Titles should never contain abbreviations, e

1170
Mazzolari, et al.
A multi-criteria meshing method applied to a shallow water model
Andrea Mazzolari, António Trigo-Teixeira and Maria Amélia V.C. Araújo
CEHIDRO, Instituto Superior Técnico, Technical University
of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal.
[email protected]
[email protected]
[email protected]
www.cerf-jcr.org
ABSTRACT
www.JCRonline.org
Mazzolari, A., Trigo-Teixeira, A., and Araújo, M.A.V.C., 2013. A multi-criteria meshing method applied to a shallow
water model. In: Conley, D.C., Masselink, G., Russell, P.E. and O’Hare, T.J. (eds.), Proceedings 12th International
Coastal Symposium (Plymouth, England), Journal of Coastal Research, Special Issue No. 65, pp. 1170-1175, ISSN
0749-0208.
In this study a multi-criteria meshing method for a 2D shallow water model is tested for a domain encompassing several
spatial and hydrodynamic scales, from the Western Atlantic Ocean shelf in front of the Iberian Peninsula to the Lima
River (Portugal) and its estuary. For the mesh design, a series of scalar meshing criteria are identified from known
physical factors of the relevant processes being studied, such as the bathymetry, the topographical length scales and the
expected characteristics of the flow motion. Computational constraints are applied as well. Each criterion is expressed
in terms of a node spacing function. The final node spacing function driving the meshing process is obtained by
merging the requirements of the previously defined criteria. An advancing front mesh generator performs the
discretization in a single, unstructured and well graded mesh, having element side lengths varying over three orders of
magnitude (from tens of km to meters). A shallow water model application tests the generated mesh, where tide
elevations and phases are imposed at the ocean boundary and freshwater inflow at the upstream boundary. The model
validation shows a general good agreement between computed and observed water elevations, currents and the
asymmetry of the tidal signal, with an underestimation of the high water levels for the most upstream station. The wet
and dry process is reproduced realistically, as the position of the wetted front matches closely the submerged areas of
time-referenced aerial images.
ADDITIONAL INDEX WORDS: Unstructured meshes, finite elements, advancing front, node spacing function, wet
and dry, Lima estuary.
INTRODUCTION
Shallow water models are often required to simulate processes
extending over several spatial and hydrodynamic scales, from the
wide ocean to coastal bays, estuaries and low lying areas exposed
to inundation. Unstructured meshes are appropriate tools for
meeting the resolution requirements of these models, where a
coarse discretization in deep waters coexists with localized
refinements wherever the involved phenomena present high
gradients, as in the case of physical forcings, steep bottom features
and detailed coastlines.
A problem often encountered in published ocean and coastal
numerical studies is the omission of the description of the mesh
generation process and of the used sizing functions, leaving the
reader uncertain about the efficiency of the proposed
discretization. However, criteria for shallow water models are well
documented and show recurrent formulations: they are often
related to the tidal wavelength (Westerink et al., 1992) the
bathymetry gradient (Bilgili et al., 2005), the coastline resolution
(Westerink et al., 1994; Blain et al., 1998; Jones and Davies,
2007), storm forcings for deep and shallow waters (Blain et al.,
1998), hydrodynamic length scales (Legrand et al., 2006), waves
coupled with meteorological tides (Araújo et al., 2011), and
localized bathymetry and altimetry features: Jones and Richards
(1992) propose the manual insertion of new nodes and the use of
diagonal swapping for the inclusion of hydraulic structures in the
____________________
DOI: 10.2112/SI65-198.1 received 07 December 2012; accepted 06
March 2013.
© Coastal Education & Research Foundation 2013
mesh. In coastal inundation studies, given a high resolution Digital
Elevation Model (DEM) which provides water and land coverage,
the proper reproduction of small scale features likely to influence
the hydrodynamics, such as channels, levees, roads, railways,
buildings, etc., is recommended (Shen et al., 2006; Blanton and
Luettich 2008; Westerink et al., 2008;). Xu et al. (2010) use a
mixed triangular and quadrangular mesh where elements covering
the low lying coastal areas coincide with the DEM grid cells. In
order to improve the representation of the modeled flood event,
specific algorithms extract significant points from a Lidar dataset
for the identification of vegetation and man-made features (Cobby
et al., 2003; Bates et al., 2003).
Alternatively, a useful tool for addressing the control of the
element characteristics in a mesh generation process is the Node
Spacing Function (NSF), which specifies how the edge length,
shape and orientation of the target element vary throughout the
domain (Frey, 1987; Lee and Hobbs, 1999). Isotropic meshes
derive from scalar NSFs, while anisotropic meshes are defined by
means of vector NSFs, where the element skewness and the
direction of elongation must also be specified. The function may
be determined from a-priori or a-posteriori criteria. A-priori
criteria follow from any field variable known prior to the
simulation. A-posteriori criteria are used in adaptive meshing
techniques (Hagen et al., 2006) where a “first try” mesh is
modified in order to minimize an error estimator of the flow
problem solution or of the governing equations.
The main aim of this paper is the definition of a multi-criteria
approach to the task of mesh generation for ocean and coastal
Journal of Coastal Research, Special Issue No. 65, 2013
A multi-criteria meshing method applied to a shallow water model
1171
domain is extended upstream until Ponte de Lima, at 24 km
distance from the river mouth, beyond the influence of the tide.
The location of the ocean boundary is similar to a previous
modeling study (Araújo et al., 2011) which gave satisfactory
results in reproducing the tide levels at the gauge in the harbor of
Viana do Castelo.
The DEM has been obtained from different data sources: for the
ocean waters, depths up to 4000 m have been digitized from the 1:
150 000 and 1: 1 000 000 nautical Portuguese charts, while deeper
bathymetry is provided by the 1’ grid resolution Global seafloor
database from the Institute of Geography and Planetary Physics
(Smith and Sandwell, 1997). The estuary bathymetry is made up
by a series of field surveys, specifically a 5 m resolution regular
grid for the lower estuary, a 25 m x 10 m resolution scatterset for
the middle estuary and a series of cross-sections for the upper
estuary and the narrower parts of the Lima river. The flood plain
altimetry was digitized from the 1: 25 000 topography map of the
Portuguese Army Geographical Institute. The merging of the
mentioned scatter sets evidenced the presence of not surveyed
areas in the intertidal region, which were object of a new survey in
2010. The final bathymetry is obtained in the form of a TIN.
Figure 1.(a) Ocean domain extension and bathymetry; (b) Lima
estuary extension and bathymetry (negative depths correspond
to points above mean sea level) and (c) central part of the
estuary, with the urban area of Viana do Castelo and the position
of the Eiffel bridge. The main and secondary river channels are
visible. Numbers correspond to the deployment sites of the 4
ADCPs.
shallow water models. The method is applied to a real domain
including part of the North Atlantic Ocean, the Western
continental shelf and shelf break of the Iberian Peninsula, and the
Lima estuary (Figure 1). The estuary has been the object of several
studies, with a focus on salinity structure and salt water intrusion
(Pinho and Vieira, 2007), or on the pollutant tracking in the lower
estuary (Vale and Dias, 2011). Rebordão and Trigo-Teixeira
(2009) performed a wet and dry simulation for a mesh of constant
element size inside the estuary; however, no quantitative results of
the flooding process are reported. None of the mentioned studies
includes far field ocean scales. In the present work the wide
number of spatial and hydrodynamic scales involved is properly
resolved by considering a series of criteria, each of them
expressed as a NSF. The superimposition of the criteria produces a
target piecewise linear NSF driving the discretization process. The
resulting mesh has been tested with the ADCIRC shallow water
model for a wet and dry study forced by astronomical tides.
METHODS
Domain and bathymetry
The domain extends from the North Atlantic Ocean to the coast
of the Western Iberian Peninsula, including the estuary of the
Lima River (Figure 1). This estuary has drawn our attention as it
represents a challenging task in terms of spatial discretization: the
lower part has a channel-like shape with the presence of a harbor.
The central part presents a complex morphology with a network of
channels and sedimentary banks covered by brackish vegetation,
exposed to the covering and uncovering of the tide. The river
floodplains adjacent to the river course are included as well up to
the 5 m bathymetric contour above Mean Sea Level (MSL), in
order to assure a buffer zone between the land boundary and the
high water spring level. The tides are semidiurnal, with a range
varying from 1.1 m on neap tides to 3.7 m on spring tides. The
Meshing criteria and mesh generation
The criteria used for the domain discretization are presented
hereafter. Each of them is related to a NSF expressed in terms of
target node element size Δx(x,y). One of the most recurrent NSF
for the ocean and continental shelf is the wavelength (λ) to grid
size (Δx) ratio (Le Provost and Vincent, 1986):
x1 ( x, y) 
 ( x, y)
r

T gh( x, y)
r
(1)
where T is the tide period, h(x,y) the water depth and r a constant.
Mesh convergence studies (Westerink et al., 1994; Luettich and
Westerink, 1995) however pointed out several shortcomings of
Eq.(1), as the inadequate resolution of steep slope features, or the
impossibility to adapt the mesh to two dimensional flow patterns
as amphidromes. Some of these limitations may be overcome by
the topographic length scale Lh (Legrand et al., 2007):
x2 ( x, y)  c1Lh ( x, y)  c1 h( x, y) h( x, y)
(2)
where h is the bathymetry gradient and c1 a coefficient of
proportionality. Eq.(2) may be combined with Eq.(1) for
increasing the mesh refinement in areas of steep bathymetry.
In near shore and estuarine regions the element size should not
only reflect the decrease in the wavelength but also be related to
the local bottom features, as the flow interaction with the
bathymetry is enhanced. The criteria proposed in open regions are
not valid anymore and must be integrated or substituted with local
ones, taking into account specific constraints. The first criterion
applied to the estuary relates the mesh resolution to the module of
the inverse of the bathymetry gradient:
x3 ( x, y)  1 h( x, y)
(3)
A lower bound of 15 m has been applied to avoid the generation
of very small elements when the local gradient is steep. According
to Eq.(3) the most refined estuarine areas are expected to occur
along the river banks or the slopes of the emerged bedforms, in
order to reproduce closely the bathymetry isolines and represent
the temporal evolution of the wetted front. The following
criterion, Δx4, addresses the resolution of the horizontal length
scales of the estuary, represented by the width of the harbor
navigation channel, the river main and secondary channels, the
river banks and adjacent floodplains. A minimum number of
Journal of Coastal Research, Special Issue No. 65, 2013
1172
Mazzolari, et al.
nodes per cross-section must be assured, as underresolution may
significantly decrease the system conveyance (Westerink et al.,
2008). Therefore the highest refinement has been attributed to the
navigation and main river channels (Δx4 = 20 m); this level of
refinement has been relaxed for the river secondary channels (Δx4
from 30 to 35 m); the refinement for banks and marshes varies
from 30 to 60 m accordingly to the distance from the river
longitudinal axis. A radial function, Δx5, has been applied from
the river mouth towards the ocean, up to a distance of 2 km, for
assuring a smooth size grading between the NSFs defined in the
estuary and in the ocean:
x5 ( x, y)  c2 ( x  xc ) 2  ( y  yc ) 2
(4)
with c2 being a coefficient of proportionality, and (xc,yc).the
coordinates of the river mouth.
For ADCIRC, the selected shallow water model, applications
with the wet and dry option should apply a severe Courant
condition (Blain et al., 2010). The related NSF can be expressed
as:
(5)
x6 ( x, y)  VMAX t 0.1
where VMAX is the module of the maximum velocity, calculated
for the same domain extension and input conditions of the present
case, at the nodes of a “first try” mesh, and Δt the computational
time step. The final NSF, to be read in the meshing algorithm
through a background unstructured mesh, is calculated as:

x f ( x, y)  max minxi ( x, y) 5 ; x6 ( x, y)
i 1

(6)
where Δx1 to Δx5 operate as a minimization condition on the final
target size function, while Δx6 sets a minimum size the mesh
elements can have.
The mesh generator discretizes the domain by an advancing front
method algorithm. Laplacian smoothing is applied for assuring
better area size transitions between adjacent elements. The
obtained mesh (Figure 2) has 63 516 nodes and 124 035 elements.
The element size varies throughout the domain from 15 km in
deep waters to 20 m inside the estuary. Areas of refinement in the
open ocean are observed in correspondence of seamounts and
along the shelf break and continental slope. Inside the estuary the
highest level of refinement is attained along the main river
channel, with spot-like node concentrations due to elevated local
bathymetry gradients. Figure 3 shows the distribution of the
element shape quality index Qi, defined as:
Figure 3. Distribution of the element quality index Qi.
Qi 
4 3 Ai
(7)
 bi2  ci2 )
where Ai is the element area, ai, bi and ci the triangle side lengths.
For an equilateral triangle Qi is equal to 1. The high number of
elements with a Qi almost attaining 1 proves the good quality of
the mesh.
(ai2
Model description
ADCIRC-2DDI is the two dimensional version of the finite
element ADvanced CIRCulation coastal ocean model (Luettich et
al., 1992), which solves the shallow water depth integrated
barotropic mass and momentum equations under the hypothesis of
incompressibility, Boussinesq and hydrostatic pressure. The
governing equations, where the free surface stresses and
atmospheric pressure gradients are neglected as not considered in
this study, are (Westerink et al., 2008):
  UH VH cos   




t  

R cos    0
(8)
Figure 2. Unstructured graded mesh derived from the node spacing function Δxf. Localized refinement is seen (a) in correspondence of
seamounts and of the shelf break, and, in a closer estuary view of the estuary (b), along the main river channel and where the
bathymetry gradient is pronounced. Numbers refer to the ADCPs locations.
Journal of Coastal Research, Special Issue No. 65, 2013
A multi-criteria meshing method applied to a shallow water model
1173
Figure 4. Comparison of the observed and computed water elevations at site 1 and site 4.
U
U
U V U  U tan 




 f V 
t R cos   R   R

g

 t   UH UH 

(   ) 

  *U
R cos  
H   
 
(9)
V
U
V V V  U tan 




 f U 
t R cos   R   R

(10)
g 
 t   VH VH 

(   ) 

  *V
R 
H   
 
where ζ is the free surface elevation relative to the geoid, H=ζ+h
the total water depth, with h being the bathymetric depth relative
to the geoid, λ and Φ the latitude and longitude coordinates, U and
V the depth averaged horizontal velocities, R the Earth radius, g
the gravity acceleration, f the Coriolis parameter, η the Newtonian
equilibrium tidal potential, α the effective earth elasticity factor, τ*
the bottom stress, dependent on the averaged velocity by a
quadratic formulation, and νt the horizontal eddy viscosity.
ADCIRC provides a wetting and drying option tested for a
number of idealized and real geometries (Blain et al., 2010).
RESULTS
A simulation of the astronomical tide is carried out in order to
assess the model capacity to reproduce, for the given domain
discretization, the hydrodynamics of the estuary. Freshwater
inflow is specified at the upstream river section, with daily
averaged flow rate values registered at Ponte da Barca, which is
the nearest hydrographic station with available data for the
analyzed period, located 17 km far from the domain upstream
boundary. The river record is made available by the Portuguese
Hydraulic Resources National Information System. Consequently
to the presence of the upstream dams of Alto Lindoso and
Touvedo, the river regime in normal conditions corresponds to the
minimum environmental flow rate. The flow rate for the present
simulation time varies between 2.73 and 15.15 m³/s. The ocean
boundary nodes have been forced with amplitude and phase values
of 13 astronomy constituents (K1, K2, L2, M2, MU2, N2, NU2,
2N2, O1, P1, Q1, S2 and T2), extracted from the Le Provost
global tide dataset (Le Provost et al., 1998). In addition, the tide
potential functions of the mentioned constituents have been
applied to the mesh interior nodes.
Free surface
The modeled tides are compared with measurements of a field
campaign which took place between the 5th and the 14th October
2006, providing time series of water elevation and velocities of 4
bottom anchored ADCPs, deployed along the main river channel
(Figures 1b and 1c): sites 1 and 2 were located respectively
downstream and upstream of the Eiffel Bridge; site 3 in the middle
estuary and site 4 along the river course, in an area still strongly
affected by the tide propagation. The free surface levels were
corrected from the influence of atmospheric pressure variations.
The ADCP at site 2 was re-deployed after a local fisherman
accidentally collected it: the measures of this instrument, being
potentially subject to errors, are not used for the validation. The
model shows a good reproduction of the tide range and phases, for
all the reference sites (Figure 4 however depicts only the results
for the most downstream and upstream sites). In particular, the
model reproduces the asymmetry of the tide curve registered at
site 4, with a steeper rising curve and a delayed falling water
profile. Also the damping of the tide range along the estuary is
well reproduced, with an average decrease between sites 1 and 4
of 0.8 m at spring tide, while the measured difference is 0.65 m.
The maximum calculated phase difference between the same sites
occurs at low water and is around 2 hours, while the averaged
registered lag is 1.5 hours. An underestimation of the water
elevation at high tide, not present for site 1, appears evident at site
4. This difference remains unaltered even if a mesh with a twofold
refinement is used, suggesting that the committed error has not to
be attributed to the mesh. A reason of the model inaccuracy may
rely in the mass balance errors of the wet and dry scheme,
especially for small scale tide only studies (Blain et al., 2010).
Depth averaged velocities
The comparison of the predicted and observed depth averaged
velocities during spring tide is illustrated in Figure 5. At site 1 the
measured series has a strong flood flow registered on the 9th
October, which might be of meteorological origin. The magnitude
of the predicted velocities reproduces generally well the observed
time series, with an optimal phase agreement. By processing the
measurements of ADCP1, the observed velocity component along
the Latitude direction shows a cut-off during the flood phase of
the spring tide, which is not noticed in the Longitudinal
component, and disappears at neap tide. This behavior may be
related to some errors in the data acquisition and explains the
overestimation of the flood current, while the ebb tide intensity is
in agreement with the observations. At site 3 the peak flood
velocity is underestimated of around 0.2 m/s, while the difference
in the ebb phase is again minimal. At site 4 the slight asymmetry
between flood and ebb tide is reflected in the modeled currents,
with a damping of the peak current of around 0.2 m/s. The model
is however able to reproduce the deformation of the tidal curve
during ebb tides.
Journal of Coastal Research, Special Issue No. 65, 2013
1174
Mazzolari, et al.
Figure 5. Depth averaged observed and computed velocities at sites 1, 3 and 4. Positive values correspond to the flood phase.
Flooded area extension
The capacity of the model in reproducing the wet and dry in the
estuary is assessed with the support of time referenced aerial
orthophotos. Figure 6 represents the position of the modeled
wetted boundary with respect to the real limits of the inundated
area, at 10:30am of September 13th 2010, time when the photos
were taken. The extension of the flooded area in the orthophotos
has been digitized manually: wherever the exact position of the
water extent was uncertain, due to the reduced depth of the water
column, the bathymetry contour 0.84 m below MSL,
corresponding to the tide level at the harbor gauge, was followed.
The parts of the estuary which are flooded in the photos, along the
main and secondary channels, result flooded also in the wet and
dry simulation. The distance between the wetted boundaries in the
model and reality is less than 50 m, a value which is comparable
with the local mesh element size. It can be concluded that the used
mesh represents accurately the underlying DEM and that the
flooding process during a tidal cycle is realistically simulated. The
elevated extension of the intertidal area in the middle estuary of
Figure 1(c), which for the simulated tide cycle is 63% of the
inundated area at high tide, may explain the presence of mass
balance errors.
CONCLUDING REMARKS
This paper describes a multi-criteria approach for the
unstructured mesh generation of ocean, coastal and estuarine
waters domains. A series of a-priori and a-posteriori criteria are
chosen, each of them expressed in terms of a NSF, covering the
whole domain or a portion of it. The final function driving the
node distribution is achieved by superimposing the previously
defined NSFs. The method is applied to mesh a tidal domain
incorporating part of the Atlantic Ocean, the Western Iberian
continental shelf, the Lima estuary, the course of the Lima River,
as far as an upstream section located beyond the tide reach, and
the adjacent low-lying floodplains. The use of a multi-criteria
method complies with several physical and empirical requirements
of shallow water models and wet and dry algorithms in the
discretization of multi-scale domains. These criteria can be
applied or adjusted for similar inundation studies. The resulting
mesh, with an element size varying over 3 orders of magnitudes,
has been included in a wet and dry tide modeling study. The
results show an overall good performance in the reproduction of
the free surface elevation, the tide range variation and current
magnitudes. The evolution of the wetting front matches with the
location of the emerged estuary bed forms, confirming the mesh
accurate reproduction of the bathymetry.
Further developments include the use of a land cover database
for an exact distribution of the roughness coefficient and the
simulation of storm surges ; a further investigation on the origin of
the model systematic errors will include a mesh convergence
study with a-posteriori error estimators and the use of alternative
wet and dry algorithms.
ACKNOWLEDGEMENT
This research was funded by the Portuguese Foundation for the
Science and Technology through the Doctoral Grant of the first
author. The authors acknowledge the ADCIRC development team
for making available the original program code; A.P. Falcão and
A. Gonçalves of Instituto Superior Técnico (Lisbon) for providing
the altimetry and intertidal bathymetry of the Lima estuary; the
Institute of Geophysics and Planetary Physics and the Portuguese
Hydraulic Resources National Information System.
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Figure 6. Comparison between the locations of the wetted fronts taken from aerial photos and from the ADCIRC wet and dry scheme,
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Journal of Coastal Research, Special Issue No. 65, 2013