Fractals/Mathematics/Vector field

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Vector field[1]

Here mainly numerical methods for time independent 2D vector fields are described.


Dictionary[edit]

  • vector function is a function that gives vector as an output
  • field : space (plane, sphere, ... )
  • field line is a line that is everywhere tangent to a given vector field
  • scalar/ vector / tensor:
    • Scalars are real numbers used in linear algebra. Scalar is a tensor of zero order
    • Vector is a tensor of first order. Vector is an extension of scalar
    • tensor is an extension of vector

Vector[edit]

Forms of 2D vector:[2]

  • [z1] ( only one complex number when first point is known , for example z0 is origin
  • [z0, z1] = two complex numbers
  • 4 scalars ( real numbers)
    • [x, y, dx , dy]
    • [x0, y0, x1, y1]
    • [x, y, angle, magnitude]
  • 2 scalars : [x1, y1] for second complex number when first point is known , for example z0 is origin


Field types[edit]

Algorithm[edit]

  • start with
    • plane (parameter plane or dynamic plane)
    • scalar function
    • vector function
  • create scalar field using scalar function ( potential)
  • create vector field from scalar field using vector function ( gradient of the potential)
  • compute:

Field line computing[edit]

Problems:

  • Field line tracing = curve sketching
  • drawing contour maps ( in computer graphic) = Numerical continuation ( in math)



Methods ( solvers) available for the field-lines[4]

  • Euler
  • RK2
  • RK3
  • RK4 - the original authors of those sampling algorithms; Herr Runge und Herr Kutta.
  None of these 4 methods generate an exact answer, but they are (from left to right) increasingly more accurate. They also take (from left to right) more and more time to finish as they require more samples for each iteration.
  You won't be able to create reliably closed curves using iterative sampling methods as small errors at any step may be amplified in successive steps. There is also no guarantee that the field-line ends up in the exact coordinate where it started.
  The Grasshopper metaball solver on the other hand uses a marching squares algorithm which is capable of finding closed loops because it is a grid-cell approach and sampling inaccuracy in one area doesn't carry over to another. 
  However the solving of iso-curves is a very different process    from the solving of particle trajectories through fields. ... 
  Typically field lines shoot to infinity rather than form closed loops. That is one reason why I chose the RK methods here, because marching-cubes is very bad at dealing with things that tend to infinity.[5]

Construction[edit]

Construction of a field line

Given a vector field and a starting point a field line can be constructed iteratively by finding the field vector at that point . The unit tangent vector at that point is: . By moving a short distance along the field direction a new point on the line can be found

Then the field at that point is found and moving a further distance in that direction the next point of the field line is found

By repeating this and connecting the points,the field line can be extended as far as desired. This is only an approximation to the actual field line, since each straight segment isn't actually tangent to the field along its length, just at its starting point. But by using a small enough value for , taking a greater number of shorter steps, the field line can be approximated as closely as desired. The field line can be extended in the opposite direction from by taking each step in the opposite direction by using a negative step .


rk4 numerical integration method[edit]

Fourth-order Runge-Kutta (RK4) in case of 2D time independent vector field


is a vector function that for each point p

p = (x, y)

in a domain assigns a vector v

where each of the functions is a scalar function:

A field line is a line that is everywhere tangent to a given vector field.

Let r(s) be a field line given by a system of ordinary differential equations, which written on vector form is:

where:

  • s representing the arc length along the field line
  • is a seed point

Given a seed point on the field line, the update rule ( RK4) to find the next point along the field line is[6]

where:

  • h is the step size
  • k are the intermediate vectors:

Visualisation of vector field[edit]

Plot types (Visualization Techniques for Flow Data) : [7]

  • Glyphs = Icons or signs for visualizing vector fields
    • simplest glyph = Line segment (hedgehog plots)
    • arrow plot = quiver plot = Hedgehogs (global arrow plots)
  • Characteristic Lines [8]
    • streamlines = curve everywhere tangential to the instantaneous vector (velocity) field (time independent vector field). For time independent vector field streaklines = Path lines = streak lines [9]
  • texture (line integral convolution = LIC)[10]
  • Topological skeleton [11]
    • fixed point extraction ( Jacobian)


LIC[edit]

input:

    • white noise
    • original vector field


LIC examples:

quiver plot[edit]

Definition

  • "A quiver plot displays velocity vectors as arrows with components (u,v) at the points (x,y)"[12]

stream plot[edit]

A stream plot uses stream lines

Example fields[edit]

Potential of Mandelbrot set[edit]

Programs[edit]

Examples[edit]

Images[edit]

Shadertoy[edit]

Videos[edit]

See also[edit]

References[edit]

  1. Vector field in wikipedia
  2. Euclidian vector in wikipedia
  3. Numerical Methods for Particle Tracing in Vector Fields by Kenneth I. Joy
  4. liruics : Introduction to Scientific Visualization - Flow Field
  5. grasshopper3d forum: field-lines-how-to-rebuild-and-make-periodic?overrideMobileRedirect=1
  6. Classification and visualisation of critical points in 3d vector fields. Master thesis by Furuheim and Aasen
  7. Flow Visualisation from TUV
  8. Data visualisation by Tomáš Fabián
  9. A Streakline Representation of Flow in Crowded Scenes from UCF
  10. lic by Zhanping Liu
  11. Vector Field Topology in Flow Analysis and Visualization by Guoning Chen
  12. matlab ref : quiver plot