Data Visualization. Fall 2017
|
|
- Leonard Francis
- 5 years ago
- Views:
Transcription
1 Data Visualization Fall 2017
2 Vector Fields Vector field v: D R n D is typically 2D planar surface or 2D surface embedded in 3D n = 2 fields tangent to 2D surface n = 3 volumetric fields When visualizing vector fields, we have to: map D to 2D screen (like with scalar fields) Then we have 1 pixel for 2 or 3 scalar values (only 1 for scalar fields) Fall 2017 Data Visualization 2
3 Vector Fields A wind tunnel model of a Cessna 182 showing a wingtip vortex Source: Fall 2017 Data Visualization 3
4 Vector Fields We can compute derived scalar quantities from vector fields and use already known methods for these: Divergence (R n R) degree of field s convergence or divergence v x v v z x y z div v = + + or equivalently div v = y lim Γ 0 1 Γ Γ ( v n ) Γ ds Fall 2017 Data Visualization 4
5 Vector Fields Divergence (R n R) Source: Eric Shaffer, Vector Field Visualization Fall 2017 Data Visualization 5
6 Vector Fields Fall 2017 Data Visualization 6 Curl (rotor, vorticity) (R n R n ) rot v = or equivalently rot v = rot v is axis of the rotation and its magnitude is magnitude of rotation y v x v x v z v z v y v x y z x y z,, Γ Γ Γ ds v 1 lim 0
7 Vector Fields Curl(rotor, vorticity) (R n R n ) 2D fluid flow simulated by NSE and visualized using vorticity Fall 2017 Source: A Simple Fluid Solver based on the FFT, J. Stam, J. of Graphics Tools 6(2), 2001, Data Visualization 7
8 Approximating Derivatives Analytical functions Ideally we are given a function in a closed form and we can (probably) compute previous derivatives analytically Sometimes the given function is too complicated We compute the derivatives numerically via differentiation Discretely sampled data The best approach is to fit a function to the data and compute analytical derivatives Numerical evaluation may be fater than analytical Fall 2017 Data Visualization 8
9 Approximating Derivatives Central-difference furmulas of Order O(h 2 ): Fall 2017 Data Visualization 9
10 Approximating Derivatives Central-difference furmulas of Order O(h 4 ): Fall 2017 Data Visualization 10
11 Numerical Integration of ODE Initial value problem: y unknown function dt integration time step y (t) = f(t, y(t)) Fast but not very accurate Higher methods available Fall 2017 Data Visualization 11
12 Numerical Integration of ODE First order Euler method: x(t + dt) =x(t) +v(x(t))dt x represents spatial position v vector field dt integration time step Fast but not very accurate Higher methods available Fall 2017 Data Visualization 12
13 Numerical Integration of ODE Second order Runge-Kutta method: x(t + dt) =x(t) + ½ (K 1 + K 2 ) where K 1 = v(x(t))dt K 2 = v(x(t) + K 1 )dt Fall 2017 Data Visualization 13
14 Numerical Integration of ODE Fourth order Runge-Kutta method (aka RK4): x(t + dt) =x(t) + 1/6 (K 1 + 2K 2 + 2K 3 + K 4 ) where K 1 = v(x(t))dt K 2 = v(x(t) + ½ K 1 )dt K 3 = v(x(t) + ½ K 2 )dt K 4 = v(x(t) + K 3 )dt Fall 2017 Data Visualization 14
15 Glyphs Icons or signs for visualizing vector fields Placed by (sub)sampling the dataset domain Attributes (scale, color, orientation) map vector data at sample points Simplest glyph: Line segment (hedgehog plots) Line from x to x + kv Optionally color map v Fall 2017 Data Visualization 15
16 Glyphs Simplest glyph: Line segment (hedgehog plots) Source: Eric Shaffer, Vector Field Visualization Fall 2017 Data Visualization 16
17 Glyphs Another variant: 3D cone and 3D arrow glyphs Shows orientation better than lines Use shading to separate overlapping glyphs Source: Eric Shaffer, Vector Field Visualization Fall 2017 Data Visualization 17
18 Glyphs Related Problems No interpolation in glyph space (unlike for scalar plots) A glyph take more space than a pixel (clutter) Complicated visual interpolation of arrows Scalar plots are dense, glyph plots are sparse Glyph positioning is important: Uniform grid Rotated grid Random samples (generally best solution) Fall 2017 Data Visualization 18
19 Characteristic Lines Important approaches of characteristic lines in a vector field: Stream line curve everywhere tangential to the instantaneous vector (velocity) field (time independent vector field) Local technique initiated from one or a few particles Trace of the direction of the flow at one single time step Curves can look different at each time for unsteady flows Represents the direction of the flow at a given fixed time Computationally expensive The scene becomes cluttered when the number of streamlines is increased Fall 2017 Data Visualization 19
20 Characteristic Lines Extensions of stream lines for time-varying data follows: Path lines trajectories of massless particles in the flow Trace of a single particle in continuous time Time dependent vector field Fall 2017 Data Visualization 20
21 Characteristic Lines Streak lines trace of dye that is released into the flow at a fixed position Curve that connects the position of all the particles which are initiated at the certain point but at different times Time lines propagation of a line of massless elements in time In steady flow a path line = streak line = stream line Fall 2017 Data Visualization 21
22 Further Reading Fall 2017 Data Visualization 22
Vector Visualization
Vector Visualization Vector Visulization Divergence and Vorticity Vector Glyphs Vector Color Coding Displacement Plots Stream Objects Texture-Based Vector Visualization Simplified Representation of Vector
More informationVector Visualization Chap. 6 March 7, 2013 March 26, Jie Zhang Copyright
ector isualization Chap. 6 March 7, 2013 March 26, 2013 Jie Zhang Copyright CDS 301 Spring, 2013 Outline 6.1. Divergence and orticity 6.2. ector Glyphs 6.3. ector Color Coding 6.4. Displacement Plots (skip)
More informationLecture overview. Visualisatie BMT. Vector algorithms. Vector algorithms. Time animation. Time animation
Visualisatie BMT Lecture overview Vector algorithms Tensor algorithms Modeling algorithms Algorithms - 2 Arjan Kok a.j.f.kok@tue.nl 1 2 Vector algorithms Vector 2 or 3 dimensional representation of direction
More informationVector Visualisation 1. global view
Vector Field Visualisation : global view Visualisation Lecture 12 Institute for Perception, Action & Behaviour School of Informatics Vector Visualisation 1 Vector Field Visualisation : local & global Vector
More informationVector Visualization. CSC 7443: Scientific Information Visualization
Vector Visualization Vector data A vector is an object with direction and length v = (v x,v y,v z ) A vector field is a field which associates a vector with each point in space The vector data is 3D representation
More informationVector Field Visualisation
Vector Field Visualisation Computer Animation and Visualization Lecture 14 Institute for Perception, Action & Behaviour School of Informatics Visualising Vectors Examples of vector data: meteorological
More informationAn Introduction to Flow Visualization (1) Christoph Garth
An Introduction to Flow Visualization (1) Christoph Garth cgarth@ucdavis.edu Motivation What will I be talking about? Classical: Physical experiments to understand flow. 2 Motivation What will I be talking
More informationVector Field Visualization: Introduction
Vector Field Visualization: Introduction What is a Vector Field? A simple 2D steady vector field A vector valued function that assigns a vector (with direction and magnitude) to any given point. It typically
More informationPart I: Theoretical Background and Integration-Based Methods
Large Vector Field Visualization: Theory and Practice Part I: Theoretical Background and Integration-Based Methods Christoph Garth Overview Foundations Time-Varying Vector Fields Numerical Integration
More informationData Visualization (CIS/DSC 468)
Data Visualization (CIS/DSC 468) Vector Visualization Dr. David Koop Visualizing Volume (3D) Data 2D visualization slice images (or multi-planar reformating MPR) Indirect 3D visualization isosurfaces (or
More informationCIS 467/602-01: Data Visualization
CIS 467/602-01: Data Visualization Vector Field Visualization Dr. David Koop Fields Tables Networks & Trees Fields Geometry Clusters, Sets, Lists Items Items (nodes) Grids Items Items Attributes Links
More informationVector Field Visualization: Introduction
Vector Field Visualization: Introduction What is a Vector Field? Why It is Important? Vector Fields in Engineering and Science Automotive design [Chen et al. TVCG07,TVCG08] Weather study [Bhatia and Chen
More information3D vector fields. Contents. Introduction 3D vector field topology Representation of particle lines. 3D LIC Combining different techniques
3D vector fields Scientific Visualization (Part 9) PD Dr.-Ing. Peter Hastreiter Contents Introduction 3D vector field topology Representation of particle lines Path lines Ribbons Balls Tubes Stream tetrahedra
More informationUsing Integral Surfaces to Visualize CFD Data
Using Integral Surfaces to Visualize CFD Data Tony Mcloughlin, Matthew Edmunds,, Mark W. Jones, Guoning Chen, Eugene Zhang 1 1 Overview Flow Visualization with Integral Surfaces: Introduction to flow visualization
More informationChapter 6 Visualization Techniques for Vector Fields
Chapter 6 Visualization Techniques for Vector Fields 6.1 Introduction 6.2 Vector Glyphs 6.3 Particle Advection 6.4 Streamlines 6.5 Line Integral Convolution 6.6 Vector Topology 6.7 References 2006 Burkhard
More informationFlow Visualisation - Background. CITS4241 Visualisation Lectures 20 and 21
CITS4241 Visualisation Lectures 20 and 21 Flow Visualisation Flow visualisation is important in both science and engineering From a "theoretical" study of o turbulence or o a fusion reactor plasma, to
More informationFlow Visualisation 1
Flow Visualisation Visualisation Lecture 13 Institute for Perception, Action & Behaviour School of Informatics Flow Visualisation 1 Flow Visualisation... so far Vector Field Visualisation vector fields
More informationFlow Visualization with Integral Surfaces
Flow Visualization with Integral Surfaces Visual and Interactive Computing Group Department of Computer Science Swansea University R.S.Laramee@swansea.ac.uk 1 1 Overview Flow Visualization with Integral
More informationOver Two Decades of IntegrationBased, Geometric Vector Field. Visualization
Over Two Decades of IntegrationBased, Geometric Vector Field Visualization Tony McLoughlin1, 1, Ronald Peikert2, Frits H. Post3, and Min Chen1 1 The Visual and Interactive Computing Group Computer Science
More information1 Mathematical Concepts
1 Mathematical Concepts Mathematics is the language of geophysical fluid dynamics. Thus, in order to interpret and communicate the motions of the atmosphere and oceans. While a thorough discussion of the
More informationLecture 1.1 Introduction to Fluid Dynamics
Lecture 1.1 Introduction to Fluid Dynamics 1 Introduction A thorough study of the laws of fluid mechanics is necessary to understand the fluid motion within the turbomachinery components. In this introductory
More informationMass-Spring Systems. Last Time?
Mass-Spring Systems Last Time? Implicit Surfaces & Marching Cubes/Tetras Collision Detection & Conservative Bounding Regions Spatial Acceleration Data Structures Octree, k-d tree, BSF tree 1 Today Particle
More informationVolume Illumination & Vector Field Visualisation
Volume Illumination & Vector Field Visualisation Visualisation Lecture 11 Institute for Perception, Action & Behaviour School of Informatics Volume Illumination & Vector Vis. 1 Previously : Volume Rendering
More informationA Broad Overview of Scientific Visualization with a Focus on Geophysical Turbulence Simulation Data (SciVis
A Broad Overview of Scientific Visualization with a Focus on Geophysical Turbulence Simulation Data (SciVis 101 for Turbulence Researchers) John Clyne clyne@ucar.edu Examples: Medicine Examples: Biology
More informationVector Visualization
Vector Visualization 5-1 Vector Algorithms Vector data is a three-dimensional representation of direction and magnitude. Vector data often results from the study of fluid flow, or when examining derivatives,
More informationFLUID MECHANICS TESTS
FLUID MECHANICS TESTS Attention: there might be more correct answers to the questions. Chapter 1: Kinematics and the continuity equation T.2.1.1A flow is steady if a, the velocity direction of a fluid
More informationSimulation vs. measurement vs. modelling 2D vs. surfaces vs. 3D Steady vs time-dependent d t flow Direct vs. indirect flow visualization
Flow Visualization Overview: Flow Visualization (1) Introduction, overview Flow data Simulation vs. measurement vs. modelling 2D vs. surfaces vs. 3D Steady vs time-dependent d t flow Direct vs. indirect
More informationPost Processing, Visualization, and Sample Output
Chapter 7 Post Processing, Visualization, and Sample Output Upon successful execution of an ADCIRC run, a number of output files will be created. Specifically which files are created depends upon how the
More informationSimulation in Computer Graphics. Particles. Matthias Teschner. Computer Science Department University of Freiburg
Simulation in Computer Graphics Particles Matthias Teschner Computer Science Department University of Freiburg Outline introduction particle motion finite differences system of first order ODEs second
More information2.29 Numerical Marine Hydrodynamics Spring 2007
Numerical Marine Hydrodynamics Spring 2007 Course Staff: Instructor: Prof. Henrik Schmidt OCW Web Site: http://ocw.mit.edu/ocwweb/mechanical- Engineering/2-29Spring-2003/CourseHome/index.htm Units: (3-0-9)
More informationmjb March 9, 2015 Chuck Evans
Vector Visualization What is a Vector Visualization Problem? A vector has direction and magnitude. Typically science and engineering problems that work this way are those involving fluid flow through a
More information21. Efficient and fast numerical methods to compute fluid flows in the geophysical β plane
12th International Conference on Domain Decomposition Methods Editors: Tony Chan, Takashi Kako, Hideo Kawarada, Olivier Pironneau, c 2001 DDM.org 21. Efficient and fast numerical methods to compute fluid
More informationChapter 1 - Basic Equations
2.20 Marine Hydrodynamics, Fall 2017 Lecture 2 Copyright c 2017 MIT - Department of Mechanical Engineering, All rights reserved. 2.20 Marine Hydrodynamics Lecture 2 Chapter 1 - Basic Equations 1.1 Description
More informationCS205b/CME306. Lecture 9
CS205b/CME306 Lecture 9 1 Convection Supplementary Reading: Osher and Fedkiw, Sections 3.3 and 3.5; Leveque, Sections 6.7, 8.3, 10.2, 10.4. For a reference on Newton polynomial interpolation via divided
More informationFLOWING FLUIDS AND PRESSURE VARIATION
Chapter 4 Pressure differences are (often) the forces that move fluids FLOWING FLUIDS AND PRESSURE VARIATION Fluid Mechanics, Spring Term 2011 e.g., pressure is low at the center of a hurricane. For your
More informationAdarsh Krishnamurthy (cs184-bb) Bela Stepanova (cs184-bs)
OBJECTIVE FLUID SIMULATIONS Adarsh Krishnamurthy (cs184-bb) Bela Stepanova (cs184-bs) The basic objective of the project is the implementation of the paper Stable Fluids (Jos Stam, SIGGRAPH 99). The final
More informationAnatomy of a Physics Engine. Erwin Coumans
Anatomy of a Physics Engine Erwin Coumans erwin_coumans@playstation.sony.com How it fits together» Terminology» Rigid Body Dynamics» Collision Detection» Software Design Decisions» Trip through the Physics
More informationENGI Parametric & Polar Curves Page 2-01
ENGI 3425 2. Parametric & Polar Curves Page 2-01 2. Parametric and Polar Curves Contents: 2.1 Parametric Vector Functions 2.2 Parametric Curve Sketching 2.3 Polar Coordinates r f 2.4 Polar Curve Sketching
More informationLecture overview. Visualisatie BMT. Fundamental algorithms. Visualization pipeline. Structural classification - 1. Structural classification - 2
Visualisatie BMT Fundamental algorithms Arjan Kok a.j.f.kok@tue.nl Lecture overview Classification of algorithms Scalar algorithms Vector algorithms Tensor algorithms Modeling algorithms 1 2 Visualization
More informationMAE 3130: Fluid Mechanics Lecture 5: Fluid Kinematics Spring Dr. Jason Roney Mechanical and Aerospace Engineering
MAE 3130: Fluid Mechanics Lecture 5: Fluid Kinematics Spring 2003 Dr. Jason Roney Mechanical and Aerospace Engineering Outline Introduction Velocity Field Acceleration Field Control Volume and System Representation
More information8. Tensor Field Visualization
8. Tensor Field Visualization Tensor: extension of concept of scalar and vector Tensor data for a tensor of level k is given by t i1,i2,,ik (x 1,,x n ) Second-order tensor often represented by matrix Examples:
More informationFlow Visualization: The State-of-the-Art
Flow Visualization: The State-of-the-Art The Visual and Interactive Computing Group Computer Science Department Swansea University Swansea, Wales, UK 1 Overview Introduction to Flow Visualization (FlowViz)
More information1. Interpreting the Results: Visualization 1
1. Interpreting the Results: Visualization 1 visual/graphical/optical representation of large sets of data: data from experiments or measurements: satellite images, tomography in medicine, microsopy,...
More informationFundamental Algorithms
Fundamental Algorithms Fundamental Algorithms 3-1 Overview This chapter introduces some basic techniques for visualizing different types of scientific data sets. We will categorize visualization methods
More informationDebojyoti Ghosh. Adviser: Dr. James Baeder Alfred Gessow Rotorcraft Center Department of Aerospace Engineering
Debojyoti Ghosh Adviser: Dr. James Baeder Alfred Gessow Rotorcraft Center Department of Aerospace Engineering To study the Dynamic Stalling of rotor blade cross-sections Unsteady Aerodynamics: Time varying
More informationMath 348 Differential Geometry of Curves and Surfaces
Math 348 Differential Geometry of Curves and Surfaces Lecture 3 Curves in Calculus Xinwei Yu Sept. 12, 2017 CAB 527, xinwei2@ualberta.ca Department of Mathematical & Statistical Sciences University of
More informationVisualization Computer Graphics I Lecture 20
15-462 Computer Graphics I Lecture 20 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 12] November 20, 2003 Doug James Carnegie Mellon University http://www.cs.cmu.edu/~djames/15-462/fall03
More informationStream Hulls: A 3D Visualization Technique for Chaotic Dynamical Systems
Stream Hulls: A 3D Visualization Technique for Chaotic Dynamical Systems Kenny Gruchalla Elizabeth Bradley Department of Computer Science, University of Colorado at Boulder, Boulder, Colorado 80309 gruchall@cs.colorado.edu
More informationChE 400: Applied Chemical Engineering Calculations Tutorial 6: Numerical Solution of ODE Using Excel and Matlab
ChE 400: Applied Chemical Engineering Calculations Tutorial 6: Numerical Solution of ODE Using Excel and Matlab Tutorial 6: Numerical Solution of ODE Gerardine G. Botte This handout contains information
More informationData Visualization. Fall 2016
Data Visualization Fall 2016 Information Visualization Upon now, we dealt with scientific visualization (scivis) Scivisincludes visualization of physical simulations, engineering, medical imaging, Earth
More informationSurfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
More informationTexture-Based Visualization of Uncertainty in Flow Fields
Texture-Based Visualization of Uncertainty in Flow Fields Ralf P. Botchen 1 Daniel Weiskopf 1,2 Thomas Ertl 1 1 University of Stuttgart 2 Simon Fraser University Figure 1: Three different advection schemes
More informationDouble Pendulum. Freddie Witherden. February 10, 2009
Double Pendulum Freddie Witherden February 10, 2009 Abstract We report on the numerical modelling of a double pendulum using C++. The system was found to be very sensitive to both the initial starting
More informationThe Level Set Method. Lecture Notes, MIT J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations
The Level Set Method Lecture Notes, MIT 16.920J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations Per-Olof Persson persson@mit.edu March 7, 2005 1 Evolving Curves and Surfaces Evolving
More informationChapter 4 FLUID KINEMATICS
Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala McGraw-Hill, 2010 Chapter 4 FLUID KINEMATICS Lecture slides by Hasan Hacışevki Copyright The McGraw-Hill Companies,
More informationCS 231. Fluid simulation
CS 231 Fluid simulation Why Simulate Fluids? Feature film special effects Computer games Medicine (e.g. blood flow in heart) Because it s fun Fluid Simulation Called Computational Fluid Dynamics (CFD)
More informationA simple example. Assume we want to find the change in the rotation angles to get the end effector to G. Effect of changing s
CENG 732 Computer Animation This week Inverse Kinematics (continued) Rigid Body Simulation Bodies in free fall Bodies in contact Spring 2006-2007 Week 5 Inverse Kinematics Physically Based Rigid Body Simulation
More informationMathematics for chemical engineers
Mathematics for chemical engineers Drahoslava Janovská Department of mathematics Winter semester 2013-2014 Numerical solution of ordinary differential equations Initial value problem Outline 1 Introduction
More informationELEC Dr Reji Mathew Electrical Engineering UNSW
ELEC 4622 Dr Reji Mathew Electrical Engineering UNSW Review of Motion Modelling and Estimation Introduction to Motion Modelling & Estimation Forward Motion Backward Motion Block Motion Estimation Motion
More informationIntroduction to Scientific Visualization
Visualization Definition Introduction to Scientific Visualization Stefan Bruckner visualization: to form a mental vision, image, or picture of (something not visible or present to the sight, or of an abstraction);
More informationDevelopment of a Maxwell Equation Solver for Application to Two Fluid Plasma Models. C. Aberle, A. Hakim, and U. Shumlak
Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models C. Aberle, A. Hakim, and U. Shumlak Aerospace and Astronautics University of Washington, Seattle American Physical Society
More informationThe State of the Art in Flow Visualization, part 1: Direct, Texture-based, and Geometric Techniques
Volume 22 (2003), Number 2, yet unknown pages The State of the Art in Flow Visualization, part 1: Direct, Texture-based, and Geometric Techniques Helwig Hauser, Robert S. Laramee, Helmut Doleisch, Frits
More informationROSE-HULMAN INSTITUTE OF TECHNOLOGY
Introduction to Working Model Welcome to Working Model! What is Working Model? It's an advanced 2-dimensional motion simulation package with sophisticated editing capabilities. It allows you to build and
More informationTopic 5-6: Parameterizing Surfaces and the Surface Elements ds and ds. Big Ideas. What We Are Doing Today... Notes. Notes. Notes
Topic 5-6: Parameterizing Surfaces and the Surface Elements ds and ds. Textbook: Section 16.6 Big Ideas A surface in R 3 is a 2-dimensional object in 3-space. Surfaces can be described using two variables.
More informationTexture Advection. Ronald Peikert SciVis Texture Advection 6-1
Texture Advection Ronald Peikert SciVis 2007 - Texture Advection 6-1 Texture advection Motivation: dense visualization of vector fields, no seed points needed. Methods for static fields: LIC - Line integral
More information14.1 Vector Fields. Gradient of 3d surface: Divergence of a vector field:
14.1 Vector Fields Gradient of 3d surface: Divergence of a vector field: 1 14.1 (continued) url of a vector field: Ex 1: Fill in the table. Let f (x, y, z) be a scalar field (i.e. it returns a scalar)
More informationUpgraded Swimmer for Computationally Efficient Particle Tracking for Jefferson Lab s CLAS12 Spectrometer
Upgraded Swimmer for Computationally Efficient Particle Tracking for Jefferson Lab s CLAS12 Spectrometer Lydia Lorenti Advisor: David Heddle April 29, 2018 Abstract The CLAS12 spectrometer at Jefferson
More informationVisualization, Lecture #2d. Part 3 (of 3)
Visualization, Lecture #2d Flow visualization Flow visualization, Part 3 (of 3) Retrospect: Lecture #2c Flow Visualization, Part 2: FlowVis with arrows numerical integration Euler-integration Runge-Kutta-integration
More informationLagrangian methods and Smoothed Particle Hydrodynamics (SPH) Computation in Astrophysics Seminar (Spring 2006) L. J. Dursi
Lagrangian methods and Smoothed Particle Hydrodynamics (SPH) Eulerian Grid Methods The methods covered so far in this course use an Eulerian grid: Prescribed coordinates In `lab frame' Fluid elements flow
More information1.2 Numerical Solutions of Flow Problems
1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian
More informationComputational Study of Laminar Flowfield around a Square Cylinder using Ansys Fluent
MEGR 7090-003, Computational Fluid Dynamics :1 7 Spring 2015 Computational Study of Laminar Flowfield around a Square Cylinder using Ansys Fluent Rahul R Upadhyay Master of Science, Dept of Mechanical
More informationLecture notes: Visualization I Visualization of vector fields using Line Integral Convolution and volume rendering
Lecture notes: Visualization I Visualization of vector fields using Line Integral Convolution and volume rendering Anders Helgeland FFI Chapter 1 Visualization techniques for vector fields Vector fields
More informationScientific Visualization Example exam questions with commented answers
Scientific Visualization Example exam questions with commented answers The theoretical part of this course is evaluated by means of a multiple- choice exam. The questions cover the material mentioned during
More informationEngineering Mathematics (4)
Engineering Mathematics (4) Zhang, Xinyu Department of Computer Science and Engineering, Ewha Womans University, Seoul, Korea zhangxy@ewha.ac.kr Example With respect to parameter: s (arc length) r( t)
More informationThis exam will be cumulative. Consult the review sheets for the midterms for reviews of Chapters
Final exam review Math 265 Fall 2007 This exam will be cumulative. onsult the review sheets for the midterms for reviews of hapters 12 15. 16.1. Vector Fields. A vector field on R 2 is a function F from
More informationVerification of Moving Mesh Discretizations
Verification of Moving Mesh Discretizations Krzysztof J. Fidkowski High Order CFD Workshop Kissimmee, Florida January 6, 2018 How can we verify moving mesh results? Goal: Demonstrate accuracy of flow solutions
More informationSPC 307 Aerodynamics. Lecture 1. February 10, 2018
SPC 307 Aerodynamics Lecture 1 February 10, 2018 Sep. 18, 2016 1 Course Materials drahmednagib.com 2 COURSE OUTLINE Introduction to Aerodynamics Review on the Fundamentals of Fluid Mechanics Euler and
More informationUsing LoggerPro. Nothing is more terrible than to see ignorance in action. J. W. Goethe ( )
Using LoggerPro Nothing is more terrible than to see ignorance in action. J. W. Goethe (1749-1832) LoggerPro is a general-purpose program for acquiring, graphing and analyzing data. It can accept input
More informationA Multi-Resolution Interpolation Scheme for Pathline Based Lagrangian Flow Representations
A Multi-Resolution Interpolation Scheme for Pathline Based Lagrangian Flow Representations Alexy Agranovsky a, Harald Obermaier a, Christoph Garth b, and Kenneth I. Joy a a University of California, Davis,
More informationLONG-TIME SIMULATIONS OF VORTEX SHEET DYNAMICS
LONG-TIME SIMULATIONS OF VORTEX SHEET DYNAMICS BY DANIEL SOLANO 1, LING XU 2, AND SILAS ALBEN 2 1 Rutgers University, New Brunswick, NJ 891, USA. 2 University of Michigan, Ann Arbor, MI 4819, USA. Abstract.
More information11/1/13. Visualization. Scientific Visualization. Types of Data. Height Field. Contour Curves. Meshes
CSCI 420 Computer Graphics Lecture 26 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 2.11] Jernej Barbic University of Southern California Scientific Visualization
More informationVisualization. CSCI 420 Computer Graphics Lecture 26
CSCI 420 Computer Graphics Lecture 26 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 11] Jernej Barbic University of Southern California 1 Scientific Visualization
More informationVisualization. Images are used to aid in understanding of data. Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [chapter 26]
Visualization Images are used to aid in understanding of data Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [chapter 26] Tumor SCI, Utah Scientific Visualization Visualize large
More informationLevel Set Method in a Finite Element Setting
Level Set Method in a Finite Element Setting John Shopple University of California, San Diego November 6, 2007 Outline 1 Level Set Method 2 Solute-Solvent Model 3 Reinitialization 4 Conclusion Types of
More informationVisualizing 3D Velocity Fields Near Contour Surfaces
Visualizing 3D Velocity Fields Near Contour Surfaces Nelson Max Roger Crawfis Charles Grant Lawrence Livermore National Laboratory Livermore, California 94551 Abstract Vector field rendering is difficult
More informationVisualization Computer Graphics I Lecture 20
15-462 Computer Graphics I Lecture 20 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 12] April 15, 2003 Frank Pfenning Carnegie Mellon University http://www.cs.cmu.edu/~fp/courses/graphics/
More informationHeight Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 12] April 23, 2002 Frank Pfenning Carnegie Mellon University
15-462 Computer Graphics I Lecture 21 Visualization Height Fields and Contours Scalar Fields Volume Rendering Vector Fields [Angel Ch. 12] April 23, 2002 Frank Pfenning Carnegie Mellon University http://www.cs.cmu.edu/~fp/courses/graphics/
More informationData Representation in Visualisation
Data Representation in Visualisation Visualisation Lecture 4 Taku Komura Institute for Perception, Action & Behaviour School of Informatics Taku Komura Data Representation 1 Data Representation We have
More informationFigure 1: Control & Simulation Loop
Pre-Lab 4 LabVIEW Tutorial Overview In this tutorial, you will be using the Control & Simulation library in LabVIEW to simulate the step response for the transfer function of a system. ATTENTION : After
More informationRealtime Water Simulation on GPU. Nuttapong Chentanez NVIDIA Research
1 Realtime Water Simulation on GPU Nuttapong Chentanez NVIDIA Research 2 3 Overview Approaches to realtime water simulation Hybrid shallow water solver + particles Hybrid 3D tall cell water solver + particles
More informationSkåne University Hospital Lund, Lund, Sweden 2 Deparment of Numerical Analysis, Centre for Mathematical Sciences, Lund University, Lund, Sweden
Volume Tracking: A New Method for Visualization of Intracardiac Blood Flow from Three-Dimensional, Time-Resolved, Three-Component Magnetic Resonance Velocity Mapping Appendix: Theory and Numerical Implementation
More informationAn Eulerian Approach for Computing the Finite Time Lyapunov Exponent (FTLE)
An Eulerian Approach for Computing the Finite Time Lyapunov Exponent (FTLE) Shingyu Leung Department of Mathematics, Hong Kong University of Science and Technology masyleung@ust.hk May, Shingyu Leung (HKUST)
More information(Discrete) Differential Geometry
(Discrete) Differential Geometry Motivation Understand the structure of the surface Properties: smoothness, curviness, important directions How to modify the surface to change these properties What properties
More informationNumerical and theoretical analysis of shock waves interaction and reflection
Fluid Structure Interaction and Moving Boundary Problems IV 299 Numerical and theoretical analysis of shock waves interaction and reflection K. Alhussan Space Research Institute, King Abdulaziz City for
More information2 Lab 2: LabVIEW and Control System Building Blocks
2 Lab 2: LabVIEW and Control System Building Blocks 2.1 Introduction Controllers are built from mechanical or electrical building blocks. Most controllers are implemented in a program using sensors to
More informationOverview. Applications of DEC: Fluid Mechanics and Meshing. Fluid Models (I) Part I. Computational Fluids with DEC. Fluid Models (II) Fluid Models (I)
Applications of DEC: Fluid Mechanics and Meshing Mathieu Desbrun Applied Geometry Lab Overview Putting DEC to good use Fluids, fluids, fluids geometric interpretation of classical models discrete geometric
More informationThe Spalart Allmaras turbulence model
The Spalart Allmaras turbulence model The main equation The Spallart Allmaras turbulence model is a one equation model designed especially for aerospace applications; it solves a modelled transport equation
More informationCompressible Flow in a Nozzle
SPC 407 Supersonic & Hypersonic Fluid Dynamics Ansys Fluent Tutorial 1 Compressible Flow in a Nozzle Ahmed M Nagib Elmekawy, PhD, P.E. Problem Specification Consider air flowing at high-speed through a
More informationSurfaces and Integral Curves
MODULE 1: MATHEMATICAL PRELIMINARIES 16 Lecture 3 Surfaces and Integral Curves In Lecture 3, we recall some geometrical concepts that are essential for understanding the nature of solutions of partial
More informationComputer Science Senior Thesis
Computer Science Senior Thesis Matt Culbreth University of Colorado May 4, 2005 0.1 Abstract Two methods for the identification of coherent structures in fluid flows are studied for possible combination
More information