8 Vector Field Topology

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1 Vector fields as ODEs What are conditions for eistence and uniqueness of streamlines? 8 Vector Field Topology For the initial value problem ( t) = v( ( t) ) i t = 0 0 a solution eists if the velocity field v is continuous. The solution is unique if the field is Lipschitz-continuous, i.e. if there is a constant M such that M v v for all in a neighborhood of. Ronald Peikert SciVis Vector Field Topology 8-1 Ronald Peikert SciVis Vector Field Topology 8-2 Vector fields as ODEs 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 in particular cellwise bi- or trilinear interpolation is Lipschitzcontinuous Consequence: Numerical vector fields do have unique streamlines, but analytic vector fields don't necessarily. : for the vector field 2/3 = ( u(, y), v(, y) ) = ( 1,3y ) v the initial value problem ( t) = v( ( t) ) i has the two solutions red blue ( t) = ( 0 + t,0) 3 () t = ( 0 + t, t ) ( 0 ) = 0 Both are streamlines seeded at the point,0. 0 Ronald Peikert SciVis Vector Field Topology 8-3 Ronald Peikert SciVis Vector Field Topology 8-4 Special streamlines ( t ) 1 It is possible that a streamline maps two different times t and t' to the same point: t = t' = There are two types of such special streamlines: stationary points: If v ( 1) = 0, then the streamline degenerates to a single point t = t R periodic orbits: If v, then the streamline is periodic: 1 0 All other streamlines are called regular streamlines. 1 ( t+ kt) = ( t) ( t, k ) R Z 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. s: convergence to a stationary point a periodic orbit Special streamlines Ronald Peikert SciVis Vector Field Topology 8-5 Ronald Peikert SciVis Vector Field Topology 8-6 1

2 Critical points A stationary point c is called a critical point if the velocity gradient J= v at c is regular (is a non-singular matri, has nonzero determinant). Near a critical point, the field can be approimated by its linearization 2 ( c + ) = +O O v J Properties of critical points: 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) 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. We define an important subclass: 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() does not change the topology of the nearby streamlines. Ronald Peikert SciVis Vector Field Topology 8-7 Ronald Peikert SciVis Vector Field Topology 8-8 Critical points in 2D Hyperbolic critical points in 2D can be classified as follows: two real eigenvalues: both positive: node source both negative: node sink different signs: saddle two conjugate comple eigenvalues: positive real parts: focus source negative real parts: focus sink Critical points in 2D In 2D the eigenvalues are the zeros of 2 + p+ q= 0 where p and q are the two invariants: p = trace( J) = λ + λ 1 2 q = det( J) = λλ 1 2 The eigenvalues are comple eactly if the discriminant 2 D= p 4q is negative. It follows: critical point types depend on signs of p,q and D hyperbolic points have either q< 0, or q> 0 and p 0 Ronald Peikert SciVis Vector Field Topology 8-9 Ronald Peikert SciVis Vector Field Topology 8-10 The p-q chart (hyperbolic types printed in red) q=p 2 /4 focus source Critical points in 2D q D<0 (comple eigenvalues) focus sink D=0 D>0 (real eigenvalues) positive trace positive discriminant Node source node source node focus source center node focus sink node sink line source shear saddle line sink p Ronald Peikert SciVis Vector Field Topology 8-11 Ronald Peikert SciVis Vector Field Topology

3 Node sink Saddle negative trace positive discriminant any trace negative determinant positive discriminant = = J A 1 A Ronald Peikert SciVis Vector Field Topology = = J A 1 A Ronald Peikert SciVis Vector Field Topology 8-14 positive trace counter-clockwise if v u y >0 negative discriminant Focus source negative trace counter-clockwise if v u y >0 negative discriminant Focus sink Ronald Peikert SciVis Vector Field Topology Ronald Peikert SciVis Vector Field Topology 8-16 positive trace between node source and focus source zero discriminant (double real eigenvalue) Node focus source Special case of node focus source: diagonal matri Star source Ronald Peikert SciVis Vector Field Topology J = = λ Ronald Peikert SciVis Vector Field Topology

4 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. 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 general zero trace counter-clockwise if v u y >0 negative discriminant Center perturbation but structurally stable if the field is divergence-free. Ronald Peikert SciVis Vector Field Topology = = J A 1 A Ronald Peikert SciVis Vector Field Topology 8-20 Other stationary points Other stationary points in 2D: If J is a singular matri, the following stationary (but not critical!) points are possible: positive trace zero determinant Line source if a single eigenvalue is zero: line source, line sink if both eigenvalues are zero : pure shear Ronald Peikert SciVis Vector Field Topology 8-21 Ronald Peikert SciVis Vector Field Topology 8-22 zero trace zero determinant Pure shear The topological skeleton The topological skeleton consists of all periodic orbits and all streamlines converging (in either direction of time) to a saddle point (separatri of the saddle), or a critical point on a no-slip boundary It provides a kind of segmentation of the 2D vector field s: Ronald Peikert SciVis Vector Field Topology 8-23 Ronald Peikert SciVis Vector Field Topology

5 The topological skeleton The topological skeleton : irrotational vector fields. An irrotational (conservative) vector field is the gradient of a scalar field (its potential). : LIC and topology-based visualization (skeleton plus a few etra streamlines). Skeleton of an irrotational vector field: watershed image of its potential field. Discussion: watersheds are topologically defined, integration required height ridges are geometrically defined, locally detectable Ronald Peikert SciVis Vector Field Topology 8-25 Ronald Peikert SciVis Vector Field Topology 8-26 The topological skeleton : topological skeleton of a surface flow Critical points in 3D Hyperbolic critical points in 3D can be classified as follows: three real eigenvalues: all positive: source two positive, one negative: 1:2 saddle (1 in, 2 out) one positive, two negative: 2:1 saddle (2 in, 1 out) all negative: sink one real, two comple eigenvalues: positive real eigenvalue, positive real parts: spiral source positive real eigenvalue, negative real parts : 2:1 spiral saddle negative real eigenvalue, positive real parts : 1:2 spiral saddle negative real eigenvalue, negative real parts : spiral sink image credit: A. Globus Ronald Peikert SciVis Vector Field Topology 8-27 Ronald Peikert SciVis Vector Field Topology 8-28 Critical points in 3D : The Lorenz attractor Types of hyperbolic critical points in 3D source spiral source 2:1 saddle 2:1 spiral saddle The other 4 types are obtained by reversing arrows The Lorenz attractor has 3 critical points: a 2:1 saddle P 0 ( 10 ( y ), 28 y z, y 8z 3) v = ( 0,0,0) at with eigenvalues { 22.83, 2.67, 11.82} two 1:2 spiral saddles P 1 and P 2 at ( 6 2, 6 2, 27) and 6 2, 6 2, 27 with eigenvalues 13.85, 0.09 ± 10.19i { } Ronald Peikert SciVis Vector Field Topology 8-29 Ronald Peikert SciVis Vector Field Topology

6 Streamlines : The Lorenz attractor stable manifold W s (P 0 ) Streamsurfaces (2D separatrices) Visualization based on 3D critical points : Flow over delta wing, glyphs (icons) for critical point types, 1D separatrices ("topological vorte cores"). Discussion: Vorte core may not contain critical points. W u (P 1 ) W s (P 0 ) image: A.Globus W u (P 2 ) Ronald Peikert SciVis Vector Field Topology 8-31 Ronald Peikert SciVis Vector Field Topology 8-32 Poincaré map of a periodic orbit in 3D: Choose a point 0 on the periodic orbit Choose an open circular disk D centered at 0 on a plane which is not tangential to the flow, and small enough that the periodic orbit intersects D only in 0 Any streamline seeded at a point D which intersects D a net time at a point D defines a mapping from to ' There eists a smaller open disk D0 D centered at 0 such that this mapping is defined for all points D 0. This is the Poincaré map. Using coordinates on the plane of D and with origin at 0, the Poincaré map can now be linearized: P where P is 22 matri. Important fact about Poincaré maps: The eigenvalues of P are independent of the choice of 0 on the periodic orbit the orientation of the plane of D the choice of coordinates for the plane A periodic orbit is called hyperbolic, if its eigenvalues lie off the comple unit circle. Hyperbolic p.o. are structurally stable. Ronald Peikert SciVis Vector Field Topology 8-33 Ronald Peikert SciVis Vector Field Topology 8-34 Hyperbolic periodic orbits in 3D can be classified as follows: Two real eigenvalues: both outside the unit circle: source p.o. both inside the unit circle: sink p.o. one outside, one inside: both positive: saddle p.o. both negative: twisted saddle p.o. Two comple conjugate eigenvalues: both outside the unit circle: spiral source p.o. both inside the unit circle: spiral sink p.o. Types of hyperbolic periodic orbits in 3D source p.o. spiral source p.o. saddle p.o. twisted saddle p.o. Types sink and spiral sink are obtained by reversing arrows. Ronald Peikert SciVis Vector Field Topology 8-35 Ronald Peikert SciVis Vector Field Topology

7 : Flow in Pelton distributor ring. Streamlines and streamsurfaces (manually seeded). Critical point of spiral saddle type and p.o. of twisted saddle type. Stable (yellow, red) and unstable (black, blue) manifolds. The topological skeleton of 3D vector fields contains 1D and 2D separatrices of (spiral) saddles. Not directly usable for visualization (too much occlusion). Alternative: only show intersection curves of 2D separatrices. Two types of saddle connectors: heteroclinic orbit: connects two (spiral) saddles homoclinic orbits: connects a (spiral) saddle with itself Idea: a 1D "skeleton" is obtained, not providing a segmentation, but indicating flow between pairs of saddles Ronald Peikert SciVis Vector Field Topology 8-37 Ronald Peikert SciVis Vector Field Topology 8-38 Comparison: icons / full topological skeleton / saddle connectors In rotational flow, a connected pair of spiral saddles can describe a vorte breakdown bubble. P 1 (2:1 spiral saddle) ideal case: W s (P 1 ) coincides with W u (P 2 ) no saddle connector P 2 (1:2 spiral saddle) Flow past a cylinder: Image credit: H. Theisel perturbed case: transversal intersection of W s (P 1 ) and W u (P 2 ) saddle connector consists of two streamlines P 1 P 2 Image credit: Krasny/Nitsche Ronald Peikert SciVis Vector Field Topology 8-39 Ronald Peikert SciVis Vector Field Topology D view Eperimental photograph of a vorte breakdown bubble Image credit: Sotiropoulos et al. If v is velocity field of a fluid: Folds must have constant mass flu. Close to P 1 or P 2 this is approimately ρv dn ρωar (density * angular velocity * cross section area * radius). It follows: cross section area ~ 1/radius Consequence: Shilnikov chaos Vorte breakdown bubble in flow over delta wing, visualization by streamsurfaces (not topology-based) Image credit: Sotiropoulos et al. Image credit: C. Garth Ronald Peikert SciVis Vector Field Topology 8-41 Ronald Peikert SciVis Vector Field Topology

8 Vorte breakdown bubble found in CFD data of Francis draft tube: Ronald Peikert SciVis Vector Field Topology

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