Numerical Methods for Geophysical Modelling Noel Barton, Paul Cleary and Nick Stokes CSIRO Mathematical and Information

1. Numerical Methods for Geophysical Modelling Noel Barton, Paul Cleary and Nick Stokes CSIRO Mathematical and Information Sciences www.cmis.csiro.au/cfd Chapman Conference Dunsborough, WA, 19-24 August 2001

2. Contents • Synopsis of three simulation tools: • Fastflo, SPH, DEM • Examples of each of them Chapman Conference Dunsborough, WA, 19-24 August 2001

3. 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

4. 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

5. 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 .(AÑ u) 6 D_jAU1_j - div (au) 7 A D_j U1_j a div u8 A_j U1_j a.u9 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.a18 D_j A_ji U1 - Ñ .(Au) 19 A_ij D_j U1AÑ u 20 A U1_i au21 A_j D_j U1_i a.Ñ u22 D_j A_j U1_i - a.Ñ u- u div a23 D_j A D_j U1_i - Ñ. (a Ñu) 24 D_j A_jk D_k U1_i - Ñ. (AÑu) 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.(Ñu) 29 D_j A_i U1_j - a (Ñ.u) - u.Ñ a30 A_i D_j U1_j a (Ñ.u)31 A_ij U1_j Au32 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 aÑu38 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)

6. 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

7. 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 EnthalpyH =

8. 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

9. Simulations using the Discrete Element Method (DEM) Collisional force model

10. Fastflo demonstration examples • flow of water through a faulted porous material • displacement of faulted rock according to linear elasticity • << see the demo >> Chapman Conference Dunsborough, WA, 19-24 August 2001

11. 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

12. SPH example • dam break near the Triunfo Pass near Los Angeles • topography obtained from US Geological Survey; region occupies about 15 km^2 • 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 • << see the video >> Chapman Conference Dunsborough, WA, 19-24 August 2001

13. 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 • << see the video >> Chapman Conference Dunsborough, WA, 19-24 August 2001

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