Numerical Methods for Engineers

Purpose of Course  showclose

Numerical methods have been used to solve mathematical expressions of engineering and scientific problems for at least 4000 years (for some historical discussion you may wish to browse the Ethnomathematics Digital Library or the MacTutor History of Mathematics Archive from St. Andrews University).*  Such methods apply numerical approximation in order to convert continuous mathematical problems (for example, determining the mechanical stress throughout a loaded truss) into systems of discrete equations that can be solved with sufficient accuracy by machine.

Numerical methods provide a way for the engineer to translate the language of mathematics and physics into information that may be used to make engineering decisions. Often, this translation is implemented so that calculations may be done by machines (computers). The types of problems that you encounter as an engineer may involve a wide variety of mathematical phenomena, and hence it will benefit you to have an equally wide range of numerical methods with which to approach some of these problems. This course will provide you with an introduction to several of those numerical methods which you may then find opportunity to practice later in the curriculum.

Let us consider a few examples in which numerical methods might offer a net benefit to the engineer.

1. A new problem results in a numerical expression involving logarithmic, polynomial, and trigonometric terms. Not only do you wish to find the zeros of this particular expression, but you need to find the zeros of 100,000 similar expressions per day.  Hence, you need an automated procedure for doing such that can be implemented on a computer.

2. Consider the manufacture of an oddly shaped part out of very expensive materials. The manufacturing process involves cutting, heating, cooling, and bending of the part. In order to optimize the manufacturing process, you wish to understand the details of the heating, cooling, and shaping processes. Modeling of these processes requires consideration of transient heat flow in three dimensions and transient deformation in three dimensions; further, the part is an oddly-shaped domain. One approach to improve understanding of the process is to numerically simulate the heating, cooling, and deformation; this simulation involves the numerical solution of systems of partial differential equations.

3. You have an empirical model of a process that predicts temperature and oxygen content as a function of time. The empirical model involves four parameters that need to be estimated from measurements under a variety of conditions. There are several nonlinear regression or optimization algorithms which might be suitable for this task.

As you progress through the curriculum, you will encounter many more problems which might benefit from analysis by numerical methods. Indeed, you may gain much insight by applying numerical methods to a variety of problems that you encounter later in the curriculum.

This course will consist of ten units: an introduction to the numerical properties of machine computations; numerical differentiation; solution of non-linear equations; linear algebra (or the solution of systems of linear equations); interpolation; regression and optimization; numerical integration; numerical solution of ordinary differential equations; numerical solution of partial differential equations; and some miscellaneous numerical tools.  Each unit is accompanied by lectures, readings, and exercises.

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Course Information  showclose

Welcome to ME101.  Below, please find general information on this course and its requirements.

Course Designers: Anonymous and Stephen Gibbs, Ph.D.

Peer Reviewers: Stephen Gibbs, Ph.D. 

Primary Resources:  This course is composed of a range of different free, online materials. However, the course references the following free, online resources from academic institutions that are key to completing this course:

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  Pay special attention to Unit 1, as this unit lays the groundwork for understanding the more advanced, exploratory material presented in the latter units. You will also need to complete:

  • Unit 1 Quizzes
  • Unit 2 Quizzes
  • Unit 3 Quizzes
  • Unit 4 Quizzes
  • Unit 5 Quizzes
  • Unit 6 Quizzes
  • Unit 7 Quizzes
  • Unit 8 Quizzes
  • Unit 9 Quizzes
  • Subunit 2.4 Assignment
  • Subunit 4.6 Assignment
  • Subunit 6.4 Assignment
  • Subunit 8.2 Assignment
  • The Final Exam

Each unit contains an assessment exercise and quizzes from the University of South Florida’s Holistic Numerical Methods Institute.  Please give time to these; they are the best way to test your knowledge and learn.

Note that you will only receive an official grade on your Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the unit multiple choice quizzes and assignments listed above.

In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.

Time Commitment: You should be able to complete this course in approximately 128 hours of study and creative effort.  Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take you 17 hours. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 2 hours) on Monday night; subunit 1.2 (a total of 3 hours) on Tuesday night; half of subunit 1.4 (about 2 hours) on Wednesday night; the remainder of subunit 1.4 (about 2 hours) on Thursday night; etc.

Tips/Suggestions: Most of the materials for this course are easy to read or study quickly; it is easy to convince yourself prematurely that you understand the material. Re-reading may be a useful technique to help better understand the material.  Most students learn this sort of material best by implementing example calculations either by hand or by machine. In fact, many students really begin to understand the underlying mathematics only after implementing numerical calculations by machine.

We encourage you to also take notes as you work through the course materials.  These notes will be useful as you prepare for your Final Exam. 



 
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Learning Outcomes  showclose

Upon successful completion of this course, the student will be able to:

  • Quantify absolute and relative errors.
  • Distinguish between round-off and truncation errors.
  • Interconvert binary and base-10 number representations.
  • Define and use floating-point representations.
  • Quantify how errors propagate through arithmetic operations.
  • Derive difference equations for first and second order derivatives.
  • Evaluate first and second order derivatives from numerical evaluations of continuous functions or table lookup of discrete data.
  • Describe situations in which numerical solutions to nonlinear equations are needed
  • Implement the bisection method for solving equations.
  • List advantages and disadvantages of the bisection method
  • Implement both Newton-Raphson and secant methods.
  • Describe the difference between Newton-Raphson and secant methods.
  • Demonstrate the relative performance of bisection, Newton-Raphson, and secant methods.
  • Define and identify special types of matrices.
  • Perform basic matrix operations.
  • Define and perform Gaussian elimination to solve a linear system.
  • Identify pitfalls of Gaussian elimination.
  • Define and perform Gauss-Seidel method for solving a linear system.
  • Use LU decomposition to find the inverse of a matrix.
  • Define and perform singular value decomposition; explain the significance of singular value decomposition.
  • Define interpolation.
  • Define and use direct interpolation to approximate data and find derivatives.
  • Define and use Newton’s divided difference method of interpolation.
  • Define and use Lagrange and spline interpolation.
  • Define regression.
  • Perform linear least-squares regression and nonlinear regression.
  • Derive and apply the trapezoidal rule and Simpson’s rule of integration.
  • Distinguish Simpson’s method from the trapezoidal rule.
  • Estimate errors in trapezoidal and Simpson integration.
  • Derive and apply Romberg and Gaussian quadrature for integration.
  • Define and distinguish between ordinary and partial differential equations.
  • Implement Euler’s methods for solving ordinary differential equations.
  • Investigate how step size affects accuracy in Euler’s method.
  • Implement and use the Runge-Kutta 2nd order method for solving ordinary differential equations.
  • Apply the shooting method to solve boundary-value problems.
  • Define Fourier series and the Fourier transform.
  • Find Fourier coefficients for a given data set or function and domain.
  • Describe the finite element method for one-dimensional problems.

Course Requirements  showclose

In order to take this course, you must:

√    Have access to a computer.

√    Have frequent broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g. Adobe Reader or FLASH (see for solutions)).

√    Have the ability to download and save files and documents to a computer.

√    Be able to download and install Scilab.*

√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    Be competent in the English language.

√    Have read the Saylor Student Handbook

√    Have completed the following courses from “The Core Program” of the Mechanical Engineering discipline: ME101, ME102, ME001/MA101, ME002/MA102, and ME003/MA221.
 
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