This document discusses the finite element method (FEM) for solving partial differential equations and boundary value problems. It provides an overview of the basic steps in FEM, including establishing the weak formulation, discretizing the domain, selecting shape functions, assembling the global stiffness matrix, applying boundary conditions, and solving the system of equations. The document also describes a Matlab FE program that will be used in the course, including the main files and plotting functions. Exercises are provided to test running the program and making advanced plots using figure and axis handles.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document summarizes an object tracking algorithm that uses Dual Tree Complex Wavelet Transform (DTCWT) feature vectors. It begins by extracting a position vector for the object in the first frame. Nine position vectors are then calculated by shifting the original position in different directions. The second frame is cropped into nine blocks based on these position vectors. DTCWT is used to extract feature vectors for the block in the first frame and nine blocks in the second frame. Block matching is performed by calculating the Manhattan distance between the first frame's feature vector and each of the second frame's nine vectors. The block with the minimum distance corresponds to the tracked object. This process is repeated over successive frames to track the moving object.
Simple Pendulum Experiment and Automatic Survey Grading using Computer VisionAnish Patel
The document describes two computer vision projects: 1) A program that analyzes video from a webcam of a simple pendulum experiment and calculates physics quantities like angle, speed, and energy. 2) A program that grades scanned survey forms by analyzing pixel density to determine filled-in answers. The pendulum program displays angle measurements over time that match expected values. The survey program accurately grades forms with different levels of noise. Both were developed in C++ using OpenCV for computer vision processing.
A landing gear assembly consists of various components viz. Lower side stay, Upperside stay, Locking actuators, Extension actuators, Tyres, and Locking pins to name a few. Each unit having a specific operation to deal with, in this project the main unit being studied is the lower brace. The primary objective is to analyse stresses in the element of the lower brace unit using strength of materials or RDM method and Finite Element Method (FEM) and compare both. Using the obtained data a suitable material is proposed for the component. The approach used here is to study the overall behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in the functioning of the element.
vFORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a simple projectile problem and also LAPACK library to solve a tridiagonal matrix problem. We use DGESV and DGTSV functions to make it possible. The given problems are built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
Comparitive Analysis of Algorithm strategiesTalha Shaikh
The document discusses various algorithm strategies including decrease and conquer, greedy approach, backtracking, and transform and conquer. It provides definitions, examples, advantages and disadvantages for each strategy. Decrease and conquer algorithms like insertion sort reduce the problem size at each step. Greedy algorithms make locally optimal choices at each step. Backtracking algorithms systematically explore all solutions using a depth first search approach.
The document discusses various techniques for edge detection and line detection in images, including:
- Canny edge detection, which uses thresholds to detect and link edges.
- Hough transforms, which detect shapes like lines and circles by counting points that agree with a shape model.
- RANSAC for line detection, which forms line hypotheses from random samples and counts supporting points.
- Techniques for thinning thick edges and detecting edge contours.
1. The document introduces hypergraphs as an extension of graphs where edges can connect more than two vertices.
2. Hypergraphs are useful for applications like image and video segmentation where they can retain more information compared to simple graphs.
3. The document provides examples of using hypergraphs for tasks like video object segmentation, multiple target tracking, multi-view reconstruction, and matching in computer vision.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document summarizes an object tracking algorithm that uses Dual Tree Complex Wavelet Transform (DTCWT) feature vectors. It begins by extracting a position vector for the object in the first frame. Nine position vectors are then calculated by shifting the original position in different directions. The second frame is cropped into nine blocks based on these position vectors. DTCWT is used to extract feature vectors for the block in the first frame and nine blocks in the second frame. Block matching is performed by calculating the Manhattan distance between the first frame's feature vector and each of the second frame's nine vectors. The block with the minimum distance corresponds to the tracked object. This process is repeated over successive frames to track the moving object.
Simple Pendulum Experiment and Automatic Survey Grading using Computer VisionAnish Patel
The document describes two computer vision projects: 1) A program that analyzes video from a webcam of a simple pendulum experiment and calculates physics quantities like angle, speed, and energy. 2) A program that grades scanned survey forms by analyzing pixel density to determine filled-in answers. The pendulum program displays angle measurements over time that match expected values. The survey program accurately grades forms with different levels of noise. Both were developed in C++ using OpenCV for computer vision processing.
A landing gear assembly consists of various components viz. Lower side stay, Upperside stay, Locking actuators, Extension actuators, Tyres, and Locking pins to name a few. Each unit having a specific operation to deal with, in this project the main unit being studied is the lower brace. The primary objective is to analyse stresses in the element of the lower brace unit using strength of materials or RDM method and Finite Element Method (FEM) and compare both. Using the obtained data a suitable material is proposed for the component. The approach used here is to study the overall behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in the functioning of the element.
vFORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a simple projectile problem and also LAPACK library to solve a tridiagonal matrix problem. We use DGESV and DGTSV functions to make it possible. The given problems are built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
Comparitive Analysis of Algorithm strategiesTalha Shaikh
The document discusses various algorithm strategies including decrease and conquer, greedy approach, backtracking, and transform and conquer. It provides definitions, examples, advantages and disadvantages for each strategy. Decrease and conquer algorithms like insertion sort reduce the problem size at each step. Greedy algorithms make locally optimal choices at each step. Backtracking algorithms systematically explore all solutions using a depth first search approach.
The document discusses various techniques for edge detection and line detection in images, including:
- Canny edge detection, which uses thresholds to detect and link edges.
- Hough transforms, which detect shapes like lines and circles by counting points that agree with a shape model.
- RANSAC for line detection, which forms line hypotheses from random samples and counts supporting points.
- Techniques for thinning thick edges and detecting edge contours.
1. The document introduces hypergraphs as an extension of graphs where edges can connect more than two vertices.
2. Hypergraphs are useful for applications like image and video segmentation where they can retain more information compared to simple graphs.
3. The document provides examples of using hypergraphs for tasks like video object segmentation, multiple target tracking, multi-view reconstruction, and matching in computer vision.
The document discusses image processing techniques including image derivatives, integral images, convolution, morphology operations, and image pyramids.
It explains that image derivatives detect edges by capturing changes in pixel intensity, and provides an example calculation. Integral images allow fast computation of box filters by precomputing pixel sums. Convolution is used to calculate probabilities as the sliding overlap of distributions. Morphology operations like erosion and dilation modify images based on pixel neighborhoods. Image pyramids create multiple resolution layers that aid in object detection across scales.
The document discusses greedy algorithms and their applications. It provides examples of problems that greedy algorithms can solve optimally, such as the change making problem and finding minimum spanning trees (MSTs). It also discusses problems where greedy algorithms provide approximations rather than optimal solutions, such as the traveling salesman problem. The document describes Prim's and Kruskal's algorithms for finding MSTs and Dijkstra's algorithm for solving single-source shortest path problems. It explains how these algorithms make locally optimal choices at each step in a greedy manner to build up global solutions.
The document discusses greedy algorithms and their properties. It describes how greedy algorithms work by making locally optimal choices at each step in the hope of reaching a globally optimal solution. Two examples are given: the activity selection problem and finding minimum spanning trees. Prim's algorithm for finding minimum spanning trees is described in detail, showing how it works by always selecting the lightest edge between the growing tree and remaining vertices.
The document discusses different types of images in Matlab including binary, grayscale, indexed, and RGB images. It also summarizes commands to convert between image types such as converting grayscale to indexed or truecolor to binary. Finally, it provides examples of how to view images, measure pixel values and distances, and crop images using the imtool command.
This paper introduces a new approach called temporal-difference (TD) search for game playing using reinforcement learning. TD search uses simulated self-play experiences to update the value function approximation for the current board position/state. Local shape features are used to evaluate board positions and are learned online through TD search rather than offline. Integrating short-term memories learned through simulation with long-term memories learned from real games results in improved performance, in a technique called Dyna-2. Experimental results show TD search improves over conventional alpha-beta search for the game of Go, achieving an Elo rating 300 points higher. However, modern Monte Carlo tree search methods have surpassed these results.
The document provides legal notices and disclaimers for an Intel presentation. It states that the presentation is for informational purposes only and that Intel makes no warranties. It also notes that Intel technologies' features and benefits depend on system configuration and may require enabled hardware, software or service activation. Performance varies depending on system configuration. The document further states that sample source code is released under the Intel Sample Source Code License Agreement and that Intel and its logo are trademarks.
This document provides a 3-sentence summary of a legal notice and disclaimer document:
The document states that any Intel technologies discussed are for informational purposes only and Intel makes no warranties regarding the information. It notes that performance may vary depending on system configuration and that source code samples are released under an Intel license agreement. The document also provides legal notices and disclaimers regarding Intel trademarks and copyright.
This document discusses using principal component analysis (PCA) and the discrete cosine transform (DCT) to recognize images from a database. It explains how bitmaps store image data, DCT compacts image energy, and PCA reduces dimensionality by finding the principal components via eigenvectors of the covariance matrix. An algorithm is proposed that uses DCT, PCA on a 3x3 block, characteristic equations to find the maximum eigenvector, and least mean square comparison to recognize queries against the database images.
The document discusses the greedy method algorithmic approach. It provides an overview of greedy algorithms including that they make locally optimal choices at each step to find a global optimal solution. The document also provides examples of problems that can be solved using greedy methods like job sequencing, the knapsack problem, finding minimum spanning trees, and single source shortest paths. It summarizes control flow and applications of greedy algorithms.
This document provides an overview of support vector machines (SVMs). It discusses how SVMs can be used to perform classification tasks by finding optimal separating hyperplanes that maximize the margin between different classes. The document outlines how SVMs solve an optimization problem to find these optimal hyperplanes using techniques like Lagrange duality, kernels, and soft margins. It also covers model selection methods like cross-validation and discusses extensions of SVMs to multi-class classification problems.
Numerical Methods in Mechanical Engineering - Final ProjectStasik Nemirovsky
Final Project for the class of "Numerical Methods in Mechanical Engineering" - MECH 309.
In this project, various engineering problems were analyzed and solved using advanced numerical approximation methods and MATLAB software.
This document provides an overview of matrix decomposition techniques for dimensionality reduction and topic modeling, specifically principal component analysis (PCA), singular value decomposition (SVD), latent semantic analysis (LSA), and non-negative matrix factorization (NMF). PCA and SVD are introduced as mathematical techniques to reduce dimensions while preserving variance/information. LSA and NMF are described as applying SVD and NMF respectively to text data to derive topic models from the latent semantic space. Examples of implementing these techniques in Python are also provided.
The document discusses image compression using the Haar wavelet basis transform. It begins by providing background on images, pixels, and file sizes. It then applies the Haar wavelet transform to compress an 8x8 sample image, representing it at decreasing resolutions of 4x4, 2x2, and 1x1 pixels. Matrices are used to represent the transform and allow reconstructing the original image. The technique provides lossless compression by retaining detail coefficients. Finally, it discusses interpreting the transform using vector spaces and scaling/wavelet basis functions.
The document discusses dimensionality reduction techniques. It begins by explaining the curse of dimensionality, where adding more features can hurt performance due to the exponential increase in the number of examples needed. It then introduces dimensionality reduction as a solution, where the data can be represented using fewer dimensions/features through feature selection, linear/non-linear transformations, or combinations. Principal component analysis (PCA) and singular value decomposition (SVD) are described as common linear dimensionality reduction methods. The document also discusses nonlinear techniques like kernel PCA and multi-dimensional scaling, as well as uses of dimensionality reduction like in image and natural language processing applications.
This document summarizes support vector machines (SVMs), a machine learning technique for classification and regression. SVMs find the optimal separating hyperplane that maximizes the margin between positive and negative examples in the training data. This is achieved by solving a convex optimization problem that minimizes a quadratic function under linear constraints. SVMs can perform non-linear classification by implicitly mapping inputs into a higher-dimensional feature space using kernel functions. They have applications in areas like text categorization due to their ability to handle high-dimensional sparse data.
This document outlines an assignment for a computer vision course. Students are asked to implement 4 vision algorithms: 2 using OpenCV and 2 using MATLAB. The algorithms are the log-polar transform, background subtraction, histogram equalization, and contrast stretching. Students must also answer 3 short questions about orthographic vs perspective projection, efficient filtering, and sensors beyond cameras for computer vision.
Image Classification And Support Vector MachineShao-Chuan Wang
This document discusses support vector machines and their application to image classification. It provides an overview of SVM concepts like functional and geometric margins, optimization to maximize margins, Lagrangian duality, kernels, soft margins, and bias-variance tradeoff. It also covers multiclass SVM approaches, dimensionality reduction techniques, model selection via cross-validation, and results from applying SVM to an image classification problem.
The document discusses algorithms and data structures using divide and conquer and greedy approaches. It covers topics like matrix multiplication, convex hull, binary search, activity selection problem, knapsack problem, and their algorithms and time complexities. Examples are provided for convex hull, binary search, activity selection, and knapsack problem algorithms. The document is intended as teaching material on design and analysis of algorithms.
Support Vector Machine (SVM) is a supervised machine learning algorithm that can be used for both classification and regression analysis. It works by finding a hyperplane in an N-dimensional space that distinctly classifies the data points. SVM selects the hyperplane that has the largest distance to the nearest training data points of any class, since larger the margin lower the generalization error of the classifier. SVM can efficiently perform nonlinear classification by implicitly mapping their inputs into high-dimensional feature spaces.
This document contains legal notices and disclaimers for an Intel presentation. It states that the presentation is for informational purposes only and that Intel makes no warranties. It also notes that performance depends on system configuration and that sample source code is released under an Intel license agreement. Finally, it provides basic copyright information.
This document provides an overview of a course on the finite element method. The course objectives are for students to learn how to write simple programs to solve problems using FEM. Assessment includes assignments, quizzes, a course project, midterm exam, and final exam. Fundamental agreements include electronic homework submission and using MATLAB or Mathematica. References on FEM are also provided. The document outlines numerical methods for solving boundary value problems and introduces weighted residual methods like the collocation method, subdomain method, and Galerkin method.
The document provides an introduction to the finite element method (FEM). It explains that FEM is a numerical method used to approximate solutions to partial differential equations. It works by dividing a complex problem into smaller, simpler parts called finite elements. This allows for the problem to be solved computationally. The document outlines the basic steps of FEM, including preprocessing (modeling, meshing), solving, and postprocessing (analyzing results). It also discusses applications, history, software, element types, meshing, convergence, and compatibility conditions of FEM.
The document discusses image processing techniques including image derivatives, integral images, convolution, morphology operations, and image pyramids.
It explains that image derivatives detect edges by capturing changes in pixel intensity, and provides an example calculation. Integral images allow fast computation of box filters by precomputing pixel sums. Convolution is used to calculate probabilities as the sliding overlap of distributions. Morphology operations like erosion and dilation modify images based on pixel neighborhoods. Image pyramids create multiple resolution layers that aid in object detection across scales.
The document discusses greedy algorithms and their applications. It provides examples of problems that greedy algorithms can solve optimally, such as the change making problem and finding minimum spanning trees (MSTs). It also discusses problems where greedy algorithms provide approximations rather than optimal solutions, such as the traveling salesman problem. The document describes Prim's and Kruskal's algorithms for finding MSTs and Dijkstra's algorithm for solving single-source shortest path problems. It explains how these algorithms make locally optimal choices at each step in a greedy manner to build up global solutions.
The document discusses greedy algorithms and their properties. It describes how greedy algorithms work by making locally optimal choices at each step in the hope of reaching a globally optimal solution. Two examples are given: the activity selection problem and finding minimum spanning trees. Prim's algorithm for finding minimum spanning trees is described in detail, showing how it works by always selecting the lightest edge between the growing tree and remaining vertices.
The document discusses different types of images in Matlab including binary, grayscale, indexed, and RGB images. It also summarizes commands to convert between image types such as converting grayscale to indexed or truecolor to binary. Finally, it provides examples of how to view images, measure pixel values and distances, and crop images using the imtool command.
This paper introduces a new approach called temporal-difference (TD) search for game playing using reinforcement learning. TD search uses simulated self-play experiences to update the value function approximation for the current board position/state. Local shape features are used to evaluate board positions and are learned online through TD search rather than offline. Integrating short-term memories learned through simulation with long-term memories learned from real games results in improved performance, in a technique called Dyna-2. Experimental results show TD search improves over conventional alpha-beta search for the game of Go, achieving an Elo rating 300 points higher. However, modern Monte Carlo tree search methods have surpassed these results.
The document provides legal notices and disclaimers for an Intel presentation. It states that the presentation is for informational purposes only and that Intel makes no warranties. It also notes that Intel technologies' features and benefits depend on system configuration and may require enabled hardware, software or service activation. Performance varies depending on system configuration. The document further states that sample source code is released under the Intel Sample Source Code License Agreement and that Intel and its logo are trademarks.
This document provides a 3-sentence summary of a legal notice and disclaimer document:
The document states that any Intel technologies discussed are for informational purposes only and Intel makes no warranties regarding the information. It notes that performance may vary depending on system configuration and that source code samples are released under an Intel license agreement. The document also provides legal notices and disclaimers regarding Intel trademarks and copyright.
This document discusses using principal component analysis (PCA) and the discrete cosine transform (DCT) to recognize images from a database. It explains how bitmaps store image data, DCT compacts image energy, and PCA reduces dimensionality by finding the principal components via eigenvectors of the covariance matrix. An algorithm is proposed that uses DCT, PCA on a 3x3 block, characteristic equations to find the maximum eigenvector, and least mean square comparison to recognize queries against the database images.
The document discusses the greedy method algorithmic approach. It provides an overview of greedy algorithms including that they make locally optimal choices at each step to find a global optimal solution. The document also provides examples of problems that can be solved using greedy methods like job sequencing, the knapsack problem, finding minimum spanning trees, and single source shortest paths. It summarizes control flow and applications of greedy algorithms.
This document provides an overview of support vector machines (SVMs). It discusses how SVMs can be used to perform classification tasks by finding optimal separating hyperplanes that maximize the margin between different classes. The document outlines how SVMs solve an optimization problem to find these optimal hyperplanes using techniques like Lagrange duality, kernels, and soft margins. It also covers model selection methods like cross-validation and discusses extensions of SVMs to multi-class classification problems.
Numerical Methods in Mechanical Engineering - Final ProjectStasik Nemirovsky
Final Project for the class of "Numerical Methods in Mechanical Engineering" - MECH 309.
In this project, various engineering problems were analyzed and solved using advanced numerical approximation methods and MATLAB software.
This document provides an overview of matrix decomposition techniques for dimensionality reduction and topic modeling, specifically principal component analysis (PCA), singular value decomposition (SVD), latent semantic analysis (LSA), and non-negative matrix factorization (NMF). PCA and SVD are introduced as mathematical techniques to reduce dimensions while preserving variance/information. LSA and NMF are described as applying SVD and NMF respectively to text data to derive topic models from the latent semantic space. Examples of implementing these techniques in Python are also provided.
The document discusses image compression using the Haar wavelet basis transform. It begins by providing background on images, pixels, and file sizes. It then applies the Haar wavelet transform to compress an 8x8 sample image, representing it at decreasing resolutions of 4x4, 2x2, and 1x1 pixels. Matrices are used to represent the transform and allow reconstructing the original image. The technique provides lossless compression by retaining detail coefficients. Finally, it discusses interpreting the transform using vector spaces and scaling/wavelet basis functions.
The document discusses dimensionality reduction techniques. It begins by explaining the curse of dimensionality, where adding more features can hurt performance due to the exponential increase in the number of examples needed. It then introduces dimensionality reduction as a solution, where the data can be represented using fewer dimensions/features through feature selection, linear/non-linear transformations, or combinations. Principal component analysis (PCA) and singular value decomposition (SVD) are described as common linear dimensionality reduction methods. The document also discusses nonlinear techniques like kernel PCA and multi-dimensional scaling, as well as uses of dimensionality reduction like in image and natural language processing applications.
This document summarizes support vector machines (SVMs), a machine learning technique for classification and regression. SVMs find the optimal separating hyperplane that maximizes the margin between positive and negative examples in the training data. This is achieved by solving a convex optimization problem that minimizes a quadratic function under linear constraints. SVMs can perform non-linear classification by implicitly mapping inputs into a higher-dimensional feature space using kernel functions. They have applications in areas like text categorization due to their ability to handle high-dimensional sparse data.
This document outlines an assignment for a computer vision course. Students are asked to implement 4 vision algorithms: 2 using OpenCV and 2 using MATLAB. The algorithms are the log-polar transform, background subtraction, histogram equalization, and contrast stretching. Students must also answer 3 short questions about orthographic vs perspective projection, efficient filtering, and sensors beyond cameras for computer vision.
Image Classification And Support Vector MachineShao-Chuan Wang
This document discusses support vector machines and their application to image classification. It provides an overview of SVM concepts like functional and geometric margins, optimization to maximize margins, Lagrangian duality, kernels, soft margins, and bias-variance tradeoff. It also covers multiclass SVM approaches, dimensionality reduction techniques, model selection via cross-validation, and results from applying SVM to an image classification problem.
The document discusses algorithms and data structures using divide and conquer and greedy approaches. It covers topics like matrix multiplication, convex hull, binary search, activity selection problem, knapsack problem, and their algorithms and time complexities. Examples are provided for convex hull, binary search, activity selection, and knapsack problem algorithms. The document is intended as teaching material on design and analysis of algorithms.
Support Vector Machine (SVM) is a supervised machine learning algorithm that can be used for both classification and regression analysis. It works by finding a hyperplane in an N-dimensional space that distinctly classifies the data points. SVM selects the hyperplane that has the largest distance to the nearest training data points of any class, since larger the margin lower the generalization error of the classifier. SVM can efficiently perform nonlinear classification by implicitly mapping their inputs into high-dimensional feature spaces.
This document contains legal notices and disclaimers for an Intel presentation. It states that the presentation is for informational purposes only and that Intel makes no warranties. It also notes that performance depends on system configuration and that sample source code is released under an Intel license agreement. Finally, it provides basic copyright information.
This document provides an overview of a course on the finite element method. The course objectives are for students to learn how to write simple programs to solve problems using FEM. Assessment includes assignments, quizzes, a course project, midterm exam, and final exam. Fundamental agreements include electronic homework submission and using MATLAB or Mathematica. References on FEM are also provided. The document outlines numerical methods for solving boundary value problems and introduces weighted residual methods like the collocation method, subdomain method, and Galerkin method.
The document provides an introduction to the finite element method (FEM). It explains that FEM is a numerical method used to approximate solutions to partial differential equations. It works by dividing a complex problem into smaller, simpler parts called finite elements. This allows for the problem to be solved computationally. The document outlines the basic steps of FEM, including preprocessing (modeling, meshing), solving, and postprocessing (analyzing results). It also discusses applications, history, software, element types, meshing, convergence, and compatibility conditions of FEM.
This document provides an introduction to numerical methods and MATLAB programming for engineers. It covers topics such as vectors, functions, plots, and programming in MATLAB. The document is divided into multiple parts that cover various numerical methods topics, including solving equations, linear algebra, functions and data, and differential equations. MATLAB code and examples are provided throughout to demonstrate numerical techniques. The overall goal is to introduce both concepts of numerical methods and MATLAB programming within an engineering context.
My mini project to be submitted to Dr. Israd Hakim Jaafar.
This is my elective course taken in my 1st semester of final year.
Hope this will allow me to work with Oil and Gas company soon
This document discusses the finite element method for 2D heat conduction problems. It introduces several common 2D elements used to discretize the domain including the Melosh element, constant strain triangle element, and isoparametric four-node element. It outlines the basic steps of the finite element method including establishing the strong and weak formulations, discretization, selecting shape functions, assembling stiffness matrices and load vectors, and solving the system of equations. Exercises are provided to calculate element stiffness matrices and implement 2D patch tests to verify the elements.
The document is a lab manual for a course on Computer Graphics and Multimedia. It contains:
1. A table of contents listing various sections like the time table, university scheme, syllabus, list of books, and list of programs.
2. The time table, university scheme, and syllabus provide details about the course schedule, assessment scheme, and topics to be covered.
3. The list of books and list of programs provide resources for students to refer to for the course and experiments to be performed in the lab.
This document describes a computer based modelling project to simulate temperature distribution across a plate. The author developed a MATLAB script and graphical user interface (GUI) to allow users to input parameters and visualize the iterative temperature corrections. Key aspects included designing the GUI, implementing a method to correct temperatures using matrix operations based on Laplace's equation, and addressing challenges in modelling a central conducting hole. The completed project allows flexible adjustment of simulation variables and outputs clear graphs of the temperature distribution.
This document provides an overview of finite element analysis (FEA). It begins by defining FEA as a numerical method originally developed for solving solid mechanics problems, but now used for multiphysics problems. It describes how FEA can be applied to various engineering areas like structure analysis, solid mechanics, dynamics, thermal analysis, and more. The document then explains the general procedure of FEA, which involves setting up a physical model, discretizing it into finite elements, choosing approximation functions, formulating equations, and obtaining results. Finally, it provides some examples of FEA simulations and discusses new developments in the field.
The document provides an overview and tutorial of the MathCAD software program. It discusses (1) what MathCAD is and its main functions, (2) how to open and navigate the MathCAD interface and toolbars, and (3) examples of how to enter equations, plot graphs, and perform calculations in MathCAD. The tutorial aims to explain the basic concepts and features of MathCAD to help readers learn how to use it to solve engineering and mathematics problems.
This document summarizes a group project on impedance matching and tuning. It discusses using transformers on transmission lines to match the signal impedance to the load impedance. It describes several methods for impedance matching - using a quarter wave transformer, L-network matching, discrete elements, single stub tuning, and double stub tuning. It provides examples of applying these different matching techniques and shows the resulting simulation plots and solutions. It also discusses the group's process for completing the project, including researching the techniques, developing the MATLAB code, and testing the results.
Problem 7PurposeBreak apart a complicated system.ConstantsC7C13.docxLacieKlineeb
Problem 7Purpose:Break apart a complicated system.Constants:C7:C13Gas-Sparge
System
Pmo794(DI/DT)^4.38DI0.36(DI2N/v)^0.115DT1.22(DIN2/g)^1.96(DI/Dt)N2.8(Q/NDI3)v8.90E-07Right Sideg9.81PM←ANSWERSQ0.00416Computed Pm917The difference between the Computed Pm and Calculated Pm
Problem 8Purpose:Calculate Wind ChillConstants:ParametersWind Speed (km/h)a13.12Air Temp oC1020304050b0.621510c-11.370d0.3965-10-20←ANSWERS-30-40QuestionsThe formula to be used in E5 such that it can be filled down and across to make the table is: ….The name for cell B6 is …To modify this worksheet for Fahrenheit you need to …..
Problem 13Purpose:Calculate square roots using Heron's MethodConstants:N225Sqrt is←ANSWERSGuessN/GuessAverageTestError10
2
Project Topic Proposal
Harita Patel
Professor Dr. Bernard Parenteau
CIS 4498
Date: 11/1/22
Project Topic Proposal
The proposed topic is cyber security. My proposal in this software development project of this class is to develop cyber security software to be a tool that protects systems against malicious attacks and online threats. The software should b able to detect and block threats that can not be detected by antivirus. The technology to be used will be defensive Artificial intelligence. Cybersecurity professional experts can utilize guarded man-made consciousness (simulated intelligence) to distinguish or stop cyberattacks. Sagacious cybercriminals use innovations like hostile computer-based intelligence and ill-disposed AI since they are harder for conventional network protection instruments to identify. Offensive AI incorporates profound fakes, bogus pictures, personas, and recordings that convincingly portray individuals or things that never occurred or don't exist. Noxious entertainers can utilize ill-disposed AI to fool machines into breaking down by giving them mistaken information. Cybersecurity professionals can utilize cautious computer-based intelligence to recognize and prevent hostile man-made intelligence from estimating, testing, and figuring out how the framework or organization's capabilities. Defensive AI can reinforce calculations, making them more challenging to break. Network protection analysts can direct more extreme weakness tests on AI models.
Artificial intelligence cautious apparatuses can precisely anticipate assault vectors, pinpoint the delicate region of the organization and frameworks, and even set it up groups for approaching occasions(Graham, Olson,& Howard, 2016). The progression of computerized data is developing a regular schedule making it progressively challenging to oversee and structure it or even to isolate what is significantly based on what is pointless. Confronted with this test, new encouraging advancement innovations are being created to bring 'information examination's to the following developmental level. Man-made consciousness (man-made intelligence), specifically, is supposed to become huge in many fields. A few types of computer-based inte.
IRJET- Stress – Strain Field Analysis of Mild Steel Component using Finite El...IRJET Journal
This document analyzes the stress-strain field of a mild steel component under uniaxial loading using the finite element method. The component is modeled in MATLAB and Autodesk Fusion 360 using three elements and four nodes. Results are obtained for tensile forces of 1000N, 3000N, and 9000N. The displacement, stress, and strain values calculated in MATLAB are approximately similar to analytical solutions. Fusion 360 provides the maximum and minimum values within the component. The analysis demonstrates that finite element modeling can accurately determine stress-strain behavior under different loads.
This document summarizes numerical methods used in various fields including engineering, crime detection, scientific computing, finding roots, and solving heat equations. It discusses how numerical methods are widely used in engineering to model systems using mathematical equations when analytical solutions are not possible. Examples of applying numerical methods include structural analysis, fluid dynamics, image processing to deblur photos, and algorithms for finding roots of equations and solving differential equations.
This document provides an introduction to solving dynamics problems using MATLAB. It discusses numerical calculations, writing scripts, defining functions, graphics, symbolic calculations, differentiation, integration, and solving equations in MATLAB. It also contains sample dynamics problems from the textbook Engineering Mechanics: Dynamics that are solved using MATLAB. The problems cover kinematics and kinetics of particles and rigid bodies, as well as vibration analysis. The goal is to illustrate how computational tools like MATLAB can be used to aid in learning dynamics, but only after mastering the fundamentals through analytical problem solving.
The document provides a lab manual for the course GE3171 - Problem Solving and Python Programming Laboratory. It includes the course objectives, list of experiments, syllabus, and programs for various experiments involving Python programming concepts like lists, tuples, conditionals, loops, functions etc. The experiments cover problems on real-life applications such as electricity billing, library management, vehicle components, building materials etc. The document demonstrates how to write Python programs to solve such problems and validate the output.
FINITE ELEMENT FORMULATION FOR CONVECTIVE-DIFFUSIVE PROBLEMS WITH SHARP GRADI...Aleix Valls
This document presents a finite element formulation using Finite Calculus (FIC) for solving convective-diffusive problems with sharp gradients. FIC modifies the governing equations by including characteristic length distances, which helps stabilize solutions. The FIC method is implemented by solving the modified governing equations using the Galerkin finite element method. Characteristic length vectors are computed based on principal curvature directions of the solution. Numerical examples demonstrate that FIC provides more accurate solutions than standard SUPG by better resolving sharp gradients.
Dynamic programming (DP) is a powerful technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of already solved subproblems. The document provides examples of how DP can be applied to problems like rod cutting, matrix chain multiplication, and longest common subsequence. It explains the key elements of DP, including optimal substructure (subproblems can be solved independently and combined to solve the overall problem) and overlapping subproblems (subproblems are solved repeatedly).
(Slides) Efficient Evaluation Methods of Elementary Functions Suitable for SI...Naoki Shibata
The document proposes efficient methods for evaluating elementary functions like sin, cos, tan, log, and exp using SIMD instructions. The methods are twice as fast as floating point unit evaluation and have a maximum error of 6 ulps. They avoid conditional branches, gathering/scattering operations, and table lookups. Trigonometric functions are evaluated in two steps - argument reduction followed by a series evaluation. Inverse trigonometric, exponential and logarithmic functions are also efficiently evaluated in a similar manner suitable for SIMD computation. Evaluation accuracy and speed are evaluated against existing methods and the code size is kept small.
This document describes how to analyze a simple pendulum using MATLAB. It contains the equation of motion for a simple pendulum, notes on defining the equation in MATLAB, instructions for defining the two equations in a script file in MATLAB, and commands for running the code to output graphs of displacement over time and a phase trajectory plot.
The document discusses applications of nanotechnology in the construction industry. It describes how nanomaterials can improve properties of materials like concrete, coatings, and glass. Some key nanomaterials discussed are nano silica, titanium dioxide, and carbon nanotubes which can enhance strength, self-cleaning abilities, and durability of construction materials. The document also outlines various nanocoatings that provide benefits like fire protection, heat insulation, corrosion protection and abrasion resistance. Reinste Nano Ventures provides several nanomaterials and nanocoatings tailored for the construction industry.
This document discusses the effects of photoanode thickness on the performance of dye-sensitized solar cells (DSSC) simulated using an equivalent circuit model. Three DSSCs were fabricated with TiO2 absorption layer thicknesses of 6 μm, 12 μm, and 18 μm. Their current-voltage characteristics were measured and fitted to an equivalent circuit model. The model shows that both the series resistance (Rs) and shunt resistance (Rsh) increase with thicker absorption layers. It also shows that increasing Rs decreases the fill factor, while cell efficiency decreases as the material thickness increases due to increased resistance.
This document proposes a five day workshop on designing low-cost nanomaterial-based sensors for internet of things applications. The workshop will be held from April 2-6, 2018 with 30 TEQIP faculty participants. It will be organized by Dr. Amit Acharyya, Dr. Sushmee Badhulika, and Dr. Ashudeb Dutta from IITH, along with one foreign or Indian expert. The workshop aims to provide an introduction to flexible substrate-based nanosensors, detection techniques, interfacing techniques, and digital system design aspects. Topics to be covered include flexible electronics, sensor fundamentals, nano sensor materials, sensor design processes, sensing interface design, applications of digital signal processing
This document discusses experimental and numerical analysis of composites with delaminations. It summarizes that composites are widely used due to properties like strength and stiffness but are prone to delamination defects. The presence of delaminations reduces stiffness and changes vibration characteristics. The study investigates the effect of different percentages of delamination on the natural frequency of composite beams, through both experimental modal testing and numerical simulation using ANSYS. The results show that natural frequency decreases with increasing delamination size, with good agreement between experimental and numerical analysis. Frequency response functions also demonstrate changes with delamination.
This document summarizes a lecture on applying finite element analysis to one-dimensional heat transfer problems. It introduces the governing differential equation of heat conduction, describes how to derive this equation from the first law of thermodynamics, and discusses applying appropriate boundary conditions. It then outlines the finite element formulation process, including discretizing the domain into elements, approximating the temperature profile within each element, and assembling the element equations to solve for temperatures at nodes. An example problem of determining the temperature distribution along an insulated rod that generates internal heat is presented to demonstrate the method.
The Visvesvaraya National Institute of Technology in Nagpur is hosting a summer school from May 15th to July 1st 2019 on research conception, techniques, and publication. The objective is to provide guidelines for writing research papers and selecting publishable topics through hands-on training and subject expert guidance. The summer school will cover topics like journal selection and evaluation and include 70% sessions for experimental or theoretical work. Students, researchers, faculty and industry professionals interested in academic research can attend. Registration is required for a minimum of one month and fees range from Rs. 2500-3000 with accommodation and food also available. The last day to register is May 10th, 2019.
The document is the syllabus for the T.E. (Mechanical Engineering) course offered by Savitribai Phule Pune University in 2015. It includes:
1. An overview of the 6 semester curriculum including subject codes, credit details, teaching schemes and examination schemes.
2. Detailed course outlines for individual subjects like Design of Machine Elements-I, Heat Transfer, Theory of Machines-II etc. including course objectives, outcomes, contents and reference books.
3. Term work requirements consisting of design projects and assignments to be completed over the semester.
In summary, the document provides a comprehensive overview of the 3rd year mechanical engineering curriculum offered by Savitribai Ph
Autodesk inventor practice part drawingsGoldi Patil
This document contains 25 detailed drawings of miscellaneous mechanical parts for practice modeling in Autodesk Inventor or other 3D CAD software. The parts range in complexity and some hints are provided for specific features. While dimensions may be missing, the goal is for students to practice their modeling skills rather than reproduce the parts exactly. Students should build the parts by creating simple sketches and additive features.
Dr. Pramod Kothmire delivered his Ph.D. defense and viva voce examination at IIT Bombay on November 11, 2020. Mr. M. M. Shah participated in and completed an online FDP on Internet of Things from December 7-11, 2020. Dr. P. P. Kothmire arranged an expert lecture on writing good research papers which was delivered by Dr. Nilaj Deshmukh on December 21, 2020.
This research article analyzes the static response of functionally graded material (FGM) plates under transverse loads for varying aspect ratios. FGM plates are modeled with power law, sigmoid law, and exponential distributions for the volume fraction. Finite element analysis is performed in ANSYS to model FGM plates with 16 layers. Parameters such as volume fraction distribution and aspect ratio are varied. The FGM plate is subjected to uniform and point loads, and the response is analyzed. Results provide insight into how the varying material properties of FGMs impact their structural response under different loading and geometric conditions.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
3. Finite Element Method
Introduction, 1D heat conduction
3
Lecture plan Finite Element Method I
wind.civil.aau.dk/lecture/7sem_finite_element/Finite_element.htm
4. Finite Element Method
Introduction, 1D heat conduction
4
Form and expectations
To give the participants an understanding of the basic elements
of the finite element method as a tool for finding approximate
solutions of linear boundary value problems. These will be
exemplified with examples within stationary heat conduction.
After the course the participants should be able to set up their
own finite element solution of linear boundary value problems.
We expect that you participate in the given lectures and
exercises and ask questions when something is unclear.
Many of the exercises are programming problems, which
should be solved on a laptop. Preferably you should work in
groups of two. Each group should bring a laptop with MatLab
installed.
We expect minor prior knowledge of scientific programming.
Basic MatLab programming will be repeated during the lectures
You should bring pen and paper, and a copy of the lecture
notes.
5. Finite Element Method
Introduction, 1D heat conduction
5
Choices we have made
The lecture material is mainly slides. The referenced book is
additional information. The book is not a beginners book, but can be
used for additional information and also for advanced Finite Element
problems. We believe the basics can be understood from the slides
and attending the lectures.
You will be programming into a rather extensive matlab toolbox
developed by the lecturers. I.e. the overview might be lost in the
beginning but you will have a working program, which also will be
used in later courses. You will program a simple Finite Element
program from nothing during the final lectures.
We do everything in matlab. You will have your own program with
the source code and all the problems this gives. We believe that in
order to understand the Finite Element method you need to do some
programming instead of just using commercial solvers
4 full lectures including exercise time and 1 self study, instead of 5
short lectures without exercises
6. Finite Element Method
Introduction, 1D heat conduction
6
Literature
Lectures will be given from the slides.
The theory and derivations are not
directly based on the chapters in the
book
The book is chosen as a good overall
Finite Element book which can be used
also for more advanced problems
During the lectures references will be
given to relevant chapters and pages
where the present subject is presented
with Cook in parenthesis. E.g. (Cook:
chapter 4). These pages are summarized
in the lecture plan.
Relatively cheap!
7. Finite Element Method
Introduction, 1D heat conduction
7
Beginners literature
The formulations are based on this book,
however, it is expensive and does not
cover more advanced issues.
What is covered in this book is essential
what we cover during the lectures and on
the slides.
References to this book will be
referenced in parenthesis with OP. E.g.
(OP: chapter 4)
8. Finite Element Method
Introduction, 1D heat conduction
8
What is the Finite Element Method (FEM)?
A nummerical approach for solving partial differential equation,
boundary value problems
Finite element method
Finite difference method
Finite volume method
Boundary element method
An approxmative solution.
Simplification of geometry
Mesh dependent
Convergence issues
Widely used within structural analysis
deformation
heat conduction
9. Finite Element Method
Introduction, 1D heat conduction
9
Why use FEM?
General approach for solving complex structures.
Static undetermined systems causes no problems
unsteady problems can be solved (not included in this course)
Few basic building blocks (elements) can simulate complex
problems
Very suiteable for computer solution.
Fast solutions
Many different load cases
10. Finite Element Method
Introduction, 1D heat conduction
10
Basic steps of the finite-element method (FEM)
1. Establish strong formulation
Partial differential equation
2. Establish weak formulation
Multiply with arbitrary field and integrate over element
3. Discretize over space
Mesh generation
4. Select shape and weight functions
Galerkin method
5. Compute element stiffness matrix
Local and global system
6. Assemble global system stiffness matrix
7. Apply nodal boundary conditions
temperature/flux/forces/forced displacements
8. Solve global system of equations
Solve for nodal values of the primary variables
(displacements/temperature)
9. Compute temperature/stresses/strains etc. within the element
Using nodal values and shape functions
11. Finite Element Method
Introduction, 1D heat conduction
11
MatLab FE-program
main.m (main program runs until "return" plot functions located at the
bottom)
Coordinates.m (defines node coordinates, done by user)
Topology.m (defines the element topology, done by user)
BoundaryConditions.m (defines the BC, done by user)
calc_globdof.m (determines the global dof numbering, done by program)
Assemblering.m (assemble the global stiffness matrix, done by program)
solvequations.m (solve the system of equations for primary dof and forces,
done by program)
Plot functions
Visualize1Dheat.m (visualize the 1D heat problem with a temperature curve)
Visualize2D (visualize all 2D problems including 2D heat conduction and 2D
structural elements)
Visualize3Dbeam_struct.m (visualize the 3D beam problem with geometry,
deformation, beam normals, node and dof numbering)
Visualize3Dbeam_struct_secForce.m (visualize the 3D beam problem with
geometry, deformation, beam normals, node numbering and section forces)
12. Finite Element Method
Introduction, 1D heat conduction
12
Exercise: Test that the program can run.
Start main.m
Run the plotting of 1D heat conduction
Plotting functions are at the bottom of the program
Output:
u = [0 2]T
K = [1 0 ; 0 1]
Hints:
Open the m-file in the editor F5 runs the program until it reaches a "return"-
command or a red circle next to the line numbers (debug mode)
To debug use F11 to advance one line (will enter sub-functions). Use F10 to
advance one line without entering sub-functions
F9 runs the marked section
Enable cell-mode. Then ctrl-enter will run the present cell. Cells are divided
by %%
13. Finite Element Method
Introduction, 1D heat conduction
13
Advanced plotting in MatLab using handles
When a plot is generated in matlab corresponding handles are
created.
a handle for the figure
a handle for the axis
a handle for each plot on the figure
In this handle every information about the plot is defined
Direct handles are created by putting a variable before the plot
The information in the handle can be accessed by
The information can be changed by
>> hp = plot([0 1],[0 1])
>> get(hp)
information is put between ' '
>> set(hp,'linewidth',3,'marker','o')
new value, is put between ' '
if it is none numerical
14. Finite Element Method
Introduction, 1D heat conduction
14
Handle for figures and axes can be accessed either by putting a
handle when generating them or by gcf (get-current-figure) or gca
(get-current-axis) as handles
Changing values works as for the plot
Position information, remember correct units
position = [lower_left_corner_x lower_left_corner_y width height]
Activating a figure for adding axes or axes for plotting
>> get(gca)
>> get(gcf)
>> hf = figure
>> get(hf)
>> ha = axes
>> get(ha)
>> set(hf,'color',[1 0 0])
>> set(ha,'fontsize',22)
>> set(hf,'units','centimeters','postion',[5 5 10 10])
>> figure(hf) >> axes(hf)
15. Finite Element Method
Introduction, 1D heat conduction
15
Exercise: Make an advanced plot using handles
figure 1 units in centimeters postion lower left corner (5,5) size (10,10)
axis units in centimeters, position lower left corner (1,1) size (8,3)
xcolor and ycolor is red, figure color is black [r,g,b]=[0,0,0]
figure 2 units in centimeters postion lower left corner (15,5) size
(10,10)
axis units in centimeters, position lower left corner (1,6) size (8,3)
xcolor and ycolor is red, figure color is black [r,g,b]=[0,0,0]
Hint
you need to change units,
position and color of the figure-
handles
you need to change the units,
position, xcolor and ycolor of the
axis-handle
16. Finite Element Method
Introduction, 1D heat conduction
16
Make video using handles
The coordinates for the part that moves are changed through the handle
(xdata,ydata) and a 'drawnow' command is made for the plot to be
updated
figure
hold on
x = 0:0.001:5;
y = sin(x);
hp1 = plot(x,y);
hp2 = plot(x(1),y(1),'ro','markersize',15);
for j=1:size(x,2)
set(hp2,'xdata',x(j),'ydata',y(j));
drawnow
end
17. Finite Element Method
Introduction, 1D heat conduction
17
Generating an avi-file (only every 100th frame is captured)
Problem:
Powerpoint cannot play the movie directly
Solution:
re-code using e.g. virtualdub
www.virtualdub.org
p=1;
for j= 1:100:size(x,2);
set(hp2,'xdata',x(j),'ydata',y(j));
drawnow;
Mov(p) = getframe(gcf);
p=p+1;
end
movie2avi(Mov,'test2.avi','Quality',100,'compression','none');
compression method: none,
divx, xvid
18. Finite Element Method
Introduction, 1D heat conduction
18
Plotting many lines in the same plot
Every plot generates many informations, i.e. many plots become
heavy
Often a FEM solution is plotted using many line pieces because we
don't know the geometry in advance. Hence, every element needs to
be plotted separately
One line, one handle independent of the number of points
By introducing NaN (Not a Number) in the coordinates the line will
be broken but still work as one line (i.e. one handle)
19. Finite Element Method
Introduction, 1D heat conduction
19
Try the following
Try to mark the lines on the three figures
figure 1: The lines can be marked separately, i.e. changed separately
figure 2: One line is drawn and can only be changed as one, but this is
not what we want. We don't want the connecting lines
figure 3: The lines are drawn separately but are marked as one, i.e. one
handle
figure
hold on
hp1=plot([0 0],[0 1])
hp2=plot([1 1],[0 1])
hp3=plot([-0.2 1.4],[0.2 0.4])
figure
hold on
hp4=plot([0 0 1 1 -0.2 1.4],[0 1 0 1 0.2 0.4])
figure
hold on
hp4=plot([0 0 NaN 1 1 NaN -0.2 1.4],[0 1 NaN 0 1 NaN 0.2 0.4])
20. Finite Element Method
Introduction, 1D heat conduction
20
Call a function given in a string variable (used in Assembling.m)
Create 2 files cube.m and square.m
cube.m
square.m
Now try
Change the first line and run the second and third again
We can call many different functions by changing a string variable
FuncName = ‘cube’
Func = str2func(FuncName);
Output = Func(5)
function out=cube(x)
out = x^3;
function out=square(x)
out = x^2;
FuncName = ‘square’
21. Finite Element Method
Introduction, 1D heat conduction
21
Ways to store data in matlab
Matrices: use square brackets [] index in parenthesis af matrix name
e.g. A(row column) columns are separated by “ “ rows by “new line”
or “;”
Advantages: when the same type and number of data should be stored,
can count over index. Should ALWAYS be initialised (A = zeros(2,2);)
Cells: use curly brackets {}
Advantages: when the same type of data are stored but the number of
data varies. Can count over index in both curly and squared brackets
Structures:use “.” in name
Advantages: When data size and type varies. You can call the data by
name instead of just by index
A = [3 4 ; 5 6]
A(2,1) = ?
B = [3 4 5; 5 6 7]
C{1} = A; C{2} = B;
C{2}(2,1) = ?
S.A = A; S.B = B; S.C = C; S.info = ‘test’
S.info(3) = ?
22. Finite Element Method
Introduction, 1D heat conduction
22
Example problem 1D stationary heat conduction (Cook:
p21-22), (OP chapter 9)
Constant area, A, thermal conductivity, k, and heat supply Q
Boundary a: constant temperature T
Boundary b: constant flux q
24. Finite Element Method
Introduction, 1D heat conduction
24
Balance or conservation equation
Heat inflow equal heat outflow
By definition
Material property or constitutive relation
one-dimensional heat equation
H + Qdx = H + dH )
dH
dx
= Q
H = Aq
25. Finite Element Method
Introduction, 1D heat conduction
25
Second order differential equation needs two boundary conditions
Possible boundary conditions: temperature or temperature gradient
(flux)
This is the strong formulation for stationary 1D heat conduction
Constant A, k ,Q with T(a)=Ta , q(b)=qb , analytical solution can be
found (Differential equations and nummerical methods 5th semester,
Zill&Cullen)
26. Finite Element Method
Introduction, 1D heat conduction
26
Step 2: Establish weak formulation
(OP: chapter 4, p56-62)
Strong form
Multiply with an arbitrary function v(x) (weight function) and integrate
over the pertinent region
27. Finite Element Method
Introduction, 1D heat conduction
27
Use integration by parts of the first term to obtain the same
derivative of the weight function and primary variable T
Weak formulation of 1D stationary heat conduction
v is not completly arbitrary since the above derivation should hold (i.e. v
should be ones differentiable and defined in the region of integration)
boundary conditions
distributed load
28. Finite Element Method
Introduction, 1D heat conduction
28
Why are we interested in the weak form?
The FE-method is based on the weak form
The FE-method is an approximate method, i.e. in the strong form we
need an approximation to the second derivative of the temperature,
where only the first direvative is needed in the weak form at the cost
of introducing an arbitrary field.
The weak form holds when discontinuities occur, the strong form
requires a modification (see OP: p60-62)
29. Finite Element Method
Introduction, 1D heat conduction
29
Step 3: Discretize over space
Original problem
Discretized problem. Define: nodes, unknown (degree-of-freedom dof)
numbering, element numbering, element topology (which nodes define
the element)
Nodes
Elements
dof
coordinate
Node number
30. Finite Element Method
Introduction, 1D heat conduction
30
Step 4: Select shape and weight functions
(OP: chapter 7, p90-94, 98-106)
The finite element is an approximative method
The discrete values Ti of the unknowns are evaluated at the nodes
The continuous field T(x) is interpolated using shape functions.
31. Finite Element Method
Introduction, 1D heat conduction
31
Approximation requirements
Convergence criteria
In the limit when the elements are infinitely small, the approximation
should be infinitely close to the exact solution
Completeness requirements
The approximation must be able to represent an arbitrary constant
temperature gradient
The approximation must be able to represent an arbitrary constant
temperature
i.e. the approximation of the temperature should include terms
corresponding to a linear polynomial
Compatibility or conforming requirement
The approximation of the temperature over element boundaries must be
continues
convergence = completeness + compatibility
32. Finite Element Method
Introduction, 1D heat conduction
32
Assuming nodal values to be known
Linear variation of temperature allows a
constant temperature gradient
Simplest one-dimensional element (OP:
p98-99)
Test
Matrix notation
shape functions
nodal values (dof)
33. Finite Element Method
Introduction, 1D heat conduction
33
Weight functions (see OP: chapter 8,p142-156)
FE-method uses the Galerkin method (OP: p152-153)
weight functions = shape functions
Transpose of a scalar field equals the field
weight functions arbitrary unknown
arbitrary field
34. Finite Element Method
Introduction, 1D heat conduction
34
Step 5: Compute element stiffness matrix
If the weak formulation holds for the entire field, it also holds for part
of the field, i.e. integration is done over one element
Insert the temperature field and arbitrary field into the weak
formulation
a and c are constants, i.e. they can be taken outside the integrals
cT cancels from the equation!!!!
36. Finite Element Method
Introduction, 1D heat conduction
36
Exercise: Compute by hand the element stiffness matrix
and force vector
Assume constant A, k, Q
q(xi) = qi, q(xj) = qj
Need the derivative of the shape functions
37. Finite Element Method
Introduction, 1D heat conduction
37
Local and global coordinates (OP: p191-193)
Integration and shape functions are defined in a local coordinate
system, i.e. along the beam axis
Often we wish to normalize the integration. In this case it is not
nessesary but for the case of argument and introduction to 2D and
numerical integration.
Global coordinate system
Local coordinate system
40. Finite Element Method
Introduction, 1D heat conduction
40
Exercise: Program the shape functions
make a function shape_1d_heat.m
function [N,B] = shape_1d_heat(x,eCoord)
% Shape functions for 1D heat element
% INPUT
% x = coordinate where N and B is evaluated [1x1]
% eCoord = [x1 y1 z1 ; x2 y2 z2] Nodal coordinates
%
% OUTPUT
% N = shape functions evaluated at coordinate x [1x2]
% B = derivative of shape functions evaluated at coordinate x [1x2]
41. Finite Element Method
Introduction, 1D heat conduction
41
step 6: Assemble global system stiffness matrix
(OP: p184-191)
Global system of equations
The global system stiffness matrix [ndof x ndof] is assembled from
all the element stiffness matrices [2x2] according to the global
numbering of the degrees-of-freedom.
42. Finite Element Method
Introduction, 1D heat conduction
42
Example: 4 dofs, 3 elements
Need information (input) about
node coordinates
element definition or topology (which nodes are used, section (A) and
material (k))
From node number the dof-number is found
From node coordinates the length is defined
43. Finite Element Method
Introduction, 1D heat conduction
43
Local dof numbering vs. global dof numbering
Local system of equations, see exercise slide 36
Local numbering
Global numbering
44. Finite Element Method
Introduction, 1D heat conduction
44
Global system of equations
In a linear system we can add stiffnesses and loads (in this case
heat supply)
Element 1: Global numbering
46. Finite Element Method
Introduction, 1D heat conduction
46
The assembling procedure is as follows
1. Determine the local stiffness matrix
2. determine the global number of dof corresponding to the local dof for
the element
3. add the components of the local stiffness matrix to the rows and
columns of the global stiffness matrix corresponding to the global
dof numbers
4. repeat 1-3 until all contributions from all elements have been added.
In MatLab this is done in assemblering.m
K(gDof,gDof) = K(gDof,gDof) + Ke;
47. Finite Element Method
Introduction, 1D heat conduction
47
In order to be able to create a global system matrix we need to give
information about
Material for each element
Section dimensions for each element
Coordinates for each node (and a numbering)
Topology for each element (which nodes are in the element)
dof numbering is given from the node numbering and number of dofs per node
(one in this case)
See calc_globdof.m for numbering of global dof
GlobDof = [nDof1 GDof1 GDof2 ...]
dof numbering of each node
nDof = total number of Dof
48. Finite Element Method
Introduction, 1D heat conduction
48
Exercise: Define the following problem in the program
Discretize into 3 elements
change coordinates.m, topology.m, material and section definitions
Explain how the structure of section and material is defined
plot the solution
49. Finite Element Method
Introduction, 1D heat conduction
49
Exercise: Enter the correct stiffness matrix into the program
Look through assemblering.m and identify the steps in slide 46
Used functions:
elemInfo: provides element type,section number,Name of stiffness function
(see slide 20), coordinates for the element, global Dof numbering
secType: provides the name of the section function giving the constitutive
relation
Modify K_1D_heat.m
Ke = K_1D_heat(eCoord,ConstRel)
Input eCoord, ConstRel
Output Ke [2x2]
Output K [4x4]
50. Finite Element Method
Introduction, 1D heat conduction
50
Step 7: Apply nodal boundary conditions
temperature/flux
Why do we need boundary conditions?
K is known from material and geometric input [ndof x ndof]
a contains the unknown temperatures [ndof x 1]
f contains the unknown loads (heat supply) [ndof x 1]
i.e. 2 x ndof unknowns but only ndof equations
We need to define ndof of the unknowns,
We define either temperature or load in each node and solve for the
remaining unknowns, i.e. for the temperature where the load is
defined and for the load where the temperature is defined.
51. Finite Element Method
Introduction, 1D heat conduction
51
The force vector, see slide 45
Q is the heat supply assumed constant, qi are the boundary nodal
fluxes
52. Finite Element Method
Introduction, 1D heat conduction
52
Exercise: Determine the boundary conditions and enter
them into the program
modify BoundaryCondition.m
Type 1 is a temperature, type 2 is a heat
supply
Calculate the load vector by hand and
enter nodal values
53. Finite Element Method
Introduction, 1D heat conduction
53
Global system with boundary conditions entered
Left hand unknowns are called primary unknowns (temperature)
First we identify the equations with primary unknowns (in this case
row 2-4)
The defined temperatures are multiplied with their respective
columns of K and transferred to the right-hand side
Step 8: Solve global system of equations
54. Finite Element Method
Introduction, 1D heat conduction
54
The reduced system can be solved
Finally, the unknown heat supplies can be determined from the
remaining equations
The reduced solutions should enter the full solution, i.e. the
boundary conditions and solved temperature should be combined in
one vector for plotting.
This is done in MatLab in solvequations.m
ured = KredFred
55. Finite Element Method
Introduction, 1D heat conduction
55
Step 9: Compute temperature/fluxes within elements
Nodal values of dofs (temperature) are given from the solution of the
global system
Internal values are determined using shape functions and nodal
values corresponding to the relevant element. See slide 32
Fluxes are determined from the derivative of the temperature. See
slide 24.
I.e. internal values are determined from nodal values multiplied with
shape functions evaluated at the respective coordinate
56. Finite Element Method
Introduction, 1D heat conduction
56
Overall steps in MatLab FE-program
Input
section, material, node coordinates, topology, boundary conditions
Generate global dof numbering
Assemble global stiffness matrix
loop over elements
element section and material gives the constitutive relation
Calculate element stiffness matrix from coordinates and constitutive relation
global numbering of element dofs
add element stiffness matrix to the system stiffness matrix according to the
global dof numbering
Setup the global system of equations by introducing boundary
conditions and solve for dofs and forces
Post processing
determine internal fields (temperature/flux) from nodal values and shape
functions
visualize results
57. Finite Element Method
Introduction, 1D heat conduction
57
Exercise: Get familiar with the program
Change boundary conditions
Change topology
Change material and sections
try with different material and sections in the same analysis
Change the discretization (element coordinates)
should all be on the x-axis, i.e. y- and z-coordinate equal zero.
Compare with analytical solution, slide 25.
Determine the temperature variation of the following problem.
Discretize into 5 elements, for the area use the height at the middle
of the element (thickness=1.0 m)
x = [0.15 0.45 0.75 1.05 1.35]
h = [0.87 0.66 0.53 0.46 0.47]