This document provides information about the Introduction to Finite Elements course taught by Professor Suvranu De. It includes contact information for the course instructor, practicum instructor, and teaching assistant. It outlines the course texts, grades which are based on homework, practicum exercises, a course project, and quizzes. It describes collaboration policies and provides details on homework, practicum exercises, the course project, course content, and a linear algebra recap.
A Powerpoint Presentation designed to provide beginners to MATLAB an introduction to the MATLAB environment and introduce them to the fundamentals of MATLAB including matrix generation and manipulation, Arrays, MATLAB Graphics, Data Import and Export, etc
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 provides an overview of matrix algebra operations in R, including vectors, matrices, and their applications in psychological data analysis. It covers vector operations like addition, multiplication, and combining vectors into matrices. Matrix topics include addition, multiplication, finding the diagonal, identity matrices, and inversion. The document also demonstrates how these operations can be used for data manipulation tasks like calculating statistics, finding test reliability, and multiple correlation analyses.
This document provides an introduction to MATLAB programming. It discusses resources for the course including the course web page and slides. It then explains what MATLAB is, how to get started using it on Windows and Linux systems, and how to get help. It also covers the MATLAB desktop environment, performing calculations on the command line, entering numeric arrays, indexing into matrices, basic plotting commands, and logical indexing.
The topic of assignment is a critical problem in mathematics and is further explored in the real
physical world. We try to implement a replacement method during this paper to solve assignment problems with
algorithm and solution steps. By using new method and computing by existing two methods, we analyse a
numerical example, also we compare the optimal solutions between this new method and two current methods. A
standardized technique, simple to use to solve assignment problems, may be the proposed method
The document provides lecture notes for a course on matrix algebra for engineers. It covers topics such as the definition of matrices, addition and multiplication of matrices, special matrices like the identity and zero matrices, transposes, inverses, orthogonal matrices, and systems of linear equations. The notes are intended to teach the basics of matrix algebra at a level appropriate for engineering students who have taken calculus. They include video links, examples, problems at the end of each section, and solutions to the problems in an appendix.
MATLAB is an interactive development environment and programming language used by engineers and scientists for technical computing, data analysis, and algorithm development. It allows users to access data from files, web services, applications, hardware, and databases, and perform data analysis and visualization. MATLAB can be used for applications in areas like control systems, signal processing, communications, and more.
This document contains a presentation on vector analysis and matrices submitted by mechanical engineering students at Sonargaon University. It includes definitions of vectors, types of vectors, vector operations of addition, subtraction, dot product and cross product. It also defines different types of matrices, matrix operations of addition and subtraction, and scalar multiplication. Applications of vectors and matrices are discussed for calculating forces, velocities, and in cryptography to encrypt data for privacy.
A Powerpoint Presentation designed to provide beginners to MATLAB an introduction to the MATLAB environment and introduce them to the fundamentals of MATLAB including matrix generation and manipulation, Arrays, MATLAB Graphics, Data Import and Export, etc
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 provides an overview of matrix algebra operations in R, including vectors, matrices, and their applications in psychological data analysis. It covers vector operations like addition, multiplication, and combining vectors into matrices. Matrix topics include addition, multiplication, finding the diagonal, identity matrices, and inversion. The document also demonstrates how these operations can be used for data manipulation tasks like calculating statistics, finding test reliability, and multiple correlation analyses.
This document provides an introduction to MATLAB programming. It discusses resources for the course including the course web page and slides. It then explains what MATLAB is, how to get started using it on Windows and Linux systems, and how to get help. It also covers the MATLAB desktop environment, performing calculations on the command line, entering numeric arrays, indexing into matrices, basic plotting commands, and logical indexing.
The topic of assignment is a critical problem in mathematics and is further explored in the real
physical world. We try to implement a replacement method during this paper to solve assignment problems with
algorithm and solution steps. By using new method and computing by existing two methods, we analyse a
numerical example, also we compare the optimal solutions between this new method and two current methods. A
standardized technique, simple to use to solve assignment problems, may be the proposed method
The document provides lecture notes for a course on matrix algebra for engineers. It covers topics such as the definition of matrices, addition and multiplication of matrices, special matrices like the identity and zero matrices, transposes, inverses, orthogonal matrices, and systems of linear equations. The notes are intended to teach the basics of matrix algebra at a level appropriate for engineering students who have taken calculus. They include video links, examples, problems at the end of each section, and solutions to the problems in an appendix.
MATLAB is an interactive development environment and programming language used by engineers and scientists for technical computing, data analysis, and algorithm development. It allows users to access data from files, web services, applications, hardware, and databases, and perform data analysis and visualization. MATLAB can be used for applications in areas like control systems, signal processing, communications, and more.
This document contains a presentation on vector analysis and matrices submitted by mechanical engineering students at Sonargaon University. It includes definitions of vectors, types of vectors, vector operations of addition, subtraction, dot product and cross product. It also defines different types of matrices, matrix operations of addition and subtraction, and scalar multiplication. Applications of vectors and matrices are discussed for calculating forces, velocities, and in cryptography to encrypt data for privacy.
Solving ONE’S interval linear assignment problemIJERA Editor
This document presents a new method called the Matrix Ones Interval Linear Assignment Method (MOILA) for solving assignment problems with interval costs. It begins with definitions of assignment problems and interval analysis concepts. Then it describes the existing Hungarian method and provides an example solved using both Hungarian and MOILA. MOILA involves creating ones in the assignment matrix and making assignments based on the ones. The document outlines algorithms for MOILA as well as extensions to unbalanced and interval assignment problems. It provides an example of applying MOILA to solve a balanced interval assignment problem and compares the solutions to Hungarian. The document introduces MOILA as a systematic alternative to Hungarian for solving a variety of assignment problem types.
The Tensor Flight Dynamics Tutor is a condensed PowerPoint presentation of my textbook Introduction to Tensor Flight Dynamics. It serves as a review of the main elements of tensor flight dynamics and can be used in the class room by professor and students. The download is unrestricted, so share it freely.
- Linear algebra is important for image recognition and other fields like physics, economics, and politics. It allows analyzing relationships between multiple variables without calculus.
- Python is a good platform for linear algebra due to libraries like NumPy that allow fast processing of multi-dimensional data like matrices. It also has simple syntax without semicolons.
- Key concepts discussed include vectors, matrices, linear transformations, abstraction, and how linear algebra solves problems in fields like quantum mechanics. Comprehensions provide a concise way to generate sets, lists, and arrays in Python.
The document discusses various techniques for parallelizing numerical algorithms like matrix multiplication. It describes how to implement matrix multiplication in parallel using techniques like block decomposition and recursive subdivision. Mesh-based algorithms like Cannon's algorithm are also covered, which rearrange matrix elements across a processor mesh to efficiently calculate matrix products in parallel. The relationships between matrices and solving systems of linear equations are also noted.
Name _______________________________ Class time __________.docxrosemarybdodson23141
Name: _______________________________ Class time: __________
Prewriting Instructions for Paper 2 (Final Paper due 4/22)
1. Your choices for Paper 2 are posted on blackboard and also listed below.
2. Choose 1 of these paper options. Notice that each choice also mentions the type of paper (comparison, etc.) My paper choice is: _________________________: paper type: _______________.
3. Read the related essay(s) in your Research and Composition textbook.
4. Thursday: write a tentative thesis for paper 2 (one sentence): ______________________________________________________________________________________________________________________________________________________________________________________________________________________.
5. Thursday: write 5 questions that you will need to answer through research to write this paper (for ex. What is the divorce rate for 2012?) Write legibly please.
1.
2.
3.
4.
5.
6. Thursday: go to the library and use the databases to locate at least three sources that will likely give you the information to answer the five questions above. At least one should be a book, at least one should be a database article. In addition, you may use your textbook, internet, or even refer to a film. Write down the all of the information about each source. You will need this information for a works cited page later or to locate the article and book again. You do not need to answer the questions right away, but if you do find the answers, take notes or make a copy of the source.
Source 1: ____________________________________________________________________________________________________________________________________________________________
Source 2: ____________________________________________________________________________________________________________________________________________________________
Source 3: ____________________________________________________________________________________________________________________________________________________________
7. Have any new questions come to mind? What are they? Write them here:
8. Have you revised your thesis? What is it? ___________________________________
_____________________________________________________________________.
9. Write a tentative first paragraph to paper 2 (this includes your thesis):
10. Turn this in Tuesday 3/25 in exchange for your last Q exercise, M&M Color Distribution.
***You need this prewriting exercise completed to receive your instructions and data for this last Q exercise and parts of this exercise will count for your attendance in a week or so.
See next page
Writing Assignment 2 Choices due on or before 4/22
Here are your choices for Writing Assignment 2 due 4/22. Additional research is required for all choices. Two visuals, tables or figures, are required. Your paper will be in MLA format with a works cited page. This paper is approximately 5 pages including a works cited page.
1. Read the essays in Chapter 8. Go .
Using Monte Carlo Simulation in Project Estimates by Akram Najjar
The PMI Lebanon is glad to announce that Akram Najjar is the speaker for the a lecture titled “Using Monte Carlo Simulation in Project Estimates” delivered on Thursday, 28 July 2016
Lecture Outline
* Why are single point estimates unreliable and what is the alternative?
* What are distributions and how do we extract random samples from them (using Excel)? Two costing examples.
* How to setup a Monte Carlo Simulation model in a spreadsheet?
* Two PM examples (in detail)
* How to statistically analyze the thousands of runs to reach reliable estimates?
Lecture Objectives
* A Project Manager usually knows how certain parameters (such as duration, resource rates or quantities) behave. However, the PM can almost never define reliable single point estimates for these parameters. The result: many projects fail due to unreliable estimates. The alternative? The PM has to use his/her knowledge of how specific parameters behave statistically. For example, the PM knows that a specific task’s duration is distributed according to the bell shaped curve OR that another is uniformly distributed (flat variation), or triangular, or Beta-PERT, etc. The PM can then use Monte Carlo Simulation (MCS) to arrive at statistically significant and robust results. Monte Carlo Simulation (MCS) is a technique that relies on two processes. Process 1 aims at developing a spreadsheet model that calculates the critical path or the total cost, etc. The calculation is setup in a single row (or Run). This row is then duplicated a large number of times (thousands). Process 2 aims at inserting Excel functions in each of the parameters (durations, costs). In each row (or Run), such functions will provide a sample drawn from a statistical distribution that properly describes the behavior of that parameter. For example, a specific duration follows a Normal (Bell) distribution with an Average A and a Standard Deviation S. The model will then generate for each run and for that duration a different value that conforms with the bell shaped curve as defined (A and S). Each of these thousands of runs will provide the PM with a different “simulation” of the duration or the total cost, etc. By statistically analyzing the thousands of results, the PM can arrive at a robust and reliable estimate. Proprietary Add On’s for Monte Carlo Simulation in Microsoft Project are available. However, it is easy, free and more flexible to use native Microsoft functions to carry out the full simulation. The talk covered all the steps needed for such simulations giving several examples
The document provides information about a course on Computer Aided Engineering Drawing. It includes 5 modules that cover topics like basic drawing principles, orthographic projections, isometric projections, development of surfaces, and multidisciplinary applications. Students will learn to generate drawings using manual techniques and CAD software. By the end of the course, students will be able to draw objects, recognize shapes through different views, develop object surfaces, create CAD drawings, and understand engineering components through diagrams. Student work will be evaluated continuously and through tests, while a semester exam will also be administered.
This document provides an introduction to MATLAB. It discusses what MATLAB is, how to perform basic matrix operations and use script files and M-files. It also covers some common MATLAB commands and functions. MATLAB can be used for applications like plotting, image processing, robotics and GUI design. Key topics covered include matrices, vectors, scalars, matrix operations, logical and relational operators, selection and repetition structures, and reading/writing data files. Plotting functions allow creating graphs and 3D surface plots. Image processing, robotics and GUI design are listed as potential application areas.
The document discusses finite element analysis (FEA) and its applications. It provides an overview of FEA, including the basic theory and principles. It explains that FEA is a numerical method for solving engineering problems by dividing a complex system into smaller pieces called finite elements. The document lists various element types and common applications of FEA, such as thermal, modal, buckling, and non-linear analyses. It also provides resources on FEA tutorials and examples involving different problem types.
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 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.
This document provides an introduction and overview of MATLAB. It discusses MATLAB basics like the command window and variables. It also covers topics like working with matrices, vectors, plotting, scripts and functions. Specific MATLAB commands covered include plot, mesh, surf, contour and more. Functions like length, dot, cross and special matrices like ones, zeros and eye are also explained.
MATLAB/SIMULINK for Engineering Applications day 2:Introduction to simulinkreddyprasad reddyvari
The document provides an introduction to MATLAB and Simulink through a presentation. It discusses what MATLAB and Simulink are, their basic functions and capabilities, and how to get started using them. The presentation covers topics such as vectors, matrices, plotting, control structures, M-files, and writing user-defined functions. The goal is to help attendees gain basic knowledge of MATLAB/Simulink and be able to explore them on their own.
This document provides the syllabus for the B.Tech Computer Science and Engineering program at Jawaharlal Nehru Technological University Hyderabad. It outlines the course structure, course codes, titles, credits, and descriptions for each semester of the 4-year program. The syllabus covers core computer science topics like programming, data structures, algorithms, databases, operating systems, as well as mathematics, science, and elective courses. Laboratory courses accompany many of the core CS courses to provide hands-on learning experiences. Upon completion of the program, students will have gained expertise in key areas of computer science along with communication and professional skills to become successful software engineers.
This document provides the syllabus for the B.Tech Computer Science and Engineering program at Jawaharlal Nehru Technological University Hyderabad. It outlines the course structure, course codes, titles, credits, and objectives over 4 years. In the first year, courses cover general subjects like Mathematics, Chemistry, and programming. In later years, courses focus on Computer Science topics such as Data Structures, Operating Systems, Databases, Networks, and electives. Students complete industry projects, a thesis, and general courses on business, communication skills, and environmental science. The document details the program's course requirements and objectives to provide students with a well-rounded education in CSE.
MATLAB DOCUMENTATION ON SOME OF THE MODULES
A.Generate videos in which a skeleton of a person doing the following Gestures.
1.Tilting his head to right and left
2.Tilting his hand to right and left
3.Walking
in matlab.
B. Write a MATLAB program that converts a decimal number to Roman number and vice versa.
C.Using EZ plot & anonymous functions plot the following:
· Y=Sqrt(X)
· Y= X^2
· Y=e^(-XY)
D.Take your picture and
· Show R, G, B channels along with RGB Image in same figure using sub figure.
· Convert into HSV( Hue, saturation and value) and show the H,S,V channels along with HSV image
E.Record your name pronounced by yourself. Try to display the signal(name) in a plot vs Time, using matlab.
F.Write a script to open a new figure and plot five circles, all centered at the origin and with increasing radii. Set the line width for each circle to something thick (at least 2 points), and use the colors from a 5-color jet colormap (jet).
G. NEWTON RAPHSON AND SECANT METHOD
H.Write any one of the program to do following things using file concept.
1.Create or Open a file
2. Read data from the file and write data to another file
3. Append some text to already existed file
4. Close the file
I.Write a function to perform following set operations
1.Union of A and B
2. Intersection of A and B
3. Complement of A and B
(Assume A= {1, 2, 3, 4, 5, 6}, B= {2, 4, 6})
Computer Science CS Project Matrix CBSE Class 12th XII .pdfPranavAnil9
The following project is based on Matrices and Determinants. It is a menu based program with data file and SQL Connectivity. The program is capable of performing all the complex functions of matrices and determinants that are mentioned in the Class 12th Math’s book. The ‘Menu’ of the program upon which it executes is as follows:
1: Generate a Random Matrix
2: Addition
3: Subtraction
4: Multiplication by a Scalar
5: Multiplication by a Matrix
6: Calculate Determinant
7: Calculate Minor
8: Calculate Cofactor
9: Calculate Adjoint
10: Transpose
11: Inversion
Unit 5 Java Programming with Linux-converted.pdfranjithunni35
This document provides an overview of applets, Java Database Connectivity (JDBC), and some example programs. It discusses:
1. What applets are and the two types (local and remote).
2. The key classes and tags used to build applets, including <applet> and the applet lifecycle methods.
3. How to pass parameters to applets using the <param> tag.
4. The Graphics class and common drawing methods like drawLine(), drawOval(), etc.
5. The 4 types of JDBC drivers and the basic steps to connect to a database using the DriverManager, Connection, Statement, and ResultSet classes.
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 binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Solving ONE’S interval linear assignment problemIJERA Editor
This document presents a new method called the Matrix Ones Interval Linear Assignment Method (MOILA) for solving assignment problems with interval costs. It begins with definitions of assignment problems and interval analysis concepts. Then it describes the existing Hungarian method and provides an example solved using both Hungarian and MOILA. MOILA involves creating ones in the assignment matrix and making assignments based on the ones. The document outlines algorithms for MOILA as well as extensions to unbalanced and interval assignment problems. It provides an example of applying MOILA to solve a balanced interval assignment problem and compares the solutions to Hungarian. The document introduces MOILA as a systematic alternative to Hungarian for solving a variety of assignment problem types.
The Tensor Flight Dynamics Tutor is a condensed PowerPoint presentation of my textbook Introduction to Tensor Flight Dynamics. It serves as a review of the main elements of tensor flight dynamics and can be used in the class room by professor and students. The download is unrestricted, so share it freely.
- Linear algebra is important for image recognition and other fields like physics, economics, and politics. It allows analyzing relationships between multiple variables without calculus.
- Python is a good platform for linear algebra due to libraries like NumPy that allow fast processing of multi-dimensional data like matrices. It also has simple syntax without semicolons.
- Key concepts discussed include vectors, matrices, linear transformations, abstraction, and how linear algebra solves problems in fields like quantum mechanics. Comprehensions provide a concise way to generate sets, lists, and arrays in Python.
The document discusses various techniques for parallelizing numerical algorithms like matrix multiplication. It describes how to implement matrix multiplication in parallel using techniques like block decomposition and recursive subdivision. Mesh-based algorithms like Cannon's algorithm are also covered, which rearrange matrix elements across a processor mesh to efficiently calculate matrix products in parallel. The relationships between matrices and solving systems of linear equations are also noted.
Name _______________________________ Class time __________.docxrosemarybdodson23141
Name: _______________________________ Class time: __________
Prewriting Instructions for Paper 2 (Final Paper due 4/22)
1. Your choices for Paper 2 are posted on blackboard and also listed below.
2. Choose 1 of these paper options. Notice that each choice also mentions the type of paper (comparison, etc.) My paper choice is: _________________________: paper type: _______________.
3. Read the related essay(s) in your Research and Composition textbook.
4. Thursday: write a tentative thesis for paper 2 (one sentence): ______________________________________________________________________________________________________________________________________________________________________________________________________________________.
5. Thursday: write 5 questions that you will need to answer through research to write this paper (for ex. What is the divorce rate for 2012?) Write legibly please.
1.
2.
3.
4.
5.
6. Thursday: go to the library and use the databases to locate at least three sources that will likely give you the information to answer the five questions above. At least one should be a book, at least one should be a database article. In addition, you may use your textbook, internet, or even refer to a film. Write down the all of the information about each source. You will need this information for a works cited page later or to locate the article and book again. You do not need to answer the questions right away, but if you do find the answers, take notes or make a copy of the source.
Source 1: ____________________________________________________________________________________________________________________________________________________________
Source 2: ____________________________________________________________________________________________________________________________________________________________
Source 3: ____________________________________________________________________________________________________________________________________________________________
7. Have any new questions come to mind? What are they? Write them here:
8. Have you revised your thesis? What is it? ___________________________________
_____________________________________________________________________.
9. Write a tentative first paragraph to paper 2 (this includes your thesis):
10. Turn this in Tuesday 3/25 in exchange for your last Q exercise, M&M Color Distribution.
***You need this prewriting exercise completed to receive your instructions and data for this last Q exercise and parts of this exercise will count for your attendance in a week or so.
See next page
Writing Assignment 2 Choices due on or before 4/22
Here are your choices for Writing Assignment 2 due 4/22. Additional research is required for all choices. Two visuals, tables or figures, are required. Your paper will be in MLA format with a works cited page. This paper is approximately 5 pages including a works cited page.
1. Read the essays in Chapter 8. Go .
Using Monte Carlo Simulation in Project Estimates by Akram Najjar
The PMI Lebanon is glad to announce that Akram Najjar is the speaker for the a lecture titled “Using Monte Carlo Simulation in Project Estimates” delivered on Thursday, 28 July 2016
Lecture Outline
* Why are single point estimates unreliable and what is the alternative?
* What are distributions and how do we extract random samples from them (using Excel)? Two costing examples.
* How to setup a Monte Carlo Simulation model in a spreadsheet?
* Two PM examples (in detail)
* How to statistically analyze the thousands of runs to reach reliable estimates?
Lecture Objectives
* A Project Manager usually knows how certain parameters (such as duration, resource rates or quantities) behave. However, the PM can almost never define reliable single point estimates for these parameters. The result: many projects fail due to unreliable estimates. The alternative? The PM has to use his/her knowledge of how specific parameters behave statistically. For example, the PM knows that a specific task’s duration is distributed according to the bell shaped curve OR that another is uniformly distributed (flat variation), or triangular, or Beta-PERT, etc. The PM can then use Monte Carlo Simulation (MCS) to arrive at statistically significant and robust results. Monte Carlo Simulation (MCS) is a technique that relies on two processes. Process 1 aims at developing a spreadsheet model that calculates the critical path or the total cost, etc. The calculation is setup in a single row (or Run). This row is then duplicated a large number of times (thousands). Process 2 aims at inserting Excel functions in each of the parameters (durations, costs). In each row (or Run), such functions will provide a sample drawn from a statistical distribution that properly describes the behavior of that parameter. For example, a specific duration follows a Normal (Bell) distribution with an Average A and a Standard Deviation S. The model will then generate for each run and for that duration a different value that conforms with the bell shaped curve as defined (A and S). Each of these thousands of runs will provide the PM with a different “simulation” of the duration or the total cost, etc. By statistically analyzing the thousands of results, the PM can arrive at a robust and reliable estimate. Proprietary Add On’s for Monte Carlo Simulation in Microsoft Project are available. However, it is easy, free and more flexible to use native Microsoft functions to carry out the full simulation. The talk covered all the steps needed for such simulations giving several examples
The document provides information about a course on Computer Aided Engineering Drawing. It includes 5 modules that cover topics like basic drawing principles, orthographic projections, isometric projections, development of surfaces, and multidisciplinary applications. Students will learn to generate drawings using manual techniques and CAD software. By the end of the course, students will be able to draw objects, recognize shapes through different views, develop object surfaces, create CAD drawings, and understand engineering components through diagrams. Student work will be evaluated continuously and through tests, while a semester exam will also be administered.
This document provides an introduction to MATLAB. It discusses what MATLAB is, how to perform basic matrix operations and use script files and M-files. It also covers some common MATLAB commands and functions. MATLAB can be used for applications like plotting, image processing, robotics and GUI design. Key topics covered include matrices, vectors, scalars, matrix operations, logical and relational operators, selection and repetition structures, and reading/writing data files. Plotting functions allow creating graphs and 3D surface plots. Image processing, robotics and GUI design are listed as potential application areas.
The document discusses finite element analysis (FEA) and its applications. It provides an overview of FEA, including the basic theory and principles. It explains that FEA is a numerical method for solving engineering problems by dividing a complex system into smaller pieces called finite elements. The document lists various element types and common applications of FEA, such as thermal, modal, buckling, and non-linear analyses. It also provides resources on FEA tutorials and examples involving different problem types.
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 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.
This document provides an introduction and overview of MATLAB. It discusses MATLAB basics like the command window and variables. It also covers topics like working with matrices, vectors, plotting, scripts and functions. Specific MATLAB commands covered include plot, mesh, surf, contour and more. Functions like length, dot, cross and special matrices like ones, zeros and eye are also explained.
MATLAB/SIMULINK for Engineering Applications day 2:Introduction to simulinkreddyprasad reddyvari
The document provides an introduction to MATLAB and Simulink through a presentation. It discusses what MATLAB and Simulink are, their basic functions and capabilities, and how to get started using them. The presentation covers topics such as vectors, matrices, plotting, control structures, M-files, and writing user-defined functions. The goal is to help attendees gain basic knowledge of MATLAB/Simulink and be able to explore them on their own.
This document provides the syllabus for the B.Tech Computer Science and Engineering program at Jawaharlal Nehru Technological University Hyderabad. It outlines the course structure, course codes, titles, credits, and descriptions for each semester of the 4-year program. The syllabus covers core computer science topics like programming, data structures, algorithms, databases, operating systems, as well as mathematics, science, and elective courses. Laboratory courses accompany many of the core CS courses to provide hands-on learning experiences. Upon completion of the program, students will have gained expertise in key areas of computer science along with communication and professional skills to become successful software engineers.
This document provides the syllabus for the B.Tech Computer Science and Engineering program at Jawaharlal Nehru Technological University Hyderabad. It outlines the course structure, course codes, titles, credits, and objectives over 4 years. In the first year, courses cover general subjects like Mathematics, Chemistry, and programming. In later years, courses focus on Computer Science topics such as Data Structures, Operating Systems, Databases, Networks, and electives. Students complete industry projects, a thesis, and general courses on business, communication skills, and environmental science. The document details the program's course requirements and objectives to provide students with a well-rounded education in CSE.
MATLAB DOCUMENTATION ON SOME OF THE MODULES
A.Generate videos in which a skeleton of a person doing the following Gestures.
1.Tilting his head to right and left
2.Tilting his hand to right and left
3.Walking
in matlab.
B. Write a MATLAB program that converts a decimal number to Roman number and vice versa.
C.Using EZ plot & anonymous functions plot the following:
· Y=Sqrt(X)
· Y= X^2
· Y=e^(-XY)
D.Take your picture and
· Show R, G, B channels along with RGB Image in same figure using sub figure.
· Convert into HSV( Hue, saturation and value) and show the H,S,V channels along with HSV image
E.Record your name pronounced by yourself. Try to display the signal(name) in a plot vs Time, using matlab.
F.Write a script to open a new figure and plot five circles, all centered at the origin and with increasing radii. Set the line width for each circle to something thick (at least 2 points), and use the colors from a 5-color jet colormap (jet).
G. NEWTON RAPHSON AND SECANT METHOD
H.Write any one of the program to do following things using file concept.
1.Create or Open a file
2. Read data from the file and write data to another file
3. Append some text to already existed file
4. Close the file
I.Write a function to perform following set operations
1.Union of A and B
2. Intersection of A and B
3. Complement of A and B
(Assume A= {1, 2, 3, 4, 5, 6}, B= {2, 4, 6})
Computer Science CS Project Matrix CBSE Class 12th XII .pdfPranavAnil9
The following project is based on Matrices and Determinants. It is a menu based program with data file and SQL Connectivity. The program is capable of performing all the complex functions of matrices and determinants that are mentioned in the Class 12th Math’s book. The ‘Menu’ of the program upon which it executes is as follows:
1: Generate a Random Matrix
2: Addition
3: Subtraction
4: Multiplication by a Scalar
5: Multiplication by a Matrix
6: Calculate Determinant
7: Calculate Minor
8: Calculate Cofactor
9: Calculate Adjoint
10: Transpose
11: Inversion
Unit 5 Java Programming with Linux-converted.pdfranjithunni35
This document provides an overview of applets, Java Database Connectivity (JDBC), and some example programs. It discusses:
1. What applets are and the two types (local and remote).
2. The key classes and tags used to build applets, including <applet> and the applet lifecycle methods.
3. How to pass parameters to applets using the <param> tag.
4. The Graphics class and common drawing methods like drawLine(), drawOval(), etc.
5. The 4 types of JDBC drivers and the basic steps to connect to a database using the DriverManager, Connection, Statement, and ResultSet classes.
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 binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Immersive Learning That Works: Research Grounding and Paths ForwardLeonel Morgado
We will metaverse into the essence of immersive learning, into its three dimensions and conceptual models. This approach encompasses elements from teaching methodologies to social involvement, through organizational concerns and technologies. Challenging the perception of learning as knowledge transfer, we introduce a 'Uses, Practices & Strategies' model operationalized by the 'Immersive Learning Brain' and ‘Immersion Cube’ frameworks. This approach offers a comprehensive guide through the intricacies of immersive educational experiences and spotlighting research frontiers, along the immersion dimensions of system, narrative, and agency. Our discourse extends to stakeholders beyond the academic sphere, addressing the interests of technologists, instructional designers, and policymakers. We span various contexts, from formal education to organizational transformation to the new horizon of an AI-pervasive society. This keynote aims to unite the iLRN community in a collaborative journey towards a future where immersive learning research and practice coalesce, paving the way for innovative educational research and practice landscapes.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
5. Course texts and references
Course text (for HW problems):
Title: A First Course in the Finite Element Method
Author: Daryl Logan
Edition: Sixth
Publisher: Cengage Learning
ISBN: 0-534-55298-6
Relevant reference:
Finite Element Procedures, K. J. Bathe, Prentice Hall
A First Course in Finite Elements, J. Fish and T. Belytschko
Lecture notes posted in LMS
6. Course grades
Grades will be based on:
1. Home works (15 %).
2. Practicum exercises (10 %) to be submitted within a
week of assignment.
3. Course project (25 %)to be submitted by December
10th (by noon)
4. Two in-class quizzes (2x25%) on 19th October, 10th
December
1. All write ups that you present MUST contain your name and RIN
2. Basic knowledge of Linear Algebra is necessary for this course.
Read Appendix A of your course text and the first lecture notes.
There will be a mini-quiz next class, the grades of which will be
included in HW1.
3. There will be NO FINAL EXAM for this course.
7. Collaboration / academic integrity
1. Students are encouraged to collaborate in the
solution of HW problems, but submit independent
solutions that are NOT copies of each other.
Funny solutions (that appear similar/same) will be
given zero credit.
Software may be used to verify the HW solutions.
But submission of software solution will result in
zero credit.
2. Groups of 2 for the projects
(no two projects to be the same/similar)
A single grade will be assigned to the group and not
to the individuals.
8. Homeworks (15%)
1. Be as detailed and explicit as possible. For full
credit Do NOT omit steps.
2. Only neatly written homeworks will be graded
3. Late homeworks will NOT be accepted.
4. Two lowest grades will be dropped (except HW
#1).
5. Solutions will be posted on LMS
9. Practicum (10%)
1. Five classes designated as “Practicum”.
2. You will need to download and install NX on your
laptops and bring them to class on these days.
3. At the end of each practicum, you will be assigned
a single problem (worth 2 points).
4. You will need to submit the practicum problem
within a week of the assignment.
5. No late submissions will be entertained.
10. Course Project (25 %)
In this project you will be required to
• choose an engineering system
• develop a mathematical model for the system
• develop the finite element model
• solve the problem using commercial software
• present a convergence plot and discuss whether
the mathematical model you chose gives you
physically meaningful results.
• refine the model if necessary.
11. • Form groups of 2 and email the TA by 24th September.
• Submit 1-page project proposal by emailing to
ifea2021fall@gmail.com latest by 8th October (in
class). The earlier the better. Projects will go on a first
come first served basis.
• Proceed to work on the project ONLY after it is
approved by the course instructor.
• Submit a one-page progress report by emailing to
ifea2021fall@gmail.com on November 5th (this will
count as 10% of your project grade)
• Submit a project report emailing to
ifea2021fall@gmail.com the TA by noon of 10th
December.
Course project (25 %)..contd.
12. Major project (25 %)..contd.
Project report:
1. Must be professional (Text font Times 11pt with
single spacing)
2. Must include the following sections:
•Introduction
•Problem statement
•Analysis
•Results and Discussions
13. Major project (25 %)..contd.
Project examples:
(two sample project reports from previous year are
provided)
1. Analysis of a rocker arm
2. Analysis of a bicycle crank-pedal assembly
3. Design and analysis of a "portable stair climber"
4. Analysis of a gear train
5.Gear tooth stress in a wind- up clock
6. Analysis of a gear box assembly
7. Analysis of an artificial knee
8. Forces acting on the elbow joint
9. Analysis of a soft tissue tumor system
10. Finite element analysis of a skateboard truck
14. Major project (25 %)..contd.
Project grade will depend on
1.Originality of the idea
2.Techniques used
3.Critical discussion
15. Cantilever plate
in plane strain
uniform loading
Fixed
boundary
Problem: Obtain the
stresses/strains in the
plate
Node
Element
Finite element
model
• Approximate method
• Geometric model
• Node
• Element
• Mesh
• Discretization
16. Course content
1. “Direct Stiffness” approach for springs
2. Bar elements and truss analysis
3. Introduction to boundary value problems: strong form, principle of
minimum potential energy and principle of virtual work.
4. Displacement-based finite element formulation in 1D: formation of
stiffness matrix and load vector, numerical integration.
5. Displacement-based finite element formulation in 2D: formation of
stiffness matrix and load vector for CST and quadrilateral elements.
6. Discussion on issues in practical FEM modeling
7. Convergence of finite element results
8. Higher order elements
9. Isoparametric formulation
10. Numerical integration in 2D
11. Solution of linear algebraic equations
17. For next class
Please read Appendix A of Logan for reading
quiz next class (10 pts on Hw 1)
19. A rectangular array of numbers (we will concentrate on
real numbers). A nxm matrix has ‘n’ rows and ‘m’
columns
34
33
32
31
24
23
22
21
14
13
12
11
M
M
M
M
M
M
M
M
M
M
M
M
3x4
M
What is a matrix?
First
column
First row
Second row
Third row
Second
column
Third
column
Fourth
column
12
M
Row number
Column number
20. What is a vector?
A vector is an array of ‘n’ numbers
A row vector of length ‘n’ is a 1xn matrix
A column vector of length ‘m’ is a mx1 matrix
4
3
2
1 a
a
a
a
3
2
1
a
a
a
21. Special matrices
Zero matrix: A matrix all of whose entries are zero
Identity matrix: A square matrix which has ‘1’ s on the
diagonal and zeros everywhere else.
0
0
0
0
0
0
0
0
0
0
0
0
4
3
0 x
1
0
0
0
1
0
0
0
1
3
3x
I
24. Properties of matrix addition:
1. Matrix addition is commutative (order of
addition does not matter)
2. Matrix addition is associative
3. Addition of the zero matrix
Matrix operations
A
B
B
A
Addition of of matrices
C
B
A
C
B
A
A
A
0
0
A
Properties
25. Matrix operations Multiplication by a
scalar
If A is a matrix and c is a scalar, then the product cA is a
matrix whose entries are obtained by multiplying each of
the entries of A by c
15
3
27
21
0
9
12
6
3
3
5
1
9
7
0
3
4
2
1
cA
c
A
26. Matrix operations
Multiplication by a
scalar
If A is a matrix and c =-1 is a scalar, then the product
(-1)A =-A is a matrix whose entries are obtained by
multiplying each of the entries of A by -1
5
1
9
7
0
3
4
2
1
1
5
1
9
7
0
3
4
2
1
-A
cA
c
A
Special case
28. Special
operations
Transpose
If A is a mxn matrix, then the transpose of A is
the nxm matrix whose first column is the first
row of A, whose second column is the second
column of A and so on.
5
7
4
1
0
2
9
3
1
A
5
1
9
7
0
3
4
2
1
A T
30. Matrix operations Scalar (dot) product of
two vectors
3
2
1
3
2
1
b
b
b
;
a
a
a
b
a
If a and b are two vectors of the same size
The scalar (dot) product of a and b is a scalar
obtained by adding the products of
corresponding entries of the two vectors
T
1 1 2 2 3 3
a b a b a b
a b
31. Matrix operations Matrix multiplication
For a product to be defined, the number of columns
of A must be equal to the number of rows of B.
A B = AB
m x r r x n m x n
inside
outside
32. If A is a mxr matrix and B is a rxn matrix, then the
product C=AB is a mxn matrix whose entries are
obtained as follows. The entry corresponding to row ‘i’
and column ‘j’ of C is the dot product of the vectors
formed by the row ‘i’ of A and column ‘j’ of B
3x3 3x2
3x2
1 2 4 1 3
A 3 0 7 B 3 1
9 1 5 1 0
3 5 1 1
C AB 10 9 notice 2 3 3
7 28 4 1
T
Matrix operations Matrix multiplication
33. Properties of matrix multiplication:
1. Matrix multiplication is noncommutative
(order of addition does matter)
It may be that the product AB exists but BA
does not (e.g. in the previous example
C=AB is a 3x2 matrix, but BA does not
exist)
Even if the product exists, the products AB
and BA are not generally the same
Matrix operations
g e n e ra l
in
B A
A B
Multiplication of
matrices
Properties
34. 2. Matrix multiplication is associative
3. Distributive law
4. Multiplication by identity matrix
5. Multiplication by zero matrix
6.
Matrix operations
C
AB
BC
A
Multiplication of
matrices
Properties
A
IA
A ;
A I
0
0 A
0 ;
A 0
CA
BA
A
C
B
AC
AB
C
B
A
T
T
T
A
B
B
A
35. 1. If A , B and C are square matrices of the
same size, and then
does not necessarily mean that
2. does not necessarily imply that
either A or B is zero
Matrix operations
Miscellaneous
properties
0
A A C
A B
C
B
0
A B
36. Inverse of a
matrix
Definition
I
A
B
B
A
If A is any square matrix and B is another
square matrix satisfying the conditions
Then
(a)The matrix A is called invertible, and
(b) the matrix B is the inverse of A and is
denoted as A-1.
The inverse of a matrix is unique
37. Inverse of a
matrix
Uniqueness
The inverse of a matrix is unique
Assume that B and C both are inverses of A
C
B
B
BI
B(AC)
C
IC
(BA)C
I
CA
AC
I
BA
AB
Hence a matrix cannot have two or more
inverses.
38. Inverse of a
matrix
Some properties
A
A
1
-1
Property 1: If A is any invertible square
matrix the inverse of its inverse is the matrix A
itself
Property 2: If A is any invertible square
matrix and k is any scalar then
1
-
1
A
k
1
A
k
39. Inverse of a
matrix
Properties
-1
1
1
A
B
B
A
Property 3: If A and B are invertible square
matrices then
1
1
1
1
1
1
1
1
1
1
A
B
AB
B
by
sides
both
ying
Premultipl
A
AB
B
A
AB
B
A
A
A
AB
(AB)
A
A
by
sides
both
ying
Premultipl
I
AB
(AB)
1
-
1
-
1
-
1
-
40. The determinant of a square matrix is a number
obtained in a specific manner from the matrix.
For a 1x1 matrix:
For a 2x2 matrix:
What is a determinant?
1 1
1 1 a
A
a
A
)
det(
;
21
12
22
11
22
21
12
11
a
a
a
a
A
a
a
a
a
A
)
det(
;
Product along red arrow minus product along blue arrow
41. Example 1
7
5
3
1
A
8
5
3
7
1
7
5
3
1
)
A
det(
Consider the matrix
Notice (1) A matrix is an array of numbers
(2) A matrix is enclosed by square brackets
Notice (1) The determinant of a matrix is a number
(2) The symbol for the determinant of a matrix is
a pair of parallel lines
Computation of larger matrices is more difficult
42. For ONLY a 3x3 matrix write down the first two
columns after the third column
Duplicate column method for 3x3 matrix
32
31
22
21
12
11
33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Sum of products along red arrow
minus sum of products along blue arrow
This technique works only for 3x3 matrices
33
21
12
32
23
11
31
22
13
32
21
13
31
23
12
33
22
11
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
)
A
det(
a
a
a
a
a
a
a
a
a
A
33
32
31
23
22
21
13
12
11
44. Finding determinant using inspection
Special case. If two rows or two columns are proportional
(i.e. multiples of each other), then the determinant of the
matrix is zero
0
8
7
2
4
2
3
8
7
2
because rows 1 and 3 are proportional to each other
If the determinant of a matrix is zero, it is called a
singular matrix
45. If A is a square matrix
Cofactor method
The minor, Mij, of entry aij is the determinant of the submatrix
that remains after the ith row and jth column are deleted from A.
The cofactor of entry aij is Cij=(-1)(i+j) Mij
31
23
33
21
33
31
23
21
12 a
a
a
a
a
a
a
a
M
33
31
23
21
12
12
a
a
a
a
M
C
a
a
a
a
a
a
a
a
a
A
33
32
31
23
22
21
13
12
11
What is a cofactor?
46. Sign of cofactor
What is a cofactor?
-
-
-
-
Find the minor and cofactor of a33
4
1
4
0
2
0
1
4
2
M 33
Minor
4
M
M
)
1
(
C 3 3
3 3
)
3
3
(
3 3
Cofactor
2
0
1
A
2
1
-
4
3
-
4
2
47. Cofactor method of obtaining the
determinant of a matrix
The determinant of a n x n matrix A can be computed by
multiplying ALL the entries in ANY row (or column) by
their cofactors and adding the resulting products. That is,
for each and
n
i
1
n
j
1
n j
n j
2 j
2 j
1 j
1 j C
a
C
a
C
a
A
)
det(
Cofactor expansion along the ith row
in
in
i2
i2
i1
i1 C
a
C
a
C
a
A
)
det(
Cofactor expansion along the jth column
49. By a cofactor along the third column
Example : evaluate
det(A)=a13C13 +a23C23+a33C33
det(A)=
1 5
1 0
3 -1
-3
2
2
= det(A)= -3(-1-0)+2(-1)5(-1-15)+2(0-5)=25
det(A)=
1 0
3 -1
-3* (-1)4 1 5
3 -1
+2*(-1)5 1 5
1 0
+2*(-1)6
50. Quadratic form
U d
k
d
T
The scalar
Is known as a quadratic form
If U>0: Matrix k is known as positive definite
If U≥0: Matrix k is known as positive semidefinite
matrix
square
k
vector
d
52. Differentiation of quadratic form
Differentiate U wrt d2
2
12
1
11
1
2
2
U
d
k
d
k
d
Differentiate U wrt d1
2
22
1
12
2
2
2
U
d
k
d
k
d
54. Outline
• Role of FEM simulation in Engineering
Design
• Course Philosophy
55. Role of simulation in design:
Boeing 777
Source: Boeing Web site (http://www.boeing.com/companyoffices/gallery/images/commercial/).
56. Another success ..in failure:
Airbus A380
http://www.airbus.com/en/aircraftfamilies/a380/
57. Drag Force Analysis
of Aircraft
• Question
What is the drag force distribution on the aircraft?
• Solve
– Navier-Stokes Partial Differential Equations.
• Recent Developments
– Multigrid Methods for Unstructured Grids
60. San Francisco Oakland Bay Bridge
A finite element model to analyze the
bridge under seismic loads
Courtesy: ADINA R&D
61. Crush Analysis of
Ford Windstar
• Question
– What is the load-deformation relation?
• Solve
– Partial Differential Equations of Continuum Mechanics
• Recent Developments
– Meshless Methods, Iterative methods, Automatic Error Control
62. Engine Thermal
Analysis
Picture from
http://www.adina.com
• Question
– What is the temperature distribution in the engine block?
• Solve
– Poisson Partial Differential Equation.
• Recent Developments
– Fast Integral Equation Solvers, Monte-Carlo Methods
63. Electromagnetic
Analysis of Packages
• Solve
– Maxwell’s Partial Differential Equations
• Recent Developments
– Fast Solvers for Integral Formulations
Thanks to
Coventor
http://www.cov
entor.com
64. Micromachine Device
Performance Analysis
From www.memscap.com
• Equations
– Elastomechanics, Electrostatics, Stokes Flow.
• Recent Developments
– Fast Integral Equation Solvers, Matrix-Implicit Multi-level Newton
Methods for coupled domain problems.
65. To actively develop advanced modeling, simulation
and imaging (MSI) technology for healthcare
through interdisciplinary collaborations with the aim
of transitioning the technology to clinical practice –
from the laboratory bench to the hospital bedside.
Research Areas:
• Virtual surgery
• Noninvasive brain imaging and neuromodulation
• Computational imaging
• Tissue manufacturing, characterization and diagnosis
http:// cemsim.rpi.edu
69. Engineering design
Physical Problem
Mathematical model
Governed by differential
equations
Assumptions regarding
Geometry
Kinematics
Material law
Loading
Boundary conditions
Etc.
General scenario..
Question regarding the problem
...how large are the deformations?
...how much is the heat transfer?
70. Engineering design
Example: A bracket
Physical problem
Questions:
1. What is the bending moment at section AA?
2. What is the deflection at the pin?
Finite Element Procedures, K J Bathe
71. Engineering design
Example: A bracket
Mathematical model 1:
beam
Moment at section AA
cm
053
.
0
AG
6
5
)
r
L
(
W
EI
)
r
L
(
W
3
1
cm
N
500
,
27
WL
M
N
3
N
W
load
at
Deflection at load
How reliable is this model?
How effective is this model?
75. Engineering design
..General scenario..
Finite element analysis
FEM analysis scheme
Step 1: Divide the problem domain into non
overlapping regions (“elements”) connected to
each other through special points (“nodes”)
Finite element model
Element
Node
76. Engineering design
..General scenario..
Finite element analysis
FEM analysis scheme
Step 2: Describe the behavior of each element
Step 3: Describe the behavior of the entire body by
putting together the behavior of each of the
elements (this is a process known as “assembly”)
79. Engineering design
Example: A bracket
Mathematical model 2:
plane stress
FEM solution to mathematical model 2 (plane stress)
Moment at section AA
cm
064
.
0
cm
N
500
,
27
M
W
load
at
Deflection at load
Conclusion: With respect to the questions we posed, the
beam model is reliable if the required bending moment is to
be predicted within 1% and the deflection is to be predicted
within 20%. The beam model is also highly effective since it
can be solved easily (by hand).
What if we asked: what is the maximum stress in the bracket?
would the beam model be of any use?
80. Engineering design
Example: A bracket
Summary
1. The selection of the mathematical model
depends on the response to be
predicted.
2. The most effective mathematical model
is the one that delivers the answers to
the questions in reliable manner with
least effort.
3. The numerical solution is only as
accurate as the mathematical model.
81. Example: A bracket
Modeling a physical
problem
...General scenario
Physical Problem
Mathematical
Model
Numerical model
Does answer
make sense?
Refine analysis
Happy
YES!
No!
Improve
mathematical
model
Design improvements
Structural optimization
Change
physical
problem
82. Example: A bracket
Modeling a physical
problem
Verification and validation
Physical Problem
Mathematical
Model
Numerical model
Verification
Validation
83. Critical assessment of the FEM
Reliability:
For a well-posed mathematical problem the numerical
technique should always, for a reasonable discretization,
give a reasonable solution which must converge to the
accurate solution as the discretization is refined.
e.g., use of reduced integration in FEM results in an
unreliable analysis procedure.
Robustness:
The performance of the numerical method should not be
unduly sensitive to the material data, the boundary
conditions, and the loading conditions used.
e.g., displacement based formulation for incompressible
problems in elasticity
Efficiency: