Finite Elements (Numerical Solution of Partial Differential Equations)

Transcription

1 Finite Elements (Numerical Solution of Partial Differential Equations) P. Bastian Universität Heidelberg Interdisziplinäres Zentrum für Wissenschaftliches Rechnen Im Neuenheimer Feld 205, D Heidelberg October 17, 2017

2 Outline Organizational Stuff Partial Differential Equations are Ubiquitous Preview: The Finite Element Method DUNE

3 Contents Organizational Stuff

4 Lecture Lecturer: Peter Bastian Office: INF 205, room 1/401 Di+Do SR B Exercise: Mi Pool 3rd Floor Lecture homepage: https: //conan.iwr.uni-heidelberg.de/teaching/finiteelements_ws2017/ Lecture notes available from the homepage Taking notes is not needed

5 Excercises Excercises (Mi computer pool room #2 on 3rd floor) organized by Linus Seelinger Office: INF 205, room Linus Seelinger is on a research stay until Nov 6 until then Marian Piatkowski marian.piatkowski@iwr.uni-heidelberg.de takes over Registration to excercises via MÜSLI system: There will be theoretical and practical excercises Practical excercises are important! They are based on the software DUNE: Acceptance to final exam (oral or written): Present two excercise solutions Selection by chance (with one refusal) Probably a little project Exercises start Wed, October 25

6 If I had Wishes Be interactive! Ask questions! There are no dumb questions! Do not miss the practical excercises. Polish your C++ knowledge... We are always looking for students to help improving/extending our software...

7 Contents Partial Differential Equations are Ubiquitous

8 Equations of Mathematical Physics I Calculus was invented for partial differential equations! I E.g. to express conservation of mass, momentum and energy in quantitative form Famous examples are: I I I I I I I Poisson (electrostatics, gravity) 1800 Euler (inviscid flow) 1757 Navier-Stokes (viscous flow) 1822/1845 Maxwell (electrodynamics) 1864 Einstein (general relativity 1915 Poisson Navier Euler Stokes Solutions in practical situtation only with modern (super) computers! Maxwell Einstein

9 Gravitational Potential (Poisson Equation) 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)

10 Star Formation Cone nebula from

11 Star Formation: Mathematical Model Euler equations of gas dynamics: 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) Plus the Poisson equation... More elaborate model requires radiation transfer, better constitutive relations, friction,... Nonlinear system of partial differential equations

12 Star Formation: Numerical Simulation (Diploma thesis of Marvin Tegeler, 2011)

13 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 (in 3d for general data)

14 Von Karman Vortex Street Re 20 (laminar) Re 200 (periodic) Re 1500 (turbulent)

15 Von Karman Vortex Street Re 20 Re 200 Re 1500

16 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

17 Application: Geo-radar Soil physics group Heidelberg Simulation: Jorrit Fahlke

18 Geothermal Power Plant z 3700m warm kalt 4000m r

19 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 w g) 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

20 Geothermal Power Plant: Results Temperature after 30 years of operation

21 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

22 Bacterial Growth and Transport in Capillary Fringe DFG Research Group 831 DyCap, Experiment by C. Haberer, Tübingen

23 Bacterial Growth and Transport in Capillary Fringe Experiment by Daniel Jost, KIT, Karlsruhe

24 Reactive Multiphase Simulation Unknowns: pressure, saturation, bacteria concentration, carbon concentration, oxygen concentration

25 Reactive Multiphase Simulation Simulation by Pavel Hron

26 Reactive Multiphase Simulation Simulation by Pavel Hron

27 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

28 Second Order Model Problems Solutions have different behavior (parabolic) (hyperbolic)

29 Contents Preview: The Finite Element Method

30 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 = Ω ( u)v dx + uv dx = Ω Ω u v dx ( u ν)v dx + uv dx = Ω Ω Ω u v + uv dx = Ω Ω Ω 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). fv dx fv dx fv dx fv dx

31 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

32 Contents DUNE

33 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

34 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.

35 Trends in Computer Architecture Moore s law Supercomputer performance (GFLOPs/s) Nature, 530 (2016), pp Power wall Power consumption is limiting factor for exascale computing Clock rate stagnates but Moore s law is still valid Memory wall Bandwidth not sufficient to sustain peak performance ILP wall Revival of vectorization in form of SIMD instructions

36 Efficient Algorithms are Key Solution of large sparse algebraic systems F (z) = 0 Consider linear case Ax = b, A R N N : Non-sparse Gauß elimination: O(N 3 ) Sparse Gauß elimination: O (N 3(d 1) d Multigrid: O(N) N Gauß elimination 2 3 N3 multigrid 1000N ,66 s 0,001 s s 0,01 s ,6 days 0,1 s years 1 s years 10 s 1 GFLOPs/s

37 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