.
13000.
14500.
13500.
13300.
15500.
15000.
14000.
2
6
3
7
4
5
6
10
7
13
9
9
10
13
13
13
14
14
20
22
22
22
30
33
23
24
-68-
5
5
6
6
7
9
9
9
10
12
13
14
14
18
19
20
20
22
21
21
29
30
32
32
30
30
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
25.
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
000
00"0
00"0
00*0
00"0
' 1L. 00 L,.. 01'09018OS 01-
-69-
0090
000
000
00'0
'000
'000
'00*0
'00*0
'00"0
00*0
00T00 0
00'100'0
OOTO0O
00T00 0
00100'0
OOTO0"O0
001000
001000
00100 0
001000
00100'0
00100.0
00100' 0
L
11
LT
LT
9z
11
0O
sz
Sz
8z
997
1
91
ST
00I00'0
00T00O 0
00TO000
OOTO0"O
OOTO0"O
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OOTO0O
00"00" 0
OOTO"
OOTO0" 0
LZ
LZ
TE
tz
'0040
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'00"0
'0040
I.E
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'ES
*-
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'EZ
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'00'0 '0000 '0090
000000000
000000000
18"6
0
I,%
91
EL
9T
v
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01
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9"
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LZ
9E
LZ
SE
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vZ
V
ST
EL
EZ
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EE
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6Z
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Appendix 1. 3.4
-70-
PRCGRAM TEA
1. 1.T.
DEPTOFCIVIL ENGINEERING,
AIN
2-DLINEAR FINITE ELEMENT FPE3UENCY DCEM
ALYSIS
OFTIDAL
MAVES
FOR SMALL SCALE GEOMETRY.
ENTER 10DIGIT F.E. GRID GEOMETRY IDENTIFICATION
CODE FOR B.C.VERSION
ENTER 10DIGIT I.D.
BEROF NOTE POINTS
NUMBER OF ELEMENTS ANrNU'
INPUT
THE NODENO.,I-CCTR. ,f-COOPD.,
INPUT
ONE
MODEILINE
N.S.L. DEPTH,
AND
MODAL
LIMER FRICTION FACTOR
ELEHENT
THE
'ORTHEELElENTAND
430E UMDBErS
THREE
NUMSER,
ELEMENT
INPUT
DIFFEFENEE 11
MAI.NODAL POINT
FREQUENCY IN RADOT:S -WL.
INPUT
CORIOLIS PARAMETER
INPUT
DRA6 COEFFICIENT
MIND
DIRECTION (INDEGREES) AND
OFNIND
SHIFT, ANGLE
WIND
PHASE
OFMIND SPEED,
AMPLITUDE
INPUT
FLUX
PRESCRIBD
NON-IERO
NORMAL
HAVE
OUNDARY
SEGMENTS
INPUT
HON
HANY
INPUT
NO.OFNODES
HERE
ELEVATION
IS PRESCRIBED
(IN RAJ)OFTHEPRESCRIBED ELEVATION AT NODE
VALUES
FORTHEHODULUSPHASE
INPUT
THE
NODE
NUER AND
ATNODE
PHASE
(IN RID) OFTHE
PRESCRIED ELEVATION
FORTHEHODUS
NUHER ANDVALUES
INPUT
THENODE
ELEVATION
ATNODE
(IN RAU)OFTHEPRESCRISED
FORTHEHODULUS PHASE
AND
VALUES
NUMER
THENODE
INPUT
ATNODE
(INRAD)OFTHEPRESCRIIED ELEVATION
FORTHENN US PHASE
VALUES
THE
NODE
W ERAND
INPUT
_______
__
PROGRAM
TEA
DEPT
OFCIVIL ENG6IEERIN6, N. I. T.
SEONETRY.
FOR SALL SCALE
OFTIDAL AVES
FREIENCY D0MIN ANALYSIS
2-0 LINEAR FINITE ELEMENT
IDENTIFICATION z EIDAYI
F.E. GRIDGEOETRY
IDENTIFICATION : SC-01
BOUNDARY
CONDITION
FREQUENCY
x 0.00014075
CORIOLIS PARMETER a 0.00010000
0.0000
•
ABOVE SURFACE
SPEED
AT 10METERS
MIND
0.00000 RADIANS
MIND PHASE SHIFT a
0.00000 DEGREES
MIND DIRECTION •
MIND DRA6 COEFFICIENT * 0.00000000
-71-
LLA
EdC3
-
C%
0. 0.
0.0.
0. WI
e~) p~-
V-L1
-m
W,
0.0
L.0
C
. 4.
W
0.0. Ir
o-
8C4C4C4C4CI
4W
0-
88888
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c
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-40 C-
e
~~
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a
C2.
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C.
4=. CD.
-
A
.
4aC'
MDIAL ELEVATIONS
NODE
1
2
3
4
5
6
7
8
9
tO
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
RODULUS
1.00000000
1.00000000
1.00000000
1.00000000
1.00100239
1.00395713
1.00278606
1.00874233
1.00641567
1.00663572
1.01004628
1.00906940
1.00869093
1.00748832
1.01123755
1.00958349
1.00940425
1.00925525
1.01037390
1.01013314
1.01006339"
1.01083832
1.01085458
1.01030048
1.01040624
1.01058496
1.01058154
1.01057978
1.01116096
1.01110617
1.01107183
1.01132440
1.01111392
1.01081104
0.00000
0.00000
0.00000
0.00000
-0.00380
-0.00162
0.00096
-0.00752
-0.00327
-0.00264
-0.00109
-0.00451
-0.00488
-0.00262
-0.00493
-0.00294
-0.00247
-0.00581
-0.00474
-0.00480
-0.00446
-0.00400
-0.00333
-0.00331
-0.00249
-0.00212
-0.00314
-0.00297
-0.0425
-0.00372
-0.00351
-0.00343
-0.00330
-0.00271
NODAL VELOCITIES
NODE
1-1IRECTION
NOULmI
S
PHASE
0.1614! 733
0.07789 T93
0.06442 079
0.11026 195
0.08233 156
0.0304 363
0.02401 071
0.06198 340
0.050601869
0.01192 018
0.049007782
0.02279 296
0.02453 620
0.03264 711
0.03512 620
0.008251685
0.01892 373
0.017531154
V
-0.89457
-0.51438
-0.11300
0.59346
-1.14076
-0.84828
-0.29799
-1.37087
-1.61163
0.28291
1.75537
0.52544
-1.91804
-1.58872
-1.54573
-2.06721
-0.91485
-1.62605
k VMB!
Y-DIRECTION
OWLUS
PHASE
0.25987739
0.17803156
0.16461041
0.17433353
0.14247863
0.10528989
0.11304919
0.02835620
0.04258825
0.07564032
0.05858621
0.02081520
0.00478852
0.03211532
0.04109181
0.04024705
0.03553314
0.01699655
.70683
1.20267
1.77377
2.29168
11.28004
1.47924
1.55365
11.66106
1.45694
11.54913
1.63318
2.76747
2.96110
1.40126
1.601:
1.32265
0.98635
.0670k
i 03 *
-74-
Itliclo
U0060M91knot II
PLIN
96e",ZITA is 10
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81141
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Appendix 1.3.5
-76-
PROSRAM
TEA
DEPT
OFCIVIL EN6INEERIN6, M. I. T.
6EONETRY.
SCALE
FORSMALL
NAVES
OFTIDAL
DOMIN ANALYSIS
FREQUENCY
2-D LINEAR FINITE ELEMENT
ENTER
10 DIGIT F.E. 6RID 6EGETRY IDENTIFICATION
ENTER
10 DI6IT I.D. CODE FOR
B.C.
VERSION
INPUT
NURDER OFELEMENTS AND
MURDER
OFNODE
POINTS
NO.,-CORD.,Y-CDOR.,
INPUT
THENODE
INPUT
ELEMENT
AND NODAL
H.S.L. DEPTH,
ONE
MIDELIE
MUER, THREE
NODE
HMBERS
FORTHE
ELENENTAM@
THEELEMENT LINEM FRICTION FACTOR
hAI. NODAL POINT DIFFERENCE
s
INPUT
FREQUENCY
INRADIANS/SEC.
CORIOLIS PAMETER
INPUT
INPUT
AMPLITUDE
OF IND SPEEU,
MIND
PNASE
SHIFT, AMLE OFVIM DIRECTION
(IN iSREES) ANDIM DAMS COEFFICIET
INPUT
FLUI PRESCRIDED
HAVE
NON-ZERO
NORMAL
HON
MANY
OUNDMY SEGMENTS
INPUT
ENDNODES
OFSOUNDMY
SEGMENT
1 NITH CORRECT
RIEIATION
INPUT END NODES OF DOUNDARY
SE61ENT
2 VITH
CORRECT
ORIENTATION
INPUT
3 MITH
CORRECT
ORIENTATION
ENDNODES
OF1OUND
MY SEGENT
INPUT
TOTAL
NO.OFDIFFERENT N S INCLUDED
INNOI-ZERO
MORMAI. FLUISEIIENTS
INPUT
YDIRECTIONS
SHIFT (IN RAD)FORI AND
PHASE
NODE
NO.,NORUS AMD
INPUT
NODE
NO.,IMOLUS AND PHASE
SHIFT (INRAD)FORI MI T DIRECTIONS
INPUT NODE
MO.
,tDUUS AND
PHASE
SHIFT (INRAD)FORI AND
T DIRECTIONS
INPUT
NODE
NO.,ODULUS ANDPHASE
SHIFT (IN RAD)FORI AMD
Y DIRECTIONS
INPUT NODE
NO.,NOULUS AWPHASE
SHIFT (I1 RA) FOR I N
ID DIRECTIONS
INPUT
NO.OFNODESERE ELEVATION
IS PRESCRIlED
INPUT
THENODE
MUR
R AND
VALUES
FORTHE IOOUS
AND
(I1 RAD)OFTHEPRESCRISED ELEVATION
ATNODE
PHASE
INPUT
THENODE
NUIDER
AIDVALUES
FORTHEHOULUS AND
OFTHEPRESCRIED ELEVATION ATiODE
PHASE
INPUT
THENODERDllER
MD VALUES
FORTHE
rODULUS
AND
PHASE
(IN RAM)
ELEVATION ATNMDE
OFTHEPRESCRIBED
INPUT
IuER MD VALUES
THENMOE
F THE
(IN RAID)
IOIIULUS AN PHASE
OFT IPRESCRIDED
ELEVATION
ATME
ti1t
RAll)
___ _
__
___ _I
I
_
__
__
P0OGRAn
TEA
DEPT OFCIVIL EGINEERING, H. I. T.
ANALYSIS OF TIDAL
NAVES
FOR SMALL SCALE GEOMETRY.
2-D LINEAR FINITE ELEMENT FREQUEHCY DOMAIN
F.E. GRID GEOMETRY IDENTIFICATION : EISAY?
BOUNDARY
CONDITION
IDENTIFICATION : C-01
FREQUENCY
a 0.00000000
-77-
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C §.
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..
A N) :0
CA
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c
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"
;I.. &.
a -0
C
-NJU
~I
0
a
I
;3
E'
~I
Cw
Y 0)
1
0.0000
0.00000
1.00421317
1.00317041
NODAL
VELOCITIES
NODE
1
2
3
4
5
66
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
FORTRAM STOP
I-DIRECTION
NODWUS
PHASE
0.11872876
0.02610023
0.00914074
0.01634801
0.01702293
0.00641079
O.OOIS07
0.02053215
0.00574442
0.01998268
0.01021991
0.00509576
0.00007838
0.01497699
0.01887863
0.00627862
0.00237505
0.00828776
0.00653017
0.00135094
0.00190716
0.01979741
0.03540456
0.02567951
0.01943137
0.00369105
0.01701885
0.02198353
0.00146240
0.03065666
0.03112854
0.04612814
0.03615336
0.06219012
0.07614501
3.14159
3.14159
0.00000
0.00000
0.00000
3.14159
3.14159
0.00000
0.00000
3.14159
3.14159
3.14159
0.00000
0.00000
3.14159
0.00000
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
0.00000
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
Y-DIRECTION
fODULUS
PHASE
0.10354253
0.07462922
0.00439589
0.01742001
0.04220178
0.04607911
0.04340251
0.00234294
0.00337163
0.01700479
0.03441213
0.00274338
0.01432924
0.01610249
0.01509755
0.02056481
0.02906004
0.00232953
0.01065585
0.02869783
0.0037544
0.01914364
0.00300588
0.02257412
0.01881449
0.04196155
0.04217280
0.04930760
0.01331787
0.0023502
0.02989864
0.00071217
0.00729552
0.05998277
3.14159
3.14159
3.14159
0.00000
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
0.00000
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
3.14159
Appendix 1.4
-82-
C
C
C
PR.CG.AM TEA
LINEAR FREQUENCY DOMAIN FINITE ELE.ENT MODEL FOR TIDAL CIRCULATION
C
C
C
C
C
C
C
C
C
C
C
C3 YRIGHT :
DEX-'- VT
OF' CIVIL ENGINEE.ING
MASSAC:USETTS INSTITUTE CF TECHNOLOGY
FOR LATION : ELEVATON/ELOCITY
EFFECTS : TIDAL MTION. WIND ST?.ESS, LINEARIZED BOTTOM FRICTIN,
AND CG~IOLIS
S3LUTION : COMACT DI?.ECT
SOUR.CE CODE : BAYOUF3V9
COMMON /SOL!/ SYSM2C,P,ELEV
DI.-ENSION ICON(600,3),M(600) .NN(450) ,.IFLSEG(600,2).ISVHT(450)
REAL*8 XORD(450) ,YORD(450),H(450),.ELFRIC(500),SVHT(450,2),
SYSNJvLC(450),SYSDC(450. 150),A(3),B(3),HS(3),SVFLP(450,4),
a
ISVFLP(450)
&
COMPLEX*16 PW(900) ,ELEVPR(450) ,FLOW(2) ,PN(450),SYSM1C(450. 150),
&
SYSM2C(450.150) ,PT(900),SYSKLC(900,3) ,P(450),ELEV(450),U(900)
COMLEXc* 16 QX,QY,TEWINXEL, TEMWINYEL, T3, T8, LEVO,
2C, ZlCCM,UNICOM
TzTWIN,TlMWINX,TEMWINY. TEMP1C, T
T
REAL*8 AREA,DEPTH,DX1,DX2,DX3,
&
DY1,DY2,DY3,ELIMAG,ELMOD,ELR.EAL,G,OEGA,PASE,PI,
&
TEM?,TXORD,TYORD,XI,X2,X3,XIMAG,XMOD,XREAL, Y, Y2
&
Y3,HTPR,LEN,XDIF,YDIF,NX,NY,QXRQXI,QYR,QYI,ZERO,UNITY,
X
DEP,WIND10,WINPHAS,WINDIR,WINDRAG,T1,T2,T4,T5,T6,T7,CORFAC
CHARACTER*10 ALPHID,BCV
MXANPD=25
MXEL=600
MXNP=450
WRITE(6,6010)
6010
FORMAT(/,' PROGRAM TEA ',/,
' DEPT OF CIVIL ENGINEERING, M. I. T. ,/,
X ' 2-D LINEAR FINITE ELEMENT FREQUENCY DOMAIN ANALYSIS ',
& 'OF TIDAL WAVES FOR SMALL SCALE GEOMETRY.',/)
PI=3.14159265
ZERO.0=O.0
UNITY=1.0
ZERCOM=DCMPLX (ZERO,ZERO)
UNICOM=DCMPLX (ZERO,UNITY)
C
C...... READ THE INPUT VARIABLES
C
WRITE (6 ,6020)
6020
FORMAT(' ENTER 10 DIGIT F.E. GRID GEOMETRY IDENTIFICATION',/)
READ(5,57) ALPHID
57
FORMAT(A10)
WRITE(6,6030)
6030
FO.MAT(' ENTER 10 DIGIT I.D. CODE FOR B.C. VERSION',/)
READ(5.57) BCV
WRITE(8,7010) ALPHID,BCV
7010
FORMAT(1X.A10)
WRITE(6,6040)
FORMAT(' INPUT NUMBER OF ELEM7NTS AND NUMBER OF NODE POINTS',/)
6040
READ(5,*) NMEL,NMNP
WRITE(8,7701) NMEL,NMNP
FORMiAT(lX,I3 1H ,I3)
7701
IF(NhMEL.GT.MXELS GOTO 115
IF(NMNP.GT.MXNP) GOTO 117
GOTO 119
WRITE(6,6050) M):EL
115
-83-
6050
FORMAT(/.' NO. OF EL--NTS SPECIFIED EXC EDS ARRAY DIMENSIONS 'o
a 'WHICH ALLOWS FOR CINLY '.1,.'ELENTS'.1,' PROGRAM EXECUTION
&
'IS STOPPED; MODIFY PROGRAM DIMENSIONING')
GOTO 1295
WRTEC6S,6055) MXN?
117
6055
k
8
FORMAT(/,' NO. OF NODES S?ECIFIED EXCEEDS ARRAY DIMENSIONING',
' WHICH ALLOWS FOR ONLY ',IS,' NODES'./.' PROGRAM EECUTION '
'IS STOPPED; MODIFY PROGRAM DIMENSIONING')
GOTO 1295
CONTINUE
WRITE(6,6060)
FORMAT(' INPUT THE NODE NO.,X-CGOC.D. ,Y-COORD., AND .NODAL',
119
600
a
' M.S.L. DEPTH,
ONE NODE/LINE'./)
7702
130
DO 130 I=1,NMNP
READ(S,*) NN(I),TXORD,TYOP.D,DEP
L=NN(I
XORD(L1 =TXORD
YORD(Li =TYOP.D
H(L1)=DEP
WRITE(8,7702) L1,TXORD,TYORD
FORMAT'(X,I4, H,,FI0.2.iH.,FiO.2)
CONTINUE
6070
WRIT(8,6070)
FORMA (' INPUT ELEMNT NUMBER, THP.EE NODE NUMBERS FOR THE ELEM-ENT '
a
,'AND THE ELEMENT LINEAR FRICTION
ACTOR' ,/)
DO 145 I=1,NMEL
READ(C5,*) N,(ICON(N.J),J=1.),ELPRIC(N)
WRITE(8,7703) N,(ICON(N,J),J=,3)
FORMAT(CX,3(I4,1H,),14)
7703
M(I)=N
CONTINUE
145
C
C...... CHECK FOR EXCESSIVE BANDWIDTH
C
IFLAG=0
MXNPD=O
DO 160 N=I,NMEL
DO 160 1=1,2
12I+1
DO 155 J=I2. 3
&M=IABS(ICON(NJ)-ICON{N,I))
IF (M.GT . MXNPD
MXNPD=MM
NCXCEED=MXNPD -JXANPD
6080
IF( M.GT.JMANPD) WRITE(6,6080) rA~NPD N ~CEED, N
FOPRMAT(' MAX. ALLOWABLE NODAL POIT 'DifFENCE OF ',13,'
IS ',
'EXCE DED BY',14,' AT ELEME.NT '.14,/," ARRAY DIMENSIONING',
'MUST BE MODIFIED : PROGRAM EXECUTION IS TERMINATED')
IF(MM'.GT.MXANPD) IFLAG=1 '
155
CONTINUE
160 CONTINUE
WRITE(6,8090) ~XtPD
6090 FCRMAT( MAX. NODAL POINT DIFFERENCE =',I,//)
IF(IFLAG.EQ.1) GO TO 1295
C
C....... IBW=BANDWIDTH OF FULL MATRIX WITH SINGLE DEGREE OF FREEDOM PER NODE
C
ISV=2*D.ND+1
IB'2=2*IBW
180 FCRMAT(' INPUT GRAVITATIONAL CONSTANT IN CONSISTENT UNITS',/)
READ5,. ) G
RI:TE (6,6: 00)
t
&
I,UT FF.UrE'NCY IN P.ADIANS/SEC.',/)
6100
FO...AT('
7705
FCRY.AT 1iX, F20.1)
RITE(.e 6110)
?.E- (5,-).OSGA
IRITE(8.7705) CHEGA
-84-
6110
6120
a
Z
FC?.AT(' INPUT CC?.ICLS PA?3.AMEE?.' ,/)
READ(5,*) CZ?.FAC
WRITE(6, 6120)
FOP.RAT(' INPUT A.PLITUDE OF WIND SP"'D, WIND PEASE SHIFT,
'ANGLE OF WIND DIRECTION (IN DEGREES)
' AND WIND DR.AG COE
CIE
T' , /)
?.EAD(5,*) VIND1O,WINRAS,W:ND IR,WIN"D?.AG
WINDIR=(?I/O180. 0) *VINDI
a
TLE'.IN=0. O012*WINDAG*WIND 10**2* (COS (WIN?HAS) +UNICH*
SIN WINPHAS))
_'.WTY-.E--WIN*SIN (WND:.)
TE':W iNX=:TEfw'I NCCS(INDIR)
6130
a
W?.ITE(6,6130)
FORMAT(' INPUT HOW MANY BOUNDA?.Y SEGMN-S HAVE NON-ZERO NC?.MAL',
' FLUX PRESCRIBED')
READ(5,*) NBSF
C
C...... INITIALIZE THE LOAD VECTOR FOR THE CONTINUITY EQUATION
C
DO 192 I=l,NMNP
192
PN(I)=ZERCOM
C
C...... LOAD VECTOR LOADING SECTION FOR NON-ZERO PRESCRIBED NRJ.XAL FLUXES
C
IF(hBSF.EQ.O) GOTO 226
C
C...... READ IN REQUIRED INFO FOR SEGMENTS WITH NON-ZERO NOP.MAL FLUX PRESCRIBED
C
DO 196 I=1,NBSF
WRITE(6.6140) I
FORMAT(' INPUT END NODES OF BOUNDARY"SEGMENT',I5,' WITH ',
6140
'CORRECT ORIENTATION')
a
READ(5,*) IFLSEG(I,) ,IFLSEG(I,2)
196
CONTINUE
WRITE(6,6145)
6145
FORMAT(' INPUT TOTAL NO.-OF DIFFERENT NODES INCLUDED IN NON-',
I
'ZERO NORMAL'FLUX SEGMENTS')
READ(5, ) NBNF
DO 200 I=I,NBNF
WRITE(6,6148)
FORMAT(' INPUT NODE NO..MODULUS AND PHASE SHIFT (IN RAD) '.
6148
e
'FOR X AND Y DIRECTIONS ')
READ(5,*) NODE.(SVFLP(NODE.J),J=1,4)
ISVFLP(I)=NODE
200
CONTINUE
DO 225 I=I.NBSF
MI=IFLSEG(I. 1)
M2=IFLSEG(I,2)
XDIF=XORD (M2)-XORD(Ml)
YDIF=YORD (M2)-YORD (M1)
LEN'= (XDIF**2+YDIF**2) **0.5
NX=-YDIF/LEN
NY= XDIF/LEN
DO 210 K=1,2
NODE=IFLSEG(I,K)
XR=SVFL? (NODE, )*COS(SVFLP (NODE,2))
QXI=SVFLP (NODE, 1)*SIN (SVFLP (NODE,2))
QYR=SVFLP (NODE,3)*COS(SVFLP (NODE,4)
QYI=SVFLP (NODE3)*SIN(SVFLP (NODE.4))
QX=DCYPLX(QXR,QXI)
QY=DCMPLX (QYR,QYI)
FLOW (K) =X*NX+QY*NY
210
CONTINUE
C
C..........LOAD THE NON-ZERO FLUX SEGM'ENT CONTR:BUTIONS INTO GLOBAL LOAD VECTR
C
-85-
~TP=LEN/6 .0
PN(M1)=PN(M:I) +TZP*(2*FLOW(1) +FLOW(2))
PN (.2)=PN(M2)+T L? (FLOW (1)+2*FLOW (2))
CONTINUE
CONTINUE
225
226
C
C ....... READ IN ELEVATION LOADING INFORMATION
C
ELEVO=ZERCOM
WRITE(6,6200)
6200
FORXAT(//,' INPUT NO. OF NODES WHERE ELEVATION IS PRESCRIBED')
R.EAD(5,*) NBL
IF(N3L.E.0) GOTO 240
DO 230 I=i,N3L
WRITE(5, 6210)
6210
FORMAT(' INPUT THE NODE NUMBER AND VALUES FOR THE MODULUS AND',
&
' PHASE (IN RAD) OF THE PRESCRIBED ELEVATION AT NODE')
READ (5,*) NODE,HTPR,PHASE
ISVHT(I) =NODE
SVHT(I,I)=HTPR
SVIT (I.2) =PHASE
X RAL=HTPR*COS (PHASE)
XIMAG=HTPR*SIN(PHASE)
ELEVPR (I)=DCMPLX (XREAL,XIMAG)
ELEVO=ELEVO+ELEVPR (I)
230
CONTI NUE
ELEVO=ELEVO/NBL
240
CONTINUE
DO 250 I=I,NBL
250
ELEVPR (I)=ELEVPR (I)-ELEVO
C
C..... PRINT INPUT VALUES
C
WRITE(6,6300)
6300
FORMAT(//,130('-')./)
WRITE(6,6010)
WRITE(6, 6310) ALPHID,BCV
FORMAT(//,1X,'F.E. GRID GEOMETRY IDENTIFICATION : ',A10,
6310
& /,1X,'BOUNDARY CONDITION IDENTIFICATION : ',A1O,//)
WRITE(6,6320) OMEGA
FORMAT(1X, 'FREQUENCY = ',F10.8)
6320
WRITE(6,6322) CORFAC
6322
FOl.MAT(IX, 'CORIOLIS PARAMETER = ',F10.8,//)
WRITE(6, 6330) WIND10O,WINPHAS,WINDIR,WINDRAG
6330
FORMAT(//,' WIND SPEED AT 10 METERS ABOVE SURFACE = ',F10.4,
&
/,' WIND PHASE SHIFT = ',F10.5,' RADIANS',
A
/,' WIND DIRECTION = ',F!0.5,' DEGREES',
/,' WIND DRAG COEFFICIENT = ',F12.8,//)
i
WRITE(6,6340) NMNP,NMEL
6340
FORMAT(' NO. OF NODE POINTS =',I3,/,' NO. OF ELEMENTS =',13,/)
WRITE(6,6090) M~N?D
WRITE(6,6350)
FO.MAT(/,1X,37('*'),10X, 'NODAL CCOR.DINATES AND DETHS',IOX,
6350
& 35(*'))
WRITE(6,6352)
6352
FORMAT(/,38X,'NODE',9XNODE', X,'X-COO.D. ',8X,'Y-COORD. 'X, 'DETH',/)
DO 270 I=1,NYMNP
LI=NN(I)
.RITE(6,6355) L1,XORD(L), YORD(L1) ,H(L1)
6355
FOR.MAT(36X,I5,6X,F10.2,6X,F10.2,6X,710.2)
CONTINUE
270
v?.ITE(66360)
6360
S
ARRY ',!CX,430('*'),
FCRMAT ((/) ,X,37('*') .CX, '''"T
//,35X, 'EL' NT',8X,'I',5X,'J',5X,'K',12X,' ' -'' T',
FRICTION FACTOR',/)
DO 257 N=1,NIYEL
-86-
=(N)
?.ITE(,655) I,(ICON(,
. =.3) J),J
6365
287
ELF.IC(I)
FORM.AT(37X,I3,8X,I3,3X,13,3X
13X,F3.7)
CONTINUE
WRITE(6,6400)
FO.MAT(6(/),1X,30('*'),10X, 'BOUNDARY CONDITIONS',1iX,30('*'))
6400
WITE(6,6410) NBSF
6410
FCO.MAT(///,1X,'NO. OF OUNDARY SEGMENITS WHICH .AVE NON-ZRO',
'NOR?.AL FLUX PRESCRI--D =',:4)
IF(N3SF.E.0) GOTO 321
VWRITE(S, 6415)
6415
FC? .A.T(//,1X, 'SEGMENTS WITH NON-ZE.O NOCMAL FL.rX P.ESCBED ,
S
'A.E: ',/,4X, 'SEGMENT NO. ',5X,'BEGINING NODE NO.',5X,
'END NODE NO.',/)
DO 305 I=I,NBSF
WRITE(6,6420) I,(IFLSEG(I,J),J=1,2)
305
6420
FO.MAT(7X,I3,2(16X,I4))
WRITE(6, 6430)
6430
FORMAT(//,iX,'PRESCRIBED FLOW VALUES FOR NODES ON',
a
' SEGMENTS ARE:')
WRITE((6,6435)
6435
FORMAT(//,6X,'NODE NO. ',9X,'X-DIRECTION',19X, 'Y-DIRECTION',1iX,
t
/,20X,2(' FLUX',6X,'PHASE',14X),'DEPTH',/)
DO 316 I=1,NBNF
NODE=ISVFLP(I)
WRITE(6,6436) NODE. (SVFLP(NODE,J),J=1,4)
FOR.MAT(6X,14.5X,2(F11.4.2X,F9.5,8X))
6436
316
CONTINUE
321
CONTINUE
WRITE(6,6440) NBL
6440
FORMAT(5(/) ' NO. OF NODES WHERE ELEVATION IS PRESCRIBED =',13)
IF(NBL.EQ.0 GOTO 2741
WRITE(6,6442)
6442
FORMAT('O'.25X,'NODE',10X. 'PRESCRIBED ELEVATION',10X,
k
'PHASE',/)
DO 2740 I=I,NBL
WRITE(6,6445) ISVHT(I),SVHT(I,1),SVHT(I.2)
6445
FORMAT(26X,I13,8X,F1.411XX,F13.5)
2740
CONTINUE
CONTINUE
2741
C
C.....
FORM THE ELEMENT MATRICES IN GLOBAL COORDINATE SYSTEM
C
NT2=NMNPs2
ILCB= (IBW-1)/2.0
ICB=ILCB+1
DO 325 I=1,NT2
DO 325 J=1.3
SYSKLC(I, J)=ZERCOM
325
DO 326 I=1,NMNP
326
SYSNULC(I) =0.0
DO 327 I=1,NMNP
DO 327 J=1.IBW2
327
SYSDC(I,J) =0.0
DO 328 I=i,NT2
328
PW(I)=ZERCOM
DO 500 N=l,h.EL
NR1=ICON(N', 1)
NR2=ICON(N,2)
NR3=ICON (N, 3)
X1=XORD (IR1)
X2=XORD (NR2)
X3=XOR (Nm3)
YI=YORD (NR)
Y2=YORD (NR2)
Y3=YORD(NR3)
S
-87-
A(1)=X3-X2
A(2)=X1-X3
A(3)=X2-X1
B()=Y2-Y3
B (2)=Y3-Y1
B(3) =Y1-Y2
A.REA=O .5*((1)*A(2)-B(2) A(i))
6500
$
IF(AREA.LE.0.0) WRITE(6,6500) N
FOPr.AT(' AP.EA OF ELMIENT' ,12,' IS LESS THAN ZERO,
' P?.CGRAM ZEXECTON IS TE.MiNATED')
IF(AREA.LE.0.0) STOP
HS (1) =2*H (N?.I) +H (.2)
+H (N?.3)
HS (2) H (NR1) +2*H (NR2) H (NF.3)
HS (3) =H (NRi) +H(NR2) +2H (N.3)
C
C...... LOAD ' . NT CONT.ISBUTION INTO GLOBAL MATRIX SYSNULC (IN LUM?EDC....... COM.PACT STORPAGE MODE)
C
T=AREA/3.0
SYSNULC(NR1) =SYSNULC(NR) +T
SYSNULC (N.2)=SYSNULC (NR.2) +T
SYSNULC(NR3) =SYSNULC(NR3)+T1
C
C...... GENERATE GLOBAL LUMIED-COMPACT STIF.FNESS MATRIX FOR MOM
C....... INCLUDES COMBINED MASS,FRICTION AND CORIOLIS MATRICES
C
T2=AREA/12.0
DO 460 I=1,3
TUM EfUATION,
IR=ICON(N,I)
IR2=2*IR
IR1=IR2-1
T3=T2*DCMPLX(4*ELFRIC(N),OMEGA*HS(I))
T4=T2*CORFAC*HS(I)
2)+T3
SYSKLC(IRI,2)=SYSKLC(IR1
SYSKLC(IR1,3)=SYSKLC(IR1,3)-T4
SYSKLC(IR2, =SYSKLC(IR2,1)+T4
SYSKLC(IR2,2 =SYSKLC(IR2,2)+T3
CONTINUE
460
C
C...... LOAD EL-'ENT CONTRIBUTIONS INTO GLOBAL MATRIX SYSD
C....... NON-SYMMETRIC STORAGE MODE)
C
T6=1.0/24.0
DO 450 I=1,3
IR=ICON(N,I)
DO 440 J=1,3
(IN COMACT
JNCC?=ICON (N, J)
440
450
JCOM?=JNCOMP+ICB-IR
T7=T6*HS(I)
SYSDC(IR,2*JCOM?-1)=SYSDC(IR,2*JCOMP-1)+B(I)*T7
SYSDC(IR,2*JCOP)=SYSDC(IR,2*JCOMP)+A(I)*T7
CONTIN'UE
CONTINUE
C
C...... ADD ELEVO COPONh~ET TO ?N LOADING VECTOR
C
T8=T1 UN ICM*OYZGA*ELEVO
DO 465 I=1,3
IR=ICON(N,I)
PN(IR)=PN(IR)+T8
465
C
C...... LOAD FW TO CONTAIN WIND:
C
T7
_--W:
ST.ESS LA INGS
7L=T
a."W :1 X* T I
DO 470 1=1,3
-88-
I?=ICON (N.I)
I.2=2IR
IR1=IR2-1
PW(I.R1)=PW (IR1) +TE7fsINXEL
PW(IP.2)=PW(!.2)+T-N"NEL
C NThNUZ
CONjTI N'UE
470
500
C
C......
C
k
INVE?.L
THE MATRIX SYSKLC
DO 910 IR=1,N~N
I?.2=2*IR
I.1=IR2-1
TEY'1C=SYSKLC (IR1, 2)*SYSKLC (:R2,2) -SYSKL.C (i2, 1)
*SYSKiLC(IR1,3)
T--P2C=SYSKLC(IR1,2)
SYSKLC(IRI,2)=SYSKLC (IR2.2) /TE.L= CP1C
SYSKC.C (IR2.2)=TE?!P2C/TEM? 1C
SYSKLC(IR2, 1)=-SYSKLC (IR2,1)/TEM 1C
SYSKC (IR1, 3) =-SYSKLC (IR1, 3)/TEM1C
CO!NTINtE.
910
C
C...... FORM THE PRODUCT PT=SYS.KLC*FW
C
DO 915 IR=1,NT2
TP 1C=Z COM
DO 912 K=1,3
KSUM=IR-2+K
IF((KSUM.LE.0) .OR.(KSUM.GT.NT2)) GOTO 912
T
1C=TEMP 1C+SYSKLC (IR,K)*PV (KSUM)
912
CONTINUE
IF(IR.GT. 1) PT(IR-1)=TMP2C
TEMP2C=TEMP C
CONTINUE
915
PT(NT2)=TEM2C
C
C...... FORM THE PRODUCT P=SYSDC*PT-PN
C
DO 920 IR=I,NMNP
JSHIFT=2*(IR-ICB)
TEMP1C=ZERCOM
DO 918 JCOCL=1,IBW2
JNCOCL=JCOCL+JSHIFT
IF((JNCOCL.LE.0).OR.(JNCOCL.GT.NT2)) GOTO 918
TEM~1C=TEMP1C+SYSDC (IR, JCOCL) *PT (JNCOCL)
CONTINUE
918
P(IR)=TEMP1C-PN(IR)
920
CONTINUE
C
C...... FORM PRODUCT SYSM1C=SYSKLCSYSDC(TRANSOSED) :SYSMiC IS STORED SIDEWAYS
C
DO 930 IR=1,NNP
JSHIFT=2*(IR-ICB)
DO 925 JCOCL=1,IBW2
JNCOCL=JCOCL+JSHIFT
SYSMC(IR, JCOCL) =ZERCOM
IF((JNCOCL.LE.0) .OR.(JNCOCL.GT.N7T2)) GOTO 925
KSUM=JCOCL-2
DO 924 KADD=1,3
KSUM=KSUM+1
IF((KSUM.LE.0).OR.(KSUM.GT.IBW'2)) GOTO 924
SYSMC (IR,JCOCL)=SYSM1C(IR,JCOCL)+SYSKLC(JNCOCL,KADD)*
SYSDC(IR,KSUM)
924
CONTINUE
925
CONTINUE
930
CONTINUE
-89-
C
C...... FORM THE PRODUCT SYSZ2C=I*O0.KE2GA*SYSNULC+G*SYSDC*SYSM1C
C
IBW3=IBV2-1
ILCB3=IBW- 1
CB3=ILCB3+ 1
DO 935 IR=1,IBW
DO 935 JCOM=1,IBW-IR
SYS2C (IR. JCOM)=ZERCOM
DO 937 IR=NMN.?+2-BW,NMN
DC 937 JCoM=hMNP~N?2+IBW, IB'W3
=ZERCOM
SYS!"2CC1., JCO.Z)
DC 950 I=.1=1 ,NP
.3E= I?.1 -IBW+ I
935
937
IF(IR3.EG.LE.0)
IRBEG=1
IR.ED=IR1IBW-1
P) IRLD=NMNP
IF(IREND.GT.NI
DO 945 I2=IP.RBEG,IP.N
KSHIFT=2* (IR1-IR2)
KBEG=1
IF(KSHIFT.LT.0) KBEG=1-KSHIFT
KEND=2* IBW-KSHIFT
IF(KSHIFT.LT. 0) K.ND=IBW2
T-P1 C=ZERCOM
DO 943 Kl=KBEG,KE-ND
K2=K1+KSHIFT
TEM2.P1C=TMP1C+SYSDC(IR1,K1)
CONTINUE
JCOMP=IR2-IR +IBW
SYSM2C(IR1,JCOMP) G*TLMP1C
) GOTQ 945
IF(JCOMP.NE.ICBP
943
945
9-0
a
SYSM1C(IR2,K2)
SYSM2C (IR1, JCOMP) =SYSM2C(IRI, JCOMP) +UNICOM*OMEGA
*SYSNULC(IR1)
CONTINUE
CONTINUE
C
C...... INCLUDE PRESCRIBED ELEVATIONS IN FINAL SYSTEM OF EQUATIONS FOR
C....... CALCULATING ELEVATIONS,
SYSFK2C*ELEV=P
C
IF(NBL.EQ.0) GOTO 990
DO 980 I=1,NBL
NODE=ISVHT (I)
C
C...... ZERO OUT THE ROW FOR THE PRESCRIBED NODE
C
DO 960 JCOM=I.IBW
960
SYS12C (NODE, JCOM) =ZERCOM
C
C...... SET DIAGONAL EQUAL TO 1
C
SYSM2C (NODE, ICB3) DCMPLX (UNITY, ZERO)
C
C...... SUBSTITUTE PRESCRIBED ELEVATION INTO P
C
980
990
C
P (NODE)=ELEVPR (I)
CONTINUE
CONTINUE
C...... SOLVE THE SYSTEM OF E;UATIONS SYS2C*ELEV--P
C
CALL EFICSOL(NNMNP,IBW3)
C
C...... CALCULATE NODAL VELOCITIES VECTOR U FR3M ELEV
C
DO 1070 I=1,NT2
1070
U(I)=Z-P.COM
-90-
:..C:M=
.N" N:
DO 1100
"r
=2 a f . ' m -. -,
.
vS,
08
I080
JCO)M=1,IW23
DO
U(IRNCO) =U (IRNCOtM) +G*SYSMiC (IRCOM, JCOM)* .EL.E (CSM)
CONTINE
C3 NTINJE
DO 1102 IR=1.N7T2
U (R)=PT (I.)-U(IR)
W.ITE(5, 6600)
:080
11i0
1102
F0RMAT(///.:X,3C(' '),.IX,'RESU.,S OF C3oMAIOTNS IOX,30(
6600
C
C...... CALCULATE AND ?P.INT MODL"US .AND PHASE FCR ELEVATION
C
WP.ITE, S. 6610)
FOP.MAT(////,X,31('-') ,8X.'NGDAL EL-VATIONS' ,7X,31('-')
6610
* //,28X,' NODE ',14X,'MODULUS ',18X,'PHASE ',)
C
C...... ADD BACK IN ELEVO TO TOTAL ELEVATION
C
DO 1246 I=1,N N?
ELEV(I)=ELEV(I) +ELrO
1246
DO 1250 I=1,.MN?
ELREAL=DEAL (ELEV (I))
ELIMAG=DMA (ELEV (I )
ELMOD=CDABS (ELEV(I)
IF((ELrEAL.EQ.0.0).AND.(ELIMAG .E5.0.0)) GOTO 1235
PHASE=ATAN2(ELIMAGELREAL)
GOTO 1240
PHASE=0.0
CONTINUE
WRITE(6. 620) I,ELMOD.PHASE
FORMAT (30X.13,14X,F12.8,14X.F9.5)
1235
1240
86620
WRITE(8.7706) I,ELMODPHASE
FO?.,MAT(1X,14.IH,F20.10,IH,,F7S.10)
7706
1250
CONTINUE
6630
WRITE(6,6630)
FORMAT(////,1X,30(-'),10X.'NODAL VELOCITIES',10X,30(-').
//,31X, 'X-DIRECTION',28X,'Y-DIRECTION'./,
C
C...... CALCULATE AND PRINT THE MODULUS AND PHASE FOR VELCCIfTY
C
&
S
S
1280
SX, 'NODE',12X. 'MODULUS' 11X.'PHASE',16X.
'MODULUS'12X,. 'PHASE' ,/
DO 1290 I=1,NMhNP
L1=NN(I) *2-1
DX1=CDABS U(L
DX2=DIMAG U(Ll)
DX3=DREAL (U (LI)
IF(DX2.EQ.0.0.AND.DX3.EQ.0.0)
DX2=ATAN2 (DX2,DX3)
GO TO 1278
LI=L1+1
DY1=CDABS (U(L)
DY2=DIXAG U(
DY3=DREAL ('J(L1)
IF(DY2.EQ.0.0.AND.DY3.EQ.0.O)
GO TO 1278
DY2=ATAN2(DY2, DY3)
1276
GO TO 1282
DX2=0.0
1278
GO TO 1280
DY2=0.0
7709
FORMAT(1X,I4,1H.,F20.10,1H ,F15.10,1H.10,H..
1282
6650
WRITE(6,6650) NN(I),DX1,DX2.DY1.DY2
FORMAT(8X, 13 OX.F12.8 6X,F95.2.12X.F2.8,7X,F9.5)
RITE( 8,7709) NN(I),DX1,DX2,DYI ,DY2
-91-
10,1H,,F15.10)
))
CONTINUE
STOP
1290
1295
SJSRC'JTNE EFICSOL fNDM,i
COMION /SOL1/ B.C,X
)
CI.LEXc*16 B(450,150),C(450),X(450)
--COMPLEX* 1 TES 31 L TEMSL2
REAL8 ZER0, UNITY
ZERl=0.0
UNITY=1.0
C
C......
C
10
ILCB (IBW- 1)/2
ICB=IL.S+1
:RCB=ILCB+2
PERFORM GA'SS ELIMINATION ON COMPACT MATRIX AND LCAD VECTOR
DO 100 IRW=1,NDM
TE'SOL1=B (IRW, IC)
C(IRW)=C (IP.W) /TESOL1
3(IRW, ICB) =DCILX (UNTITY, ZERO)
DO 10 JNMw=IRCB,I3W
B (IRW, JNMR) =B (IRW. JNMP".) /TEMSOL1
CONTINUE
IRWB=IR+W 1
IRW&=IR+IL CB
DO 90 IRWZ=IRWB,IRWE
IR.MVE=ICB- (IRWZ-IRW)
TLMSOL2=B (IRWZ, IRMVE)
B(IRWVZ, IRMV) =DCMPLX (ZERO., ZERO)
IROCHIRMEE+ 1
DO 40 JCOZ=IROCHIB
40
a
B(IRWZ.JCOZ) =(IRWZ.JCOZ) -TF.SOL2* (
B(IRW,JCOZ+(IRWZ-IRW)))
CONTINUE
C(IRWZ) =C (IRWZ) -TSOL2*C (IRW)
CONTINUE
CONTINUE
90
100
C
C...... SOLVE FOR UNKNPW'NS USING BACK-SUBSTITUTION
C
ISV=NDM
200
CONTINUE
X(ISV) =C (ISV)
DO 300 JSV1. ILCB
KSV=ISV+JSV
300
301
IF(KSV.GT.NDM) GOTO 301
X(ISV)=X(ISV)-B ISVICB+JSV) *X(KSV)
CONTINUE
CONTIIUE
ISV=ISV-1
IF(ISV.GE.1) GOTO 200 RETURN
END
-92-
Appendix 2:
-93-
Program RENUMB
2.1.
inputs/outputs
2.2
listing
Appendix
-94-
2.1
Program RENUIB renumbers grid node numbers in an efficient manner in
order to optimize the maximum nodal point difference between connecting
nodes, allowing program TEA to be run more efficiently.
RENUMB should be
run after a grid has been set up or after some grid modification has been
done.
Input requirements for RENUMB are exactly the same as for program
TEA.
Hence a complete input file set up for TEA can be run with RENUMB.
The un-renumbered grid input file is read in on TAPE 5.
output file will be output onto TAPE 6.
The renumbered
We note that only node numbers
change when running RENUMB and that element numbers remain the same.
Finally RENUMB outputs renumbering information on TAPE 8.
shown in Appendix 1.3.3 has been run with RENUMB and is
-95-
The example
listed here.
-96-
oooco7.00*
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0000O00.0
000001000O
000cO00*0
000001000O~
00000100*0
000OO00*0
00000100*0
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'r
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Is
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Ir
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-L6-
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at
Appendix
-98-
2.2
frcgra~
for
a,a::c rC :er4 t will give the following inOut
variables
for program tea :
a2Pi~.~bcY
nmel nmn
cc
node x-cbord
y-coord
elt iconi icon2 icon3 epth
CC
CCCCCCCCCCCccccCccCCCCCC
ccCCe
dmIensicn next(600) xrd (600),yord(60) ,de (600) , icon (800,3) ,j
ht(600), nint (00), fric(600)
c aracter=10 al:id,bcv
read(5,1000) albh:d
read(5,1000) bcv
1C00 for-a:(a:0)
wrlte(6,1001) albhid
write(6,1001) bcv
1001 format(Ix,ai0)
read(5,*) nmel,nmnp
write (6,2001) nmel nmnp
cc
ec
cc
2001 format (1x,13,lh,,i3)
do 10 i=1,nmno
read(5,*) next(i),xcrd(i),yord(i),dep(i)
nint(next (i))=i
10 continue
read(5, ) (n,(icon(n.j),j=1,3),fric(n),n=1,nmel)
cc
call renum(nmnp,nmel,icon,jnt,idiff,ndiff)
cc
mabwh=ndiff+l
do 13 li=1,nmnp
do 11 i=l,nmnp
:j nt(i)
If (ii.ne.1) go to 11
11
13
write(8,3001) next(ii),xord(i),yord(i),dep(i)
continue
continue
do 12 1= ,nmel
12=icon (i,2
13=icon (,3)
nt( 12)
] 3= nt 13)
write (6,4001)
i,j1,j2,j3.fric(i)
12 continue
3001 format(lx,i4,1h,,ff0.2,lh,,10.2,h,f10.4)
4001 format(1x,4(14,1h,),f12.8)
read (5,*) g
write (6,5001)
5001 format(1x,f10.5)
read(5,*) omega
write(6,5003) omega
5003 format(x,f15.10)
read(5,*) corfac
write (8,5003) corfac
read(5,*) vwn1,win2,win3,win4
write(6,5005) ;inl,win2,win3,win4
5005 format(lx,3(flS.10,','),f15.10)
read(5,*) nbsf
write (6,5006) nbsf
5006 format(lx,il0)
if(nbsf.eq.0) goto 5100
do 5040 i=1,nbsf
read(5,*) nl,n2
;nl=1 nt(nl
n2=] nt (n2)
-99-
-rite(6,5030) n ,4 .2
5030 format(1x,l10,,' 110)
5040 continue
read(5,*) nbnf
bnf
•rite(6,5006)
do 5070 i=1,:bnf
read(5,*) node,svl,sv2,sv3,sv4
j node=jn=(node)
rit e(o,5065) lnode,svi,sv2,sv3,sv4
5055 form t(x,i0, (',',f15.0))
5370 continue
nb1
5100 read(5,)
:z te(6,5006) nbl
if(nbl.ec.0) goto 9999
do 5300 i=:,nDl
read(5.*) node, htpr,phase
]node=int(node)
write(&,5250) jnode,ht~r,Dhase
5250 formna(xz,i10,2(', ,fl.10))
5300 continue
cc
9999 stop
end
CCCCC
ccccc
subroutine renum(nodes,1ments,icon,
nt,idiff,ndiff)
dimension icon(600,3), nt(600),jt(2400),memlt(4800)
dimension Imem(600), ix5)j ti(5) jt2(5),jt315),jt4(5)
dimension x(5),jntx(5)
write (8,19)
do 100 i=,lments
il=i
12=+600
13= 1+600*2
14=1+600*3
t(i1)=icon(1i I1
t(i2) =icon(i,2
i-t(i3)=icon(l,3
0t(i4)=0
!9 format(lhb,'input data'/lhO,'no.',3x,'.',3x,'2.'.3x.'3.'/lh0)
1x=1ments/4+1
do 150 i=1,lx
!00 continue
:p=0
dO 160 ii=1,4
ix(ii)i*Ip
l1(10)=
it2(ii)=
t(i+1P)
(i+ip+600)
43(ii)= t~(i+lp+1200)
]t4(ii)=J (i+p+1800)
1p=lp+1x
160 continue
write (8,2) (ix(ii),jtl(ii),jt2(ii),.jt3(ii),jt4(ii),ii=1,4)
150 continue
2 format(lh,4(515,3x))
ccc
call setu (nodes, lments,jt,mej t,Jmenm,nt
dff,ndiff)
:rite(8,2b) idiff
20 format(lh0,' maximum difference idiff=',i5)
call c:-tnn(nodes,lments,jt,memt,jmem.jntidiff,ndiff)
rite (8,29)
29 for at(1,,' results od nodes after renumbering')
30
write (8:. 30)
fors(1t0,9x, 'j ,2
nx=odes/5+1
,'jnt(j)')
do 200 j=1,nx
-100-
db 210
j=1,5
t(J+np)
]ntxtl)=nl
nD=nD-nx
2.0 cbnitnue
rte (8,31) (jx(jj),jn
200 continue
31 for at(Ih, 5(5x,215))
(jj) ,jj= ,5)
wrT:e (8,32)
do 300 j=,1x
db 310 lj=1,4
ix(i 1)ji +-P )
.2(]])=]t(+1D+600)
]t3(Q.)= t(3+j-+1200)
310 continue
vrite (8,33) (ix(jj),Jnt(jt1(jj)),jnt(jt2(jj)),jnt(jt3(jj)),
vjj=1,4)
300 continue
32 format(lbO,9x ',3x,'11',3x,'12',3x,'i3'/lhO)
33
format(lh,4(5x, i5s))
return
end
cccC
subroutine setup(nodes,lments,jt,memjt,jmem,jnt,idiff,ndiff)
cccc
dimension jt(2400),memjt(4800),Jmem(600),Jnt(600)
idiff=0
l,nodes
do 10
10 jmem(J =o
o 60 ~=1.1ments
do 50 1=1,4
nti =jt(0o,(1-t1)+*
if (nti.eq.0)
jsub=(Jnti-1)*8
go to 60
do 40 ii=1 ,4
if(ii.eq.i) go to 40
jit= t(o00o(Ii-)+J)
it (ijt.eq.0) go to 50
mem1 mem jnti)
if (meml.eq.0) go to 30
do 20 ii=1,meml
if( memjt(jsub+iii).eq.jjt) go to 40
20
continue
30
Jmem(inti)= mem( nti)+1
40
50
60
continue
continue
continue
memt jsub+ mem( nti))=It
.gt.idif) idiff=iabs(jnti-jjt)
if (labs(jnt -jjt
return
end
ccccC
subroutine optnum(nodes,lments,jt,memjt,=em,jnt,id!ff,ndilf)
ccccC
dimension jt(2400),memjt(4800),Jmem(600),jnt(600)
dimension newjt(600), joint(600)
njts=nodes
m nmax=idiff
do 60 ik=l,nodes
do 20 l1,nodes
20
joint( )=0
newjt (
=0
-101-
ax0=O
i=1
nevit(1)=ik
joit(ik)=1
k=1
k4=lmem(nejt(i))
if(k4.eq.0) go to 45
su= (ne'j t()-)*8
.k4
A=
co 40
k5=mem(n
.. ( oint
40
45
50
55
60
sub+
5) .g.
) go to 40
k=k-,.
ne~t(k)=k5
ioin (k5)=k
idi f=iabs (I-k)
if(ndiff.ge.minmax) go to 60
if(ndiff .gt. max) max=ndiff
ccntinue
fCI(k.eq.njts) go to 50
i=i+1
go to 30
minmax=max
do 55 j=1,modes
jnt(j) joint(j)
continue
nrdif=ndiff+l1
100
'rite(8,100) ndiff,nmdiff
format(bO, 'maximum difference after renumbering ndiff',5, 'newm
xbwb ', i5)
retrn
end
-102-
Appendix 3:
-103-
Massachusetts Bay Example
3.1
input for Massbay tidal case
3.2
3.3
input for Massbay steady current case
output for Massbay tidal case
3.4
output for Massbay steady current case
Appendix
-104-
3.1
M.ASzXAC
B-:-F3Vi0
224,
1
2,
3,
4,
5,
5,
7,
8,
9,
10,
11
12,
:3,
14,
15,
16,
17,
:8,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54,
55,
S6,
57,
58,
59,
60,
61,
62,
63,
140
-21654.0000,
-17685.0000,
-1958.0C0,
-13462.0000,
-10753.000.,
-14224.0000,
-14001.0000,
-18955.000,
-24415.0000,
-24829.0000,
-3CSS.0C3C00,
-5588.0000,
-7420.C0000,
-10160.CC0000,
-11462.CO00,
-14288.0000,
-17621.0000,
-23876.00C0,
-1365.0000,
1016.C0000,
-1334.0000,
-1715.0000,
-6255.0000,
-8604.0000,
-11239.0000,
-13335.0000,
-23304.0000,
-14446.0000,
3905.0000,
5556.0000,
10065.0000,
5334.0000,
2953.0000,
1588.0000,
-3016.0000,
-5874.0000,
-8541.0000,
-9366.0000,
-9906.0000,
8033.0000,
13018.0000,
17621.0000,
13335.0000,
9610.0000,
6255.0000,
1683.0000,
-2604.0000,
-5884.0000,
-6223.0000,
-6191.0000,
21971.0000,
16955.0000,
12922.0000,
7874.0000,
2477.CO30,
-1969.0000,
-3302.0000,
26384.0000,
22606.0000,
15675.0000,
9938.0000,
4191.0000,
-254.0000,
68040.C000,
70549.0000,
54643.0000,
76531.0000,
71215 .0000,
66326.0000,
61786.0000,
59531 .0000,
50833.0000,
65024.0000,
80677.0000,
74517.0000,
66834.0000,
62548.0000,
59055.0000,
57817.0000,
53023.0000,
55817.0000,
82804.0000,
75184.0000,
68358.0000,
62167.0000,
60738.0000,
59055.0000,
56198.0000,
53721.0000,
51435.0000,
49371.0000,
86265.0000,
80137.0000,
73660.0000,
68898.0000,
62421.0000,
55817.0000,
57245.0000,
57023.0000,
56293.0000,
52737.0000,
48419.0000,
90297.0000,
81820.0000,
73724.0000,
67247.0000,
61055.0000,
55372.0000,
50102.0000,
52292.0000,
52959.0000,
48768.0000,
44006.0000,
66104.0000,
59435.0000,
53435.0000,
47625.0000,
42228.0000,
48228.0000,
38068.0000,
58452.0000,
51562.0000,
45800.0000.
41656.0000,
35862.0000,
32226.0000,
-105-
5. 0000
5.0000
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5.0000
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34.0000
30 0000
15.0000
5. 0000
10.0000
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5.0000
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15.0000o
62. 0000
50. 0000
60.0000
40.0000
25.0000
30.0000
15.0000
10.0000
10.0000
20.0000
55.0000
65.0000
75.0000
76.0000
60.0000
45.0000
35.0000
23.0000
20.0000
15.0000
15.0000
75.0000
80.0000
55.0000
85.0000
71.0000
42.0000
30.0000
34.0000
25.0000
10.0000
50.0000
80.0000
78.0000
50.0000
25.0000
25.0000
10.0000
30.0000
85.0000
70.0000
37.0000
22.0000
10.0C00
64,
3,
73,
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75,
76,
739,
40,
81,
82,
83,
84,
85,
86,
827,
88,
89,
90,
91,
92,
93,
94,
95,
96,
97,
98,
99,
100,
101,
102,
103,
104,
105,
106,
107,
108,
109,
110,
111,
112,
113,
114,
115,
116,
117,
118,
119,
120,
121,
122,
123,
124,
125,
126,
129,
30480.0000,
51245.0000,
25226.0000,
18352.0000,
12446.0000,
7049.0000,
3239.0000,
34544.0000,
30798.0000,
24892.0000,
20257.0000,
14224.0000,
303. 0000,
5207.0000,
37433.0000,
33719.0000,
27496.0000,
20066.0000,
14732.0000,
10065.0000,
6096.0000,
1475.0000,
40799.0000,
36735.0000,
33941.0000,
26988.0000,
21146.0000,
16129.0000,
11970.0000,
8255.0000,
3250.0000,
5969.0000,
40545.0000,
44577.0000,
33150.0000,
36576.0000,
27877.0000,
23146.0000,
44037.0000,
40577.0000,
36195.0000,
17812.0000,
14542.0000,
10287.0000,
7938.0000,
39053.0000,
35433.0000,
29655.0000,
25718.0000,
20003.0000,
15399.0000,
11557.0000,
44037.0000,
45942.0000,
41815.0000,
36703.0000,
31845.0000,
26321.0000,
20013.0000,
14478.0000,
10795.0000,
48000.0000,
42736.0000,
38132.0000,
32925.0000,
27178.0000,
20479.0000,
16764.0000,
13399.0003,
11239.0000,
3150.0000,
25554.0000,
44323.0000,
37052.0000,
3639.0000,
33687.0000,
29749.0000,
24733.0000,
19653.0000,
39465.0000,
31052.0000,
29337.0000,
26289.0000,
22479. 0000,
18225.0000,
16764.0000,
16462.0000,
33814.0000,
27464.0000,
25527.0000,
23463.0000,
20574.0000,
16732.0000,
13367.0000,
14542.0000,
13012.0000,
13430.0000,
28067.0000,
27464.0000,
19431.0000,
22416.0000,
16510.0000,
14288.0000,
12509.0000,
9081.0000,
10097.0000,
12033.0000,
19558.0000,
13875.0000,
10573.0000,
6287.0000,
7366.0000,
3239.0000,
4953.0000,
23844.0000,
19050.0000,
14034.0000,
8954.0000,
4699.0000,
381.0000,
794.0000,
-1048.0000,
667.0000,
12954.0000,
8192.0000,
3520.0000,
-1016.0000,
-4699.0000,
-5017.0000,
-500.0000,
-2953.0000,
-106-
30.0000
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53.0000
48.0000
40.0000
30.0000
15.0000
5.0000
10.0000
30.0000
30.0000
46.0000
40.'0000
45.0000
43.0000
37.0000
25.0000
5.0000
5.0000
38.0000
35.0000
35.0000
32.0000
33.0000
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15.0000
27.0000
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29.0000
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5.0000
5.0000
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22.0000
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21. COC
20.0000
17.0000
5.0000
5.0000
130,
.31,
132,
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'34,
137,
:38,
:39,
140,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,.
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
38.
37,
38,
39,
40,
41,
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-107-
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'00
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'860
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'951
'961
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'861
'361
'16
'051
'681
'881
Appendix
-111-
3.2
MASEXDAC
B-02:F3V10
224, 140
1,
-21654.0000,
2,
-17685.0000,
3,
-19558.0000,
4,
-13462.0000,
5,
-10763.0000,
-14224.CO00,
6,
7,
-14001.0000,
8,
-18655.0000,
9,
-24416.0000,
10,
-24829.0000,
11,
-8055.0000,
12,
-5588.0000,
13,
-7420.0000,
14,
-10160.0000,
15,
-11462.0000,
16,
-14288.0000,
17,
-17621.0000,
18,
-23876.0000,
19,
-1365.0000,
20,
1016.0000,
21,
-1334.0000,
22,
-1715.0000,
23,
-6255.0000,
24,
-8604.0000,
25,
-11239.0000,
26,
-13335.0000,
27,
-23304.0000,
28,
-14446.0000,
29,
3905.0000,
30,
5556.0000,
31,
10065.0000,
32,
5334.0000,
33,
2953.0000,
34,
1588.0000,
35,
-3016.0000,
36,
-5874.0000,
37,
-8541.0000,
38,
-9366.0000,
39,
-9906.0000,
40,
8033.0000,
41,
13018.0000,
42,
17621.0000,
43,
13335.0000,
44,
9610.0000,
45,
6255.0000,
46,
1683.0000,
47,
-2604.0000,
48,
-5884.0000,
49,
-6223.0000,
50,
-6191.0000,
51,
21971.0000,
52,
16955.0000,
12922.0000,
53,
7874.0000,
54,
2477.0000,
55,
-1969.0000,
56,
57,
-3302.0000,
58,
26384.0000,
59,
22605.0000,
15875.C00,
60,
61,
938.0000,
4191.0000,
62,
63,
-254.0000,
68040.0000,
70549.0000,
64643.0000,
76581.0000,
71215.0000,
65326.0000,
61786.0000,
59531.00003,
60833.0000,
65024.0000,
80677.0000,
74517.0000,
66834.0000,
62548.0000,
59055.0000,
57817.0000,
53023.0000,
55817.0000,
82804.0000,
75184.0000,
68358.0000,
62167.0000,
60738.0000,
59055.0000,
56198.0000,
53721.0000,
51435.0000,
49371.0000,
86265.0000,
80137.0000,
73660.0000,
68898.0000,
62421.0000,
55817.0000,
57245.0000,
57023.0000,
56293.0000,
52737.0000,
48419.0000,
90297.0000,
81820.0000,
73724.0030,
67247.0O00,
61055.0000,
55372:0000,
50102.0000,
52292.0000,
52959.0000,
48768.0000,
44006.0000,
66104.0000,
59436.0000,
53435.0000,
47625.C000,
42228.0000,
46228.0000,
38058. 000,
'58452.0000,
51562.000,
46800. 0000,
41655.0003,
35862.0000,
32226.0:03,
-112-
S.0000
5.0000
10.0000
5.0000
34.0000
34. 000
30.0000
15.0000
5.000
5.C000
13 .C00
45 .0000
40.0000
30.0000
25.0000
25.0000
5.0000
15.0000
15.0000
62.0000
50.0000
60.0000
40.0000
25.0000
30.0000
15.0000
10.0000
10.0000
20.0000
55.0000
65.0000
75.0000
76.0000
60.0000
45.0000
35.0000
23. 0000
20.0000
15.0000
15.0000
75.0000
80 .003
55. 0000
85.0000
71.0000
42.0000
30.0000
34.0000
25.0000
10.0000
50.0000
80.0000
78.0000
50.0000
25.0000
25.0000
10.0000
30.0000
85.0000
70.0000
37.0000
22.0000
0.o0000
64,
65,
65,
67,
68,
69,
70,
7:1,
72,
73,
74,
75,
76,
77,
78,
79,
80,
81,
82,
83,
84,
85,
85,
87,
88,
89,
90,
91,
92,
93,
94,
95,
96,
97,
98,
99,
100,
101,
102,
103,
104,
105,
106,
107,
108,
109,
110,
111,
112,
113,
114,
115,
116,
117,
118,
119.
120,
121,
122,
123,
124,
125,
128,
127,
128,
129,
30480.0000,
25225.0000,
18352.0000,
12446.0000,
7049.0OCOO,
3239.0000,
34544.0000,
30798.0000,
24892.0000,
20257.0000,
14224.0000,
9303. 000,
5207.0000,
37433.0000,
33719.0000,
27496.0000,
20066.0000,
14732.0000,
10065.0000,
6096.0000,
1475.0000,
40799.0000,
36735.0000,
33941.0000,
26988.0000,
21146.0000,
16129.0000,
11970.0000,
8255.0000,
3250.0000,
5969.0000,
40545.0000,
44577.0000,
33150.0000,
36576.0000,
27877.0000,
23146.0000,
17812.0000,
14542.0000,
10287.0000,
7938.0000,
39053.0000,
35433.0000,
29655.0000,
25718.0000,
20003.0000,
15399.0000,
11557.0000,
44037.0000,
45942.0000,
41815.0000,
36703.0000,
31845.0000,
26321.0000,
20013.0000,
14478.0000,
10795.0000,
48000.0000,
42736.0000,
38132.0000,
32925.0000,
27178.0000,
20479.0000,
16764.0000,
13399.0000,
11239.0000,
51245 .0000,
44037.0000,
40577.0000,
36195.0000,
31560.0000,
25654.0000,
44323.0000,
37052.0000,
36639.C0000,
33587.00 0,
29749.0000,
247:3.0000,
19653.0000,
39465.0000,
31052.0000,
29337.0000,
26289.0000,
22479.0000,
18225.0000,
16764.0000,
16462.0000,
33814.0000,
27464.0000,
25527.0000,
23463.0000,
20574.0000,
16732.0000,
13367.0000,
14542.0000,
13012.0000,
13430.0000,
28067.0000,
27464.0000,
19431.0000,
22416.0000,
16510.0000,
14288.0000,
12509.0000,
9081.0000,
10097.0000,
12033.0000,
19558.0000,
13875.0000,
10573.0000,
6287.0000,
7366.0000,
3239.0000,
4953.0000,
23844.0000,
19050.0000,
14034.0000,
8954.0000,
4699.0000,
381.0000,
794.0000,
-1048.0000,
667.0000,
12954.0000,
8192.0000,
3620.0000,
-1016.0000,
-4699.0000,
-5017.0000,
-3588.0000,
-5080.0000,
-2953.0000,
-113-
30.0000
50. 00C
62.0000
50.0000o
25.0000
10.0000
22.0000
21.0000
50.0000
58.0000
47. CC0000
33.0000
5.0000
32.0000
60.0000
60.0000
55.0030
40.0000
30.0000
10.0000
5.0000
55.0000
30.0000
55.0000
53.0000
48.0000
40.0000
30.0000
15.0000
5.0000
10.0000
30.0000
30.0000
46.0000
40.0000
45.0000
43.0000
37.0000
25.0000
5.0000
5.0000
38.0000
35.0000
35.0000
32.0000
33.0000
22.0000
5.0000
20.0000
15.0000
27.0000
31.0000
29.0000
20.0000
25.OoCO
15.0000
5.0000
5.0000
18.0000
22.0000
22.0000
21 .COCO
20.0000
17. 0000
5.0000
5.0000
130,
131,
132,
133,
134,
:38,
'39,
140,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54,
55,
49086.0000,
44154.0000,
39592.0000,
33909.0000,
27432.0000,
21939.0000,
15939.0000,
49879.0000,
44926.0000,
39656.0000,
33560. 0000,
29,
40,
29,
30,
29,
19,
20,
19,
12,
19,
19,
11,
11,
4,
4,
5,
2,
4,
6,
2,
2,
3,
1,
2,
41,
31,
41,
30,
20,
30,
32,
20,
21,
20,
12,
20,
13,
12,
5,
12,
6,
5,
14,
6,
6,
7,
3,
7,
3,
8,
3,
9,
10,
3,
43,
42,
43,
31,
32,
31,
44,
32,
32,
33,
21,
32,
22,
21,
13,
21,
23,
13,
14,
13,
24,
14,
15,
14,
14,
7,
16,
7,
7,
8,
8,
17,
8,
18,
9,
8,
18,
27,
43,
51,
43,
44,
44,
53,
44,
45,
44,
33,
33,
34,
34,
22,
22,
35,
23,
22,
41,
6921.0000,
2159.0000,
-2032.0000,
-5429.0000,
10160.0000,
-9335 .0000,
-6985.0000,
-762.OCO0,
-5525 .000,
-8445.0000,
10668.0000.
5.0000
5.0000
10.0000
15.0000
5 0000
10.0000
5.0000
5.0000
5.0000
5.0300
5.0000
0.005000 5 .I"I0 00
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0. 005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0 .005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0. 005000
0.005000
0.005000
0.005000
0. OSC
0.005000
0 005000
0.005000
41,
30,
30,
20,
12,
12,
12,
5,
5,
6,
3,
42,
31,
31,
31,
32,
21,
21,
13,
13,
13,
14,
6,
7,
8,
9,
51,
42,
43,
43,
44,
33,
33,
22,
22,
23,
23,
24,
15,
15,
16,
16,
17,
18,
17,
52,
52,
52,
53,
45,
45,
33,
34,
35,
-114-
55,
57,
58,
59,
60,
51,
52,
53,
54,
65,
65,
57,
68,
69,
70,
71,
72,
73,
74,
75,
76,
77,
78,
79,
80,
81,
82,
83,
84,
85,
86,
87,
88,
89,
90,
91,
92,
93,
94,
95,
96,
97,
98,
99,
100,
101,
102,
103,
104,
105,
106,
107,
108,
109,
110,
111,
112,
113,
114,
115,
116,
117,
1-18,
119,
120,
121,
23,
23,
24,
24,
24,
15,
16,
15,
51,
52,
52,
53,
45,
34,
34,
35,
35,
36,
37,
37,
25,
25,
26,
26,
58,
59,
59,
60,
53,
53,
60,
60,
54,
46,
46,
47,
47,
48,
48,
38,
38,
64,
65,
70,
65,
65,
66,
66,
61,
61,
55,
55,
56,
56,
49,
39,
77,
77,
71,
71,
72,
73,
73,
73,
67,
67,
36,
24,
37,
25,
15,
16,
26,
17,
52,
59,
53,
45,
46,
46,
47,
47,
48,
48,
48,
38,
38,
26,
28,
17,
59,
65,
60,
66,
60,
54,
61,
54,
55,
55,
56,
56,
49,
49,
38,
39,
28,
65,
71,
71,
72,
66,
73,
67,
67,
62,
62,
57,
57,
50,
50,
50,
78,
71,
79,
72,
73,
80,
74,
67,
68,
62,
0.005000
35,
35,
36,
37,
25,
25,
25,
26,
53,
58,
59,
54,
54,
45,
46,
34,
47,
35,
36,
48,
37,
38,
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28,
64,
64,
65,
65,
59,
60,
66,
61,
61,
54,
55,
46,
56,
47,
49,
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39,
70,
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77,
71,
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72,
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66,
67,
61,
62,
55,
57,
56,
49,
85,
78,
78,
79,
79,
79,
80,
74,
74,
68,
O. 05s00
0CO0500C
0.005000
0.005000
0.005000
0.005000
0.005300
0.03500
0.00500
0.005300
0.0050330
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0..0.CSO
0.005000
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0.005000
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0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
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0.005000
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0.005000
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0.005000
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0.005300
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
0.005000
-115-
-9TT11-
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-1.83382
Appendix
-131-
3.4
TEA
DEPT OF CIVIL ENGINm?.RNG, M. I. T.
2-D L NEA FINITE ELfEENT FRE-UEN-CY DC.AIN ANALYSIS OF TIDAL vAVELS FOR
?.SSGRA.
SMALL SCALE G-.MTRY.
:s"ER 10 DIGIT F.E. GRID GE-OMETRY IDENTIFICATION
NT
-ER 10 DIGIT I.D. CODE FOR B.C. VERSION
I.\P'JT Nr!BR OF ELEMENTS AND h'.3XER OF NODE POINTS
IN? T THE NODE NO.,X-COORD. ,Y-CCCRD., AND NODAL M.S.L. DEPTH. ONE NZDE/L.NE
IN?'T ELEM-ET N%"ER. TH?.EE NODE N.UMBERS FOR THE EL.-TAND T-:E ELEMENT LIEAP.
.AX. NODAL POINT DIFFERENCE z
a
FRICTION FACTOR
3
NPJUT FREUENCY IN R.ADIANS/SEC.
INPUT CORIOLIS PARAMETER
IhPUT AMPLITUDE OF VIND SPEED. VIND PHASE SHIFT, ANGLE OF 'IND DIRECTION
(IN DEOREZS)
AND WIND DRAG COEFFICIENT
INPUT HOW MANY SOUNDARY SEGENTS HAVE NON-ZERO NORMAL FLUX PRESCRIBED
IN.UT
INP.UT
IN,.T
INPUT
INPUT
INPUT
INhU T
INPUT
INfPUT
INPUT
INPUT
NO.
THE
THE
THE
THE
THE
THE
THE
THIE
THE
THE
OF NODES WHERE ELEVATION IS PRESCRIBED
NODE NUMBER AND VALUES FOR THE MODULUS
NODE NU'SmER AND VALUES FOR THE MODULUS
NODE NUMBER AND VALUES FOR THE M7ODULUS
NODE NUMBER AND VALUES FOR THE MODULUS
NODE NUMBER AND VALUES FOR THE MODULUS
NODE NUMBER AND VALUES FOR THE MODULUS
NODE NUMBER AND VALUES FOR THE MODULUS
NODE NUMBER AND VALUES FOR THE MODULUS
NODE NUMBER AND VALUES FOR THE MODULUS
NODE NUMBER AND VALUES FOR THE MODULUS
AND
AND
AND
AND
AND
AND
AND
AND
AND
AND
PHASE IN AD) OF THE PR.ESCRIBED ELEVATION AT
PHASE IN RAD) OF THE PRESCRIBED ELEVATION AT
PHASE IN AD) OF THE P.ESCRIBED ELEVATION AT
PHASE (N AD) OF THE PRESCRIBED ELEVATION AT
PHASE IN RAD) OF THE PRESCRIBED ELEVATION AT
PHASE IN RADI OF THE PRESCRIBED ELEVATION AT
PHASE IN RAD OF THE PRESCRIBED ELEVATION AT
PHASE IN RAD 0 THE PRESCRIBED ELEVATION AT
PHASE IN RAD OF THE PRESCRIBED ELEVATION AT
PHASE IN RAD OF THE PRESCRIBED ELEVATION AT
PROGRAM TEA
DEPT OF CIVIL ENGINEERING. N. I. T.
2-D LINEAR FINITE ELDE=T FRE UENCY DOMAIN ANALYSIS OF TIDAL WAVES FOR S
I
F.E. GRID GEOMETRY IDENTIFICATION : MASEXAC
BCU''DARY CONDITION IDENTIFICATION : B-02:F3VI
FREUEN..Y 2 0.00000000
CCRIOLIS PARAMETER
0.0001000
VIND
VIND
W:
V
W:%'D
SPEED AT 10 METERS ABOVE SURFACE a
PHASE SHIFT u
0.00000 RADIANS
DIRECTION *
0.00000 DEGREES
DRAG CCFCIENT a
0.00000000
NO. OF NODE POINTS
0.0000
=140
-132-
SCALE GEOMETRY.
.ALL
NODE
NODE
NODE
NKDE
NODE
NODE
NODE
NODE
NODE
NODE
NO.
-ELCMNTS =224
CDAL PS'NT DI- -C.*EC =
. ,C
13
'******
NODE
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
38
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
55
D=.."*s'*'
lE?T.HS
NGCAL
Lc=,'NA="A
COCRDINATES ANS
X-COORD.
-21654.00
-:75685.00
-19558.00
-13452.00
-10763 00
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-:8955.00
-24416.00
-24829.00
-8065.00
-5588.00
-7420 00
-10160.00
-11462.00
-14288.00
-17821.00
-23876.00
-1365.00
1016.00
-1334.00
-1715.00
-6255.00
-8604.00
-11239.00
-13335.00
-23304.00
-14446.00
3905.00
5556.00
10065.00
6334.00
2953.00
1688.00
-3016.00
-5874.00
-8541.00
-9366.00
-9206.00
8033.00
13018.00
17621.00
13335.00
9510.00
6255.00
1683.00
-2604.00
-5884.00
-6223.00
-6:91.00
21971.00
16955.00
12922.00
7874.00
2477 00
-:989.00
-133-
= '*''*'
Y-CC0RD.
DETH
6B040 00
7C549.C0
64843.00
75581.c0
71215.00
65326.00
6:785.00
59531.00
60833.00
65024.00
80677.00
74517.00
66534.00
62548.00
59055.00
57817.00
53:23.00
55817.00
82804.00
75184.00
68358.00
62167.00
6:738.00"
59055.00
56198.00
53721.00
51435.00
49371.00
86266.00
80137.00
73680.00
68898.00
62421.00
55817.00
57245.00
57023.00
56223.00
52737.00
48419.00
90297.00
81820.00
73724.00
67247.00
61055.00
55372.00
50102.00
52292.00
52959.00
487C8.00
44006.00
66104.00
59436.00
53435.00
47625.00
42228 0C
4e^28.00
*
5 00
5 00
10.00
5 00
34.00
34.00
30.00
15.00
5 00
5.00
10 00
46.00
4: 00
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25.00
25.00
5.00
15.00
15.00
62.00
50.00
60.00
43.00
25.00
30.00
15.00
10.00
10 00
20.00
55.00
65.00
75.00
76.00
60.00
45.00
35.00
23.00
20.00
15.00
15.00
75.00
80.00
55.00
85.00
71.00
42.00
30.00
34.00
25.00
10.00
50.00
80 00
78 00
50.00
25 00
25 00
57
58
59
60
61
62
63
64
55
66
57
53
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
-3302.00
25384.00
22508.00
15875 00
9933.00
4;91.00
-254.00
30480.00
25226.CO
18352 00
12446.00
7049.00
3239.00
34544.00
30798.00
24892.00
20257.00
14224.00
9303.00
5207.00
37433.00
337:9.00
27495.00
20066.00
14732.00
10065.00
6096.00
1475.00
40799.00
36735.00
33941.00
26988.00
21146.00
16129.00
11970.00
8255.00
3250.00
6969.00
40545.00
44577.00
33150.00
35576.00
27877.00
23146.00
17812.00
14542.00
10287.00
7938.00
39053.00
35433.00
29655.00
25718.00
20003 00
15399.00
11557.00
3£2C8.00
52452.00
5:562 00
4600.00
4:65.00
35:2.00
3-225.00
5:245.00
00
44^37
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35195 00
3:560.00
2554.00
44323.00
37052.00
35339.00
33587 00
29749 00
24733.00
19553.00
39455.00
31052.00
23337.00
25289.00
22479.00
18225.00
16764.00
16462.00
33814.00
27454.00
25527.00
23463.00
20574.00
16732.00
13367.00
14542.00
13012.00
13430.00'
28067.00
27454.00
19431.00
22416.00
16510.00
14288.00
12509.00
9081.00
1C097.00
12033.00
19558.00
13575.00
10573.00
6287.00
7366.00
3239.00
4;53.00
10.00
30 00
85.00
70.00
37.00
22.00
10.00
30.00
50 00
62.00
53 00
25.00
10.00
22.00
2: 00
50.00
58.00
47.00
33.00
5.00
32.00
60.00
60.00
55.00
40.00
30.00
10.00
5.00
55.00
30.00
55.00
53.00
48.00
40.00
30.00
15.00
5.00
10.00
30.00
30.00
46.00
40.00
45.00
43.00
37.00
25.00
5.00
5.00
38.00
35.00
35.00
32.00
33.00
22 00
5.00
112
44037.00
23844.00
20.00
113
114
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41815 00
35703.00
19050.00
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29.00
26321 00
20013.00
14478.00
10795.00
48000.00
42735.00
381.00
794.00
-1049.00
657 00
12954.00
8192.00
20.00
25.00
15 00
5.00
5 00
18.00
117
118
119
120
121
122
-134-
:23
124
125
126
127
128
129
:30
131
:32
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134
135
136
:37
138
139
140
6921.00
2159.00
-2032.00
-5429 C0
-10160.00
-9335.00
-5985.00
-762.00
-5525.00
-8446.00
-10668.00
*s****
-LESNT
AP.RAY
a.
I
I.E--NT
J
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
40
29
29
19
19
19
11
4
4
2
2
2
41
41
30
20
20
20
12
12
5
6
6
3
3
3
3
42
31
31
32
32
32
21
21
13
13
:4
-135-
29
30
19
20
12
11
4
5
2
6
3
1
31
30
20
32
21
12
13
6
6
14
7
7
8
9
10
43
43
32
44
33
21
22
13
23
14
24
22.00
22.00
21.00
20.00
17.C0
5.00
5.00
5 00
5.CO
10.00
15.00
5.00
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27432.00
21939.00
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49879.00
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39656.00
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41
41
30
30
20
12
12
12
6
5
6
3
42
31
31
31
32
21
21
13
13
13
14
8
7
8
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51
42
43
43
44
33
33
22
22
23
23
a********=***u***********s
'ELEMENFP.CTIrON FACTOR
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0.0050000
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39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
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80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
1C3
104
14
14
7
7
8
8
8
18
51
43
44
44
44
33
22
22
22
23
23
24
24
24
15
16
16
51
52
52
53
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34
34
35
35
38
37
37
25
25
26
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58
59
59
60
53
53
60
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54
46
46
47
47
48
48
38
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64
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65
65
66
66
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15
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17
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16
16
18
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27
43
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53
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33
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23
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BOUhDARY CONDITIONS
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RESULTS CF COM UTATIONS
PHASE
0.00000
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0.00000
0.00000
0.00000
0.00000
0.0000
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* ******SS***S **
NCDAL ELEVATIONS
NODE
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
MODULUS
0.04239742
0.04146851
0.03871197
0.04140596
0.03409668
0.03367290
0.03430751
0.03695301
0.04199020
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0.04012124
0.03313332
0.03328799
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0.03938028
0.03370254
0.03071956
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0.03291786
-139-
PHASE
0.00000
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