Topics in Flow Visualization Lecture 15 April 14, 2020 Outline - - PowerPoint PPT Presentation

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Topics in Flow Visualization Lecture 15 April 14, 2020 Outline - - PowerPoint PPT Presentation

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CS53000 - Spring 2020 Introduction to Scientific Visualization Topics in Flow Visualization Lecture 15 April 14, 2020 Outline Vortices Flow separation and attachment Lagrangian Coherent Structures CS530 / Spring 2020 : Introduction to

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CS53000 - Spring 2020

Introduction to Scientific Visualization

Lecture

Topics in Flow Visualization

April 14, 2020

15

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Outline

Vortices Flow separation and attachment Lagrangian Coherent Structures

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortices

Intuitive notion

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http://www.cse.ohio-state.edu/~jiang/Vortex/ieeeVis02.ppt

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortices

Intuitive notion

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http://www.cse.ohio-state.edu/~jiang/Vortex/ieeeVis02.ppt

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Introduction

A vortex is the rotating motion of a multitude of material particles around a common center

  • H. J. Lugt, The dilemma of defining a vortex, Recent Developments in Theoretical and Experimental Fluid

Mechanics, Springer, 1979

A vortex exists when its streamlines, mapped onto a plane normal to its core, exhibit a circular or spiral pattern, under an appropriate reference frame (→ self referential!)

  • S. K. Robinson, Coherent motions in the turbulent boundary layer, Ann. Rev. Fluid Mech., vol. 23, 1991

A vortex is comprised of a central core region surrounded by swirling streamlines

  • L. M. Portela, Identification and characterization of vortices in the turbulent boundary layer. Ph.D. thesis,

Stanford University, 1997

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortices

Vortex core

Line-type center of swirling motion (skeleton) Region surrounded by swirling streamlines

Vortex region

Includes surrounding streamlines

Vortex boundary

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Region-based Definitions

Simple criteria

High vorticity magnitude (quantifies flow rotation) Low pressure (swirling motion around region of low pressure) High helicity: , norm. hel.: (vorticity in flow direction)

Thresholds on these quantities yield regions

Bounded by isosurfaces Visualized by volume rendering

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⇤ = ⇥ ⇤ v

h = ⌅ · ⌅ v

hn = ⌅ · ⌅ v |⌅ ||⌅ v|

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Region-based Definitions

Lambda2 criterion

Derived from Navier-Stokes equation Idea: split Jacobian into symmetric (S) and antisymmetric (Q) parts: Constraint on medium eigenvalue of : Correspond to local minima of pressure Widely used in practice (standard)

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  • J. Jeong, F. Hussain, On the Identification of a Vortex, J. Fluid Mechanics, 1995

J = ⇤ v

S = J + JT 2

S2 + Q2

λ2 ≤ 0

Q = J − JT 2

<latexit sha1_base64="c4d4R13GKNk/a9ygIJMIfF/dwF0=">AB/nicbVDLSgNBEOyNrxhfq+LJy2AQvBh2o6AXIehFckogL0jWMDuZTYbMPpiZFcKy4K948aCIV7/Dm3/jJNmDJhY0FXdHe5EWdSWda3kVtZXVvfyG8WtrZ3dvfM/YOWDGNBaJOEPBQdF0vKWUCbilO5Gg2Hc5bvju6nfqRCsjBoqElEHR8PA+YxgpW+uZRHd2gnicwSaroHFUfGmlSTvtm0SpZM6BlYmekCBlqfOrNwhJ7NAEY6l7NpWpJwEC8UIp2mhF0saYTLGQ9rVNMA+lU4yOz9Fp1oZIC8UugKFZurviQT7Uk58V3f6WI3kojcV/O6sfKunYQFUaxoQOaLvJgjFaJpFmjABCWKTzTBRDB9KyIjrLNQOrGCDsFefHmZtMol+6JUrl8WK7dZHk4hM4AxuoAL3UIMmEjgGV7hzXgyXox342PemjOymUP4A+PzB87MlBw=</latexit>
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Lambda2 criterion

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Region-based Definitions

Isosurface of Lambda2 (-0.1) Picture by M. Rütten, DLR Göttingen

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortex Core Line Extraction

Topology

Line-type separatrix of spiral saddle is a vortex core

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  • M. Tobak, D. J. Peake, Topology of 3D separated flow, Ann. Rev. Fluid Mech., vol. 14, 1982
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortex Core Line Extraction

Predictor/Corrector algorithm

Correlation between vorticity direction and pressure minimum along vortex core Vorticity direction (predictor) Pressure minimum in normal plane (corrector)

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  • D. Banks, B. Singer, Vortex tubes in turbulent flows: Identification,

representation, Reconstruction, IEEE Visualization 1994

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortex Core Line Extraction

Predictor/Corrector algorithm

Combined with thresholding in normal plane to yield tubes

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  • D. Banks, B. Singer, Vortex tubes in turbulent flows: Identification,

representation, Reconstruction, IEEE Visualization 1994

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortex Core Line Extraction

First-order method

Piece-wise linear interpolation Line-type center of swirling motion extracted as intersection of spiral saddle’s separatrix with cell interior

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  • D. Sujudi, R. Haimes, Identification of swirling flow in 3D vector fields, AIAA Paper 95-1715, 1995

Sketch by M. Roth

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortex Core Line Extraction

First-order method

Vortex cores have zero curvature Disconnected segments Correct in linear flows

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  • D. Sujudi, R. Haimes, Identification of swirling flow in 3D vector fields, AIAA Paper 95-1715, 1995

Sketch by M. Roth

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortex Core Line Extraction

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  • D. Kenwright, R. Haimes, Vortex identification - applications in aerodynamics: A case study, IEEE Visualization 1997
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Vortex Core Line Extraction

Parallel Operator: previous methods rely on

the parallelism of two vector fields Sujudi/Haimes (first-order)

Velocity parallel to acceleration

Parallel Operator applied point-wise yields continuous lines

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  • R. Peikert, M. Roth, The Parallel Vectors Operator - A Vector Field Visualization Primitive, IEEE Visualization 1999

⌅ v/ /⌅ a

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Galilean Invariance

Fundamental principle in physics (Newton) Vortex core lines should be independent of particular inertial reference frame Streamline-based methods depend on particular frame of reference Region-based definitions (e.g., ) are galilean invariant

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λ2

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Galilean Invariance

Idea: derive line-type information from

region-type scalar criterion by extract ridge / valley lines

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  • J. Sahner, T. Weinkauf, H.-C. Hege, Galilean Invariant Extraction and Iconic Representation of Vortex Core Lines, Eurographics 2005.
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CS530 / Spring 2020 : Introduction to Scientific Visualization.

Time-Dependent Vortices

  • Lagrangian Averaged Vorticity Deviation:
  • Add up vorticity deviation from regional average

along path lines

  • Convex regions surround large local maxima are

(resilient) Lagrangian vortices

20 Advanced Flow Visualization April 13, 2020

  • G. Haller, A. Hadjighasem, M. Farazmand, F. Huhn, Defining Coherent Vortices

Objectively from the Vorticity, J. Fluid Mech. 795, 2016.

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21 CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

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21 CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Definition(s) of Flow Separation

Flow abruptly leaves or returns to solid body (2D/3D phenomenon) Occurs along separation / attachment lines Critical in low speed flight configurations (takeoff, landing)

Reduced lift Control issues Beneficial for delta wings and fighter aircrafts

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Definition(s) of Flow Separation

Visualization of separation / attachment lines

On embedded surface Visualization vs. extraction of lines Based on analysis of wall streamlines in shear stress vector field (no slip boundary conditions) No formal characterization “Streamlines tend to accumulate” Heuristics needed

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Definition(s) of Flow Separation

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  • D. Kenwright, C. Henze, C. Levit, Feature Extraction of Separation and Attachment Lines, IEEE TVCG 5(2), 1999
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Topological Approach

In 2D: separation and attachment points are critical points of tangential velocity

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  • J. Helman, L. Hesselink, Visualizing Vector Field Topology in Fluid Flow Data Sets,

IEEE Computer Graphics and Applications 11(3), 1991

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Topological Approach

Topological approach

In 3D: separatrices on surfaces integrated from saddle points are closed separation / attachment lines

Issues

Separation / attachment lines can be open... ... although some disagree (Haller et al.) ... but not always critical points present on the surface in CFD simulation data!

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  • A. Globus, C. Levit, T. Lasinski, A Tool for Visualizing the Topology of 3D Vector Fields, IEEE Visualization 1991
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Topological Approach

Work by Surana, Haller and others argue that flow separation is indeed a topological structure

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Topological Approach

Work by Surana, Haller and others argue that flow separation is indeed a topological structure

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Topological Approach

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  • A. Surana, O. Grunberg, & G. Haller, Exact

theory of three-dimensional flow

  • separation. Part I. Steady separation,
  • J. Fluid. Mech., 564 (2006) 57-103.
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020 30

Superimposition of wall streamlines on oil flow patterns from experiment

Texture-based Methods

  • W. De Leeuw, H.-G. Pagendarm, F. Post, B. Walter, Visual Simulation of Experimental Oil-Flow Visualization by

Spot Noise Images from Numerical Flow Simulation, 6th EG Workshop on Scientific Visualization, 1995

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Modified Spot Noise algorithm

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Texture-based Methods

Standard Spot Noise

  • W. De Leeuw, H.-G. Pagendarm, F. Post, B. Walter, Visual Simulation of Experimental Oil-Flow Visualization by

Spot Noise Images from Numerical Flow Simulation, 6th EG Workshop on Scientific Visualization, 1995

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Modified Spot Noise algorithm

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Texture-based Methods

Spot Noise + convergence encoding

  • W. De Leeuw, H.-G. Pagendarm, F. Post, B. Walter, Visual Simulation of Experimental Oil-Flow Visualization by

Spot Noise Images from Numerical Flow Simulation, 6th EG Workshop on Scientific Visualization, 1995

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Modified Spot Noise algorithm

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Texture-based Methods

Spot Noise + convergence encoding + advection

  • W. De Leeuw, H.-G. Pagendarm, F. Post, B. Walter, Visual Simulation of Experimental Oil-Flow Visualization by

Spot Noise Images from Numerical Flow Simulation, 6th EG Workshop on Scientific Visualization, 1995

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020 34

Cell-wise feature line extraction

Basic observation: separation / attachment lines present in two linear flow patterns

Kenwright’s Method

saddle node

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Cell-wise feature line extraction

Idea: extract intersection of those lines with each cell in piecewise linear flow

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Kenwright’s Method

saddle node

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020 36

Cell-wise feature line extraction

Algorithm: for each cell Compute position and type of critical point (linear extrapolation outside cell) If saddle point or node compute intersection(s) of line directed by eigenvectors with cell If intersection found determine feature type (eigenvalues) and add to list

Kenwright’s Method

  • D. Kenwright, Automatic Detection of Open and Closed Separation and

Attachment Lines, IEEE Visualization 1998

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Cell-wise feature line extraction

Limitations Disconnected line segments (Jacobian is cell-wise constant) False positives (e.g. parallel flow) Problems with curved lines (nonlinear pattern)

Kenwright’s Method

  • D. Kenwright, Automatic Detection of Open and Closed Separation and Attachment Lines, IEEE Visualization 1998
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Point-wise method

Observation: in saddle and node patterns separation and attachment lines correspond to position where Velocity is parallel to eigenvector of Jacobian Velocity is an eigenvector of Jacobian Flow curvature is zero (streamline = straight line)

Kenwright’s Method

= ⇧ v × J⇧ v ||⇧ v||3 = ⇧ v × ⇥⇧ v ||⇧ v||3 = 0

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Point-wise method

Method: search for zero flow curvature For each vertex

Compute point-wise jacobian Compute curvature

Extract zero isoline of curvature Yields connected line segments (closed curves) Post-processing required to filter out false positives (e.g., flow parallelism

  • f feature line)

Kenwright’s Method

  • D. Kenwright, Chris Henze, Creon Levit, Feature Extraction of Attachment and Separation Lines, IEEE TVCG 5(2), 1999
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Kenwright’s Method

Point-wise method

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Shortcomings of Topology

In time-dependent setting

Topology concerned with streamlines (Euler) Limit sets and infinite-time integral curves (asymptotic convergence) Time treated as parametric domain Lagrange: interested in behavior of particle motion

  • ver time

application cases cover a finite time interval

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Shortcomings of Topology

In 3D flows

Connection between topology and flow features of interest is often unsatisfactory

Flow attachment / separation manifolds Vortices More complex flow structures for little is known ahead of time (exploratory flow visualization)

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Shortcomings of Features

Lack of formal criteria Conflicting definitions False positives No quantification of associated uncertainty

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Finite-Time Lyapunov Exponent

Lagrangian analysis

characterize the coherence of the behavior of (massless) particles in the flow incorporate transient nature of phenomenon

For Visualization

well defined (formal expression) -> algorithms

no a priori knowledge is required -> offline processing provide quantitative information naturally encodes uncertainty

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CS530 / Spring 2020 : Introduction to Scientific Visualization.

Finite-Time Lyapunov Exponent

Measure local variation of pathline trajectories after a finite time interval Intuitive interpretation: maximum stretching of infinitesimal volume element along the flow

45 Advanced Flow Visualization April 13, 2020

σ∆t(t, x) := 1 ∆tln

  • λmax(Dx φ∆t(t, x))
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CS530 / Spring 2020 : Introduction to Scientific Visualization.

Finite-Time Lyapunov Exponent

Measure local variation of pathline trajectories after a finite time interval Intuitive interpretation: maximum stretching of infinitesimal volume element along the flow

45 Advanced Flow Visualization April 13, 2020

σ∆t(t, x) := 1 ∆tln

  • λmax(Dx φ∆t(t, x))
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SLIDE 48

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Finite-Time Lyapunov Exponent

Measure local variation of pathline trajectories after a finite time interval Intuitive interpretation: maximum stretching of infinitesimal volume element along the flow

45 Advanced Flow Visualization April 13, 2020

σ∆t(t, x) := 1 ∆tln

  • λmax(Dx φ∆t(t, x))
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CS530 / Spring 2020 : Introduction to Scientific Visualization.

Finite-Time Lyapunov Exponent

Measure local variation of pathline trajectories after a finite time interval Intuitive interpretation: maximum stretching of infinitesimal volume element along the flow

45 Advanced Flow Visualization April 13, 2020

σ∆t(t, x) := 1 ∆tln

  • λmax(Dx φ∆t(t, x))
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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Finite-Time Lyapunov Exponent

The flow map… …maps a particle seeded at position x to its position after advection along the flow

Derivative (Jacobian) of this map describes the

local variations of the flow map

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φ : I ⊆ I R × U ⊆ Mn − → Mn

⌅⌥ ⇥(t, x) ⌅t

  • t=τ

= ⌥ v(, x)

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Finite-Time Lyapunov Exponent

Local variations of the flow map: Stretching magnitude: Maximal stretch: LCS extracted as ridges of FTLE

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φ(x + dx) = φ(x) + rφ|x dx + O(| |dx| |2)

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kφ(x + dx) φ(x)k2 ⇡ krφ dxk2 = hrφ dx, rφ dxi = D (rφ)T rφ dx, dx E

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λmax ⇣ (rφ)T rφ ⌘

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Finite-Time Lyapunov Exponent

Local variations of the flow map: Stretching magnitude: Maximal stretch: LCS extracted as ridges of FTLE

47

φ(x + dx) = φ(x) + rφ|x dx + O(| |dx| |2)

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kφ(x + dx) φ(x)k2 ⇡ krφ dxk2 = hrφ dx, rφ dxi = D (rφ)T rφ dx, dx E

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Cauchy-Green strain tensor

λmax ⇣ (rφ)T rφ ⌘

<latexit sha1_base64="6fN6xIsk7XfjqQ0h/7erKyJra6k=">ACKHicbVDLSgMxFM3Ud31VXboJFkE3ZaYKurPoxqVCq0Knljtpg3NZIbkjliGfo4bf8WNiCLd+iWm7Szq40DC4ZxzSe4JEikMu7IKczNLywuLa8UV9fWNzZLW9s3Jk414w0Wy1jfBWC4FIo3UKDkd4nmEAWS3wb9i7F/+8C1EbGq4yDhrQi6SoSCAVqpXTrzpQ13oJ1F8Dj0JQ/xIL8VBL8pCd8Lbo9PLyv06lGZ8R2qexW3AnoX+LlpExyXLVLb34nZmnEFTIJxjQ9N8FWBhoFk3xY9FPDE2B96PKmpQoiblrZNEh3bdKh4axtkchnaizExlExgyiwCYjwJ757Y3F/7xmiuFpKxMqSZErNn0oTCXFmI5box2hOUM5sASYFvavlPVA0PbdGW4P1e+S+5qVa8o0r1+rhcO8/rWCa7ZI8cEI+ckBq5JFekQRh5Ii/knXw4z86r8+mMptGCk8/skB9wvr4Bz72nAw=</latexit>
slide-53
SLIDE 53

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Finite-Time Lyapunov Exponent

Spectral norm of jacobian measures maximum stretch around considered position FTLE quantifies hyperbolic divergence Interpretation: large values of FTLE for forward advection: presence

  • f an underlying repelling manifold

large values of FTLE for backward advection: presence of an underlying attracting manifold

48

σ∆t(t, x) := 1 ∆tln

  • λmax(Dx φ∆t(t, x))
slide-54
SLIDE 54

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Finite-Time Lyapunov Exponent

Interpretation:

Both attracting and repelling manifolds constitute Coherent Lagrangian Structures

49

slide-55
SLIDE 55

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Finite-Time Lyapunov Exponent

Interpretation:

Both attracting and repelling manifolds constitute Coherent Lagrangian Structures

49

slide-56
SLIDE 56

CS530 / Spring 2020 : Introduction to Scientific Visualization.

FTLE Computation

  • Integrate path lines from a regular grid of

seed points from time to time

  • Record final position (= )
  • Use finite differences to compute derivative
  • f flow map

50 Advanced Flow Visualization April 13, 2020

xijk

<latexit sha1_base64="iG2KzEGIloF0azpQEqpbtyF/VE=">AB+XicbVDLSsNAFL2pr1pfUZduBovgqiRV0GXRjcsK9gFtKJPpB07mYSZSbGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b/APjxqyiRhLZIxCPZ9bGinAna0kxz2o0lxaHPacef3OZ+Z0qlYpF40LOYeiEeCRYwgrWRBrbdD7Ee+0H6lA1S9jJBnbVqTlzoFXiFqQKBZoD+6s/jEgSUqEJx0r1XCfWXoqlZoTrNJPFI0xmeAR7RkqcEiVl86TZ+jMKEMURNI8odFc/b2R4lCpWeibyTynWvZy8T+vl+jg2kuZiBNBVkcChKOdITyGtCQSUo0nxmCiWQmKyJjLDHRpqyKcFd/vIqadr7kWtfn9ZbdwUdZThBE7hHFy4gbcQRNaQGAKz/AKb1ZqvVjv1sditGQVO8fwB9bnD14glCI=</latexit>

t0

<latexit sha1_base64="r4TK2bpUlOS9+KYXmxMgjhtyG1s=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeCF48V7Qe0oW62m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GNzO/cS1EbF6wEnC/YgOlQgFo2ile+y7/XLFrbpzkFXi5aQCORr98ldvELM04gqZpMZ0PTdBP6MaBZN8WuqlhieUjemQdy1VNOLGz+anTsmZVQYkjLUthWSu/p7IaGTMJApsZ0RxZJa9mfif10xvPYzoZIUuWKLRWEqCcZk9jcZCM0ZyoklGlhbyVsRDVlaNMp2RC85ZdXSatW9S6qtbvLSv0xj6MIJ3AK5+DBFdThFhrQBAZDeIZXeHOk8+K8Ox+L1oKTzxzDHzifPw5ojb0=</latexit>

t0 + T

<latexit sha1_base64="4k0xVSgOU47EmCzpfJeH2O5f98=">AB7nicbVBNS8NAEJ34WetX1aOXxSIQkmqoMeCF48V+gVtqJvtpl262YTdiVBCf4QXD4p49fd489+4bXPQ1gcDj/dmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR23TJxqxpslrHuBNRwKRvokDJO4nmNAokbwfju5nfuLaiFg1cJwP6JDJULBKFqpjX2XJGv1R2K+4cZJV4OSlDjnq/9NUbxCyNuEImqTFdz03Qz6hGwSfFnup4QlYzrkXUsVjbjxs/m5U3JulQEJY21LIZmrvycyGhkziQLbGVEcmWVvJv7ndVMb/1MqCRFrthiUZhKgjGZ/U4GQnOGcmIJZVrYWwkbU0Z2oSKNgRv+eV0qpWvKtK9eG6XHvM4yjAKZzBXhwAzW4hzo0gcEYnuEV3pzEeXHenY9F65qTz5zAHzifP8iBjqQ=</latexit>

φt0+T

t0

(xijk)

<latexit sha1_base64="y3UTHSM7npbDGAqOENEuRmjYORk=">ACDXicbVC7TsMwFHV4lvIKMLJYFKQipCopSDBWYmEsUl9SU4LjOq2p40S2g6i/ALv8LCAEKs7Gz8DU6bAVqOZOvonHt17z1exKhUlvVtLCwuLa+sFtaK6xubW9vmzm5LhrHApIlDFoqOhyRhlJOmoqRTiQICjxG2t7oMvPb90RIGvKGkekF6ABpz7FSGnJNQ+daEjdRLlWepP9J4207ARIDT0/eUjdhN6N0mPXLFkVawI4T+yclECOumt+Of0QxwHhCjMkZde2ItVLkFAUM5IWnViSCOERGpCuphwFRPaSyTUpPNJKH/qh0I8rOF/dyQokHIceLoyW1TOepn4n9eNlX/RSyiPYkU4ng7yYwZVCLNoYJ8KghUba4KwoHpXiIdIKx0gEUdgj178jxpVSv2aV6fVaq3eZxFMA+OABlYINzUANXoA6aAINH8AxewZvxZLwY78bHtHTByHv2wB8Ynz/c0pwo</latexit>

1 2h   φt0+T

t0

(x(i+1)jk) φt0+T

t0

(x(i−1)jk) φt0+T

t0

(xi(j+1)k) φt0+T

t0

(xi(j−1)k) φt0+T

t0

(xij(k+1)) φt0+T

t0

(xij(k−1))   ⇡ rφt0+T

t0

(xijk)

<latexit sha1_base64="Ulz/DlINLWea5VbjSft7K3s9XCU=">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</latexit>
slide-57
SLIDE 57

CS530 / Spring 2020 : Introduction to Scientific Visualization.

FTLE Computation

  • Integrate path lines from a regular grid of

seed points from time to time

  • Record final position (= )
  • Use finite differences to compute derivative
  • f flow map

50 Advanced Flow Visualization April 13, 2020

xijk

<latexit sha1_base64="iG2KzEGIloF0azpQEqpbtyF/VE=">AB+XicbVDLSsNAFL2pr1pfUZduBovgqiRV0GXRjcsK9gFtKJPpB07mYSZSbGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b/APjxqyiRhLZIxCPZ9bGinAna0kxz2o0lxaHPacef3OZ+Z0qlYpF40LOYeiEeCRYwgrWRBrbdD7Ee+0H6lA1S9jJBnbVqTlzoFXiFqQKBZoD+6s/jEgSUqEJx0r1XCfWXoqlZoTrNJPFI0xmeAR7RkqcEiVl86TZ+jMKEMURNI8odFc/b2R4lCpWeibyTynWvZy8T+vl+jg2kuZiBNBVkcChKOdITyGtCQSUo0nxmCiWQmKyJjLDHRpqyKcFd/vIqadr7kWtfn9ZbdwUdZThBE7hHFy4gbcQRNaQGAKz/AKb1ZqvVjv1sditGQVO8fwB9bnD14glCI=</latexit>

t0

<latexit sha1_base64="r4TK2bpUlOS9+KYXmxMgjhtyG1s=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeCF48V7Qe0oW62m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GNzO/cS1EbF6wEnC/YgOlQgFo2ile+y7/XLFrbpzkFXi5aQCORr98ldvELM04gqZpMZ0PTdBP6MaBZN8WuqlhieUjemQdy1VNOLGz+anTsmZVQYkjLUthWSu/p7IaGTMJApsZ0RxZJa9mfif10xvPYzoZIUuWKLRWEqCcZk9jcZCM0ZyoklGlhbyVsRDVlaNMp2RC85ZdXSatW9S6qtbvLSv0xj6MIJ3AK5+DBFdThFhrQBAZDeIZXeHOk8+K8Ox+L1oKTzxzDHzifPw5ojb0=</latexit>

t0 + T

<latexit sha1_base64="4k0xVSgOU47EmCzpfJeH2O5f98=">AB7nicbVBNS8NAEJ34WetX1aOXxSIQkmqoMeCF48V+gVtqJvtpl262YTdiVBCf4QXD4p49fd489+4bXPQ1gcDj/dmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR23TJxqxpslrHuBNRwKRvokDJO4nmNAokbwfju5nfuLaiFg1cJwP6JDJULBKFqpjX2XJGv1R2K+4cZJV4OSlDjnq/9NUbxCyNuEImqTFdz03Qz6hGwSfFnup4QlYzrkXUsVjbjxs/m5U3JulQEJY21LIZmrvycyGhkziQLbGVEcmWVvJv7ndVMb/1MqCRFrthiUZhKgjGZ/U4GQnOGcmIJZVrYWwkbU0Z2oSKNgRv+eV0qpWvKtK9eG6XHvM4yjAKZzBXhwAzW4hzo0gcEYnuEV3pzEeXHenY9F65qTz5zAHzifP8iBjqQ=</latexit>

φt0+T

t0

(xijk)

<latexit sha1_base64="y3UTHSM7npbDGAqOENEuRmjYORk=">ACDXicbVC7TsMwFHV4lvIKMLJYFKQipCopSDBWYmEsUl9SU4LjOq2p40S2g6i/ALv8LCAEKs7Gz8DU6bAVqOZOvonHt17z1exKhUlvVtLCwuLa+sFtaK6xubW9vmzm5LhrHApIlDFoqOhyRhlJOmoqRTiQICjxG2t7oMvPb90RIGvKGkekF6ABpz7FSGnJNQ+daEjdRLlWepP9J4207ARIDT0/eUjdhN6N0mPXLFkVawI4T+yclECOumt+Of0QxwHhCjMkZde2ItVLkFAUM5IWnViSCOERGpCuphwFRPaSyTUpPNJKH/qh0I8rOF/dyQokHIceLoyW1TOepn4n9eNlX/RSyiPYkU4ng7yYwZVCLNoYJ8KghUba4KwoHpXiIdIKx0gEUdgj178jxpVSv2aV6fVaq3eZxFMA+OABlYINzUANXoA6aAINH8AxewZvxZLwY78bHtHTByHv2wB8Ynz/c0pwo</latexit>

1 2h   φt0+T

t0

(x(i+1)jk) φt0+T

t0

(x(i−1)jk) φt0+T

t0

(xi(j+1)k) φt0+T

t0

(xi(j−1)k) φt0+T

t0

(xij(k+1)) φt0+T

t0

(xij(k−1))   ⇡ rφt0+T

t0

(xijk)

<latexit sha1_base64="Ulz/DlINLWea5VbjSft7K3s9XCU=">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</latexit>

expensive!!

slide-58
SLIDE 58

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020 51

Adaptive approximation of flow map 


Lagrangian Flow Analysis

slide-59
SLIDE 59

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020 51

Adaptive approximation of flow map 


Lagrangian Flow Analysis

slide-60
SLIDE 60

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Incremental Flow

Adaptive resolution of fine detail

52

slide-61
SLIDE 61

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

Incremental Flow

Adaptive resolution of fine detail

52

slide-62
SLIDE 62

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

FTLE-based 3D Visualization

53

Delta wing Delta wing (cropped)

Direct Volume Visualization of FTLE+ and FTLE-:

slide-63
SLIDE 63

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

FTLE-based 3D Visualization

User-guided PDF

54

Section plane orthogonal to main flow direction Delta Wing

slide-64
SLIDE 64

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

FTLE-based 3D Visualization

User-guided PDF

54

Section plane orthogonal to main flow direction Delta Wing Pathlines colored according to PDF

slide-65
SLIDE 65

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020

FTLE-based 3D Visualization

55

ICE train Backward stream surfaces from FTLE+ ridges

slide-66
SLIDE 66

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020 56

Turbulent jet into steady medium

slide-67
SLIDE 67

CS530 / Spring 2020 : Introduction to Scientific Visualization. Advanced Flow Visualization April 13, 2020 56

Turbulent jet into steady medium

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SLIDE 68

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Subdivision Scheme

  • Multiresolution sampling ( )
  • Subdivision operator S:
  • Compare and to assess

approximation quality

57 Advanced Flow Visualization April 13, 2020

grid points are f l

i := f

i h

  • ,

i = 0,...,2l

where h := 1

2l

nverges p

(S f l)2i := f l

i

(S f l)2i+1 := 1 2

  • f l

i + f l i+1

  • ˜

f l

i = (S f l−1)i.

and f l

i

4-point Refinement

slide-69
SLIDE 69

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Subdivision Scheme

  • Multiresolution sampling ( )
  • Subdivision operator S:
  • Compare and to assess

approximation quality

57 Advanced Flow Visualization April 13, 2020

grid points are f l

i := f

i h

  • ,

i = 0,...,2l

where h := 1

2l

nverges p

(S f l)2i := f l

i

(S f l)2i+1 := 1 2

  • f l

i + f l i+1

  • ˜

f l

i = (S f l−1)i.

and f l

i

  • 4-point Refinement
slide-70
SLIDE 70

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Subdivision Scheme

  • Multiresolution sampling ( )
  • Subdivision operator S:
  • Compare and to assess

approximation quality

57 Advanced Flow Visualization April 13, 2020

grid points are f l

i := f

i h

  • ,

i = 0,...,2l

where h := 1

2l

nverges p

(S f l)2i := f l

i

(S f l)2i+1 := 1 2

  • f l

i + f l i+1

  • ˜

f l

i = (S f l−1)i.

and f l

i

  • 4-point Refinement
slide-71
SLIDE 71

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Example

58 Advanced Flow Visualization April 13, 2020

points of f l where di < ε points of f l where di > ε points of f l+1 taken from prediction ˜ f l+1 points of f l+1 computed using function evaluation

4-point Refinement

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SLIDE 72

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Smooth Subdivision

  • 4-point subdivision scheme

59 Advanced Flow Visualization April 13, 2020

1 256 ·
  • 9
  • 9

1

  • 9

1 81 81 81

  • 9

1

  • 9
  • 9

1

  • 9
  • 9

face point 81 edge midpoint

  • 1

9 9

  • 1
1 16 ·

4-point Refinement

slide-73
SLIDE 73

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Adaptive Sampling

60 Advanced Flow Visualization April 13, 2020

Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications, Garth et al., IEEE Vis 2007

4-point Refinement

slide-74
SLIDE 74

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Results

61 Advanced Flow Visualization April 13, 2020

Dataset time-dep. Resolution Evaluations

  • rel. l2-error
  • rel. l∞-error

TDELTA wing no 1024×1153 12% 5.32·10−5 7.38·10−3 TDELTA slice no 2048×1024 7% 6.64·10−5 9.18·10−3 Cylinder yes 20482 9% 6.79·10−8 9.12·10−6 Can dataset yes 1283 14% 9.43·10−7 1.33·10−1 ICE train no 2892 ×65 5% 3.81·10−4 3.37·10−1 TDELTA box no 257×321×65 25% 9.59·10−6 2.6·10−3

4-point Refinement

slide-75
SLIDE 75

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Results

62 Advanced Flow Visualization April 13, 2020

Dataset time-dep. Resolution Evaluations

  • rel. l2-error
  • rel. l∞-error

TDELTA wing no 1024×1153 12% 5.32·10−5 7.38·10−3 TDELTA slice no 2048×1024 7% 6.64·10−5 9.18·10−3 Cylinder yes 20482 9% 6.79·10−8 9.12·10−6 Can dataset yes 1283 14% 9.43·10−7 1.33·10−1 ICE train no 2892 ×65 5% 3.81·10−4 3.37·10−1 TDELTA box no 257×321×65 25% 9.59·10−6 2.6·10−3

4-point Refinement

slide-76
SLIDE 76

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Ridge-based Refinement

  • Extract / detect ridges at each intermediate

resolution and discard empty cells

63 Advanced Flow Visualization April 13, 2020

Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction, Filip Sadlo, Ronald Peikert, IEEE Visualization 2007

Filtered AMR

slide-77
SLIDE 77

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Dense Adaptive Refinement

64 Advanced Flow Visualization April 13, 2020

(a) (b)

Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction, Filip Sadlo, Ronald Peikert, IEEE Visualization 2007

Filtered AMR

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SLIDE 78

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Ridge-based Refinement

  • Results

65 Advanced Flow Visualization April 13, 2020

direct adaptive initial grid 193x193x97 (3613153 nodes) 13x13x7 (1183 nodes) final grid 193x193x97 (3613153 nodes) 298964 nodes flow map [s] 19953.51 2350.21 FTLE [s] 10.73 30.73 ridge extr. [s] 278.46 2337.16 total [s] 20242.74 4930.72

Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction, Filip Sadlo, Ronald Peikert, IEEE Visualization 2007

Filtered AMR

slide-79
SLIDE 79

CS530 / Spring 2020 : Introduction to Scientific Visualization.

Results

66 Advanced Flow Visualization April 13, 2020

Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction, Filip Sadlo, Ronald Peikert, IEEE Visualization 2007

Filtered AMR