This document discusses issues related to unstructured mesh generation. It begins by introducing unstructured grids and their application of graph theory. It then discusses methods for generating unstructured meshes, including Delaunay triangulation, Voronoi diagrams, and non-triangulation methods. The document also covers data structures for storing unstructured mesh connectivity and algorithms for ordering and partitioning unstructured meshes.
This document discusses different techniques for image segmentation, which is the process of partitioning an image into meaningful regions or objects. It covers several main methods of region segmentation, including region growing, clustering, and split-and-merge. It also discusses techniques for finding line and curve segments in an image, such as using the Hough transform or edge tracking procedures. Finally, it provides examples of applying these segmentation techniques to extract regions, straight lines, and circles from images.
The document describes various computer graphics output primitives and algorithms for drawing them, including lines, circles, and filled areas. It discusses line drawing algorithms like DDA, Bresenham's, and midpoint circle algorithms. These algorithms use incremental integer calculations to efficiently rasterize primitives by determining the next pixel coordinates without performing floating point calculations at each step. The midpoint circle algorithm in particular uses a "circle function" and incremental updates to its value to determine whether the next pixel is inside or outside the circle boundary.
The document discusses different methods for representing segmented image regions, including:
1) Representing regions based on their external (boundary-based) characteristics or internal (pixel-based) characteristics.
2) Common boundary representation methods are boundary following algorithms, chain codes, and polygon approximation.
3) Chain codes represent boundaries as sequences of line segments coded by direction. Polygon approximation finds the minimum perimeter polygon to capture a boundary shape using the fewest line segments.
Image segmentation involves partitioning an image into regions, linear structures, or shapes. There are several main methods of region segmentation including region growing, clustering, and split and merge. Region growing starts with seed pixels and grows regions by adding similar neighboring pixels. Clustering groups pixels into clusters to minimize differences within clusters. Common clustering algorithms include K-means, ISODATA, and histogram-based clustering. Edge detection finds boundaries between regions by looking for changes in intensity values. Popular edge detectors include Sobel, Canny, and zero-crossing operators. Line and curve segments can be found from edge images using tracking or the Hough transform, which accumulates votes for parameter values of lines and curves in an image.
Image segmentation involves partitioning an image into regions, linear structures (line segments, curve segments), or 2D shapes (circles, ellipses). Common region segmentation methods include region growing, clustering, and split-merge. Region growing starts with seed pixels and grows regions based on similarity. Clustering groups pixels into clusters to minimize dissimilarity within clusters. The Hough transform detects lines or curves by accumulating votes in a parameter space based on edge pixels.
Image segmentation is a computer vision task that involves dividing an image into multiple segments or regions, where each segment corresponds to a distinct object, region, or feature within the image. The goal of image segmentation is to simplify and analyze an image by partitioning it into meaningful and semantically relevant parts. This is a crucial step in various applications, including object recognition, medical imaging, autonomous driving, and more.
Key points about image segmentation:
Semantic Segmentation: This type of segmentation assigns each pixel in an image to a specific class, essentially labeling each pixel with the object or region it belongs to. It's commonly used for object detection and scene understanding.
Instance Segmentation: Here, individual instances of objects are separated and labeled separately. This is especially useful when multiple objects of the same class are present in the image.
Boundary Detection: Some segmentation methods focus on identifying the boundaries that separate different objects or regions in an image.
Methods: Image segmentation can be achieved through various techniques, including traditional methods like thresholding, clustering, and region growing, as well as more advanced techniques involving deep learning, such as using convolutional neural networks (CNNs) and fully convolutional networks (FCNs).
Challenges: Image segmentation can be challenging due to variations in lighting, color, texture, and object shape. Overlapping objects and unclear boundaries further complicate the task.
Applications: Image segmentation is used in diverse fields. For example, in medical imaging, it helps identify organs or abnormalities. In autonomous vehicles, it aids in identifying pedestrians, other vehicles, and obstacles.
Evaluation: Measuring the accuracy of segmentation methods can be complex. Metrics like Intersection over Union (IoU) and Dice coefficient are often used to compare segmented results to ground truth.
Data Annotation: Creating ground truth annotations for segmentation can be labor-intensive, as each pixel must be labeled. This has led to the development of datasets and tools to facilitate annotation.
Semantic Segmentation Networks: Deep learning architectures like U-Net, Mask R-CNN, and Deeplab have significantly improved the accuracy of image segmentation by effectively learning complex patterns and features.
Image segmentation plays a fundamental role in understanding and processing images, enabling computers to "see" and interpret visual information in ways that mimic human perception.
Image segmentation is a computer vision task that involves dividing an image into meaningful and distinct segments or regions. The goal is to partition an image into segments that represent different objects or areas of interest within the image. Image segmentation plays a crucial role in various applications, such as object detection, medical imaging, autonomous vehicles, and more.
Perimetric Complexity of Binary Digital ImagesRSARANYADEVI
Perimetric complexity is a measure of the complexity of binary pictures. It is defined as the sum of inside and outside perimeters of the foreground, squared, divided by the foreground area, divided by . Difficulties arise when this definition is applied to digital images composed of binary pixels. In this article we identify these problems and propose solutions. Perimetric complexity is often used as a measure of visual complexity, in which case it should take into account the limited resolution of the visual system. We propose a measure of visual perimetric complexity that meets this requirement.
This document discusses issues related to unstructured mesh generation. It begins by introducing unstructured grids and their application of graph theory. It then discusses methods for generating unstructured meshes, including Delaunay triangulation, Voronoi diagrams, and non-triangulation methods. The document also covers data structures for storing unstructured mesh connectivity and algorithms for ordering and partitioning unstructured meshes.
This document discusses different techniques for image segmentation, which is the process of partitioning an image into meaningful regions or objects. It covers several main methods of region segmentation, including region growing, clustering, and split-and-merge. It also discusses techniques for finding line and curve segments in an image, such as using the Hough transform or edge tracking procedures. Finally, it provides examples of applying these segmentation techniques to extract regions, straight lines, and circles from images.
The document describes various computer graphics output primitives and algorithms for drawing them, including lines, circles, and filled areas. It discusses line drawing algorithms like DDA, Bresenham's, and midpoint circle algorithms. These algorithms use incremental integer calculations to efficiently rasterize primitives by determining the next pixel coordinates without performing floating point calculations at each step. The midpoint circle algorithm in particular uses a "circle function" and incremental updates to its value to determine whether the next pixel is inside or outside the circle boundary.
The document discusses different methods for representing segmented image regions, including:
1) Representing regions based on their external (boundary-based) characteristics or internal (pixel-based) characteristics.
2) Common boundary representation methods are boundary following algorithms, chain codes, and polygon approximation.
3) Chain codes represent boundaries as sequences of line segments coded by direction. Polygon approximation finds the minimum perimeter polygon to capture a boundary shape using the fewest line segments.
Image segmentation involves partitioning an image into regions, linear structures, or shapes. There are several main methods of region segmentation including region growing, clustering, and split and merge. Region growing starts with seed pixels and grows regions by adding similar neighboring pixels. Clustering groups pixels into clusters to minimize differences within clusters. Common clustering algorithms include K-means, ISODATA, and histogram-based clustering. Edge detection finds boundaries between regions by looking for changes in intensity values. Popular edge detectors include Sobel, Canny, and zero-crossing operators. Line and curve segments can be found from edge images using tracking or the Hough transform, which accumulates votes for parameter values of lines and curves in an image.
Image segmentation involves partitioning an image into regions, linear structures (line segments, curve segments), or 2D shapes (circles, ellipses). Common region segmentation methods include region growing, clustering, and split-merge. Region growing starts with seed pixels and grows regions based on similarity. Clustering groups pixels into clusters to minimize dissimilarity within clusters. The Hough transform detects lines or curves by accumulating votes in a parameter space based on edge pixels.
Image segmentation is a computer vision task that involves dividing an image into multiple segments or regions, where each segment corresponds to a distinct object, region, or feature within the image. The goal of image segmentation is to simplify and analyze an image by partitioning it into meaningful and semantically relevant parts. This is a crucial step in various applications, including object recognition, medical imaging, autonomous driving, and more.
Key points about image segmentation:
Semantic Segmentation: This type of segmentation assigns each pixel in an image to a specific class, essentially labeling each pixel with the object or region it belongs to. It's commonly used for object detection and scene understanding.
Instance Segmentation: Here, individual instances of objects are separated and labeled separately. This is especially useful when multiple objects of the same class are present in the image.
Boundary Detection: Some segmentation methods focus on identifying the boundaries that separate different objects or regions in an image.
Methods: Image segmentation can be achieved through various techniques, including traditional methods like thresholding, clustering, and region growing, as well as more advanced techniques involving deep learning, such as using convolutional neural networks (CNNs) and fully convolutional networks (FCNs).
Challenges: Image segmentation can be challenging due to variations in lighting, color, texture, and object shape. Overlapping objects and unclear boundaries further complicate the task.
Applications: Image segmentation is used in diverse fields. For example, in medical imaging, it helps identify organs or abnormalities. In autonomous vehicles, it aids in identifying pedestrians, other vehicles, and obstacles.
Evaluation: Measuring the accuracy of segmentation methods can be complex. Metrics like Intersection over Union (IoU) and Dice coefficient are often used to compare segmented results to ground truth.
Data Annotation: Creating ground truth annotations for segmentation can be labor-intensive, as each pixel must be labeled. This has led to the development of datasets and tools to facilitate annotation.
Semantic Segmentation Networks: Deep learning architectures like U-Net, Mask R-CNN, and Deeplab have significantly improved the accuracy of image segmentation by effectively learning complex patterns and features.
Image segmentation plays a fundamental role in understanding and processing images, enabling computers to "see" and interpret visual information in ways that mimic human perception.
Image segmentation is a computer vision task that involves dividing an image into meaningful and distinct segments or regions. The goal is to partition an image into segments that represent different objects or areas of interest within the image. Image segmentation plays a crucial role in various applications, such as object detection, medical imaging, autonomous vehicles, and more.
Perimetric Complexity of Binary Digital ImagesRSARANYADEVI
Perimetric complexity is a measure of the complexity of binary pictures. It is defined as the sum of inside and outside perimeters of the foreground, squared, divided by the foreground area, divided by . Difficulties arise when this definition is applied to digital images composed of binary pixels. In this article we identify these problems and propose solutions. Perimetric complexity is often used as a measure of visual complexity, in which case it should take into account the limited resolution of the visual system. We propose a measure of visual perimetric complexity that meets this requirement.
This document proposes a new method for corner detection in images using difference chain coding as a measure of curvature. The method involves extracting a one-pixel thick boundary from the image, chain encoding it to determine slope, smoothing the boundary to remove noise, and calculating difference codes to determine points of high curvature change, which indicate corners. Preliminary results show the method is simple, efficient, and performs comparably to standard corner detection techniques like Harris and Yung.
This document discusses algorithms for drawing 2D graphics primitives like lines, triangles, and circles in computer graphics. It begins by introducing basic concepts like coordinate systems, pixels, and graphics APIs. It then covers algorithms for drawing lines, including the slope-intercept method, DDA algorithm, and Bresenham's line drawing algorithm, which uses only integer calculations for better performance. Finally, it briefly mentions extending these techniques to draw other shapes like circles and curves, as well as filling shapes.
Q1Perform the two basic operations of multiplication and divisio.docxamrit47
Q1
Perform the two basic operations of multiplication and division to a complex number in both rectangular and polar form, to demonstrate the different techniques.
· Dividing complex numbers in rectangular and polar forms.
· Converting complex numbers between polar and rectangular forms and vice versa.
Q2
Calculate the mean, standard deviation and variance for a set of ungrouped data
· Completing a tabular approach to processing ungrouped data.
Q3
Calculate the mean, standard deviation and variance for a set of grouped data
· Completing a tabular approach to processing grouped data having selected an appropriate group size.
Q4
Sketch the graph of a sinusoidal trig function and use it to explain and describe amplitude, period and frequency.
· Calculate various features and coordinates of a waveform and sketch a plot accordingly.
· Explain basic elements of a waveform.
Q5
Use two of the compound angle formulae and verify their results.
· Simplify trigonometric terms and calculate complete values using compound formulae.
Q6
Find the differential coefficient for three different functions to demonstrate the use of function of a function and the product and quotient rules
· Use the chain, product and quotient rule to solve given differentiation tasks.
Q7
Use integral calculus to solve two simple engineering problems involving the definite and indefinite integral.
· Complete 3 tasks; one to practise integration with no definite integrals, the second to use definite integrals, the third to plot a graph and identify the area that relates to the definite integrals with a calculated answer for the area within such.
Q8
Use the laws of logarithms to reduce an engineering law of the type y = axn to a straight line form, then using logarithmic graph paper, plot the graph and obtain the values for the constants a and n.
· See Task.
Q9
Use complex numbers to solve a parallel arrangement of impedances giving the answer in both Cartesian and polar form
· See Task.
Q10
Use differential calculus to find the maximum/minimum for an engineering problem.
· See Task.
Q11
Using a graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trig formulae.
· See Task.
Q12
Use numerical integration and integral calculus to analyse the results of a complex engineering problem
· See Task.
Level of Detail in
Solution
s: Need to show work leading to final answer
Need
Question 1
(a) Find:
(4 + i2)
(1 + i3)
Use the rules for multiplication and division of complex numbers in rectangular form.
(b) Convert the answer in rectangular form to polar form
(c) Repeat Q1a by first converting the complex numbers to polar form and then using the rules for multiplication and division of complex numbers in polar form.
(d) Convert the answer in polar form to rectangular form.
Question 2
The following data within the working area consists of measurements of resistor values from a producti ...
This document proposes a hardware implementation of a fixed-function 3D graphics pipeline for mobile applications. It presents the design of modules for vertex transformation, rasterization, texture mapping, and data transmission. Simulation results show the design can render 3D objects with color, textures, and different rendering modes. The design was fabricated in a 130nm technology and achieved a core power consumption of 1.768mW. Future work could involve replacing the fixed-function pipeline with programmable shaders to improve flexibility.
The document provides information about computer graphics concepts including:
1. Summarizing questions and answers about 3D triangles, rotation matrices, vector operations, splines, and computer graphics techniques like environment mapping and anti-aliasing.
2. Explaining modifications made to the active edge list algorithm to enable scan conversion of different triangle types like smoothly shaded, textured, and environment mapped triangles.
3. Deriving the 4x4 projection matrix that maps a 3D object point to its shadow point on a plane, to create planar shadows.
This document discusses algorithms for drawing straight line segments on a digital display. It describes how line segments are defined by their endpoint coordinates and how those coordinates are converted to integer pixel positions. It then explains how the slope-intercept equation can be used to calculate the slope and y-intercept of a line from its endpoints. Finally, it introduces the digital differential analyzer (DDA) algorithm, which uses incremental steps in x or y based on the slope to efficiently calculate pixel positions along the line segment.
Grid generation and adaptive refinementGoran Rakic
This document discusses various methods for generating computational grids for solving partial differential equations (PDEs). It describes algebraic methods like transfinite interpolation that use known functions to transform domains, as well as PDE-based methods like Thompson's elliptic PDE grid that define grid requirements mathematically. It also covers unstructured grids, adaptive refinement, and moving grids that adjust grid resolution based on solution features.
The document discusses computer graphics concepts like points, pixels, lines, and circles. It begins with definitions of pixels and how they relate to points in geometry. It then covers the basic structure for specifying points in OpenGL and how to draw points, lines, and triangles. Next, it discusses algorithms for drawing lines, including the digital differential analyzer (DDA) method and Bresenham's line algorithm. Finally, it covers circle drawing and introduces the mid-point circle algorithm. In summary:
1) It defines key computer graphics concepts like pixels, points, lines, and circles.
2) It explains the basic OpenGL functions for drawing points and lines and provides examples of drawing simple shapes.
3) It
Unit-V discusses curves and fractals. Curves are continuous maps from a one-dimensional space to an n-dimensional space. Curves can be represented explicitly, implicitly, or parametrically. Splines are commonly used curves that are constructed by specifying control points and interpolating between them. Bezier curves are another type of curve defined by control points, where the first and last points are the endpoints and the other points control the tangents. Both splines and Bezier curves can be subdivided recursively to render smooth curves.
This document discusses edge linking and boundary detection techniques. It begins by describing how ideal edge detection would yield only boundary pixels, but in reality noise and other factors mean not all edges are detected. Edge detection is followed by linking procedures to assemble edge pixels into meaningful boundaries. Local processing analyzes pixel neighborhoods to link similar pixels based on gradient strength and direction. The Hough transform is introduced as a global processing technique that links points by determining if they lie on a specified shape curve through a parameter space representation.
This document discusses gate level minimization and optimization of Boolean functions. It covers cost criteria for logic circuits including literal cost, gate input cost, and gate input cost with NOTs. Common techniques for Boolean function optimization are also described, including algebraic manipulation and Karnaugh maps. Karnaugh maps allow visual grouping of minterms to minimize gate inputs and simplify Boolean expressions. The concepts of implicants, prime implicants, and essential prime implicants are introduced in the context of Karnaugh map analysis. Don't care conditions are also discussed.
This document provides an overview of different techniques for representing polygon meshes and parametric curves. It discusses explicit representations of polygon meshes using vertex and edge lists, as well as parametric cubic curves including Hermite, Bezier, and B-spline curves. It describes how each technique represents the geometry and constraints, and properties such as continuity and invariance. Key topics covered include representations of polygon meshes, Hermite curves defined by endpoints and tangents, Bezier curves defined by control points, and B-spline curves having local control and smooth joins.
Efficient Solution of Two-Stage Stochastic Linear Programs Using Interior Poi...SSA KPI
The document describes efficient solution methods for two-stage stochastic linear programs (SLPs) using interior point methods. Interior point methods require solving large, dense systems of linear equations at each iteration, which can be computationally difficult for SLPs due to their structure leading to dense matrices. The paper reviews methods for improving computational efficiency, including reformulating the problem, exploiting special structures like transpose products, and explicitly factorizing the matrices to solve smaller independent systems in parallel. Computational results show explicit factorizations generally require the least effort.
This document discusses line drawing algorithms in computer graphics. It begins by explaining the concept of rasterization and how lines are approximated on a discrete pixel grid. It then covers several common line drawing algorithms, including the digital differential analyzer (DDA) algorithm and Bresenham's algorithm. The DDA algorithm uses incremental integer calculations to determine each new pixel along the line segment. Bresenham's algorithm only uses integer arithmetic by tracking the error between the ideal line segment and the pixel grid. It provides faster performance over DDA by avoiding floating-point operations. The document provides detailed explanations, mathematical formulas, and examples of each algorithm.
This document summarizes the finite difference method for numerically solving heat transfer problems. The method involves establishing a nodal network to discretize the domain, deriving finite difference approximations of the governing heat equation at each node, developing a system of simultaneous algebraic equations relating all nodal temperatures, and solving the system of equations using numerical techniques like matrix inversion or iterative methods. Examples are provided to illustrate the finite difference approximations, formation of the algebraic system, and solution via the Jacobi and Gauss-Seidel iteration methods.
- Dimensionality reduction techniques assign instances to vectors in a lower-dimensional space while approximately preserving similarity relationships. Principal component analysis (PCA) is a common linear dimensionality reduction technique.
- Kernel PCA performs PCA in a higher-dimensional feature space implicitly defined by a kernel function. This allows PCA to find nonlinear structure in data. Kernel PCA computes the principal components by finding the eigenvectors of the normalized kernel matrix.
- For a new data point, its representation in the lower-dimensional space is given by projecting it onto the principal components in feature space using the kernel trick, without explicitly computing features.
This document discusses double patterning lithography techniques. It introduces how optical lithography is approaching its limits and double patterning is needed for smaller feature sizes. It describes the double patterning process and challenges including feature distortion and decreased yield. The document outlines techniques for polygon cutting, priority search trees, and decomposing conflict graphs into tri-connected components to solve the layout splitting problem. Experimental results on test cases including a 320k polygon design show the method achieves 3-10x speedup.
This document summarizes an investigation of using a dual tree algorithm and space partitioning trees to approximate matrix multiplication more efficiently than the naive O(MDN) approach under certain conditions. It presents an algorithm that organizes the row vectors of the left matrix and column vectors of the right matrix into ball trees, then performs a dual tree comparison to estimate the product matrix entries. For this to provide better complexity than naive multiplication, the vectors must fall into clusters proportional to D^τ for some τ > 0. However, uniformly distributed vectors would result in exponentially small expected cluster sizes, limiting the practical applicability of this approach. Future work is needed to address this issue.
This document proposes a new method for corner detection in images using difference chain coding as a measure of curvature. The method involves extracting a one-pixel thick boundary from the image, chain encoding it to determine slope, smoothing the boundary to remove noise, and calculating difference codes to determine points of high curvature change, which indicate corners. Preliminary results show the method is simple, efficient, and performs comparably to standard corner detection techniques like Harris and Yung.
This document discusses algorithms for drawing 2D graphics primitives like lines, triangles, and circles in computer graphics. It begins by introducing basic concepts like coordinate systems, pixels, and graphics APIs. It then covers algorithms for drawing lines, including the slope-intercept method, DDA algorithm, and Bresenham's line drawing algorithm, which uses only integer calculations for better performance. Finally, it briefly mentions extending these techniques to draw other shapes like circles and curves, as well as filling shapes.
Q1Perform the two basic operations of multiplication and divisio.docxamrit47
Q1
Perform the two basic operations of multiplication and division to a complex number in both rectangular and polar form, to demonstrate the different techniques.
· Dividing complex numbers in rectangular and polar forms.
· Converting complex numbers between polar and rectangular forms and vice versa.
Q2
Calculate the mean, standard deviation and variance for a set of ungrouped data
· Completing a tabular approach to processing ungrouped data.
Q3
Calculate the mean, standard deviation and variance for a set of grouped data
· Completing a tabular approach to processing grouped data having selected an appropriate group size.
Q4
Sketch the graph of a sinusoidal trig function and use it to explain and describe amplitude, period and frequency.
· Calculate various features and coordinates of a waveform and sketch a plot accordingly.
· Explain basic elements of a waveform.
Q5
Use two of the compound angle formulae and verify their results.
· Simplify trigonometric terms and calculate complete values using compound formulae.
Q6
Find the differential coefficient for three different functions to demonstrate the use of function of a function and the product and quotient rules
· Use the chain, product and quotient rule to solve given differentiation tasks.
Q7
Use integral calculus to solve two simple engineering problems involving the definite and indefinite integral.
· Complete 3 tasks; one to practise integration with no definite integrals, the second to use definite integrals, the third to plot a graph and identify the area that relates to the definite integrals with a calculated answer for the area within such.
Q8
Use the laws of logarithms to reduce an engineering law of the type y = axn to a straight line form, then using logarithmic graph paper, plot the graph and obtain the values for the constants a and n.
· See Task.
Q9
Use complex numbers to solve a parallel arrangement of impedances giving the answer in both Cartesian and polar form
· See Task.
Q10
Use differential calculus to find the maximum/minimum for an engineering problem.
· See Task.
Q11
Using a graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trig formulae.
· See Task.
Q12
Use numerical integration and integral calculus to analyse the results of a complex engineering problem
· See Task.
Level of Detail in
Solution
s: Need to show work leading to final answer
Need
Question 1
(a) Find:
(4 + i2)
(1 + i3)
Use the rules for multiplication and division of complex numbers in rectangular form.
(b) Convert the answer in rectangular form to polar form
(c) Repeat Q1a by first converting the complex numbers to polar form and then using the rules for multiplication and division of complex numbers in polar form.
(d) Convert the answer in polar form to rectangular form.
Question 2
The following data within the working area consists of measurements of resistor values from a producti ...
This document proposes a hardware implementation of a fixed-function 3D graphics pipeline for mobile applications. It presents the design of modules for vertex transformation, rasterization, texture mapping, and data transmission. Simulation results show the design can render 3D objects with color, textures, and different rendering modes. The design was fabricated in a 130nm technology and achieved a core power consumption of 1.768mW. Future work could involve replacing the fixed-function pipeline with programmable shaders to improve flexibility.
The document provides information about computer graphics concepts including:
1. Summarizing questions and answers about 3D triangles, rotation matrices, vector operations, splines, and computer graphics techniques like environment mapping and anti-aliasing.
2. Explaining modifications made to the active edge list algorithm to enable scan conversion of different triangle types like smoothly shaded, textured, and environment mapped triangles.
3. Deriving the 4x4 projection matrix that maps a 3D object point to its shadow point on a plane, to create planar shadows.
This document discusses algorithms for drawing straight line segments on a digital display. It describes how line segments are defined by their endpoint coordinates and how those coordinates are converted to integer pixel positions. It then explains how the slope-intercept equation can be used to calculate the slope and y-intercept of a line from its endpoints. Finally, it introduces the digital differential analyzer (DDA) algorithm, which uses incremental steps in x or y based on the slope to efficiently calculate pixel positions along the line segment.
Grid generation and adaptive refinementGoran Rakic
This document discusses various methods for generating computational grids for solving partial differential equations (PDEs). It describes algebraic methods like transfinite interpolation that use known functions to transform domains, as well as PDE-based methods like Thompson's elliptic PDE grid that define grid requirements mathematically. It also covers unstructured grids, adaptive refinement, and moving grids that adjust grid resolution based on solution features.
The document discusses computer graphics concepts like points, pixels, lines, and circles. It begins with definitions of pixels and how they relate to points in geometry. It then covers the basic structure for specifying points in OpenGL and how to draw points, lines, and triangles. Next, it discusses algorithms for drawing lines, including the digital differential analyzer (DDA) method and Bresenham's line algorithm. Finally, it covers circle drawing and introduces the mid-point circle algorithm. In summary:
1) It defines key computer graphics concepts like pixels, points, lines, and circles.
2) It explains the basic OpenGL functions for drawing points and lines and provides examples of drawing simple shapes.
3) It
Unit-V discusses curves and fractals. Curves are continuous maps from a one-dimensional space to an n-dimensional space. Curves can be represented explicitly, implicitly, or parametrically. Splines are commonly used curves that are constructed by specifying control points and interpolating between them. Bezier curves are another type of curve defined by control points, where the first and last points are the endpoints and the other points control the tangents. Both splines and Bezier curves can be subdivided recursively to render smooth curves.
This document discusses edge linking and boundary detection techniques. It begins by describing how ideal edge detection would yield only boundary pixels, but in reality noise and other factors mean not all edges are detected. Edge detection is followed by linking procedures to assemble edge pixels into meaningful boundaries. Local processing analyzes pixel neighborhoods to link similar pixels based on gradient strength and direction. The Hough transform is introduced as a global processing technique that links points by determining if they lie on a specified shape curve through a parameter space representation.
This document discusses gate level minimization and optimization of Boolean functions. It covers cost criteria for logic circuits including literal cost, gate input cost, and gate input cost with NOTs. Common techniques for Boolean function optimization are also described, including algebraic manipulation and Karnaugh maps. Karnaugh maps allow visual grouping of minterms to minimize gate inputs and simplify Boolean expressions. The concepts of implicants, prime implicants, and essential prime implicants are introduced in the context of Karnaugh map analysis. Don't care conditions are also discussed.
This document provides an overview of different techniques for representing polygon meshes and parametric curves. It discusses explicit representations of polygon meshes using vertex and edge lists, as well as parametric cubic curves including Hermite, Bezier, and B-spline curves. It describes how each technique represents the geometry and constraints, and properties such as continuity and invariance. Key topics covered include representations of polygon meshes, Hermite curves defined by endpoints and tangents, Bezier curves defined by control points, and B-spline curves having local control and smooth joins.
Efficient Solution of Two-Stage Stochastic Linear Programs Using Interior Poi...SSA KPI
The document describes efficient solution methods for two-stage stochastic linear programs (SLPs) using interior point methods. Interior point methods require solving large, dense systems of linear equations at each iteration, which can be computationally difficult for SLPs due to their structure leading to dense matrices. The paper reviews methods for improving computational efficiency, including reformulating the problem, exploiting special structures like transpose products, and explicitly factorizing the matrices to solve smaller independent systems in parallel. Computational results show explicit factorizations generally require the least effort.
This document discusses line drawing algorithms in computer graphics. It begins by explaining the concept of rasterization and how lines are approximated on a discrete pixel grid. It then covers several common line drawing algorithms, including the digital differential analyzer (DDA) algorithm and Bresenham's algorithm. The DDA algorithm uses incremental integer calculations to determine each new pixel along the line segment. Bresenham's algorithm only uses integer arithmetic by tracking the error between the ideal line segment and the pixel grid. It provides faster performance over DDA by avoiding floating-point operations. The document provides detailed explanations, mathematical formulas, and examples of each algorithm.
This document summarizes the finite difference method for numerically solving heat transfer problems. The method involves establishing a nodal network to discretize the domain, deriving finite difference approximations of the governing heat equation at each node, developing a system of simultaneous algebraic equations relating all nodal temperatures, and solving the system of equations using numerical techniques like matrix inversion or iterative methods. Examples are provided to illustrate the finite difference approximations, formation of the algebraic system, and solution via the Jacobi and Gauss-Seidel iteration methods.
- Dimensionality reduction techniques assign instances to vectors in a lower-dimensional space while approximately preserving similarity relationships. Principal component analysis (PCA) is a common linear dimensionality reduction technique.
- Kernel PCA performs PCA in a higher-dimensional feature space implicitly defined by a kernel function. This allows PCA to find nonlinear structure in data. Kernel PCA computes the principal components by finding the eigenvectors of the normalized kernel matrix.
- For a new data point, its representation in the lower-dimensional space is given by projecting it onto the principal components in feature space using the kernel trick, without explicitly computing features.
This document discusses double patterning lithography techniques. It introduces how optical lithography is approaching its limits and double patterning is needed for smaller feature sizes. It describes the double patterning process and challenges including feature distortion and decreased yield. The document outlines techniques for polygon cutting, priority search trees, and decomposing conflict graphs into tri-connected components to solve the layout splitting problem. Experimental results on test cases including a 320k polygon design show the method achieves 3-10x speedup.
This document summarizes an investigation of using a dual tree algorithm and space partitioning trees to approximate matrix multiplication more efficiently than the naive O(MDN) approach under certain conditions. It presents an algorithm that organizes the row vectors of the left matrix and column vectors of the right matrix into ball trees, then performs a dual tree comparison to estimate the product matrix entries. For this to provide better complexity than naive multiplication, the vectors must fall into clusters proportional to D^τ for some τ > 0. However, uniformly distributed vectors would result in exponentially small expected cluster sizes, limiting the practical applicability of this approach. Future work is needed to address this issue.
Similar to mesh generation techniqure of structured gridpdf (20)
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
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2. • There are several methods to develop the structured meshes:
Algebraic methods, Interpolation methods, and methods based
on solving partial differential equations.
• In algebraic methods, the transformation is done analytically
when the boundaries are rather simple and regular.
• For irregular geometries, the coordinates of the mesh (interior)
nodes are obtained by numerical interpolation between the
prescribed boundary data. One such method is transfinite
interpolation.
• PDE methods are classified as elliptic, parabolic or hyperbolic,
depending on the characteristics of the grid generation equations,
which are generally transformed onto a rectangular domain and
solved along with the governing equations of the problem. 2
3. Algebraic Mapping
• Consider the simply connected domain shown in Fig. 3.2.1
whose sides AB, BC, CDand DA are given by the equations f1(x,
y) = 0, f2 (x, y) = 0, f3 (x, y) = 0 and f4 (x, y) = 0, respectively.
• Without loss of generality, one can map the curves AB and CD
onto the lines = 0 and = 1.0, using a transformation of the
form
η = f1 (x, y)/{f1 (x, y) – f3 (x, y)}
• Similarly it can be assumed that
ξ = f4 (x, y)/{f4 (x, y) – f2 (x, y)}
3
4. A B
=1
A‘ B‘
=0
C
=1
C‘
D‘ =1
D
=0
=0
=0 =1
Algebraic Mapping
Fig. 3.2.1 Algebraic mapping (a) Physical domain in x-y system with
regular boundaries (b) Rectangular body in the (ξ-η) transformed plane.
(a) (b)
4
5. D
B
C
A
=0
=1
=1
=0
• In order to apply the algebraic mapping technique for a domain
between the suction surface AB and pressure surface CD of a
turbomachine blade cascade (Fig. 3.2.2), one may first
approximate the lines AB and CD in terms of analytical
functions: f1 and f3 and follow the other steps as suggested
earlier.
Fig. 3.2.2 Algebraic mapping of a simple blade cascade 5
6. Transfinite Interpolation
• Apply unidirectional interpolation in – direction (or –
direction) between the boundary grid data given on the curves
= 0 and = 1 (or = 0 and = 1) and obtain the coordinates
x’p y’p for every interior and bondary point.
• Calculate the mismatch between the interpolated and the actual
coordinates on the = 0 and = 1 (or = 0 and = 1)
boundaries.
• Linearly interpolate the difference in the boundary point
coordinates in (or direction and find the correction to be
applied to the coordinates of every interior point.
6
7. • For applying the transfinite interpolation to a cascade geometry,
consider the four – sided geometry (ABCD) (Fig. 3.2.3).
• It is desired to generate ξ – constant and η – constant lines within
ABCD which upon transformation would become equi–spaced
orthogonal grid lines inside a rectangular domain of size 1 x 1.
• The first task to be completed is the placement of grid points on
the boundary (Fig. 3.2.3). Here, in order to get a rectangular
grid, the number of points on opposite sides should be equal.
• Also, if some idea is available regarding the nature of gradients
in the problem, the boundary points can be located so as to
resolve the high gradient regions, i.e. the regions where the inter
row spacing domain merges with the blade surfaces.
7
8. Fig. 3.2.3 Transfinite interpolation for an aerofoil cascade identification of
boundaries 8
10. • Now, let us apply linear interpolation in the ξ – direction
between two grid points which lie on the sane η = constant
line. Since the total range of ξ variation is from 0 top 1, the
linear interpolation formulae for the coordinates of an
interior point P are written as
x’p = (1 – ξ) xE + ξxF; y’p = (1 – ξ) yE + ξyF
where E and F are the two boundary grid points.
• Carrying out the same operation for all the η – constant lines
including the boundaries (η = 0 and η = 1), the interpolated
points (marked by x) will appear as in Figure 3.2.4.
• It may be noted that on the boundaries AB and CD, the grid
points obtained by unidirectional interpolation between the
coordinated of the corner nodes (A, B) or (C, D), do not
coincide with actual grid points shown as dots (.). 10
12. • In order to remove this anomaly, the difference between the
actual points and the interpolated points should be subtracted on
the η = 0 and η = 1 boundary curves.
• Moreover, some corrections need to be applied to the
coordinates of point P which have been obtained by
unidirectional interpolation in the ξ – direction, along an η =
constant line.
• Introducing such corrections brings in the influence of the
boundary grid point data in the η – direction also, for
determining the coordinates of point P. considering two grid
points G and H corresponding to the ξ = constant line on which
P lies, the corrections for the coordinates of point P are
∆xP = (1 – η) ∆xG + η∆xH; ∆yP = (1 – η) ∆yG + η∆yH (3.2.1)
12
13. • Here ∆xG, ∆xH, ∆yG, ∆yH, are the corrections for the boundary
points given by
∆xG = x’G – xG; ∆yG = y’G – yG;
∆xH = x’H – xH; ∆yH = y’H – yH; (3.2.2)
• In Eq. 3.2.2, the coordinates indicated with prime are those
obtained by unidirectional interpolation in ξ – direction and
those without prime are the actual boundary grid point data. The
final values of the coordinates of P (after interpolation in both ξ
and η directions)are obtained as
xp = x’p – ∆xp yp = y’p – ∆yp
• Performing the above sequence of operations for every interior
point gives the mesh in the physical domain as shown in Fig.
3.2.6. The corresponding transformed mesh is as usual a
rectangular grid. 13
15. • It is important to note that unidirectional interpolation can be
done first in the η direction along each ξ = constant curve. In that
case, corrections will have to be applied for matching actual grid
points and the interpolated points along the boundaries AD (ξ =
0) and BC (ξ = 1). The final grid obtained by both the above
approaches will be exactly the same.
15
16. Domain Vertex Method
• Domain vertex method is also an interpolation method,
preferably used for generating multi-block structured grids. They
make use of tensor products of unidirectional interpolation
functions for two or three dimensions.
• Relation between physical (x, y) and transformed (ξ, η)
coordinates, in two dimensions, is given by:
• Here, suffix i indicates the physical coordinate directions. N and
M represent node numbers in the direction of the coordinates.
• are the unidirectional functions and
denotes tensor product.
, 1,2, , 1,2
, , 1,2, 1,2,3,4
N M
i iNM
i N iN
x x i N M
x x i N
N M
and ,
N
16
17. • The following functions are known as Blending functions:
• An example of this transformation from the physical to
transformed domain in two dimensions is shown in Fig. 3.2.7.
1 1 1
2 2 1
3 2 2
4 1 2
ˆ ˆ
1 1
ˆ ˆ
1
,
ˆ ˆ
ˆ ˆ
1
N
17
Fig. 3.2.7 Two dimensional domain
18. 1
2
3
4
5
6
7
8
1 1 1
1 1
1
1 1
, ,
1 1
1
1
N
18
• Similarly in three dimensions, relation between physical and
transformed coordinates is given by
• Blending functions:
ˆ ˆ ˆ , 1,2,3 , , 1,2
or
, , , 1,2,3 1,....,8
i N M P iNMP
i N iN
x x i N M P
x x i N
19. 19
Fig. 3.2.8. Transformation of Three Dimensional Domain using Domain
Vertex Method
• An example of this transformation from the physical to
transformed domain in three dimensions is shown in Fig. 3.2.8.
20. Exercise Problems
• Using transfinite interpolation method generate grids for
The region bounded by
• Using transfinite interpolation method generate grid for any
triangular region.
20
2 2
2 2
2 2 2 2
3 3
1, 1
4 9 4 9
3 3
1, 1
9 4 9 4
x x
y y
x y x y
21. Summary of Lecture 3.2
Methods for algebraic mesh generation and with transfinite
interpolation are illustrated for structured grids. A domain vertex
method popular for finite element methods is also presented.
21
END OF LECTURE 3.2
24. Mesh Generation Strategy
• Mesh generation is an important pre-processing step in CFD of
turbomachinery, quite analogous to the development of solid
modeling that has been discussed in the earlier module for
building the physical model of the computational domain.
• Two contrasting methodologies are developed for mesh
generation: one, the multi-block structured mesh and the other,
fully unstructured mesh using tetrahedra, hexahedra, prisms and
pyramids.
• The former method of structured mesh generation produces the
highest quality meshes from the point of view of solver accuracy
but does not scale well on PC clusters.
• By contrast, fully unstructured meshes are fast to generate and
automate the scale well on clusters, but do not allow solvers to
deliver their highest quality solutions. 3
25. • Further, numerical tolerancing issues arise within the CAD
system and are often exacerbated while imported from the
modeler to the mesh generating tool. In the process, due to
greatly differing scales within the geometry and lack of
numerical compatibility between various geometrical
representations, the model looses “water-tightness” and
necessitates substantial “cleaning”.
• The CSG and BREP paradigms discussed in the previous
module are also applicable while developing mesh generation
algorithms and provide the required water-tightness to the
geometry.
• Most CFD analysis codes, whether commercially available or
developed in-house, follow the same (BREP) paradigm.
4
26. • In order to solve the differential equations numerically, the
continuous physical domain needs to be identified with a large
set of discrete locations called nodes.
• The number of these discrete data points should be so large that
the characteristic variations in the flow properties, determined
after solving the differential equations by the numerical method,
should be as close to the “exact solution” or “bench-mark
solution” as possible.
• A method should be developed to mark the nodes in a fashion
that is demanded by the numerical method that is to be used for
solving the differential equation.
• The popular mesh generation methods are: structured,
unstructured and hybrid.
5
27. Structured Mesh Generation
• For the implementation of numerical methods such as the finite
difference, each node in the computational domain must have
easily identifiable neighboring nodes. A grid or mesh that
satisfies this demand is the structured mesh.
• Implementation of numerical methods on structured meshes
using Cartesian or cylindrical polar grid system is possible only
for simple rectangular or axi-symmetric geometries.
• In general, the generation of structured mesh for a complex flow
domain involves automatic discretization methodology with
boundary fitting coordinates and with coordinate transformations
as discussed.
6
28. • The basic steps in the methods of generating structured meshes
for complex geometries are:
– mapping of the complex physical domain on to a simple
computational domain;
– usage of body fitting coordinates
– transformation of lengths, areas, volumes and all vector
quantities(e.g. velocity).
7
29. • The mapping transformations should preferably be
– smooth
– conformal and
– controlled for grid spacing.
• Iso-parametric mapping of sub-domains enables creation of
multi-block structured grids. The sequence of mapping
determines whether the final mesh is a pseudo rectangular, O-
type, C-type or H-type.
8
30. FIG. 3.1.1 Pseudo rectangular mesh
• Figure 3.1.1. demonstrates the method of generating the pseudo
rectangular mesh for a physical region ABCD, bounded by lines
x = 0.5 and y = √(1-x2).
• The co-ordinates ξ and η are body conforming.
9
31. • Using the transformation,
the domain ABCD in x-y plane (3.1.1 (a)) is mapped on to ξ-η
plane as a unit square. Note that this transformation is not unique
and we may have used suitable alternative transformations as
well.
• The grid formed by the intersection of ξ = constant and
η=constant lines in the physical domain shows the body
conforming nature of these coordinates (Fig. 3.1.1 (b)).
• As we noticed in Lecture 1.2 (refer Fig. 1.2.5), the
turbomachinery flow geometries are multiply connected
domains, for which three basic grid configurations: O-type, C-
type and H-type are widely used. For a given geometry, any
one of these configurations can be obtained by suitable mapping.
2
0.5
(1 )
y
and x
x
10
32. • Consider the multiply connected domain shown in Fig. 3.1.2.
For the same geometry, different grid configurations (O, C or H)
are generated by adopting slightly different methodologies. This
is described in the following.
• O-type Meshing
Introduce a branch cut and identify points (A,B,C,D) on
either side of the branch cut as shown in Fig. 3.1.2 (a). Then,
by mapping AB on A’B’, BC onto B’C’, CD on to C’D’ and
DA onto D’A’, O-type grid is obtained.
The object boundary (AB ) and the external boundary (CD)
become opposite sides of the transformed domain. The two
sides of the branch-cut (BC and AD) are also mapped onto
two opposite sides of the rectangular domain. Now, a grid
constructed by ξ = constant and η=constant lines in the
physical domain is O-type, as shown in Fig. 3.1.2(c).
The O-type meshes generated by this method for NACA
airfoil and a turbomachinery blade are given in Figs. 3.1.3
and 3.1.4 respectively.
11
33. (a) (b)
Fig. 3.1.2 O-type Grid Generation, (a) basic branch-cut scheme, (b) Cartesian
grid in ξ-η plane (c) O-grid in the physical plane
12
34. • An O-type mesh for symmetric NACA aerofoil is shown in Fig.
3.1.3.
Fig. 3.1.3 O-type mesh for symmetric NACA aerofoil
13
35. Fig. 3.1.4: O type mesh for a turbomachinery blade
14
36. • C-type Meshing
For C-type meshing of the same multiply connected domain,
a branch-cut, as shown in Fig. 3.1.5 (a) is introduced and two
points A and B are identified where the branch-cut meets the
outer boundary. Points C and D are suitably selected on the
external boundary and mapping is carried out with AB onto
A’B’, BC onto B’C’, CD onto C’D’ and DA onto D’A’.
Note that the forward sweep of the branch-cut (AP), the
object surface (PQ) and the reverse sweep of the branch-cut
(QB) comprise one side A’B’ of the transformed region. The
object surface is mapped onto the patch P’Q’ on this side. It
can be seen that in this transformation, the η-constant lines in
the interior envelop the object and the branch-cut, thus
forming a C-type configuration.
Figure 3.1.6 shows a C-type mesh for a turbomachinery
blade. 15
37. Fig. 3.1.5 C-type Mesh Generation, (a) basic branch-cut scheme, (b) Cartesian
grid in ξ-η plane (c) C-grid in the physical plane
16
39. • H-type Meshing
For H-type configuration, two branch-cuts are introduced on
either side of the object and the upper and lower portions
(ABCD and EFGH) are separately mapped onto A’B’C’D’ and
E’F’G’H’ in the proper sequence (Fig. 3.1.7). Here, the object
reduces to a line P’Q’ in the middle of the transformed
domain.
It is evident form the examples that by choosing the mapping
configuration, different types of grids can be generated for the
same geometry. The appropriate choice depends on the nature
of the problem to be solved. For complex domains with many
objects, it may be necessary to map different regions
separately, using local transformations. A variety of grid
layouts such as the overlaid grids and embedded grids can be
achieved through such procedures (Fig. 3.1.7).
Figure 3.1.8 shows a H-type mesh for a turbomachinery
blade.
18
40. Fig. 3.1.7 H-type Mesh Generation, (a) basic branch-cut scheme, (b) Cartesian
grid in ξ-η plane (c) H-grid in the physical plane
19
42. Summary of Lecture 3.1
Mesh generation strategies for structured mesh are discussed.
The methods for different types of meshes such as O, C and H
type grids are presented.
21
END OF LECTURE 3.1
44. Boundary Discretization
• All types of grid generation methods (structured and
unstructured) involve boundary discretization as its first step.
The distribution of grid density largely depends on the
distribution of the nodes on the boundaries.
• In Module 2, Fig. 2.2.6 (d) shows the discretization of the
boundary of the airfoil for various segments. A large number of
boundary nodes are obtained in this process. This boundary
curve is regarded as the base curve for the generation of the first
grid layer.
• In the method of successive layer grid generation, a fine layer of
grid is first generated in vicinity of these so generated boundary
nodes.
2
45. • A structured grid is thus generated by starting from the boundary
curve and marching in successive layers. Simple algebraic
relations are derived based on analytical geometry
considerations for ensuring the cell orthogonality and providing
cell areas.
• Next, quadrilateral cells are constructed on outward normals at
all grid points of the base curve.
• The outer surface of these cells is treated as the base curve for
generation of the next grid layer. This marching process is
continued till a prescribed number of (say, N) layers are formed.
• The procedure for obtaining the outward normals and the cell
area are explained in the following section.
3
46. Construction of Outward Normal
• Consider a base curve Γi as shown in Fig. 3.3.3, which has
discretized points such as a, b and c. At point b, bm and bn are
unit vectors along the outward normals of the edges ab and bc
respectively. The outward normal at the point b of the curve Γi is
along the ray bo, which is the bisector of angle mbn. The co-
ordinates of the points m, n and o are derived as:
(3.3.2)
(3.3.3)
(3.3.4)
where, li
ab and li
bc are lengths of the edge ab and bc on Γi
respectively.
, ,
, ,
and
, ,
2 2
i i
b ab b a b ab b a
m m i i
ab ab
i i
b bc b c b bc b c
n n i i
bc bc
m n m n
o o
x l y y y l x x
x y
l l
x l y y y l x x
x y
l l
x x y y
x y
4
47. Cell Area
• Consider that a cell should be constructed about an edge (ab).
The cell area is a function of the length of the specific edge (l1
ab)
on the boundary curve (Γ1) and a distance parameter, δi, i = 1 to
N.
• The distance parameter δ determines the normal dimension of
the cell. It is formulated with the help of an exponential
stretching function (given in Eq. (3.3.1)):
where L = δN, the value of L is chosen as a function of the
characteristic dimension in question e.g., chord length for an
airfoil.
1
1
1
1
S N i
N
i
S
e
L e
5
48. • The cell formed over the edge ab on Γi will have an area Aabed.
• This area is obtained by giving chosen weightages to the lengths
of edges on the boundary curve (Γ1) and the base curve (Γi). The
area Aabed is given by:
Aabed = (li
abε+ l1
ab(1 – ε))hi (3.3.5)
where ε is called the cell size control factor and hi = (δ – δi-1). A
suitable value for ε (between 0 and 1) is selected to get an
appropriate cell size distribution within the domain.
6
50. Cell Construction
• The cell above the edge ab of Fig. 3.3.3 is formed by knowing
the co-ordinates of the points a, b, d, o and the prescribed cell
area, Aabed.
• Here e is the grid point in the new layer and it corresponds to the
point b on the base curve.
• While the co-ordinates of the point o are calculated from Eqs.
(3.3.2 to 3.3.4), the co-ordinates of point d are obtained during
the cell formation on the preceding edge of ab.
• As triangles dbe and dbo in Fig. 3.3.3 are similar, the ratio of
areas Adbe and Adbo is equal to the ratio of the lengths lbe and lba.
8
51. • Thus, the co-ordinates of the point e are estimated as
(xe, ye) = (fxo + (1 – f )xb, fyo + (1 – f )yb) (3.3.6)
where
f = (lbe/lbo) = (Adbe/Adbe) = (Aabed – Aabd) /Abod
• The initial cells of every layer are constructed as rectangular
cells.
Method for Damping Grid Oscillations
• The above grid generation method produces grid oscillations.
Dampening grid oscillations in the present method is done by
averaging the normal distances.
• For instance, the point e in Fig. 3.3.3 is relocated to e', such that
the distance between b and e' is the arithmetic mean of the
distances lad , lba and laf . 9
52. • In this manner all the points in the new grid layer are relocated
repeatedly for a few iterations, typically five times.
• For situations such as sharp corners, where there is a danger of
grid line intersection, the cell (hi) used in Eq. (3.6.1) is varied
linearly to facilitate gradual turning of grid lines. Thus while
smoothing the grid, small variations in the cell area are allowed.
10
Fig. 3.3.4: Grid oscillations in orthogonal grid without damping (∆ = 0.1, N
= 10, s = –5 and ε = 0.25).
53. • The above method of grid generation is named as successive
layer grid generation scheme.
• This method has been applied for obtaining orthogonal grid over
an ellipse having a major axis of 2 units and a minor axis of 0.2
units.
• Figures 3.3.4 and 3.3.5a show the resulted grid without and with
grid smoothing respectively.
• In Fig. 3.3.4, small grid oscillations are found from sixth layer,
which amplify from one layer to the next.
• It is found that the grid in Fig. 3.3.5 obtained using ε = 0.25 is
smooth even for large normal distances from the surface.
11
54. • Figure 3.3.5 shows the enlarged views of the grid near the
training edge obtained with two more values of ε, zero and one
respectively.
• It is seen that as ε increases, the grid lines near the rear
stagnation point tend to diverge as the cell areas increase from
inner layers to the outer.
• Figure 3.3.6 shows the grid generated around a 90o sharp corner
with ∆ = 1, N = 20 s = –1 and ε = 0.5, which demonstrates the
capability of the method to discretize open geometries having
sharp corners.
• The case of possible grid line intersection with controlled cell
area variations for the corner (cavity) problem, shown in Fig.
3.3.7.
12
55. Fig. 3.3.5: Grid around an ellipse with damping (∆ = 0.1, N = 10, s = –5) (a)
ε = 0.25 (b) ε = 0.0 (c) ε = 1.
13
56. Fig. 3.3.6: Grid exterior to a corner
Fig. 3.3.7: Grid inside the corner of a cavity
14
57. Exercise Problem
• Generate a grid over NACA0012 using successive layer method.
Summary of Lecture 3.3
The methodology for another algebraic method named
successive grid generation is described. Technique for damping
the oscillations is also presented with different examples.
15
END OF LECTURE 3.3
65. Differential Equation Based Schemes
• As stated in the previous lecture, another important and most
widely used method of structured mesh generation is based on
solving partial differential equations. These techniques may be
based on any one of the following schemes depending on the
characteristics of the grid generation equations. They are:
– Hyperbolic PDE based schemes
– Elliptic PDE based schemes
– Parabolic PDE based schemes
2
66. Elliptic PDE Based Methods
• In this lecture, Elliptic PDE based grid generation schemes will
only be discussed. These methods are particularly useful for
confined physical domains of turbomachinery flows.
• The elliptic PDEs for grid generation describe the variation of
the body fitting coordinates (ξ, η, ζ ) in the interior of the
physical domain, with prescribed values or slopes at the
boundary.
• In the computational domain, however, the physical coordinates
(x, y, z) are treated as the unknown variables on the grid formed
by the ξ=constant, η= constant and ζ=constant lines. They are
determined by numerically solving the transformed grid
generation equations.
3
67. Elliptic Solvers
• Consider the transformation functions, which are solutions of an
elliptic Dirichlet boundary value problem. The mathematical
problem is given by
(3.4.1)
with fixed , , on the boundaries. P, Q and R are source
functions, which can be used to grid point controlling.
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
( , , )
( , , )
( , , )
P x y z
x y z
Q x y z
x y z
R x y z
x y z
4
68. Laplace Solvers
• The condition: P=Q=R=0, results in to uniform distribution of
the points. The system then becomes the Laplacian.
• The Laplacian grid generation system satisfies the maximum and
the minimum principles , i.e. both maximum and minimum for
, , occur only at the boundary.
• The solutions of the Laplacian operator, , , are either
harmonic, sub-harmonic or super-harmonic. Therefore they are
very smooth functions. Further, the , , have continuous
derivatives of all orders. This makes the solution of the
transformed governing equation accurate
5
69. Two dimensional Transformation Functions
• Let us consider the body fitting coordinate transformation, of the
form = (x,y) and = (x,y). Therefore, one can write,
(3.4.2)
• Considering the reverse transformation x=x( , ) and y=y( , ),
(3.4.3)
• Combining the above two, we have
(3.4.4)
( , )
( , )
x y
x y
x y
x x
dx dy
x y d dx
dx dy
x y d dy
( , )
( , )
x x
x x dx d
y y
y y dy d
x y
x y
x x
y y J
y x
y x
1
1
| |
6
70. Two dimensional Transformation Functions
• One can therefore write the two dimensional transformation
functions as
(3.4.5)
• Note that the Jacobian, J, represents local area scaling factor and
should not become zero.
( , )
( )
( , )
( , )
( , )
( , )
( )
( , )
( , )
( , )
x
y
f y
y f y f
f
x y J
x f
x f x f
f
x y J
7
71. Extension to Three Dimensions
• Extending the arguments to three dimensions, one can derive the
relationships between the derivatives of the Cartesian
coordinates (x,y,z) and the curvilinear coordinates ( , , ) in the
form:
(3.4.6)
1
x y z
x y z
x y z
x x x
y y y
z z z
8
72. Elliptic Solvers in 2D
(3.4.7)
2 2 2
2 2
2 2 2
2 2
2 2
2 2
2 2
2 2
2 0
2 0
1
1
x x x x x
P Q
y y y y y
P Q
x x y y
P
x x y y
Q
2 2 2 2
,
x y x y
x x y y
9
73. Demonstration
• Consider a planar region, as shown in Fig. 3.4.1, in which a
structured grid has to be generated
Fig. 3.4.1 Physical domain 10
74. • Generate a rectangular (ξ, η) = (0,1)x(0,1) uniformly discretized,
as shown in Fig. 3.4.2, (ξi = i*Δξ, ηj=Δη, Δξ, Δη are the uniform
step lengths in ξ, η directions, respectively ) plane given by
Fig. 3.4.2 Uniform grid in computation domain
11
75. Algorithm
• Map the boundaries (from physical to computational) as shown
in Fig. 3.4.3.
Fig. 3.4.3 Mapping of physical to computational domain
• Due to the mapping of the physical boundaries over the
boundaries of the computation domain, x, y are known along the
boundaries of the computational domain. 12
76. • Therefore, once x, y are computed in the interior of the
computational domain, the required grid in the physical domain
is established.
• To obtain x, y in the interior of the computational domain, solve
(using the boundary values of x and y as boundary conditions)
(3.4.8)
for x, y over the uniformly discretized computational domain
2 2 2
2 2
2 2 2
2 2
2 0
2 0
x x x x x
P Q
y y y y y
P Q
13
77. Multiply Connected Domain
• The algorithm described in the earlier slides works for simply
connected domains.
• For multiply connected domains, for example like annular
regions shown in Fig. 3.4.4, artificial boundaries can be
introduced to convert them in to simply connected regions.
Fig. 3.4.4 Introduction of artificial boundary 14
78. Boundary Conditions on the Artificial Boundaries
• Dirichlet boundary conditions over the artificial boundaries in
the multiply-connected regions may lead to non-smooth grid
lines as shown in Fig. 3.4.5.
Fig. 3.4.5 Non smooth grid lines over artificial boundary 15
79. Periodic Boundary Conditions on Artificial Cuts
• However Periodic boundary conditions over artificial cuts
generate smooth grid lines as shown in Fig. 3.4.6.
Fig. 3.4.6 Smooth grid over artificial boundaries
16
80. Grid Lines Attraction and Repulsion
• In general, grid points are attracted in the convex regions and
repulsive in the concave regions as shown in the Fig. 3.4.7.
17
Fig. 3.4.7 Grid point attraction and repulsion
81. Exercise Problems
• Repeat the exercise problems of Lecture 3.2 using PDE method.
• Generate uniform grid in square and cubical, rectangular and
cubical region.
• Generate uniform grid in cylindrical and spherical regions.
• Two-dimensional region bounded by circle of radius r = 1.
• Three-dimensional region bounded by sphere of radius r = 1.
• Annular region in 2D bounded by r = a and r = b with a < b.
• Annular region in 3D bounded by r = a and r = b with a < b.
18
82. Summary of Lecture 3.4
• Mesh generation schemes by solving hyperbolic, elliptic and
parabolic partial differential equation methods are presented.
The methods are explained through examples.
19
END OF LECTURE 3.4