Bridging the Gap: Harnessing advanced Computational Libraries for coupled multi-physics problems
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1 Bridging the Gap: Harnessing advanced Computational Libraries for coupled multi-physics problems Marc Spiegelman 1,2 and Cian R Wilson 1 Plus enormous contributions from PETSc, FEniCS and AMCG groups 1 Lamont-Doherty Earth Observatory, Columbia University, New York 2 DEES/DAPAM, Columbia University, New York BTG, Princeton, 1 2 October 2012
2 Introduction Motivation Computation is essential for exploring non-linear multi-physics problems in Geosciences Major advances in hardware, software and algorithms make complex problems more accessible However, still a considerable gap between Geosciences and Computational Sciences/Math Some Existing Barriers: Overall Complexity of both software libraries and models Lack of transparency/reproducibility/reusability of current models Language/Training issues between computation and Earth Sciences Spiegelman and Wilson TerraFERMA BTG
3 Introduction Motivation Computation is essential for exploring non-linear multi-physics problems in Geosciences Major advances in hardware, software and algorithms make complex problems more accessible However, still a considerable gap between Geosciences and Computational Sciences/Math Some Existing Barriers: Overall Complexity of both software libraries and models Lack of transparency/reproducibility/reusability of current models Language/Training issues between computation and Earth Sciences We need better tools for exploring complex (and simple) models and making advanced computation accessible to more general users Spiegelman and Wilson TerraFERMA BTG
4 Target Application : multi-physics models of magmatic plate boundaries image courtesy of the Neptune project (wwwneptunewashingtonedu) Central to Solid Earth Geosciences: Essential for understanding Seismic, Volcanic and Tsunamagenic Natural Hazards global tectonics and geochemical cycling Spiegelman and Wilson TerraFERMA BTG
5 Target Application : multi-physics models of magmatic plate boundaries image courtesy of the Neptune project (wwwneptunewashingtonedu) Central to Solid Earth Geosciences: Essential for understanding Seismic, Volcanic and Tsunamagenic Natural Hazards global tectonics and geochemical cycling Multi-physics: Tightly coupled non-linear systems Coupled fluid-solid mechanics Complex solid rheologies Coupled thermodynamics/geodynamics Spiegelman and Wilson TerraFERMA BTG
6 Target Application : multi-physics models of magmatic plate boundaries image courtesy of the Neptune project (wwwneptunewashingtonedu) Central to Solid Earth Geosciences: Essential for understanding Seismic, Volcanic and Tsunamagenic Natural Hazards global tectonics and geochemical cycling Multi-physics: Tightly coupled non-linear systems Coupled fluid-solid mechanics Complex solid rheologies Coupled thermodynamics/geodynamics Considerable uncertainty in equations, constitutive relations and coupling Spiegelman and Wilson TerraFERMA BTG
7 Model Requirements User Flexibility: Choice of equations, geometry, elements, constitutive relations, coupling Wide choice of solvers/preconditioners for coupled non-linear problems Ability to rapidly compose a range of models from simple process models to regional geodynamic models All choices available at or near run time Spiegelman and Wilson TerraFERMA BTG
8 Model Requirements Infrastructure: Residual monitoring for the full coupled problem Model reproducibility: Transparent options system Regression tested Generic model services: Standardized I/O Checkpointing Monitoring and detectors Parallel, scaleable, open source, version controlled, regression tested free Spiegelman and Wilson TerraFERMA BTG
9 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation Au = f F(u) = 0 u n+1 = f(u n ) Spiegelman and Wilson TerraFERMA BTG
10 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation Au = f F(u) = 0 u n+1 = f(u n ) Spiegelman and Wilson TerraFERMA BTG
11 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation Au = f F(u) = 0 u n+1 = f(u n ) Spiegelman and Wilson TerraFERMA BTG
12 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation Au = f F(u) = 0 u n+1 = f(u n ) Spiegelman and Wilson TerraFERMA BTG
13 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation Au = f F(u) = 0 u n+1 = f(u n ) P E T S c Spiegelman and Wilson TerraFERMA BTG
14 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation F E ni C S Au = f F(u) = 0 u n+1 = f(u n ) P E T S c Spiegelman and Wilson TerraFERMA BTG
15 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation F E ni C S G m sh Au = f F(u) = 0 u n+1 = f(u n ) P E T S c Spiegelman and Wilson TerraFERMA BTG
16 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation S P u D F E ni C S G m sh Au = f F(u) = 0 u n+1 = f(u n ) P E T S c Spiegelman and Wilson TerraFERMA BTG
17 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation S P u D F E ni C S G m sh Au = f F(u) = 0 u n+1 = f(u n ) P E T S c TerraFERMA Transparent Finite Element Rapid Model Assembler Spiegelman and Wilson TerraFERMA BTG
18 Application: Fluid Migration in Subduction Zones ( ( )) vs + v T s 2η = p +ϕ 0 ϕk 2 v s = ϕ 0 P ζ K µ P + P ( ) K ζ = µ k p + Γ ρ ρ f ϕ 0 Dϕ Dt = (1 ϕ 0ϕ) P ζ + Γ ϕ 0 Spiegelman and Wilson TerraFERMA BTG
19 Example: Infinite Prandtl Thermal Convection 2 µ v + v T 2 + p T k = 0, v = 0, T v T + T =0 t Ra Spiegelman and Wilson TerraFERMA BTG
20 Example: Infinite Prandtl Thermal Convection 2 µ v + v T 2 + p T k = 0, v = 0, T v T + T =0 t Ra Spiegelman and Wilson TerraFERMA BTG
21 Example: Infinite Prandtl Thermal Convection 2 µ v + v T 2 + p T k = 0, v = 0, T v T + T =0 t Ra Spiegelman and Wilson TerraFERMA BTG
22 Example: Infinite Prandtl Thermal Convection 2 µ v + v T 2 + p T k = 0, v = 0, T v T + T =0 t Ra Let u = (v, p, T ) u i+1 = u i αj(u i ) 1 r (u i ) Spiegelman and Wilson TerraFERMA BTG
23 Example: FEniCS Equation Description Given function u = (v, p, T ), test function u t = (v t, p t, T t ) and trial function u a = (v a, p a, T a ): Weak form of residual, r(u): rv = r p = r T = Ω Ω Ω [( v t + v T t 2 ) ( ) ] v + v T : 2µ v t p (v t ) z T, 2 p t v, [ T t ((T T n ) + tv θ T θ ) + t ] Ra T t T θ r =rv + r p + r T Weak form of Jacobian, J (u): J = r (u) Spiegelman and Wilson TerraFERMA BTG
24 Example: FEniCS Equation Description Given function u = (v, p, T ), test function u t = (v t, p t, T t ) and trial function u a = (v a, p a, T a ): Weak form of residual, r(u): r v = (inner(sym(grad(v t)), 2*mu*sym(grad(v))) - div(v t)*p - T*v t[1] )*dx r p = p t*div(v)*dx r T = (T t*((t - T n) + dt*inner(v theta, grad(t theta))) + (dt/ra)*inner(grad(t t), grad(t theta)) )*dx r = r v + r p + r T Weak form of Jacobian, J (u): J = derivative(r, u, u a) Spiegelman and Wilson TerraFERMA BTG
25 Example: PETSc Block Preconditioning & Solvers J(u i )δu = r(u i ) J = K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T B 1 B 2 0 A System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Velocity 1 (preonly,mg) Velocity 2 (preonly,mg) Spiegelman and Wilson TerraFERMA BTG
26 Example: PETSc Block Preconditioning & Solvers J(u i )δu = r(u i ) J = K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T B 1 B 2 0 A System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Velocity 1 (preonly,mg) Velocity 2 (preonly,mg) Stokes Advection-Diffusion Spiegelman and Wilson TerraFERMA BTG
27 Example: PETSc Block Preconditioning & Solvers J PC (u i ) 1 J(u i )δu = J PC (u i ) 1 r(u i ) J = J PC = K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T B 1 B 2 0 A K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T 2 M 0 B 1 B 2 0 A System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Velocity 1 (preonly,mg) Velocity 2 (preonly,mg) Spiegelman and Wilson TerraFERMA BTG
28 Example: PETSc Block Preconditioning & Solvers J PC (u i ) 1 J(u i )δu = J PC (u i ) 1 r(u i ) J = J PC = K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T B 1 B 2 0 A K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T 2 M 0 B 1 B 2 0 A System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Velocity 1 (preonly,mg) Velocity 2 (preonly,mg) Spiegelman and Wilson TerraFERMA BTG
29 Example: PETSc Block Preconditioning & Solvers J PC (u i ) 1 J(u i )δu = J PC (u i ) 1 r(u i ) J = J PC = K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T B 1 B 2 0 A K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T 2 M 0 B 1 B 2 0 A System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Velocity 1 (preonly,mg) Velocity 2 (preonly,mg) Spiegelman and Wilson TerraFERMA BTG
30 Example: PETSc Block Preconditioning & Solvers J PC (u i ) 1 J(u i )δu = J PC (u i ) 1 r(u i ) J = J PC = K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T B 1 B 2 0 A K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T 2 M 0 B 1 B 2 0 A System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Velocity 1 (preonly,mg) Velocity 2 (preonly,mg) Spiegelman and Wilson TerraFERMA BTG
31 Example: PETSc Block Preconditioning & Solvers J PC (u i ) 1 J(u i )δu = J PC (u i ) 1 r(u i ) J = J PC = K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T B 1 B 2 0 A K 11 K 12 G 1 C 1 K 21 K 22 G 2 C 2 G T 1 G T 2 M 0 B 1 B 2 0 A System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Velocity 1 (preonly,mg) Velocity 2 (preonly,mg) Spiegelman and Wilson TerraFERMA BTG
32 Generic Structure of PDE Based Numerical Models Options System Constitutive PDEs Eqs Meshes/ Geometry Discrete Eqs Solvers Visualization Error Est Validation S P u D F E ni C S G m sh Au = f F(u) = 0 u n+1 = f(u n ) P E T S c TerraFERMA Transparent Finite Element Rapid Model Assembler Spiegelman and Wilson TerraFERMA BTG
33 Example: TerraFERMA Options Spiegelman and Wilson TerraFERMA BTG
34 Example: TerraFERMA Options Spiegelman and Wilson TerraFERMA BTG
35 Example: Setting Equations in TerraFERMA Spiegelman and Wilson TerraFERMA BTG
36 Example: Setting Equations in TerraFERMA Spiegelman and Wilson TerraFERMA BTG
37 Example: Solver Options in TerraFERMA Spiegelman and Wilson TerraFERMA BTG
38 Example: Solver Options in TerraFERMA System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Spiegelman and Wilson TerraFERMA BTG
39 Example: Solver Options in TerraFERMA System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Spiegelman and Wilson TerraFERMA BTG
40 Example: Solver Options in TerraFERMA System (fgmres, fieldsplit) Temperature (gmres, ilu) Stokes (fgmres, fieldsplit) Pressure (cg,sor) Velocity (preonly,fieldsplit) Spiegelman and Wilson TerraFERMA BTG
41 Example: Benchmark (Blankenbach, 1989) Nu v rms q 1 q 2 T e z e Ra = 10 4 isoviscous Ra = 10 5 isoviscous Ra = 10 6 isoviscous Ra = 10 4 T -dep viscosity Benchmark Benchmark Benchmark Benchmark Spiegelman and Wilson TerraFERMA BTG
42 Current Benchmarks Incompressible 2D Convection (Blankenbach et al, 1989) Incompressible 3D Convection (Busse et al, 1994) Cylindrical Laminar Plumes (Vatteville et al, 2009) Compressible 2D Convection (King et al, 2009) Kinematic Subduction Zones (van Keken et al, 2008) Linearized Free Surface Evolution (Kramer et al, 2012) Non-linear magma waves (Simpson and Spiegelman, 2011) Cylindrical Convection (Rhodri Davies: Spiegelman and Wilson TerraFERMA BTG
43 Fluid Migration Buoyancy Buoyancy and Pore Pressure Spiegelman and Wilson TerraFERMA BTG
44 Fluid Migration Buoyancy Buoyancy and Pore Pressure Spiegelman and Wilson TerraFERMA BTG
45 Fluid Migration Buoyancy Buoyancy and Pore Pressure Spiegelman and Wilson TerraFERMA BTG
46 Fluid Migration Buoyancy Buoyancy and Pore Pressure Spiegelman and Wilson TerraFERMA BTG
47 Fluid Migration Buoyancy Buoyancy and Pore Pressure Spiegelman and Wilson TerraFERMA BTG
48 Conclusions Many important problems in Earth Sciences can be described by non-linear coupled systems of partial differential equations Much of the complexity modelling these systems stems from the nature of multi-physics where small changes in the coupling between fields or constitutive relations can lead to radical changes in behavior and the resulting demands on discretizations and solvers Many established single-physics models have locked in solution strategies that are unsuited or difficult to extend to the more challenging non-linear behaviour of a multi-physics system We require a significant increase in flexibility for users for both composing problems (from simple model problems to more realistic coupled simulations) and changing solver strategies Several advanced, open-source computational libraries mean that it is now possible to provide much greater flexibility to the user TerraFERMA is a model building framework that gives access to these libraries while also providing a transparent description of the model Spiegelman and Wilson TerraFERMA BTG
49 Spiegelman and Wilson TerraFERMA BTG
50 Equations (Katz et al, 2007) Solid flow for solid velocity, v s, and dynamic pressure, p : ( ( )) vs + v T s 2 η = p + ϕ 0 ϕ k 2 v s = ϕ 0 P ζ Fluid flow for pore pressure, P, and porosity, ϕ: ( K µ P + P ζ = K µ k p Dϕ Dt = ( ) P 1 ϕ 0 ϕ ζ + Γ ϕ 0 ) + Γ ρ ρ f ϕ 0 Constitutive relations: η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
51 Equations (Katz et al, 2007) Solid flow for solid velocity, v s, and dynamic pressure, p : ( ( )) vs + v T s 2 η = p + ϕ 0 ϕ k 2 v s = ϕ 0 P ζ Fluid flow for pore pressure, P, and porosity, ϕ: ( K µ P + P ζ = K µ k p Dϕ Dt = ( ) P 1 ϕ 0 ϕ ζ + Γ ϕ 0 ) + Γ ρ ρ f ϕ 0 Constitutive relations: η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
52 Equations (Katz et al, 2007) Solid flow for solid velocity, v s, and dynamic pressure, p : ( ( )) vs + v T s 2 η = p + ϕ 0 ϕ k 2 v s = ϕ 0 P ζ Fluid flow for pore pressure, P, and porosity, ϕ: ( K µ P + P ζ = K µ k p Dϕ Dt = ( ) P 1 ϕ 0 ϕ ζ + Γ ϕ 0 ) + Γ ρ ρ f ϕ 0 Constitutive relations: η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
53 Equations (Katz et al, 2007) Solid flow for solid velocity, v s, and dynamic pressure, p : ( ( )) vs + v T s 2 η = p + ϕ 0 ϕ k 2 v s = ϕ 0 P ζ Fluid flow for pore pressure, P, and porosity, ϕ: ( K µ P + P ζ = K µ k p Dϕ Dt = ( ) P 1 ϕ 0 ϕ ζ + Γ ϕ 0 ) + Γ ρ ρ f ϕ 0 Constitutive relations: η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
54 Equations (Katz et al, 2007) Solid flow for solid velocity, v s, and dynamic pressure, p : ( ( )) vs + v T s 2 η = p + ϕ 0 ϕ k 2 v s = ϕ 0 P ζ Fluid flow for pore pressure, P, and porosity, ϕ: ( K µ P + P ζ = K µ k p Dϕ Dt = ( ) P 1 ϕ 0 ϕ ζ + Γ ϕ 0 ) + Γ ρ ρ f ϕ 0 Constitutive relations: η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
55 Equations (Katz et al, 2007) Solid flow for solid velocity, v s, and dynamic pressure, p : ( ( )) vs + v T s 2 η = p 2 v s = 0 Fluid flow for pore pressure, P, and porosity, ϕ: K µ P + P ζ = K µ k + Γ ρ ρ f ϕ 0 Dϕ Dt = P ζ + Γ ϕ 0 Constitutive relations: η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
56 Solid Flow Velocity streamlines 50km T = T s T = zt 0 /50 v i = 0 600km v = v 0 T = T 0 T = T s + (T 0 T s)erf ( z/ κt age) Spiegelman and Wilson TerraFERMA BTG
57 Equations Buoyancy and Pore Pressure: Constitutive relations: K µ P + P ζ = K µ k + Γ ρ ρ f ϕ 0 Dϕ Dt = P ζ + Γ ϕ 0 η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
58 Equations Buoyancy (zero compaction length approximation): Dϕ Dt = K µ k + Γ ( ) ρ + 1 ϕ 0 ρ f Buoyancy and Pore Pressure: Constitutive relations: K µ P + P ζ = K µ k + Γ ρ ρ f ϕ 0 Dϕ Dt = P ζ + Γ ϕ 0 η = η (T, ɛ, ϕ), ζ = ηϕ m, K = ϕ n Spiegelman and Wilson TerraFERMA BTG
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