Transcript Slide 1

A Fully Conservative 2D Model
over Evolving Geometries
Ricardo Canelas
Master degree student
IST
14.12.09
Teton Dam 1976
Objectives:
To develop a 2D fully conservative model for the
propagation of discontinuous flows over evolving
geometries
Tackle the domain definition and mesh
generation issues, define a database structure
susceptible to be well articulated with the
discretization procedure.
Gmsh
(http://www.geuz.org/gmsh/)
Winged Edge Data Structure
Gmsh can use a simple scripting language
as I/O
Use Gmsh as routine to:
generate the initial mesh
Possible to integrate in another code the
tasks of generating meshing domains,
outputting results, and generating new
meshes
generate subsequent refined meshes using a background
mesh technique that is built according to a non-real time
evaluation of the spatial variation of hydrodynamic variables
(height and velocities) and of the morphological parameters
(slopes)
Merging of altimetric information (DTM) with the “flat” mesh
Efficient terrain surface discretization using a “parametric”
space: a simple projection in Cartesian coordinates
One condition: the DTM is a regular grid, for fast interpolation
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Example on an idealized surface
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Domain definition
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Gmsh Delaunay triangulation
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DTM70merging
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Final surface
Generation of anisotropic meshes with background
meshing techniques on Gmsh
Calibrate initial mesh characteristic lenghts(generalized size of
na element around a point) on each node according to
defined criteria:
spacial variation of hidrodinamic variables (height and
velocities) and morphological parameters (slopes)
Example: linear variation of charactistic lenghts acording to
third coordinate
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Original mesh
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Refined mesh (Delaunay triangulation)
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Mesh topology data structure
Winged Edge data structure
Basic element is edge
Highly redundant
Constant time queries
Relatively small memory
requirements
Define discretization procedure for
an uncoupled solution
Use of Flux-vector Splitting Finite Volume Method:
- Evaluate the PDEs, in their integral form over any
discrete cell and balance the fluxes through the cell
edges, in an attempt to estimate the real continuous
solution.
- Flux-vector Splitting considers a linear separation of
the flux
Complete Conservation Equations
Total mass
Momentum in x direction
Momentum in y direction
Sediment mass in transport layer
Closure equations needed for hb, ub, τb, Cb* and Λ – granular dynamics and numerical simulation
Equation discretization
Full system in a compact form
Were
is the independant conservative variables vector
is the primitive variables vector
and
are the flux vectors in x and y direction
is the source terms vector
Trough proper integration,
and evaluation of the integral form,
the final expression for the computation of flux trough
element edges becomes
[Ferreira, 2009]
Geomorphology – code integration in the uncoupled case
Development of 2D code that allows the computation of bed
and lateral erosion and the integration of debris volume
derived from geotechnical failure in the flow, compatible with
the FVM nature of the hydrodynamical code is of major
importance in this work.
Bed erosion – Formulation
Equation for the mass conservation of sediments in the bed
Closure equations (derived from granular dinamics)
Equilibrium concentration
Adaptation lenght
Bed erosion – discretization problems
ΔZb is evaluated in the conservation equation for the bed of the system, at the
barycenter of each element
Free surface level remais constant,
velocities are computed again in each
cell to acomodate volume change
Time step
ΔZb1
ΔZb2
Compatibility problems in the edges due to diferential erosion in adjacent elements – must
devise a conservative way to force compatibility
Geotechnical failiure
Geotechnical failiure represents a big contribuition of solid
material to the flow in the case of dam break, and should be
evaluated carefully.
The initial aproach will be comparing each element maximum
gradient with the critical value and performing a rotation of
the element on a normal to the line of maximum gradient,
fixed on the lowest node of the element.
Geotechnical Failiure model
Θ=i-icrit
i>icrit
ΔZ1
ΔZ2
Geotechnical Failiure model
Compatibilized elements
Volume to integrate on
the flow on the next time
Model Limitations
-Instantaneous failiure and colapse;
-Accuracy dependant on element size, computed not
regarding this fact;
Advantages
-Easy to implement;
-Low computing load
Model Validation
-Actual case study with results produced by a 1D
model is available for direct comparison