method of finite elements by dr. mojsilovic

7/28/2019 Method of finite elements by Dr. Mojsilovic

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Swiss Federal Institute of Technology

e o o n

Held by Dr. N. Mojsilovi Assistant: Jianjun Qin, HIL E Lectures homepage: http://w

Course book: Finite Elemen

Method of Finite Elements I

Oral examination, 30 minute


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ourse v

26.02.2010 Introduction

e ements; as c mat ema *05.03.2010 Basic concep 12.03.2010 Displacemen

elements *19.03.2010 Formulation . .

*16.04.2010 Isoparametr. . russ e emen

quadrilateral elements



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ourse

30.04.2010 Element mat * 7. .2 1 Beam elem

elements

14.05.2010 Plate elemen *21.05.2010 Shell eleme

static analysis 04.06.2010 Conver enc

completeness, accuracy o

elementsMethod of Finite Elements I


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eve

MFE is the confluence of thstructural anal sis, variatiocomputer

1950s, M.J. Turner at Boeingeneral): Direct Stiffness M Academia: J.H. Argyris, R.W

e ement , H. . Mart n an popularisation

s, e os an e eu

Commercial finite element



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n ro uc on o Elem

dependencies between guantities are formulated

element and than extend

As a result we obtain difor integral equations for analytical solution is notso ve us ng some nume

MFE is based on the phyo serve oma n, us redegrees of freedom; more


,


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eps n

Continuum is discretized i Elements are connected at oun ar es

State of deformation, stresescr e y n erpo a oncorresponding values in th

has a great influence on th



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as c y

Direct MFE: analogue to Variational MFE: based

stationarity of a functionpo en a energy or comp

Residual MFE: based ona are use o escr e

Energy Balance MFE: ba,

thermodynamic problem



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o e ng o e



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o e ng o e

MFE is only a way of solving

quality of the mathematical

Thus, mathematical model m T e c osen mat emat ca moresponse can be predicted wmeasure on e response o mathematical model

The most effective mathemathe one that gives the require


accuracy an at east costs


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Complex physical problem mo


mat emat ca mo e


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Detailed reference mode


or ana ys s


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o ce o ma ema ca moresponse

The most effective mathemat

answers with the least amoun Any solution (including MFE

limited to information contai

bad input bad output (garb Assessment of accuracy is baresults from very comprehen

it has to be based on experienMethod of Finite Elements I


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as a oo



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as a oo

Practical a lication reobtained by MFE are rel

,robust this implies tha

response quantity signi

of the obtained results (



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a r

A matrix is an array of orderedconsists of mn numbers arran

thus the matrix is of order m x onl one row (m = 1) or one colvector



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a r

Example: Ax=b, where A is maarra of unknowns and b an ar



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a r

The transpose of the mxn matrix Ainterchanging the rows and column

= , it o ows t at t e numequal and that aij = aji. Because m = n=

Note, symmetry implies that A is sq

square matrix need not be symmetri quare matrix wit on y zero e emwhere they are unity, is called an ide



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an e

For symmetric banded matri,

If the half-bandwidth, mA, ofnonzero elements onl on thdenote it as a diagonal matri


S i F d l I tit t f T h l


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pera ons w

Matrix equality: A(mxp) = B

an p = q an aij = ij

on: mxp an nxm = n and p = q. Thus

=

scalar c by multiplying all ele= , .




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pera ons w

Multiplication: Two matricmulti lied onl if = n. Thus

=

r

ij

Inversion: The inverse of m

inverse matrix exist than we A matrix with an inverse ismatrix wit out an inverse is




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pera ons w

Multiplication C=AB




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pera ons w

Inversion: AA-1 = A-1A = I




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pera ons w

Distributive law does hold

Associative law does hold

AB = CB does not im l th

Special rule for the transpo

(AB)T




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pera ons w

Matrices can be subdivided

manipulations Partitionin lines must run matrix




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gy

e race

The trace of a matrix A is de

The trace of a matrix is a sca

Some rules:tr A+B = tr

tr(cA) =tr(AB) =




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gy

e race

The trace of a matrix A, tr(A




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e e erm n

The determinant of a matrix A

The determinant of a matrix i

n

i 1

e=

=

where A1j is the (n-1)x(n-1) mthe 1st row and the th column fr

if A= a then detA=aMethod of Finite Elements I



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e e erm n

The determinant of a matrix is

n

i 1

e=

=




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e e erm n

The determinant of a matrix us

n

j

i

a11

det1)det( AA=

=




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e e erm n


2n

j+

2 2

1

e e ji

a=

=




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e e erm n


3n

j+

3 3

1

e ej ji

a=

=




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e e erm n

It is convenient to decompose= T

L is a lower triangular matrixe ual to 1 and D is a dia onal m

1

21

31

L l

l l

=

Thus the determinant of matr

=det A




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e e erm n

LDL decomposition: A=LDLT




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ome se u D rm

e = e

det(A-1) =

det(I




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ens

A set of quantities that obey

those of another

example: tensor of order 0 is a

Bathe: An entity is called a s=

frame and nine components tthese com onents are related tij=pikpjltkl, P being a rotation m




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ress

=

z

xzxyxx

zz

Mohrs circles (for example: p




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Variational operator Variations of deformation are

the equilibrium and are consisteof the s stem Some rules:

( )uddu

=

xx

=

a a

udxudx


0 0


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method of finite elements by dr. mojsilovic

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