Numerik 2 Motivation

Transcription

1 Numerik 2 Motivation P. Bastian Universität Heidelberg Interdisziplinäres Zentrum für Wissenschaftliches Rechnen Im Neuenheimer Feld 368, D Heidelberg Peter.Bastian@iwr.uni-heidelberg.de April 15, 2014 P. Bastian (IWR) Numerik 2 April 15, / 33

2 Outline 1 Organizational Stuff 2 Partial Differential Equations are Ubiquitous 3 Preview: The Finite Element Method 4 DUNE P. Bastian (IWR) Numerik 2 April 15, / 33

3 Organizational Stuff Contents 1 Organizational Stuff P. Bastian (IWR) Numerik 2 April 15, / 33

4 Organizational Stuff Lecture Lecturer: Peter Bastian Office: INF 368, room Room and time (note change of room on Wednesday): Wed (INF 329, SR26), Fr 9-11 (INF 350 / OMZ R U014) Enter Otto Meyerhoff Center (building 350) from the east side! Lecture homepage: numerik2_ss2014/ Lecture notes available from the homepage I will extend the lecture notes during the course (>chapter 10) P. Bastian (IWR) Numerik 2 April 15, / 33

5 Organizational Stuff Excercises Excercises designed by Pavel Hron (and myself) Office: INF 368, room Student tutor: René Hess Excercises are managed via the MUESLI system: view/326 Please register if you have not yet done so! Time to be determined via doodle: Currently available: Mo 14-16, Wed (preferred) There will be theoretical and practical excercises Practical excercises use the software DUNE: P. Bastian (IWR) Numerik 2 April 15, / 33

6 Contents 2 Partial Differential Equations are Ubiquitous P. Bastian (IWR) Numerik 2 April 15, / 33

7 Gravitational Potential Find function Ψ(x) : Ω R, Ω = R 3 such that: x1 x 1 Ψ(x)+ x2 x 2 Ψ(x)+ x3 x 3 Ψ(x) = Ψ(x) = Ψ(x) = 4πGρ(x) G: gravitational constant, ρ: mass density in kg/m 3 Force acting on point mass m at point x: F (x) = m Ψ(x) P. Bastian (IWR) Numerik 2 April 15, / 33

8 Star Formation Cone nebula from P. Bastian (IWR) Numerik 2 April 15, / 33

9 Star Formation: Mathematical Model Euler equations of gas dynamics plus gravity: t ρ + (ρv) = 0 t (ρv) + (ρvv T + pi ) = ρ Ψ t e + ((e + p)v) = ρ Ψ v Ψ = 4πG ρ (mass conservation) (momentum conservation) (energy conservation) (gravitational potential) Constitutive relation: p = (γ 1)(e ρ v 2 /2) More elaborate model requires radiation transfer, better constitutive relations, friction,... Nonlinear system of partial differential equations P. Bastian (IWR) Numerik 2 April 15, / 33

10 Star Formation: Numerical Simulation (Diploma thesis of Marvin Tegeler, 2011). P. Bastian (IWR) Numerik 2 April 15, / 33

11 Flow of an Incompressible Fluid (Incompressible) Navier-Stokes Equations: v = 0 t v + (vv T ) ν v + p = f (mass conservation) (momentum conservation) ρ is independent of pressure No compression work, isothermal situation Pressure is independent variable Existence of solutions is Millenium Prize Problem P. Bastian (IWR) Numerik 2 April 15, / 33

12 Von Karman Vortex Street Re 20 (laminar) Re 200 (periodic) Re 1500 (turbulent) P. Bastian (IWR) Numerik 2 April 15, / 33

13 Von Karman Vortex Street Re 20 Re 200 Re 1500 P. Bastian (IWR) Numerik 2 April 15, / 33

14 Propagation of Electromagnetic Waves (Macroscopic)Maxwell equations: E = t B H = j + t D D = ρ B = 0 (Faraday) (Ampère) (Gauß) (Gauß for magnetic field) Constitutive relations: D = ɛ 0 E + P B = µ 0 (H + M) (D: electric displacement field, P: polarization) (H: magnetizing field, M: magnetization) Linear, first-order hyperbolic system P. Bastian (IWR) Numerik 2 April 15, / 33

15 Application: Geo-radar Soil physics group Heidelberg Simulation: Jorrit Fahlke P. Bastian (IWR) Numerik 2 April 15, / 33

16 Geothermal Power Plant z 3700m warm kalt 4000m r P. Bastian (IWR) Numerik 2 April 15, / 33

17 Geothermal Power Plant: Mathematical Model Coupled system for water flow and heat transport: t (φρ w ) + {ρ w u} = f u = k µ ( p ρ wg) t (c e ρ e T ) + q = g q = c w ρ w ut λ T (mass conservation) (Darcy s law) (energy conservation) (heat flux) Nonlinearity: ρ w (T ), ρ e (T ), µ(t ) Permeability k(x) : 10 7 in well, in plug Space and time scales: R=15 km, r b =14 cm, flow speed 0.3 m/s in well, power extraction: decades P. Bastian (IWR) Numerik 2 April 15, / 33

18 Geothermal Power Plant: Results Temperature after 30 years of operation P. Bastian (IWR) Numerik 2 April 15, / 33

19 Geothermal Power Plant: Results 1.2e e e+07 Extracted Power / Watt 1.05e+07 1e e+06 9e e+06 8e e+06 7e Time / days Extracted power over time P. Bastian (IWR) Numerik 2 April 15, / 33

20 Bacterial Growth and Transport in Capillary Fringe DFG Research Group 831 DyCap, Experiment by C. Haberer, Tübingen P. Bastian (IWR) Numerik 2 April 15, / 33

21 Partial Differential Equations are Ubiquitous Bacterial Growth and Transport in Capillary Fringe Experiment by Daniel Jost, KIT, Karlsruhe P. Bastian (IWR) Numerik 2 April 15, / 33

22 Reactive Multiphase Simulation Unknowns: pressure, saturation, bacteria concentration, carbon concentration, oxygen concentration Simulation by Pavel Hron P. Bastian (IWR) Numerik 2 April 15, / 33

23 Reactive Multiphase Simulation Simulation by Pavel Hron P. Bastian (IWR) Numerik 2 April 15, / 33

24 Reactive Multiphase Simulation Simulation by Pavel Hron P. Bastian (IWR) Numerik 2 April 15, / 33

25 Second Order Model Problems Poisson equation: gravity, electrostatics (elliptic type) u = f u = g u ν = j in Ω on Γ D Ω on Γ N = Ω \ Γ D Heat equation (parabolic type) t u u = f in Ω Σ, Σ = (t 0, t 0 + T ) u = u 0 at t = t 0 u = g on Ω Wave equation (sound propagation) (hyperbolic type) tt u u = 0 in Ω P. Bastian (IWR) Numerik 2 April 15, / 33

26 Partial Differential Equations are Ubiquitous Second Order Model Problems Solutions have different behavior (parabolic) P. Bastian (IWR) (hyperbolic) Numerik 2 April 15, / 33

27 Preview: The Finite Element Method Contents 3 Preview: The Finite Element Method P. Bastian (IWR) Numerik 2 April 15, / 33

28 Preview: The Finite Element Method What is a Solution to a PDE? Strong form: Consider the model problem u + u = f in Ω, u ν = 0 on Ω Assume u is a solution and v is an arbitrary (smooth) function, then ( u + u)v dx = fv dx Ω Ω ( u)v dx + uv dx = fv dx Ω Ω Ω u v dx ( u ν)v dx + uv dx = fv dx Ω Ω Ω Ω u v + uv dx = fv dx Ω Ω a(u, v) = l(v) Weak form: Find u H 1 (Ω) s. t. a(u, v) = l(v) for all v H 1 (Ω). P. Bastian (IWR) Numerik 2 April 15, / 33

29 Preview: The Finite Element Method The Finite Element (FE) Method Idea: Construct finite-dimensional subspace U H 1 (Ω) Partition domain Ω into elements t i : 0 t 1 t 2 t 3 1 Ω = (0, 1), T h = {t 1, t 2, t 3 } Construct function from piecewise polynomials, e.g. linears: 0 1 U h = {u C 0 (Ω) : u ti is linear } Insert in weak form: U h = span{φ 1,..., φ N }, u h = N j=1 x jφ j, then u h U h : a(u h, φ i ) = l(φ i ), i = 1,..., N Ax = b P. Bastian (IWR) Numerik 2 April 15, / 33

30 DUNE Contents 4 DUNE P. Bastian (IWR) Numerik 2 April 15, / 33

31 DUNE Challenges for PDE Software Many different PDE applications Multi-physics Multi-scale Inverse modeling: parameter estimation, optimal control Many different numerical solution methods, e.g. FE/FV No single method to solve all equations! Different mesh types: mesh generation, mesh refinement Higher-order approximations (polynomial degree) Error control and adaptive mesh/degree refinement Iterative solution of (non-)linear algebraic equations High-performance Computing Single core performance: Often bandwidth limited Parallelization through domain decomposition Robustness w.r.t. to mesh size, model parameters, processors Dynamic load balancing in case of adaptive refinement P. Bastian (IWR) Numerik 2 April 15, / 33

32 DUNE DUNE Software Framework Distributed and Unified Numerics Environment Domain specific abstractions for the numerical solution of PDEs with grid based methods. Goals: Flexibility: Meshes, discretizations, adaptivity, solvers. Efficiency: Pay only for functionality you need. Parallelization. Reuse of existing code. Enable team work through standardized interfaces. P. Bastian (IWR) Numerik 2 April 15, / 33

33 DUNE AMG Weak Scaling Results AAMG in DUNE is Ph. D. work of Markus Blatt BlueGene/P at Jülich Supercomputing Center P 80 3 degrees of freedom ( finest mesh), CCFV Poisson problem, 10 8 reduction AMG used as preconditioner in BiCGStab (2 V-Cycles!) procs 1/h lev. TB TS It TIt TT P. Bastian (IWR) Numerik 2 April 15, / 33