a fully conservative 2d model over evolving geometries ricardo canelas master degree student ist...
DESCRIPTION
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 StructureTRANSCRIPT
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A Fully Conservative 2D Model over Evolving Geometries
Ricardo CanelasMaster degree student
IST 14.12.09
Teton Dam 1976
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Objectives:
To develop a 2D fully conservative model for the propagation of discontinuous flows
over evolving geometries
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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
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Use Gmsh as routine to:
generate the initial mesh
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)
Gmsh can use a simple scripting language as I/O
Possible to integrate in another code the tasks of generating meshing domains, outputting results, and generating new meshes
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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
Domain definition Gmsh Delaunay triangulation
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DTM merging
Final surface
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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)
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Example: linear variation of charactistic lenghts acording to third coordinate
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Mesh topology data structure
Winged Edge data structure
Basic element is edgeHighly redundantConstant time queriesRelatively small memory
requirements
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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
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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
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Equation discretization
Full system in a compact form
Wereis 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
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Trough proper integration,
and evaluation of the integral form,
the final expression for the computation of flux trough element edges becomes
[Ferreira, 2009]
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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.
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Bed erosion – Formulation
Equation for the mass conservation of sediments in the bed
Closure equations (derived from granular dinamics)
Equilibrium concentration
Adaptation lenght
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Bed erosion – discretization problems
ΔZb is evaluated in the conservation equation for the bed of the system, at the barycenter of each element
Compatibility problems in the edges due to diferential erosion in adjacent elements – must devise a conservative way to force compatibility
ΔZb1ΔZb2
Time step
Free surface level remais constant, velocities are computed again in each cell to acomodate volume change
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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.
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Geotechnical Failiure model
i>icrit
Θ=i-icrit
ΔZ1
ΔZ2
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Geotechnical Failiure model
Compatibilized elements Volume to integrate on the flow on the next time
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Model Limitations
-Instantaneous failiure and colapse;-Accuracy dependant on element size, computed not regarding this fact;
Advantages
-Easy to implement;-Low computing load
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Model Validation
-Actual case study with results produced by a 1D model is available for direct comparison