modelling & computer applications in engineering geng2140€¦ · a.v. dyskin, cre, uwa...

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


What is the unit about?

  Numerical methods   Packages (Excel, MathLab)   Numerical errors   Sensitivity analysis   Computer simulations


Topics

 Computer arithmetic. Truncation and roundoff errors

 Matrices and linear equations

  Ill-conditioned matrices  Sensitivity analysis


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.


Teaching material

Lecture notes and assignments – from http://undergraduate.csse.uwa.edu.au/units/GENG2140/

WebCT Lectopia


Rules

Room 1.48 E-mail: [email protected]

 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


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


Points to learn  Types of error  Precision  Catastrophic cancellation  Numerical differentiation  Sensitivity of numerical differentiation to

errors  How to choose the increment  Unstable algorithms


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)


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]


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


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)


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


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)


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


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<<


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′


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


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


Moments of inertia for different shapes

x

y

1 m

1 m

3

1

0


x 1 m

4

1

0


y

1 m


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


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


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


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


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


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


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


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


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:


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


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


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


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:


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


Stresses at different angles p

p

p

p p

p

p

p


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


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.


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


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

⎛ ⎞⎜ ⎟

= ⎜ ⎟⎜ ⎟⎝ ⎠


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


Solving systems of linear algebraic equations

 Gaussian eliminations  Method of inverse matrix


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


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


Case 2 of detA=0 Example

⎪⎩

⎪⎨

⎧

=++

=++

=+−

4422321

zyxzyxzyx

0412321111

det =

−

=A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

→⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

→⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

111

000230111

211

230230111

421

412321111

Mechanical interpretation: Mechanism

No solutions


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


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

modelling & computer applications in engineering geng2140€¦ · a.v. dyskin, cre, uwa...

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