Vector Field Topology 8-1 Ronald Peikert SciVis 2007 - Vector Field Topology
Vector fields as ODEs What are conditions for existence and uniqueness of streamlines? • For the initial value problem ( ) ( ) ( ) ( ) i = = x t x x t v x t 0 0 ( ) a solution exists if the velocity field is continuous. v x • The solution is unique if the field is Lipschitz-continuous, i.e. if q p there is a constant M such that ( ) ( ) ( ) ( ) ′ ′ − ≤ − v x v x M x x ′ x for all in a neighborhood of x . 8-2 Ronald Peikert SciVis 2007 - Vector Field Topology
Vector fields as ODEs Lipschitz-continuous is stronger than continuous (C 0 ) but weaker than continuously differentiable (C 1 ). Important for scientific visualization: • piecewise multilinear functions are Lipschitz-continuous p p • in particular cellwise bi- or trilinear interpolation is Lipschitz- continuous Consequence: Numerical vector fields do have unique streamlines, but analytic vector fields don't necessarily but analytic vector fields don t necessarily. 8-3 Ronald Peikert SciVis 2007 - Vector Field Topology
Vector fields as ODEs Example: for the vector field ( ) ( ) ( ) ( ) ( ) = = 2 / 3 v x u x y , , v x y , 1, 3 y the initial value problem ( ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) i 0 = = x x x t v x t 0 has the two solutions ( ) ( ) ( ( ) ) = = + + x x t t x x t t , 0 0 red 0 ( ) ( ) = + 3 x t x t t , blue 0 Both are streamlines seeded at ( ( ) ) x 0 ,0 0 ,0 the point p . 8-4 Ronald Peikert SciVis 2007 - Vector Field Topology
Special streamlines ( ) ( ) It is possible that a streamline x t maps two different times t and t' to the same point: ( ) ( ) = = x t x t ' x 1 There are two types of such special streamlines: = v x ( ( ) ) 0 • stationary points: If y p , , then the streamline degenerates to g 1 1 a single point ( ) ( ) = ∈ R x t x t 1 ( ) ≠ 0 • periodic orbits: If v x , then the streamline is periodic: 1 ( ) ( ) ( ) + = ∈ R ∈ Z x t kT x t t , k All other streamlines are called regular streamlines. 8-5 Ronald Peikert SciVis 2007 - Vector Field Topology
Special streamlines Regular streamlines can converge to stationary points or periodic Regular streamlines can converge to stationary points or periodic orbits, in either positive or negative time. However, because of the uniqueness, a regular streamline cannot contain a stationary point or periodic orbit. Examples: convergence to E l t • a stationary point • a periodic orbit 8-6 Ronald Peikert SciVis 2007 - Vector Field Topology
Critical points A stationary point x c is called a critical point if the velocity gradient ( ) = ∇ at x c is regular (is a non-singular matrix, has nonzero J v x d determinant). i ) Near a critical point, the field can be approximated by its li linearization i ti ( ) ( ) + = + 2 v x x Jx O x c Properties of critical points: P ti f iti l i t • in a neighborhood, the field takes all possible directions • critical points are isolated (as opposed to general stationary points, e.g. points on a no slip boundary) 8-7 Ronald Peikert SciVis 2007 - Vector Field Topology
Critical points Critical points can have different types, depending on the eigenvalues of J , more precisely on the signs of the real parts of the eigenvalues of the eigenvalues. We define an important subclass: We define an important subclass: A critical point is called hyperbolic if all eigenvalues of J have A critical point is called hyperbolic if all eigenvalues of J have nonzero real parts. The main property of hyperbolic critical points is structural stability: Adding a small perturbation to v ( x ) does not change the topology of the nearby streamlines. 8-8 Ronald Peikert SciVis 2007 - Vector Field Topology
Critical points in 2D Hyperbolic critical points in 2D can be classified as follows: • two real eigenvalues: – both positive: both positive: node source node source – both negative: node sink – different signs: g saddle • two conjugate complex eigenvalues: – positive real parts: focus source – negative real parts: i l f focus sink i k 8-9 Ronald Peikert SciVis 2007 - Vector Field Topology
Critical points in 2D In 2D the eigenvalues are the zeros of + + = 2 x px q 0 where p and q are the two invariants: h d th t i i t ( ) = − = − λ + λ p trace( ) J 1 2 = = = = λλ λλ q q det( ) det( ) J J 1 2 The eigenvalues are complex exactly if the discriminant = − 2 D p p 4 q q is negative. It follows: • critical point types depend on signs of p,q and D < < > > ≠ ≠ q q 0 or 0, or q q 0 and 0 and p p 0 0 • hyperbolic points have either h b li i t h ith 8-10 Ronald Peikert SciVis 2007 - Vector Field Topology
Critical points in 2D The p-q chart (hyperbolic types printed in red) q q q=p 2 /4 D=0 D>0 D<0 (real (complex eigenvalues) eigenvalues) focus source focus sink node focus node focus node focus node focus center source sink node sink node source shear line source line sink p saddle saddle 8-11 Ronald Peikert SciVis 2007 - Vector Field Topology
Node source • positive trace • positive determinant • positive discriminant p Example ⎛ ⎞ ⎛ ⎞ 0.425 0.43125 0.5 0 = = = = − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J A A A A − ⎝ ⎠ ⎝ ⎠ 0.1 1.075 0 1 8-12 Ronald Peikert SciVis 2007 - Vector Field Topology
Node sink • negative trace • positive determinant • positive discriminant p Example − − − ⎛ ⎞ ⎛ ⎞ 0.425 0.43125 0.5 0 = = = = − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J A A A A − − ⎝ ⎠ ⎝ ⎠ 0.1 1.075 0 1 8-13 Ronald Peikert SciVis 2007 - Vector Field Topology
Saddle • any trace • negative determinant • positive discriminant p Example − − ⎛ ⎞ ⎛ ⎞ 0.43375 1.07812 0.25 0 = = = = − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J A A A A − ⎝ ⎠ ⎝ ⎠ 0.25 1.15 0 1 8-14 Ronald Peikert SciVis 2007 - Vector Field Topology
Focus source • positive trace counter-clockwise if ∂ ∂ − ∂ ∂ > • positive determinant v x u y 0 • negative discriminant g Example − − ⎛ ⎞ ⎛ ⎞ 1.48 1.885 0.5 1 = = = = − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J A A A A − ⎝ ⎠ ⎝ ⎠ 1.04 0.48 1 0.5 8-15 Ronald Peikert SciVis 2007 - Vector Field Topology
Focus sink • negative trace counter-clockwise if ∂ ∂ − ∂ ∂ > • positive determinant v x u y 0 • negative discriminant g Example − − ⎛ ⎞ ⎛ ⎞ 1.48 1.885 0.5 1 = = = = − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J A A A A − − − ⎝ ⎠ ⎝ ⎠ 1.04 0.48 1 0.5 8-16 Ronald Peikert SciVis 2007 - Vector Field Topology
Node focus source • positive trace between node source • positive determinant and focus source • zero discriminant (double real eigenvalue) ( g ) Example − ⎛ ⎞ ⎛ ⎞ 1.25 0.5625 0.5 0 = = = = − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J A A A A − ⎝ ⎠ ⎝ ⎠ 1 0.25 1 0.5 8-17 Ronald Peikert SciVis 2007 - Vector Field Topology
Star source Special case of node focus source: diagonal matrix Example ⎛ ⎞ ⎛ ⎞ 2 0 1 0 = = = = λ λ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J ⎝ ⎠ ⎝ ⎠ 0 2 0 1 8-18 Ronald Peikert SciVis 2007 - Vector Field Topology
Nonhyperbolic critical points If the eigenvalues have zero real parts but are nonzero (eigenvalues are purely imaginary), the critical point is the boundary case between focus source and focus sink. This type of critical point is called a center. yp p Depending on the higher derivatives, it can behave as a source or as a sink. Because a center is nonhyperbolic, it is not structurally stable in B t i h b li it i t t t ll t bl i general perturbation b t t but structurally stable if the field is divergence-free. t ll t bl if th fi ld i di f 8-19 Ronald Peikert SciVis 2007 - Vector Field Topology
Center • zero trace counter-clockwise if ∂ ∂ − ∂ ∂ > • positive determinant v x u y 0 • negative discriminant g Example − − ⎛ ⎞ ⎛ ⎞ 0.98 1.885 0 1 = = = = − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ J J A A A A − ⎝ ⎠ ⎝ ⎠ 1.04 0.98 1 0 8-20 Ronald Peikert SciVis 2007 - Vector Field Topology
Other stationary points Other stationary points in 2D: If J is a singular matrix, the following stationary (but not critical!) If J i i l t i th f ll i t ti (b t t iti l!) points are possible: • if a single eigenvalue is zero: line source, line sink • if b th if both eigenvalues are zero : i l pure shear h 8-21 Ronald Peikert SciVis 2007 - Vector Field Topology
Recommend
More recommend
