Mech 374 Numerical Methods In Engineering

MECH 374 Numerical Methods in Engineering Department of Mechanical Engineering, Hong Kong University of Science and Technology (http://teaching.ust.hk/~mech374/)

Course Description: The use of numerical methods for the analysis simulation, and design of engineering processes and systems has been increasing at a rapid pace in recent years. This course is intended for teaching numerical methods for engineering students at the senior level as well as at the beginning graduate level. The course will have three important objectives: (1) to teach the basic theories and fundamentals of numerical methods; (2) to help the students to acquire skills to implement these methods for computer solution; and finally (3) to provide an environment where the students can familiarize themselves with many today’s popular commercial software systems and their use in the solution of engineering problems. On the first objective, the following fundamental aspects will be covered: analysis of errors, roots of equations, linear and algebraic equations, optimizations, curve-fitting and approximation, numerical differentiation and integration, ordinary differential equations, and partial differential equations. On the second objective, computer programming basics as well as certain specific computer languages such as MATLAB will be introduced. On the last objective, the students will learn how to use MATLAB and Excel VBA to implement their own numerical methods. This course is structured as a 3+1 credits course, with 3 lecture credits and 1 for the lab.

Instructor: Dr. Kai Tang Dept. of Mech. Engineering; E-mail: [email protected]; Tel: 2358-8656; Room: 2544; Office hours: Any time (just come to see me or email me to set up an appointment time if you like).

Textbook: • •

S.C. Chapra and R.P. Canale, "Numerical Methods for Engineers", 5th Edition, McGraw Hill, 2006. Research papers

Grade Policy: • • • •

Homework Lab projects Mid-term exam Final-exam

Time and place:

5% 25% 30% 40%

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Lecture:

Monday/Friday

10:30 - 11:50,

Room 4505

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Lab class:

Monday

09:30 - 10.20,

Room 4225c

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Lab Practice: Monday

14:00 - 17:00,

Room 4225c

Lab TA: •

Zhanqing CHEN Email: [email protected]; Office hours: Monday 14:0017:00, Room 4225c (or other time by appointment).

Announcements

1. The Mid-term Exam is on Monday, Oct. 20th, 2008. Here are the details: Place: Room 2404 (Lift # 17/18) Date and time: Monday, Oct. 20th, 2008, 9:30am - 12:30pm (3 hrs) Format: Closed book, but you are allowed to bring a total of 5 A4 size "cheat" sheets (10 sides) on which you can write whatever you like. 2. The Lab on Monday Oct. 20th is now moved to 10:30am Noon on Friday, Oct. 24th. That is, on Friday, Oct. 24th, 10:30am - Noon, instead of going to the lecture, you go to Lab in room 4225C.

Syllabus and Schedule: Part One: Modeling, Computers, and Error Analysis

(week 1-1) Chapter 1: Mathematical Modeling and Engineering Problem Solving (week 1-1) 1.1 A Simple Mathematical Model 1.2 Conservation Laws and Engineering Download Classnote Chapter 2: Programming and Software 2.1 Packages and Programming

(week 1-1)

2.2 Structured Programming 2.3 Modular Programming 2.4 Excel 2.5 MATLAB 2.6 Other Languages and Libraries Download Classnote Chapter 3: Approximations and Round-Off Errors (week 1-2) 3.1 Significant Figures 3.2 Accuracy and Precision 3.3 Error Definitions 3.4 Round-Off Errors Download Classnote Chapter 4: Truncation Errors and the Taylor Series (week 2-1) 4.1 The Taylor Series 4.2 Error Propagation 4.3 Total Numerical Error 4.4 Blunders, Formulation Errors, and Data Uncertainty Download Classnote Summary

Part Two: Roots of Equations Chapter 5: Bracketing Methods 5.1 Graphical Methods 5.2 The Bisection Method 5.3 The False-Position Method 5.4 Incremental Searches and Determining Initial Guesses Download Classnote

(week 2-1) (week 2-2)

Chapter 6: Open Methods (week 2-2) 6.1 Simple Fixed-Point Iteration 6.2 The Newton-Raphson Method 6.3 The Secant Method 6.4 Multiple Roots 6.5 Systems of Nonlinear Equations Download Classnote Chapter 7: Roots of Polynomials 1) 7.1 Polynomials in Engineering and Science 7.2 Computing with Polynomials 7.3 Conventional Methods 7.4 Müller’s Method 7.5 Bairstow’s Method 7.6 Other Methods

(week 3-

7.7 Root Location with Libraries and Packages Download Classnote Chapter 8: Engineering Applications: Roots of Equations (week 3-1) 8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 8.2 Open-Channel Flow (Civil/Environmental Engineering) 8.3 Design of an Electric Circuit (Electrical Engineering) 8.4 Vibration Analysis (Mechanical/Aerospace Engineering) Download Classnote Summary

Part Three: Linear Algebraic Equations Chapter 9: Gauss Elimination 2) 9.1 Solving Small Numbers of Equations 9.2 Naïve Gauss Elimination 9.3 Pitfalls of Elimination Methods 9.4 Techniques for Improving Solutions 9.5 Complex Systems 9.6 Nonlinear Systems of Equations 9.7 Gauss-Jordan Download Classnote

(week 3-1) (week 3-

Chapter 10: LU Decomposition and Matrix Inversion (week 4-1) 10.1 LU Decomposition 10.2 The Matrix Inverse 10.3 Error Analysis and System Condition Download Classnote Chapter 11: Special Matrices and GaussSeidel (week 4-1) 11.1 Special Matrices 11.2 Gauss-Seidel 11.3 Linear Algebraic Equations with Libraries and Packages Download Classnote Chapter 12: Engineering Applications: Linear Algebraic Equations (week 4-2) 12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) 12.2 Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) 12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering) 12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering) Download Classnote Summary

Part Four: Optimization

(week 4-2)

Chapter 13: One-Dimensional Unconstrained Optimization (week 4-2) 13.1 Golden-Section Search 13.2 Quadratic Interpolation 13.3 Newton’s Method Download Classnote Chapter 14: Multidimensional Unconstrained Optimization (week 5-1) 14.1 Direct Methods 14.2 Gradient Methods Download Classnote Chapter 15: Constrained Optimization 15.1 Linear Programming 15.2 Nonlinear Constrained Optimization 15.3 Optimization with Packages Download Classnote

(week 5-2)

Chapter 16: Engineering Applications: Optimization (week 6-1) 16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering) 16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) 16.4 Mountain Bike Design (Mechanical/Aerospace Engineering) Download Classnote Summary

Mid-term review Homework solutions - chaps 1-5, chaps 6-11, chaps 12-16 Part Five: Curve Fitting Chapter 17: Least-Squares Regression 17.1 Linear Regression 17.2 Polynomial Regression 17.3 Multiple Linear Regression 17.4 General Linear Least Squares 17.5 Nonlinear Regression Download Classnote Chapter 18: Interpolation (week 7-2) 18.1 Newton’s Divided-Difference Interpolating Polynomials 18.2 Lagrange Interpolating Polynomials 18.3 Coefficients of an Interpolating Polynomial 18.4 Inverse Interpolation 18.5 Additional Comments 18.6 Spline

(week 6-1) (week 6-2) (week 7-1) (week 7-1)

Interpolation (week 8-1) Download Classnote Chapter 19:

Fourier Approximation (week 8-1) 19.1 Curve Fitting with Sinusoidal Functions 19.2 Continuous Fourier Series 19.3 Frequency and Time Domains 19.4 Fourier Integral and Transform 19.5 Discrete Fourier Transform (DFT) 19.6 Fast Fourier Transform (FFT) 19.7 The Power Spectrum 19.8 Curve Fitting with Libraries and Packages Download Classnote

(week 8-2)

Chapter 20: Engineering Applications: Curve Fitting (week 8-2) 20.1 Linear Regression and Population Models (Chemical/Bio Engineering) 20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) 20.3 Fourier Analysis (Electrical Engineering) 20.4 Analysis of Experimental Data (Mechanical/Aerospace Engineering) Download Classnote Summary

Part Six: Numerical Differentiation and Integration Chapter 21: Newton-Cotes Integration Formulas 21.1 The Trapezoidal Rule 21.2 Simpson’s Rules 21.3 Integration with Unequal Segments 21.4 Open Integration Formulas 21.5 Multiple Integrals Download Classnote Chapter 22: Integration of Equations 22.1 Newton-Cotes Algorithms for Equations 22.2 Romberg Integration 22.3 Gauss Quadrature 22.4 Improper Integrals Download Classnote

(week 8-2)

(week 9-1)

(week 9-2)

Chapter 23: Numerical Differentiation (week 10-1) 23.1 High-Accuracy Differentiation Formulas 23.2 Richardson Extrapolation 23.3 Derivatives of Unequally Spaced Data 23.4 Derivatives and Integrals for Data with Errors 23.5 Numerical Integration/Differentiation with Libraries and Packages Download Classnote

Chapter 24: Engg. Applications: Numerical Integration and Differentiation (week 10-1) 24.1 -Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) 24.2 -Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) 24.3 -Root-Mean-Square Current by Numerical Integration (Electrical Engineering) 24.4 -Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) Download Classnote Summary

Part Seven: Ordinary Differential Equations

(week 10-1)

Chapter 25: Runge-Kutta Methods 1) 25.1 Euler’s Method 25.2 Improvements of Euler’s Method 25.3 Runge-Kutta Methods 25.4 Systems of Equations 25.5 Adaptive Runge-Kutta Methods Download Classnote Chapter 26: Stiffness and Multistep Methods 26.1 Stiffness 26.2 Multistep Methods Download Classnote

(week 10-2, 11-

(week 11-1)

Chapter 27: Boundary-Value and Eigenvalue Problems (week 11-2) 27.1 General Methods for Boundary-Value Problems 27.2 Eigenvalue Problems 27.3 ODEs and Eigenvalues with Libraries and Packages Download Classnote Chapter 28: Engineering Applications: Ordinary Differential Equations (week 12-1) 28.1 -Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) 28.2 Predator-Prey Models and Chaos (Civil/Environmental Engineering) 28.3 Simulating Transient Current for an Electric Circuit (Electrical Engineering) 28.4 The Swinging Pendulum (Mechanical/Aerospace Engineering) Download Classnote Summary

Part Eight: Partial Differential Equations (a demo) Chapter 29: Finite Difference: Elliptic Equations 29.1 The Laplace Equation

(week 12-1) (week 12-2)

29.2 Solution Techniques 29.3 Boundary Conditions 29.4 The Control-Volume Approach 29.5 Software to Solve Elliptic Equations Download Classnote Chapter 30: Finite Difference: Parabolic Equations (week 13-1) 30.1 The Heat Conduction Equation 30.2 Explicit Methods 30.3 A Simple Implicit Method 30.4 The Crank-Nicolson Method 30.5 Parabolic Equations in Two Spatial Dimensions Download Classnote Chapter 31: Finite-Element Method 31.1 The General Approach 31.2 Finite-Element Application in One Dimension 31.3 Two-Dimensional Problems 31.4 Solving PDEs with Libraries and Packages Download Classnote

(week 13-2)

Chapter 32: Engineering Applications: Partial Differential Equations 32.1 -One-Dimensional Mass Balance of a Reactor (Chemical/BioEngineering) 32.2 Deflections of a Plate (Civil/Environmental Engineering) 32.3 Two-Dimensional Electrostatic Field Problems (Electrical Engineering) 32.4 -Finite-Element Solution of a Series of Springs (Mechanical/Aerospace Engineering) Download Classnote

Final review Homework solutions - chaps 17, chaps 18-25, chaps 26-32 Final Exam

(week 14)