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Multi-Scale Behaviour in the Geo- Science II: Introduction to Continuum Mechanics and Geological Applications by Hans Mühlhaus The Australian Computational Earth Systems Simulator (ACcESS)

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Overview Introduction 1D relationships Statics and kinematics Stress, Strain, Stretching, Spin, Objective Variables, Constitutive Relationships, stress equilibrium Level set method Outline, upwinding (Taylor Galerkin), two step methods, examples Exercises

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Displacement, Strain and Stretching L 0 (initial length) u 2 (displacement) L=L 0 +u 2 (length after application of force f 2 ) f 2 (Force) Strain : x2x2 x1x1 Stretching : Where

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1D Force and Stress Equilibrium x1x1 x2x2 The sum of all vertical forces must vanish for force equilibrium: Constitutive Relationship (Hookes law) Thus

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1D Constitutive Relations and Balance Equations More Constitutive Relationships: Newtonian Creep Insert into stress equilibrium: Considering the definition of the material time derivative:

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1D Constitutive Relations and Balance Equations x1x1 x2x2 C is a Concentration, R is a Reaction Term (e.g. Mass Source), q i Flux of Concentration x3x3 Assumption: Thus: Example 2: Heat Equation Heat source (radioactive decay) Heat capacity Thermal conductivity Change of concentration due to change of size

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Governing Equations Einsteins summation convention: Stress Equilibrium: Heat Equation: Stretching:

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Governing equations Melt viscosity for magma with 0.65% water content Here we use: with Heat Equation Stress Equilibrium

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Satellite Image of Volcanic Event

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2 nd talk: Volcano modelling Montserrat, West Indies

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Intraplate: Hotspots Anomalous areas of volcanism Mantle plumes –Ocean: low-viscosity basaltic magmas, Hawaiian Islands –Continental: high silica (high viscosity) rhyolites, Yellowstone Little information on magma source

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Volcano Facts 1511 known eruptions in last years deaths in last 400 years Biggest eruption: Yellowstone, USA (2500km 3 ) Potential problem: Vesuvius, Italy Poorly understood natural phenomena with approximately 30 eruptions in any given year. Volcanoes also produce many natural resources such as important minerals and metals.

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Generic Volcano Magma chamber at depth (5 – 60km) Plumbing from chamber to surface not well constrained 800 to 1200 degree C Changes in regional stress, earthquakes, can cause the volcano to erupt New eruption from exertion of magma forces, increased gas pressure or both Long term activity governed by rate of supply of new magma Different styles of volcanoes relate to different hazards

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Physical properties of magma Magma = melt + crystals + gas. Melt: Temperature о С, pressure MPa Crystals: size m, number density up to m -3, fraction up to 95 % Gas: H 2 O %, CO %, mass fraction % Melt viscosity Pas Bulk viscosity depends upon: Chemical composition - more SiO 2 higher viscosity Temperature - higher temperature lower viscosity Water content - higher content lower viscosity Crystal content - higher content higher viscosity

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Dome Growth Styles Axisymmetrical lava domePlaty lava dome Ross Griffiths & Jonathan Fink Lobate lava domeSpiny lava dome

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t=0 H t>0 r=0 r=R z=0 D/2

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The Level Set Method: Presentation Implicit representation of the interface by the zero level set of a smooth function φ φ is usually a signed distance function At each time step, φ is updated solving the advection equation:

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The Level Set Method: Solving the advection equation (1/4) Explicit Implicit Taylor Galerkin Test: A gaussian is advected in a constant 1D velocity field.

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The level set method …continued Advection dominated pdes need to require special treatment…..upwinding etc Taylor-Galerkin:

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The level set method …continued 2-step alternative to Taylor-Galerkin upwinding (very effective in the presence of diffusion terms….):

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Formulation Finley PDE: Example : Momentum and Heat equation Davies, M., Gross, L., Mühlhaus, H.–B., 2004, Scripting High Performance Earth Systems Simulations on the SGI Altix 3700, Proc. 7th Intl Conf. on High Performance Computing and Grid in Asia Pacific Region,

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EScript for i in range(numDim):\par for j in range(numDim):\par tau += stress[i,j] * stress[i,j]\par tau = sqrt(0.5 * tau + small)\par map["tau"] = tau\par \par # tau_Y\par \par # release memory\par # power law\par Xi_P1 = (tau / tau_0) ** (1 - n1)\par Xi_P2 = (tau / tau_Ystep) ** (1 - n2)\par Xi_P = Xi_P1 * Xi_P2 / (Xi_P1 + Xi_P2)\par map["Xi_P"] = Xi_P\par \par # release memory\par del tau_Ystep \par \par # melting temperature\par T_M = T_M0 + gamma * p\par map["T_M"] = T_M\par \par

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The Level Set Method: Solving the advection equation (2/4) Taylor Galerkin: The gaussian keeps its shape. Implicit: The gaussian is deformed in the direction of the velocity field.

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Level set applied to a cantilever beam Presentation of the test case: Constitutive relationship: Stress equilibrium:

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Level set applied to a cantilever beam

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Influence of contributions to stress rate: Oldroyd stress rate:

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Level set applied to a cantilever beam Accuracy of the method: Conservation of volume

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The Level Set Method: Solving the advection equation (3/4) Previous test: No topological change in the solution Need for a new test with: and New test: shearing flow Mesh: 100x100 Courant Number: steps forward 1000 steps with -v

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The Level Set Method: Solving the advection equation (4/4) The shape gets noisy… Problem: φ looses its distance function property Reinitialisation needed!

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The Level Set Method: Reinitialisation (1/3) Idea: Rebuild a signed distance function ψ from the distorted function φ Requirements: The interface must not be changed ψ must represent a distance function Solution: Solve to steady state the equation: Rewritten as: with Interpretation: The distance information is carried by w, a unit vector pointing away from the interface.

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The Level Set Method: Reinitialisation (2/3) 1D 2D 3D

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The Level Set Method: Reinitialisation (3/3) Same test as before, with reinitialisation

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The Level Set Method: Benchmarks (1/2) Axisymmetrical case: A fluid is submitted to a centrifugal force Interest: The analytical steady state is known (grey line) Parameters: mesh:20x20, density of air: 0 kg/m 3, density of fluid: 10 3 kg/m 3 Results:

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The Level Set Method: Benchmarks (2/2) Rayleigh-Taylor instability

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Level set cont. : Merger of small and large bubbles Parameters: Surface tension: Calculation, includes inertia, implicit, Courant Number=0.5, msh:30 by 45 8 node quads

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Level set cont. : Merger of small and large bubbles

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t=0 H t>0 r=0 r=R z=0 D/2

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Axisymmetric volcano

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Axisymmetric volcano (T= )

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Exercises h0 R z r A cylindrical container of radius R is filled initially to height h0 with an incompressible fluid of density and viscosity. The container is then rotated around his axis at a constant spin Determine the steady state position of the free surface of the fluid.

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Exercise5 Infinite vent: Hagen-Poiseulle flow Stress Equilibrium Heat Equation Dimensionless form Here D z r

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