Numerical Methods for Geophysical Modelling

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Numerical Methods for Geophysical Modelling Noel
Barton, Paul Cleary and Nick Stokes CSIRO
Mathematical and Information Sciences www.cmis.csi
ro.au/cfd
Chapman Conference Dunsborough, WA, 19-24 August
2001
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• Contents
• Synopsis of three simulation tools
• Fastflo, SPH, DEM
• Examples of each of them

Chapman Conference Dunsborough, WA, 19-24 August
2001
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• Fastflo
• Status 50-60 person-years of development by
  CSIRO distributed internationally by NAG.
• Description
• general purpose (2D/3D) PDE solver using finite
  elements
• high level command language for coding of
  timestepping or nonlinearities, graphics, control
  of the computations,
• selection of sparse matrix solvers (direct and
  iterative)
• flexible (equations, geometry, algorithms, free
  boundaries)
• Applications porous media flow with heat,
  stress and chemical reactions elastic waves in
  oilfields formation of ore deposits

Chapman Conference Dunsborough, WA, 19-24 August
2001
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Mesh generation in Fastflo
triangular mesh generator linear and
quadratic approx 2D triangles,
quadrilaterals 3D tetrahedra, hexahedra
interface to third-party software
isoparametric elements deformable boundaries
block mesh generator axisymmetry
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Derivative expressions
1 D_j A D_j U1 - Ñ.(a Ñ u) 2
A U1 au 3 A_j
D_j U1 a.Ñ u 4 D_j A_j
U1 - Ñ. (au) 5 D_j A_jk D_k
U1 - Ñ .(A Ñ u) 6 D_jAU1_j -
div (au) 7 A D_j U1_j a
div u 8 A_j U1_j a.u 9
D_j A_k D_k U1_j - div (a.Ñ u)
10 D_j A_j D_k U1_k - div (a div u)
11 D_j A_jk U1_k - div (Au)
12 A_jk D_j U1_k div (Au)
13 D_i A U1 - Ñ (au) 14 A
D_I U1 aÑ u 15 A_i U1
au 16 D_i A_j D_j U1 - Ñ (a.Ñ u)
17 D_j A_j D_i U1 - a.Ñ (Ñ u)
- (Ñ u) Ñ.a 18 D_j A_ji U1
- Ñ .(Au) 19 A_ij D_j U1 AÑ u 20
A U1_i au 21 A_j D_j U1_i
a.Ñ u 22 D_j A_j U1_i -
a.Ñ u- u div a 23 D_j A D_j U1_i
- Ñ. (a Ñ u) 24 D_j A_jk
D_k U1_i - Ñ. (A Ñ u) 25 D_i A
D_j U1_j - Ñ (a Ñ.u) 26 D_i A_j
U1_j - Ñ(a.u) 27 D_j A D_i
U1_j (Ñ a) .Ñ(div u)-Ñ (div au)
28 A_j D_i U1_j a.(Ñ u) 29
D_j A_i U1_j - a (Ñ.u) - u.Ñ a 30
A_i D_j U1_j a (Ñ.u) 31 A_ij
U1_j Au 32 D_i A_jk D_j U1_k -
Ñ.(AÑ u) 33 D_j A_jk D_i U1_k
34 D_j A_ik D_k U1_j 35 D_j A_ij
D_k U1_k 36 D_j A_ik D_j U1_k 37
D_j A_k D_j U1_k - div aÑ u 38 D_j
A_i D_j U1 - div aÑ u
38 expressions hard-wired into the package
D_j A D_j U1 - Ñ.(a Ñ u)
A_j D_j U1_i a.Ñ u
D_i A D_j U1_j - Ñ (a Ñ.u)
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• Smoothed Particle Hydrodynamics (SPH)
• Status 15 person-years of development by
  CSIRO in-house code made available through
  contracts.
• Description
• (2D/3D) CFD solver based on particle method
  (field variables represented by point-based
  kernel approximation Lagrangian method in that
  points can move)
• suitable for free surface flows, splashing,
  impacts
• additional physical effects (e.g. heat
  transfer, rheology) can be readily included
• Applications high pressure diecasting,
  injection moulding, two-phase flow in electric
  furnaces

Chapman Conference Dunsborough, WA, 19-24 August
2001
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SPH - Smoothed Particle Hydrodynamics
SPH is a particle based method for modelling heat
and fluid flows Particle equations of motion are
derived from the Navier-Stokes equations using
smoothing or interpolation
Continuity Equation
Momentum Equation
Energy Equation
Equation of state
where
Enthalpy H
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• Discrete Element Method
• Status 15 person-years of development by
  CSIRO in-house code made available through
  contracts web-based mill simulation
  pre-processor about to be released.
• Description
• discrete element solver for rapid granular flows
  in complex geometries
• disks/superquadrics in 2D, spheres in 3D
• additional physics under development, especially
  breakage
• Applications widely applied to grinding mills
  materials handling mixing sampling separating

Chapman Conference Dunsborough, WA, 19-24 August
2001
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Simulations using the Discrete Element Method
(DEM) Collisional force model
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• Fastflo demonstration examples
• flow of water through a faulted porous material
• displacement of faulted rock according to linear
  elasticity
• ltlt see the demo gtgt

Chapman Conference Dunsborough, WA, 19-24 August
2001
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• Fastflo demonstration examples issues that
  could be included
• coupling of flow and elasticity
• flow in faults linked to elastic movement
• some resistance to compression in the faults
• sliding friction in the faults (Bingham fluid?)
• inclusion of other physics e.g. heat, chemical
  reactions, 3D, time-dependence
• other possibilities mantle convection, lava
  flows, magma chamber convection,
  elasto-viscoplastic flow, slab subduction

Chapman Conference Dunsborough, WA, 19-24 August
2001
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• SPH example
• dam break near the Triunfo Pass near Los Angeles
• topography obtained from US Geological Survey
  region occupies about 15 km2
• SPH simulations, coarse resolution 60,000
  fluid particles (corresponding to fluid particle
  separation of 6 m), 80 hrs CPU (500 MHz
  processor) for 300 sec simulation
• other possible applications tsunamis, volcanic
  eruptions, lava flows, kimberlites, lava flow,
  elasto- viscoplastic flow, direct simulation of
  flow through porous media
• ltlt see the video gtgt

Chapman Conference Dunsborough, WA, 19-24 August
2001
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• Discrete Element Example
• breakage occurs when forces within particles
  exceed a threshold new particles are generated
  (how to do this is a topic of current research)
• breakage of particles under stress 2D
• breakage of particles in a tumbling cube 3D
• other related applications excavation,
  materials handling, sampling, grinding
• ltlt see the video gtgt

Chapman Conference Dunsborough, WA, 19-24 August
2001
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• Summary
• Synopsis of three simulation tools with
  possible application in geodynamics
• Fastflo, SPH, DEM
• An example of each has been shown.
• Wed be happy to provide further information
  contact us on
• www.cmis.csiro.au/cfd

Chapman Conference Dunsborough, WA, 19-24 August
2001