Swiss Federal Institute of Technology

Swiss Federal Institute of Technology

The Finite Element Method and the Analysis of Systems with Uncertain Properties

Prof. Dr. Michael Havbro Faber Swiss Federal Institute of Technology

ETH Zurich, Switzerland


Contents of Today's Lecture

• Some preliminary considerations

• An overview of the lecture

• Introduction to the use of finite element

- Physical problem, mathematical modeling and finite element solutions- Finite elements as a tool for computer supported design and assessment

• Basic mathematical tools (vectors, tensors and matrix calculus)


Preliminary considerations

• Aspects of engineering modelling

Decisions Real world Consequences




Decisions

Solution of model Consequences

Real world

Model of real world

Choice of model

Change of world

Improve knowledge

Solution strategy




Input Filter Output

Uncertainties

UncertainFor given decisions




Decisions


Real world

Model of real world

Choice of model

Change of world

Improve knowledge

Solution strategy

Choice of scale- Indicators of the state of the real world

Inherent uncertainty

Epistemic uncertainty




Decisions


Real world

Model of real world

Choice of model

Change of world

Improve knowledge

Solution strategy

Choice of scale- Indicators of the state of the real world

Inherent uncertainty

Epistemic uncertainty

Optimization of decisions by maximizationof expected utility – a function of risk


An overview of the lecture


Introduction to the use of finite element

• Physical problem, mathematical modeling and finite element solutions

- we are only working on the basisof mathematic models !

- choice of mathematical model is crucial !

- mathematical models must be reliable and effective

Physical problem

Mathematical modelgoverned by differential equations andassumptions on- geometry- kinematics- matrial laws- loading- boundary conditions- etc.

Finite elememnt solutionChoice of - finite elements- mesh density- solution parametersRepresentation of- loading- boundary conditions- etc.

Assessment of accuracy of finite elementSolution of mathematical model

Interpretation of results

Design improvements

Refinement of analysis

Impr

ove

mat

hem

atic

alm

odel

Cha

nge

phys

ical

prob

lem



• Reliability of a mathematical model

The chosen mathematical model is reliable if the required response is known to be predicted within a selected level of accuracy measured on the response of a very comprehensive mathematical model

• Effectiveness of a mathematical model

The most effective mathematical model for the analysis is surely that one which yields the required response to a sufficient accuracy and at least costs



• Example

Complex physical problem modeled by a simple mathematical model



• Example

Detailed reference model – 2-D plane stress model – for FEM analysis



• Example

Comparison between simple and more refined model results

Reliability and efficiencymay be quantified !



• Observations

Choice of mathematical model must correspond to desired responsemeasures

The most effective mathematical model delivers reliable answers with the least amount of efforts

Any solution (also FEM) of a mathematical model is limited to information contained in the model – bad input – bad output

Assessment of accuracy is based on comparisons with results from very comprehensive models – however, in practice often based on experience



• Observations

Some times the chosen mathematical model results in problems such as singularities in stress distributions

The reason for this is that simplifications have been made in the mathematical modeling of the physical problem

Depending on the response which is really desired from the analysis this may be fine – however, typically refinements of the mathematical model will solve the problem



• Finite elements as a tool for computer supported design and assessment

FEM forms a basictool framework in researchand applications coveringmany different areas

- Fluid dynamics- Structural engineering- Aeronautics- Electrical engineering- etc.



• Finite elements as a tool for computer supported design and assessment

The practical application necessitates that solutions obtained by FEM are reliable and efficient

however

also it is necessary that the use of FEM is robust – this implies that minor changes in any input to a FEM analysis should not change the response quantity significantly

Robustness has to be understood as directly related to the desired type of result – response


Basic mathematical tools

• Vectors and matrixes

11 1 1

1

1

1 1

2 2,

i n

i ii

m mn

m m

a a a

a a

a a

x bx b

x b

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Ax b

A

x b

is the transpose of if there is

(square matrix)and (symmetrical matrix)

T

ij ji

m na a

==

T

A AA = A

1 0 0 00 1 0 0

is a unit matrix0 0 1 0

0 0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

I



• Banded matrixes

A A

Symmetric banded matrixes

0 for , 2 1 is the bandwidth

3 2 1 0 02 3 4 1 01 4 5 6 10 1 6 7 40 0 1 4 3

ija j i m m= > + +

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

AA 2m =



• Banded matrixes and skylines

3 2 0 0 02 3 0 1 00 0 5 6 10 1 6 7 40 0 1 4 3

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A

1m +A



• Matrix equality

( ) ( ) if and only if and and

ij ij

m p n qm n p qa b

× = ×= ==

A B



• Matrix addition

( ) and ( ) can be added if and only if and and

ij ij ij

n q n qm n p q

c a b

× ×= =

= +

A B

C = A + B



• Matrix multiplication with a scalar

A matrix is multiplied by a scalar by multiplying all elements of with

ij ij

cc

cb ca==

AA

B A



• Multiplication of matrixes

1

Two matrixes ( ) and ( ` ) can be multiplied only if

, ( )m

ij ir rjr

p m n q m n

c a b p q=

× × =

=

= ×∑

A B

C BA

C



• Multiplication of matrixes

The commutative law does not hold, i.e.

, unless and commute

The distributive law holds, i.e.

( )

The associative law holds, i.e.

( ) ( )

≠

= + = +

= = =

AB BA A B

E A B C AC BC

G AB C A BC ABC

, does not imply that however does hold for special cases (e.g. for )

Special rule for the transpose of matrix products

( )T T T

= ==

=

AB CB A CB I

AB B A



• The inverse of a matrix

1

1 1

The inverse of a matrix is denoted

If the inverse matrix exist then there is:

the matrix is said to be non-singular

The inverse of a matrix product:

( )

−

− −

−

= =

=1 -1 -1

A A

AA A A I

A

AB B A



• Sub matrixes

11 12

21 22

11 12 13

21 22 23

31 32 33

A matrix may be sub divided as:

a a aa a aa a a

a a

a a

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

A

A

A



• Trace of a matrix

1

The trace of a matrix ( ) is defined through:

( )n

iii

n n

tr a=

×

=∑

A

A



• The determinant of a matrix

11 1

1

1

st th

The determinant of a matrix is defined throughthe recurrence formula

det( ) ( 1) det

where is the ( -1) ( -1) matrix obtained by eliminating the

1 row and the column from the mat

nj

j ji

j

a

n n

j

+

=

= −

×

∑A A

A

[ ]11 11

rix and where there is

if ,deta a= =

A

A A



• The determinant of a matrix

It is convenient to decompose a matrix by the socalled Cholesky decomposition

where is a lower triangular matrix with all diagonal elements equal to 1 and is a diagonal matrix with componen

T=

A

A LDLL

D

1

ts then the determinant of the matrix canbe writtes as

det

ii

n

iii

d

d=

=∏

A

A

21

31 32

1 0 01 0

1L l

l l

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

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