An Introduction to Flow Visualization (1) Christoph Garth

Transcription

1 An Introduction to Flow Visualization (1) Christoph Garth

2 Motivation What will I be talking about? Classical: Physical experiments to understand flow. 2

3 Motivation What will I be talking about? Modern: Numerical experiments (CFD) to understand flow. 3

4 Motivation What will I be talking about? Modern: Numerical experiments (CFD) to understand flow. 4

5 Motivation What will I be talking about? Flow Visualization Model Flow Data Visual Representation Insight + Understanding Simulation (CFD) or Experiment Human Visual System 5

6 Motivation What will I be talking about? Scalar Fields Scientific Visualization Tensor Fields Vector Fields Geometric Methods Topological Visualization Feature-Based Texture-Based (time permitting) 6

7 Motivation Flow visualization is an old discipline... Leonardo DaVinci, ca

8 Motivation Point of departure: Vector fields are integral constituents in many scientific application domains Astrophysics, Fusion, Combustion, Medicine, Fluid Flow,... Flow & vector field visualization is an ingredient in answering typical science questions How & why are things transported? Where do things go? How & why do things mix? 8

9 Vector Fields So... what is a vector field? Simple answer: a map v : I D IR n I IR, D IR n that returns a direction and speed for every point in a domain. 9

10 Vector Fields Simplest way to visualize vector fields: draw arrows or glyphs. 10

11 Vector Fields Simplest way to visualize vector fields: draw arrows or glyphs. 11

12 Vector Fields Simplest way to visualize vector fields: draw arrows or glyphs. Glyphs do not describe and provide insight into transport or mixing... 12

13 Vector Fields Vector fields induce a family of ordinary differential equations (ODEs): dx dt = v(t, x) ODE x(t 0 )=x 0 initial condition Any solution for a given x0, t0 is called integral curve, trajectory or orbit. 13

14 Vector Fields Graphically: v v (t 0,x 0 ) v 14

15 Vector Fields Drawing trajectories is a simple way to visualize vector fields and convey some insight. Intuition: idealized particles, embedded in a flow. 15

16 Vector Fields Drawing trajectories is a simple way to visualize vector fields and convey some insight. Intuition: idealized particles, embedded in a flow. 16

17 Vector Fields Do trajectories always exist? Existence and uniqueness of integral curves if and only if v is continuous in t, and Lipschitz-continuous in x v(t, x) v(t, y) <L x y x, y D, t I Holds for virtually all vector fields from simulations. 17

18 Vector Fields 18

19 Vector Fields 19

20 Vector Fields Integral curves are typically computed forward in time, but can also be computed backward in time. v v v (t 0,x 0 ) Find the curve that ends at x0, t0. 20

21 Vector Fields 21

22 Vector Fields 22

23 Vector Fields 23

24 Vector Fields Integral curves provide a simple answer to two basic (but important!) questions: Where do things go? Where are things coming from? Modern analysis of transport and mixing is based on study of integral curves (more later). 24

25 Numerical Integration How does one compute integral curves? Closed form solutions (analytic) to integral curve ODE are not available in general. One has to use numerical approximation methods, called ODE solvers or numerical integration schemes. Basic ODE solver description: given, computes a discrete approximation of (t). v x 25

26 Numerical Integration Euler scheme: simplest method, fast, inaccurate, unstable (x 0,t 0 ) (x 1,t 1 ) (x 2,t 2 ) (x 3,t 3 ) (x 4,t 4 ) x n+1 = x n + tv(x n,t n )+O(( t) 2 ) 26

27 Numerical Integration Runge-Kutta, 2nd order: do Euler step first, then improve k1 k1 = v(x n,t n ) (x 0,t 0 ) k 2 (x 1,t 1 ) k2 = v(x n t k 1,t n t) x n+1 = x n + t k 2 + O(( t ) 3 ) (2 stages) 27

28 Numerical Integration k1 = v(x n,t n ) k2 = v(x n t k 1,t n t) x n+1 = x n + t k 2 + O(( t ) 3 ) can also be written as a table: 0 ½ ½ 0 1 Butcher tableau 28

29 Numerical Integration Runge-Kutta schemes exist for arbitrary orders 3rd (3 stages), 4th (4 stages), 5th (6 stages),... Textbook on Numerical ODEs 0 1/2 1/2 1/2 0 1/ /6 1/3 1/3 1/6 RK4 Butcher tableau 29

30 Numerical Integration t How to choose? In principle, need to make sure that no information is missed t needs to be small enough such that nothing is missed. 30

31 Numerical Integration Adaptive stepsize control, chooses observed vector field complexity. t based on Popular schemes: Runge-Kutta-Fehlberg (RKF / RK45) six stage method simultaneous order 4 and order 5 approximations comparison allows error estimate, increase or decrease in step size in response Many visualization systems (e.g., VTK) implement RKF. 31

32 Short Detour: Vector Field Interpolation

33 Vector Field Interpolation Numerical integrators treat as a black box: put in position x and time t, get vector back. But, vector fields are typically given in discrete form over some computational mesh, with vectors associated to points. v 33

34 Vector Field Interpolation Interpolation can be used to reconstruct a continuous mesh from the discrete data: linear interpolation over triangles or tetrahedra trilinear interpolation over hexahedra or voxels non-linear interpolation for other mesh/cell types Hard problem for irregular meshes: which cell contains x? 34

35 Vector Field Interpolation Typical use case for spatial data structures: octree, kd-tree,... But, modern simulation meshes are complex: Many points and cells (1M 10B) Adaptive resolution cells are smaller where vector field more complex strong variation in cell size (factor 1M is not unusual) Non-linear cell types (hex, pyramids, prisms) Holes and complex boundaries 35

36 Vector Field Interpolation 36

37 Vector Field Interpolation Good cell location data structure is essential in such cases for acceptable performance when computing integral curves integral curves / particles x 1000 integration steps per curve x 4 vector field evaluations per step (RK4) x 1 ms per evaluation (large mesh) = 4,000s (1h 7min) 37

38 Numerical Integration Integration efficiency depends on lookups of : fast spatial data structure for mesh types that require it reduce the number of lookups, i.e. integration steps? v How does the choice of numerical integration scheme figure into this? Higher approximation order same error for larger steps Lower vector field complexity can take larger + fewer steps Adaptive stepsize control is the key to efficiency. 38

39 Numerical Integration One more caveat: Visualization requires graphical output... so must be constrained to guarantee truthful output: t? More steps, less efficiency...? 39

40 Numerical Interpolation Thankfully, solution is simple: Numerical integration scheme with dense output: compute a polynomial description of the curve in between the solution points xn. faithful graphical output: sample polynomial in between the xn. Example: Dormand-Prince scheme (DOPRI5) 5th-order adaptive dense Runge-Kutta scheme 40

41 Vector Field Glyphs For completeness: Glyph techniques 41

42 Vector Field Glyphs Glyphs are very simple to implement and understand, but provide very little understanding. Other problems: Displaying a vector requires two visual attributes: direction and magnitude Drawing a glyph requires more than one pixel: Visual collisions, glyphs overlap. Challenging to visualize a large vector field. 42

43 Streamlines and Pathlines Direct Integral Curve Visualization Streamlines ( Pathlines ( v v is constant in time) varies with time) 43

44 Streamlines and Pathlines Animated Particles (also convey speed) 44

45 Streamlines and Pathlines Hybrid: Pathlets Small integral curve segments around moving particle position. Convey speed and orientation. 45

46 Streamlines and Pathlines Other variants:... Tubes Ribbons 46

47 Time and Streak Lines Physical Experiment Délery,

48 Time and Streak Lines Time Lines: mimic line of smoke or dye timeline T = 1 T = 2 T = 3 Set of adjacently particles at an instant of time 48

49 Time and Streak Lines Streak Lines: mimic flow of smoke or dye from a nozzle Set of all fluid particles that have gone through a common point at some time in the past. 49

50 Time and Streak Lines Time Lines vs. Streak Lines 50