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Development of a new computational mathematics education for themechanical engineering program at Chalmers University of Technology
Mikael Enelund and Stig Larsson
Chalmers University of Techology
Goteborg, Sweden
– p.1/23
Background
Mechanical Engineering at Chalmers: member of CDIO since 2000
mechanics and strength of materials
set up model, solve equations on the computer, analyze the simulations
interaction between courses
need for tools for realistic desing-build projects
Chemical Engineering and Bio Engineering at Chalmers: computationally orientedmath courses since 1999
full integration of numerical and symbolical aspects
constructive approach
programming
applications early in the curriculum
integration with chemistry courses
– p.2/23
New math courses for the Mech Engg program
Ongoing development:
Course materials for computational mathematics to supplement traditional textbooks
Introductory course in Matlab programming
Computer-oriented projects and exercises that will be used simultaneously in themathematics courses and the courses in mechanics and thermodynamics
To be launched in the academic year 2007-08
– p.3/23
The computer
the computer — a computational tool
new conditions for engineering work
new possibilities for teaching
not fully reflected in the education and textbooks neither in mathematics norengineering courses
applications — numerical computation — mathematics
constructive (and more understandable) mathematics
general equations, not just simplified special equations
applications already in the beginning of the education
more advanced applications later in the education
– p.4/23
Matlab
Matlab = “matrix laboratory”
numerical matrix computation
interactive calculator
tools for graphics and graphical user interfaces
programming environment
toolboxes for many scientific fields
students write their own programs for various algorithms
forces understanding of mathematical concepts (function, matrix, ...)
attitude: natural to use mathematics on the computer
a long tradition to use Matlab among teachers and researchers at Chalmers
fun!
Replacements:
Comsol Script
Gnu Octave
– p.5/23
Computation
symbolic computation:Z
y
0
dx
1 + x2=
h
arctan xi
y
0
= arctan(y)
by hand, Mathematica, Maple, Matlab, ...
theory
modeling: set up and re-write equations
exact solution of simple special equations
– p.6/23
Computation
symbolic computation:Z
y
0
dx
1 + x2=
h
arctan xi
y
0
= arctan(y)
by hand, Mathematica, Maple, Matlab, ...
theory
modeling: set up and re-write equations
exact solution of simple special equations
numerical computation:Z
1
0
dx
1 + x2≈
X
n
hn
1 + x2n
= 0.78
(by hand), Matlab, Mathematica, Maple, ...
approximate solution of general equations
– p.6/23
Computation
symbolic computation:Z
y
0
dx
1 + x2=
h
arctan xi
y
0
= arctan(y)
by hand, Mathematica, Maple, Matlab, ...
theory
modeling: set up and re-write equations
exact solution of simple special equations
numerical computation:Z
1
0
dx
1 + x2≈
X
n
hn
1 + x2n
= 0.78
(by hand), Matlab, Mathematica, Maple, ...
approximate solution of general equations
We do not use symbolic programs.
– p.6/23
General versus special equation
general equations
algebraic equation: f(x) = 0
ordinary differential equation: u′(t) = f(t, u(t))
partial differential equation: −∇ · (a(x)∇u(x)) = f(x)
special equations
algebraic equation: x2 + ax + b = 0, x = − a
2±
q
( a
2)2 − b
ordinary differential equation: u′′ + ω2u = 0, u(t) = A cos(ωt) + B sin(ωt)
partial differential equation: −∆u = 0, u(x) = −1/(4π|x|)
general equation: numerical solution
special equation: symbolic solution
– p.7/23
Literature
K. Eriksson, D. Estep, and C. Johnson, “Applied Mathematics – Body and Soul”,Springer 2003
software
lecture notes
This textbook has been used in the Chemical Engineering program but will be replaced by
standard textbooks plus lecture notes from 2006.
– p.8/23
Constructive mathematics
real number = decimal expansion
solve f(x) = 0, construct decimal expansion = real number x ∈ R
Constructive proof:
1. bisection algorithm gives xn
2. convergence (decimal expansion) xn → x
3. x is a solution f(x) = 0
4. uniqueness
Example: f(x) = x2 − 2 = 0, x = 1.4142 . . . =:√
2
Intermediate value theorem.
A continuous function f : [a, b] → R assumes all values between f(a) and f(b).
Proof. If y is an intermediate value, then solve f(x) − y = 0 by the bisection algorithm.
– p.9/23
Constructive mathematics
system of algebraic equations: f(x) = 0
algorithm: Newton’s method
application: statics, thermodynamical equilibrium, ...
system of ordinary differential equations: u′ = f(x, u)
algorithm: Euler’s method
applications: dynamics, thermodynamics, control theory, ...
partial differential equations: −∇ · (a∇u) = f
algorithm: finite element method
applications: solid mechanics, elasticity, heat transfer, fluid flow, ...
– p.10/23
Suggested curriculum in year 1
Period 1
Programming in Matlab (4.5 ECTS)
Introduction to mathematics (7.5 ECTS)
Introduction to mechanical engineering
Period 2
Analysis and linear algebra A (7.5 ECTS)
Thermodynamics (7.5 ECTS)
Introduction to mechanical engineering (7.5 ECTS), continued
Period 3
Analysis and linear algebra B (7.5 ECTS)
Mechanics and solid mechanics I (7.5 ECTS)
Period 4
Analysis and linear algebra C (7.5 ECTS)
Mechanics and solid mechanics II (7.5 ECTS)
– p.11/23
Years 2 and 3
Mathematical statistics
Transforms and differential equations (elective)
These are not considered in the present project.
Several engineering courses use computational methods:
Mechanics
Mechatronics
Machine design
Machine design project
Manufacturing
Finite element methods (elective)
Control theory
Fluid mechanics
– p.12/23
Mathematics/mechanics common projects
General goals:
cross fertilization
holistic view of both subjects
modern way of working: modeling, simulation, and analysis
allow more complex applications
visualize phenomena
working with realistic engineering problems
better understanding of mathematical concepts
motivate the study of mathematics
– p.13/23
Examples of projects
Write a Matlab code to determine the value of the parameter A so that the magnitude of thestress is less than half the yield stress. Plot the deformed truss and the stress distribution.
������������������������
������������������������
L
LLL
A B C
Area= 2A
Area= A
Q
– p.14/23
Examples of projects
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1
−0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1.5
−1
−0.5
0
0.5
1
1.5
2
– p.15/23
Examples of projects
Calculate the stress concentration factor. Determine whether the increased stresses near theholes are correlated or not. Symmetries should be used. Use Matlab and the PDE toolbox.
σ0 σ0
h
R
a cbφD
x, u1
y, u2
– p.16/23
Examples of projects
0 5010015020025030035040045050055060065070075080085090095010001050110011501200125013001350
0
185
370
Color: sxx Vector field: (u,v) Displacement: (u,v)
0
0.5
1
1.5
2
2.5
– p.17/23
Examples of projects
0 5010015020025030035040045050055060065070075080085090095010001050110011501200125013001350
0
185
370
Color: sxx Vector field: (u,v) Displacement: (u,v)
0
0.5
1
1.5
2
2.5
– p.18/23
Examples of projects
0 5010015020025030035040045050055060065070075080085090095010001050110011501200125013001350
0
185
370
Color: sxx
0
0.5
1
1.5
2
2.5
– p.19/23
Specific goals
Truss problem:
Understand fundamental principles of static analysis
Introduction to the finite element technique in structural mechanics
Programming: from problem definition to working code
Mathematical treatment of large linear systems of equations
Plate problem:
By visualization students develop intuition about stress distribution
Motivate the need to study the equations of elasticity
Introduction to the finite element method
Introduction to error estimation and adaptive mesh refinement
– p.20/23
Conclusions
Experience from the Chemical Engineering program and the Mechanics courses
Students appreciate to work with realistic models
This kind of course has the potential to increase the interest for mathematics
Strengthens the connection between applications and mathematics
Possibility to solve the complete problem: from modeling and solution of equations tosimulation and comparison with physical reality
Important to emphasize symbolic hand calculation and programming concepts
– p.21/23
Parallel developments
Umeå university (pilot project)
Royal Institute of Technology, Stockholm (pilot project)
– p.22/23
Further information
My homepagehttp://www.math.chalmers.se/˜stig
Body and Soul
http://www.bodysoulmath.org
– p.23/23
