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A.V. Dyskin, CRE, UWA
Modelling & Computer Applications in Engineering
GENG2140 Pantazis C. Houlis
Notes by Prof. Arcady Dyskin
A.V. Dyskin, CRE, UWA GENG2140 Slide 2
Objectives
Create mathematical and numerical models of simple but realistic engineering systems
Solve models using a computer and critically assess results Understand when an engineering system may be treated as
linear and when non-linear treatment is necessary as well as when the system can by considered probabilistic and when statistical methods are required
Use mathematical software to efficiently analyse and solve problems in engineering
Apply the knowledge of basic science and engineering fundamentals; and undertake problem identification, formulation and solution
A.V. Dyskin, CRE, UWA GENG2140 Slide 3
What is the unit about?
Numerical methods Packages (Excel, MathLab) Numerical errors Sensitivity analysis Computer simulations
A.V. Dyskin, CRE, UWA GENG2140 Slide 4
Topics
Computer arithmetic. Truncation and roundoff errors
Matrices and linear equations
Ill-conditioned matrices Sensitivity analysis
A.V. Dyskin, CRE, UWA GENG2140 Slide 5
Additional literature Forsythe, G.E., M.A. Malcolm & C.B. Moler. 1977.
Computer Methods for Mathematical Computations. Prentice-Hall, Inc.
Woodford, C. and C. Phillips. 1997. Numerical Methods with Worked Examples. Chapman and Hall, London
Montgomery, D.C., Runger, G.C. and Hubele, N.F. 2001. Engineering Statistics. John Wiley & Sons, Inc.
S.M. Ross, 1987. Introduction to Probability and statistics for engineers and scientists. John Wiley & Sons, Inc., New York, London, Sydney, Toronto.
I.M. Sobol, 1994. A Primer for the Monte-Carlo Method. CRC Press.
A.V. Dyskin, CRE, UWA GENG2140 Slide 6
Teaching material
Lecture notes and assignments – from http://undergraduate.csse.uwa.edu.au/units/GENG2140/
WebCT Lectopia
A.V. Dyskin, CRE, UWA GENG2140 Slide 7
Rules
Room 1.48 E-mail: pantazis.houlis@uwa.edu.au
Have a copy of the submitted assignment Remember the name of your tutors Make sure that all your assignments are
marked Keep the marked assignments until the end
of the semester
A.V. Dyskin, CRE, UWA GENG2140 Slide 8
Errors
Types of errors Computer arithmetic. Truncation and roundoff
errors Example. Numerical differentiation Example. Unstable algorithm
Absolute error rrar −=Δ
Relative error a
rrr rr
Δ≈
Δ=δ
Here r is the exact value, ra is an approximate value
A.V. Dyskin, CRE, UWA GENG2140 Slide 9
Points to learn Types of error Precision Catastrophic cancellation Numerical differentiation Sensitivity of numerical differentiation to
errors How to choose the increment Unstable algorithms
A.V. Dyskin, CRE, UWA GENG2140 Slide 10
Example. Compression of a layered sample Uniaxial loading of
layered material (glass layers) stress vs strain - glass1
01020304050607080
0 0.002 0.004 0.006 0.008
strain
stre
ss (M
Pa)
Stress-strain curve (courtesy Glen Snowen)
A.V. Dyskin, CRE, UWA GENG2140 Slide 11
Tangential modulus εΔ
σΔ≈
ε
σ=εddE )(
-20000
-10000
0
10000
20000
30000
40000
50000
60000
70000
80000
0 10 20 30 40 50 60 70 80
ε [10-6]
E(ε) [MPa]
A.V. Dyskin, CRE, UWA GENG2140 Slide 12
Types and sources of errors Human/Faulty equipment errors (can be corrected)
• Checks and verifications Errors of measurements
• Systematic – Calibration
• Random – Repeated measurements – Statistical treatment
Truncation/Roundoff errors • Computer arithmetic • Small • Double precision computations
A.V. Dyskin, CRE, UWA GENG2140 Slide 13
Computer arithmetic. Truncation and roundoff errors
The floating-point arithmetic
Real numbers are usually represented in computers by floating-point numbers F. They are characterised by: the number base β, the precision t and the exponent range [L, U].
UeLtiddddx ie
tt ≤≤=−β≤≤β⋅⎟⎟⎠
⎞⎜⎜⎝
⎛
β++
β+
β±= ,,,1,10,2
21 …
If d1≠0 (for x≠0), then the floating-point number system F is normalised. The integer e is called the exponent
( )ttddf β++β= …1 is the mantissa (fraction)
A.V. Dyskin, CRE, UWA GENG2140 Slide 14
Precision PC
Real (real*4) mode β = = = = −2 24 123 123, , ,t U LDouble precision (real*8) mode, β = = = = −2 53 1023 1023, , ,t U L
Real (real*4) mode β = = = = −10 7 38 38, , ,t U LDouble precision (real*8) mode, β = = = = −10 16 308 308, , ,t U L
Any real number x is replaced in a computer by the closest number, fl(x), from F
The relative error in rounding: t
xxxfl −β≤
− 1
21)(
In decimal system it would approximately correspond to
A.V. Dyskin, CRE, UWA GENG2140 Slide 15
Influence of small errors
Catastrophic cancellation • Loss of accuracy due to subtraction of close numbers
0.123456-0.123455=0.000001 (only one significant digit left)
• Numerical differentiation
Unstable algorithms Sensitive models
• Ill conditioned systems (next two chapters)
A.V. Dyskin, CRE, UWA GENG2140 Slide 16
Numerical differentiation
Finite difference x
xfxxfxfΔ
−Δ+≈ʹ′
)()()(
Choice of Δx: the smaller the better?
Derivative x
xfxxfxfx Δ
−Δ+=ʹ′
→Δ
)()(lim)(0
Other approximations
xxxfxf
xxxfxfxf
x Δ
Δ−−≈
Δ
Δ−−=ʹ′
→Δ
)()()()(lim)(0
xxxfxxf
xxxfxxfxf
x Δ
Δ−−Δ+≈
Δ
Δ−−Δ+=ʹ′
→Δ 2)()(
2)()(lim)(
0
A.V. Dyskin, CRE, UWA GENG2140 Slide 17
Example 1. Function with random errors of amplitude 0.1
x
y = 1 + sin ( ) + errorsxπ
10
2.2
1
1.4
1.8
lL
1x
Δ = 0.1x
Δ = 0.01x
010
5
0
5
10
15Derivative
Lxl <<Δ<<
A.V. Dyskin, CRE, UWA GENG2140 Slide 18
Example 2
RAND()1.02 ∗+= xy
x y 0.1 0.2 0.3 Exact0 0.03401 0
0.1 0.033396 -0.00613 0.20.2 0.1349 1.015037 0.504452 0.40.3 0.123053 -0.11847 0.448285 0.296812 0.60.4 0.195969 0.729159 0.305346 0.54191 0.80.5 0.323187 1.272176 1.000668 0.627623 10.6 0.44743 1.242435 1.257305 1.081257 1.20.7 0.562628 1.151978 1.197206 1.222196 1.40.8 0.739883 1.772547 1.462262 1.388987 1.60.9 0.878518 1.38635 1.579449 1.436958 1.81 1.033344 1.54826 1.467305 1.569052 2
Approximation for yʹ′
A.V. Dyskin, CRE, UWA GENG2140 Slide 19
Plots
0
2
4
6
0 0.5 1 1.5 2 2.5 X
Yy(x)
Exact y'(x)
Δx=0.1 Δx=0.2
Δx=0.3
A.V. Dyskin, CRE, UWA GENG2140 Slide 20
Unstable algorithms. Example: Moment of inertia
x
y
Non-homogeneous material
1)( −= xexρ
1 m
1 m
2
1
0
12 EdxexI xy ≡= ∫ −
∫ ρ=1
0
2 )( dxxxI y
Relative density distribution
A.V. Dyskin, CRE, UWA GENG2140 Slide 21
Moments of inertia for different shapes
x
y
1 m
1 m
3
1
0
13 EdxexI xy ≡= ∫ −
x 1 m
4
1
0
14 EdxexI xy ≡= ∫ −
y
1 m
A.V. Dyskin, CRE, UWA GENG2140 Slide 22
General case
∫ >= −1
0
1 0dxexE xnn
Integration by parts gives the following direct recurrent formula
632120.011,1 01 ≈−=−= − eEnEE nn
x 1 m
y
1 m
A.V. Dyskin, CRE, UWA GENG2140 Slide 23
Calculations Computer with β=10 and t=6 n
Recurrent formula Exact
2 0.264242 0.2642413 0.207274 0.2072774 0.170904 0.1708935 0.14548 0.145336 0.12712 0.1268027 0.11016 0.1123848 0.11872 0.1009329 -0.06848 0.091612
A.V. Dyskin, CRE, UWA GENG2140 Slide 24
Analysis of recurrent computation of inertia moments
Δ×−=Δ×−=
Δ×−=Δ××−×−=
Δ×+=Δ×+×−=
Δ−=
Δ+=
362880!9
6207274.023264242.0312264242.02367879.021
367879.063212.0
999
3
2
1
0
exactexact EEE
EEEE
Initial roundoff error 610368.0 −×≈Δ
The error of 9-th step is 0.133>E9exact
This algorithm is unstable – error accumulation
11 −−= nn nEE
A.V. Dyskin, CRE, UWA GENG2140 Slide 25
Summary Errors
• Measurements • Computational
– Computer arithmetic – Truncation and roundoff errors – Controlled by precision - formula
Catastrophic cancellation • Subtraction of close numbers • Numerical differentiation
– Amplifies errors – Choice of step (not too large, not too small)
Unstable algorithms • Inertia moment example • Simple methods could lead to catastrophic accumulation of errors
A.V. Dyskin, CRE, UWA GENG2140 Slide 26
Matrices and linear equations Matrix operations Example of applications of matrices.
Stresses Stress transformations under coordinate
rotations
Principal stresses and principal directions Systems of linear algebraic equations Inverse matrix
A.V. Dyskin, CRE, UWA GENG2140 Slide 27
Points to note Matrix – a table of numbers with special operations Stress is a matrix (it is not just force per area) Matrix can represent the effect of co-ordinate rotations
• Change of stress matrix due to co-ordinate rotations • Principal stresses and principal co-ordinate axes
– How to find – Engineering significance
System of linear equations • One solution • Many solutions • No solutions • Engineering significance
Methods of solution • Gaussian elimination • Method of inverse matrix
A.V. Dyskin, CRE, UWA GENG2140 Slide 28
Matrix operations Addition A+B Multiplication AB Transpose AT
Determinant det(A) Inverse matrix A-1
Eigenvalues and eigenvectors Matrix norm (equivalent to the length of vector) Condition number (indicator of sensitivity of the
system of linear equations to random errors) Matrix rank
A.V. Dyskin, CRE, UWA GENG2140 Slide 29
Matrix addition and multiplication
A 1 4 7
2 5 8
3 6 9
B 1 2 1
0 3 1
1 4 2
C A B = C 0 6 6
2 8 7
4 10 7
A 1 4 7
2 5 8
3 6 9
B 1
2 1
0 3 1
1 4 2
= . A B 0 0 0
3 9 15
3 12 21
= . B A 6 42 19
6 51 23
6 60 27
BAAB ≠
=TA
1
2
3
4
5
6
7
8
9
A
1
4
7
2
5
8
3
6
9Matrix transpose
A.V. Dyskin, CRE, UWA GENG2140 Slide 30
Inverse matrix
IAAAA == −− 11
A is a square matrix
The inverse matrix exists if and only if det(A)≠0
If det(A)=0, matrix A is called singular
Inverse matrix A-1:
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−−=
nnnn
n
n
aaa
aaaaaa
A
21
22121
11211
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−−=
100
010001
Iwhere I is the identity matrix:
A.V. Dyskin, CRE, UWA GENG2140 Slide 31
Determinant and inverse
B 1
2 1
0 3 1
1 4 2
Determinant: = B 3
= B 1 0.667 0 0.333
0.333 1 0.333
1 2 1
=.B B 11
0
0
0
1
0
0
0
1
=.B 1 B
1
0
0
0
1
0
0
0
1
A.V. Dyskin, CRE, UWA
A 2 1
1 2
x
y Ax = λx = λ
Ax - λx = 0 Ax – (λΙ )x = 0 (A – λΙ )x = 0
λΙ λ
0 0 λ
Α - λΙ 2 - λ
1 1
2 - λ
det(Α - λΙ ) = 0 ⇒ λ2 - 4λ + 3 = 0 ⇒ λ1 = 1 , λ2 = 3
2 - λ1
1 1
2 - λ1
x
y
0
0 = ⇒ ⇒ x + y = 0 ⇒
x
y
0
0 = 1
1 1 1
1
-1 x1 =
2 - λ2
1 1
2 - λ2
x
y
0
0 = ⇒ ⇒ x - y = 0 ⇒
x
y
0
0 = -1
1 1 -1
1
1 x2 =
Ax1 = λ1x1
Ax2 = λ2x2 ⇒ A (x1 x2) = (x1 x2)
λ1
0 0 λ2
⇒ (x1 x2) -1 A (x1 x2) = λ1
0 0 λ2
Eigenvalues and eigenvectors
1 -1
1 1
2 1
1 2
1 -1
1 1
1 0
0 3 = ⇒
-1
A.V. Dyskin, CRE, UWA GENG2140 Slide 33
Eigenvalues and eigenvectors
M 1 2 3
2 5 8
3 8 9
= eigenvals ( ) M 0.192 1.294 16.102
R eigenvecs ( ) M = R 0.957 0.282 0.07
0.175 0.752 0.635
0.232 0.596 0.769
= . . T R M R 0.192 0 0
0 1.294 1.776 10 15
0 1.554 10 15
16.102
A.V. Dyskin, CRE, UWA GENG2140 Slide 34
Stress
n
F A
Surface element
Force is a vector Direction of surface element is
represented by its normal vector
Stress is a matrix
AreaForceStress = ?
High School
A.V. Dyskin, CRE, UWA GENG2140 Slide 35
Stress matrix σz
x
y
z
σy
σx
τzy
τzx
τyx
τyz
τxy
τxz σ
σ τ ττ σ τ
τ τ σ
σ σ
σ σ
σ σ
=
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
≡
≡
≡
x xy xz
yx y yz
zx zy z
x xx
y yy
z zz
Stress matrix (tensor)
τ τ τ τ τ τxy yx xz zx yz zy= = =, ,Symmetry:
A.V. Dyskin, CRE, UWA GENG2140 Slide 36
Stress in the given direction (on the given surface element)
Stress acting on the surface element
z
y
x
F=(Fx, Fy, Fz)
n=(nx, ny, nz)
A
S FA
n n n
SFA
n n n
S FA
n n n
xx
x x y yx z zx
yy
x xy y y z zy
zz
x xz y yz z z
= = + +
= = + +
= = + +
σ τ τ
τ σ τ
τ τ σ
A.V. Dyskin, CRE, UWA GENG2140 Slide 37
Stresses at different angles p
p
p
p p
p
p
p
A.V. Dyskin, CRE, UWA GENG2140 Slide 38
Coordinate rotation
x
y
z
i j
k
ix
iy
iz
σ σʹ′
Rotation matrix
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=
zyx
zyx
zyx
Rkkkjjjiii
where ⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=
zzz
yyy
xxxTR
kjikjikji
σʹ′=RσRT
Stress transformation
- transposed matrix
A.V. Dyskin, CRE, UWA GENG2140 Slide 39
Example. Fault in Earth’s crust
x
y
z
N
12 MPa
6 MPa
9 MPa
MPa⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−
−
=σ
9000120006
20° 60°
zf
yf
xf
The fault dips 60 ° in the dip direction 20° North-West. The friction angle is ϕ= 30°, the cohesion C=0.
A.V. Dyskin, CRE, UWA GENG2140 Slide 40
Stresses at the fault Step 1
x
y
z
x ʹ′
20°
z ʹ′ y ʹ′
20°
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−=
100020cos20sin020sin20cos
R
x ʹ′
y ʹ′
z ʹ′ zf
yf
xf
60° ⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−
=
60cos060sin01060sin060cos
fRStep 2
( ) ( ) 2.430tan276.7944.1232
231 =<=σ+σ=τ ff The fault is stable
σ1:=RσRT
σf:=Rfσ1RfT
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−−
−−
=σ
9000298.11928.10928.1702.6
1
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−
−−−
=σ
276.767.1995.067.1298.11964.0995.0964.0425.8
f
A.V. Dyskin, CRE, UWA GENG2140 Slide 41
Principal stresses σI≥ σII≥ σIII
Principal directions Three special orientations of the axes at which there are no shear stresses
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
σ
σ
σ
=σ
III
II
I
000000
σz
x
y
z
σy
σx
τzy
τzx
τyx
τyz
τxy
τxz
Eigenvectors Eigenvalues
Principal directions ( principal axes)
xx xy xz
yx yy yz
zx zy zz
σ τ τ
σ τ σ τ
τ τ σ
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎝ ⎠
A.V. Dyskin, CRE, UWA GENG2140 Slide 42
Example
10 MPa
5 MPa
x
y
z
MPa⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
=σ
1050500000
σ
0
0
0
0
0
5
0
5
10
=eigenvals ( )σ
0
2.071
12.071
=eigenvecs ( )σ
1
0
0
0
0.924
0.383
0
0.383
0.924
MPaprinc⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
=σ
071.120000000071.2 I II III
II
I III θ
θ=cos-10.924=22.48°
σII
σI
σIII
A.V. Dyskin, CRE, UWA GENG2140 Slide 43
Solving systems of linear algebraic equations
Gaussian eliminations Method of inverse matrix
A.V. Dyskin, CRE, UWA GENG2140 Slide 44
Solution
Gaussian elimination
⎩⎨⎧
=+−
=+
22412yx
yxExample
⇒⎩⎨⎧
=
=+
4412
yyx
Gaussian elimination The first equation is multiplied with 2 and added to the second one
Back substitutions
⎩⎨⎧
=
=+
112
yyx
⇒⎩⎨⎧
=
=
102
yx
10
=
=
yx
Matrix form
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
2412
A ⎟⎟⎠
⎞⎜⎜⎝
⎛=21
BMatrix of the system
Vector of right parts ⎟⎟
⎠
⎞⎜⎜⎝
⎛
−=
224112
)|( bAAugmented matrix
⎟⎟⎠
⎞⎜⎜⎝
⎛→⎟⎟⎠
⎞⎜⎜⎝
⎛→⎟⎟⎠
⎞⎜⎜⎝
⎛→⎟⎟⎠
⎞⎜⎜⎝
⎛→⎟⎟⎠
⎞⎜⎜⎝
⎛
− 101001
101002
111012
414012
212412
detA=8≠0
A.V. Dyskin, CRE, UWA GENG2140 Slide 45
Case 1 of detA=0 Example
⎪⎩
⎪⎨
⎧
=++
=++
=+−
3422321
zyxzyxzyx
0412321111
det =
−
=A
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
→⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
→⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
→⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
→⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
03/13/4
0003/2103/501
03/11
0003/210111
011
000230111
111
230230111
321
412321111
x=4/3-5/3 z
y=1/3-2/3 z Mechanical interpretation: statically indeterminate system
Infinite number of solutions
A.V. Dyskin, CRE, UWA GENG2140 Slide 46
Case 2 of detA=0 Example
⎪⎩
⎪⎨
⎧
=++
=++
=+−
4422321
zyxzyxzyx
0412321111
det =
−
=A
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
→⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
→⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
111
000230111
211
230230111
421
412321111
Mechanical interpretation: Mechanism
No solutions
A.V. Dyskin, CRE, UWA GENG2140 Slide 47
Method of inverse matrix Matrix form BAX =
is the vector of unknowns ⎟⎟⎠
⎞⎜⎜⎝
⎛=yx
X
If the inverse matrix exists, the solution of the linear system can be expressed in the matrix form
BAX 1−=
A method of finding inverse matrix
i-th column of the inverse matrix is a solution of ⎟
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
=
0...1...0
Axi-th place
A.V. Dyskin, CRE, UWA GENG2140 Slide 48
Summary In many packages operations written in a matrix form run faster Matrix multiplication is not commutative AB≠BA Inverse matrix A-1 exists only if detA≠0 Stress is a (symmetric) matrix with diagonal elements
representing normal stresses, non-diagonal elements – shear stresses
Change with co-ordinate rotation: σʹ′=RσRT
In principal directions only normal (principal) stresses exist. Principal directions and stresses are eigenvectors and eigenvalues of the stress matrix. An application: failure analysis
Singular matrix of the system • Many solutions – statically indetermined system • No solutions – mechanism
Inverse matrix is used to solve systems with the same matrix but varying right hand parts
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