MECH 4740  Numerical Methods in Engineering

Department of Mechanical Engineering, Hong Kong University of Science and Technology

(http://teaching.ust.hk/~mech4740/)

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: mektang@ust.hk; 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:

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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  
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Chapter 2:     Programming and Software                                                                     (week 1-1)
2.1 Packages and Programming  
2.2 Structured Programming  
2.3 Modular Programming  
2.4 Excel  
2.5 MATLAB  
2.6 Other Languages and Libraries  
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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  
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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  
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Summary

Part Two: Roots of Equations                                                                  (week 2-1)

Chapter 5:    Bracketing Methods                                                                                      (week 2-2)
 5.1 Graphical Methods  
5.2 The Bisection Method  
5.3 The False-Position Method  
5.4 Incremental Searches and Determining Initial Guesses  
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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  
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Chapter 7:    Roots of Polynomials                                                                                     (week 3-1)
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  
7.7 Root Location with Libraries and Packages  
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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)  
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Summary

Part Three: Linear Algebraic Equations                                                (week 3-1)

Chapter 9:    Gauss Elimination                                                                                         (week 3-2)
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  
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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  
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Chapter 11:    Special Matrices and Gauss-Seidel                                                        (week 4-1)
 11.1 Special Matrices  
11.2 Gauss-Seidel  
11.3 Linear Algebraic Equations with Libraries and Packages  
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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)  
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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  
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Chapter 14:    Multidimensional Unconstrained Optimization                                    (week 5-1)
 14.1 Direct Methods  
14.2 Gradient Methods  
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Chapter 15:    Constrained Optimization                                                                        (week 5-2)
 15.1 Linear Programming  
15.2 Nonlinear Constrained Optimization  
15.3 Optimization with Packages  
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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)  
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Summary

Mid-term review                                                                                     (week 6-1)

Homework solutions - chaps 1-5, chaps 6-10, chaps 11-15                    (week 6-2)

Part Five: Curve Fitting                                                                          (week 7-1)

Chapter 17:    Least-Squares Regression                                                                       (week 7-1)
 17.1 Linear Regression  
17.2 Polynomial Regression  
17.3 Multiple Linear Regression  
17.4 General Linear Least Squares  
17.5 Nonlinear Regression  
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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 Interpolation                                                                                                      (week 8-1)
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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)                                                                              (week 8-2)
19.6 Fast Fourier Transform (FFT)  
19.7 The Power Spectrum  
19.8 Curve Fitting with Libraries and Packages  
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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)  
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Summary

Part Six: Numerical Differentiation and Integration                             (week 8-2)

Chapter 21:    Newton-Cotes Integration Formulas                                                       (week 9-1)
 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  
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Chapter 22:    Integration of Equations                                                                            (week 9-2)
 22.1 Newton-Cotes Algorithms for Equations  
22.2 Romberg Integration  
22.3 Gauss Quadrature  
22.4 Improper Integrals  
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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  
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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)  
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Summary

Part Seven: Ordinary Differential Equations                                          (week 10-1)

Chapter 25:    Runge-Kutta Methods                                                                                  (week 10-2, 11-1)
 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  
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Chapter 26:    Stiffness and Multistep Methods                                                               (week 11-1)
 26.1 Stiffness  
26.2 Multistep Methods  
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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  
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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)  
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Summary

Part Eight: Partial Differential Equations (a demo)                                    (week 12-1)

Chapter 29:    Finite Difference: Elliptic Equations                                                          (week 12-2)
 29.1 The Laplace Equation  
29.2 Solution Techniques  
29.3 Boundary Conditions  
29.4 The Control-Volume Approach  
29.5 Software to Solve Elliptic Equations  
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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  
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Chapter 31:    Finite-Element Method                                                                                  (week 13-2)
 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  
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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)  
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Final review                                                                                                (week 14)  

 

Homework solutions - chaps 16-20, chaps 21-25, chaps 26-30, chaps 31-35

 

Final Exam