minimization of the expected compliance as an …migucarr/cursos/ma38b... · 2004-01-20 · dome...

MINIMIZATION OF THE EXPECTEDCOMPLIANCE AS AN ALTERNATIVEAPPROACH TO MULTILOAD TRUSS

OPTIMIZATION.

Felipe Alvarez Daziano

Miguel Carrasco Briones

Universidad de Chile

Facultad de Ciencias Fısicas y Matematicas

Departamento de Ingenierıa Matematica

Optimal design of trusses. – p. 1/46

� Optimization of trusses� Standard minimum compliance truss design.� Instabilities and the standard multiload model.� Random Perturbations: minimizing theexpected compliance.� Numerical examples.


Optimization of trusses


Introduction

� Definition.� Problem: find the best structure capable to carry aexternal nodal force.� Constraints: mechanical equilibrium , total volumeand others . . .� Minimize the compliance.


Minimum compliance truss design

� ��

� � ��(

�

)

��

�� ! � � "$# % � � � % "'& �

Design variables� �

:vector of bar volumes (topology).� � : nodal positions (geometry).#( � )� is called Compliance.Optimal design of trusses. – p. 5/46

Matematical formulation

� � � * ,+ - number of bars.� � � * &+ .

number of nodes.� / grades of fredom.� � � * 0

nodal load vector.� � � * 0

nodal displacements.

� ��1324 25

� � 6��

�� 87� � ":9 % � � � % "<; �

� � � = � � * > �? @� ACB # � A � D

� � �� * 0 E 0� stiffness matrix.


Stiffness Matrix

� �� ACB #

� A � A �� where � A �� F AG A �� ( H A �� H A �� * 0 E 0

Remark

� A is dyadic (� A � I A I A ).� F A: Young modulus.� G A ��

: length of bar

J �

� H A �� : cosines/sines vector.


ground structure approach

� geometric variable K � � �

is very difficult

solution: ground structure

Nodal positions are fixed in the referenceconfiguration ( is fixed).

We use a mesh full of nodes and bars.

f


ground structure approach

� geometric variable K � � �

is very difficult

� solution: ground structure� Nodal positions are fixed in the referenceconfiguration (� is fixed).� We use a mesh full of nodes and bars.

f


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−1

−0.5

0

0.5

1

1.5

2

2.5

3

X

Y

f

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−1

−0.5

0

0.5

1

1.5

2

2.5

3

X

Y

f

Typically a large number of bar vanish at the optimum.


� � ��ML NO �L NP

� � ��(

�

)

��

�� !

#( � )� is independent of the choice � satisfying� � � � � � ��Optimal design of trusses. – p. 10/46

Minimax Formulation

(

Q

) � ��RL N P � ST#U AU = � V � A VXW � V D �

Dual of

� Q �

is the design problem� � �

.advantages� reduction of the dimention.� The proof of equivalence between

� Q �

and

� � �

don’t use the dyadic structure of matrix

� A �

disadvantages� Lost of differentiability of the objective function.


Instabilities and the standard multiload model


Example

The design problem

� � �

may produce insatisfactoryresults in respect to mechanical stability

1 2 3

4 5 6


Multiload model

Standard multiload model

� ��ML NO

�Y

ZB # [ Z � Z �� Z

�� Z � � Z� \ � � � � �� ]

�� We minimize a weighted average of the compliances.


Random Perturbations: minimizing theexpected compliance.


Random load model

Let

^ � * 0

be a perturbation on the load vector

��

_ � ^� � � �`a

acb# ( � �ed ^ � V if

� � �f and � � * 0

such that� � � � � � �ed ^�d g otherwise

_ih � * 0� j � * 0 � � k � *l =d g D� m j � * � �

Results to be proper, lower semicontinuous andconvex.


Random load model

For heach

��

_ �on� � � h � * 0� j � * 0 � � k � *l =d g D� mpj � * � �

^ q k _ � ^� � �

is measurable.We asume that

^

is a random variable correspondingto an uncertain nodal load perturbation

^h �r � � k � * 0� j � * 0 � �

s q k ^ � s �


Random load model

Minimum expected compliance design problem

(

�+ t

) � � ��ML u O v'w x _ � ^� � �yRemark:

" � � z � � � � � � = � � � � � > � � * 0 D

v'w x _ � ^� � � y � {�|_ � ^� � �} t ��~ � d g t = " �� D

if

" � � � * 0

it may ocurr that

v w x _ � ^� � � y � d g whenevert = �d ^ � " � DC� @ K �is not a feasible member.


Example: discrete perturbation

1.

t # = ^ � � D �

if

@ � �

and

@

otherwise.We obtein the clasical single load model.

2.

t ( with finite support

�i�� t ( � � = ^ #� � � �� ^ Y D �

� �+ t ( � � ��L NO

�Y

ZB # [ Z � Z � Z

�� Z � � Z� J � � � � �� ]

�� f �

� �+ t ( �is the standard multiload model where[ Z � t � ^ � ^ Z�

and

� Z � �ed ^ Z� \ � � � � �� ]��


Continuos case


Continuous case

Theorem 0.1. Let

^h r k * 0

be a continuous random variablewith mean vector

v � ^ � � @

and Variance–covariance matrix� S� � ^ � � � � � with

�� * 0 E Y �Then the corresponding minimum expected compliance designproblem

� �+ t �

is given by

� �+ t � � ��ML NO

� � ��

mean load

d � �� S��

� � � �

variance��

� � � � � � �

� � �f �


Continuous case

Remark

� � and

#( �� S��

are independent of thechoice � and

�

satisfaying

� � � � � � �� and

� � � � � � �

respectively.Remark � Z� \ � � � � �� ]� columns of

�

Defining

�� # � � � �� Y � � * 0 � Y� # �� A � � � S � � � A� � A� � � �� A � � * 0 � Y � # � E 0 � Y� # ��

� � � � � � AB #

� A � � A�

Then� �+ t �

may be written as a multiload model.Optimal design of trusses. – p. 22/46

Example

Random one dimentional preturbation

^ � �} � where} � * 0� v � � � � @� � S� � � � � � ( �Then

v � ^ � � @

and

� S � � ^ � � �} }

� �+ t'� � � � ��L NO

� � �� d � �} ��

��

� � � � � � �}

�� f �


example

Random multidimentional independent

perturbation

^ � Y ZB # � Z} Z� where} Z� * 0� v � � Z� � @

and

� S� � � Z� � � ( Z�

� �+ t�� ML NO

� � �� d �

ZB # � Z} Z �� Z

��

� � � � � Z � � Z} Z \ � � � � �� ]

��


Stochastic model including variance

We consider

� ��ML NO [ v$w x _ � ^� � � y d ¡ � S� x _ � ^� � � y

Example:

^�¢ � @� � � � �Taking for simplicity [ � @

and

¡ �

� �+ t

var

� � ��ML NO

� �� S�� £� � ( � d � � � � �

��

� � � � � � �

��

Apparent harder to solve. Optimal design of trusses. – p. 25/46

Numerical results


Toy example

0 0.5 1 1.5 2 2.5−0.5

0

0.5

1

1.5

1 2 3

4 5 6

0 0.5 1 1.5 2 2.5−0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5−0.5

0

0.5

1

1.5

(a)

0 0.5 1 1.5 2 2.5−0.5

0

0.5

1

1.5

(b)

Toy example, Ben-Tal and Nemirowski 1997 – p. 27/46

Problem compl.mean

maxcompl.

mincompl.

Standardsingle load

¤ d g d g

(a) 0.025 0.053 0.013

(b) 0.023 0.051 0.012

Toy example, Ben-Tal and Nemirowski 1997 – p. 28/46

Dome

Dome 3D – p. 29/46

Dome

Dome 3D – p. 30/46

Dome

Dome 3D – p. 31/46

Electricity mast

−6 −4 −2 0 2 4 6 8 10 12

0

5

10

15

−6 −4 −2 0 2 4 6 8 10 12

0

5

10

15

−6 −4 −2 0 2 4 6 8 10 12

0

5

10

15

Electricity mast , Achtziger et al. 1992 – p. 32/46

Michel 3D

Michel 3D – p. 33/46

Michel 3D

Michel 3D – p. 34/46

Cantilever

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 −0.2

0

0.2

−0.1

0

0.1

0.2

0.3

0.4

Y

X

Z

Cantilever , Achtziger 1997 – p. 35/46

Cantilever , Achtziger 1997 – p. 36/46

Conclusions

� Classical model (one load) doesn’t get goodresults .� Random model helps to avoid this problem.� It’s equivalent to multiload one, but anotherinterpretations in weight factors holds.� We have to know the influence of eachperturbation.� Increase the dimention of the problem.

Optimal truss design – p. 37/46


OPTIMIZATION.







0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−1

−0.5

0

0.5

1

1.5

2

2.5

3

X

Y

f

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−1

−0.5

0

0.5

1

1.5

2

2.5

3

X

Y

f


stability result

Lemma 0.1. Let

� � � and

¥

be such that

¥ ?#( � S T Z � Z� Zwith

� � � � � Z � � Z,

\ � � � � �� ]. For each� � � � �¦ = � #� � � �� Y� W � #� � � �� W � Y D� there exists � � * 0

satisfying

� � � � � � �

and moreover¥? #( � )� .

fi

fkf1

Conv{

−f

} ,..., f kf1 ,

− −,...,

i

f

Figure 1: Convex Hull of §©¨ª ª ª ¨ «¨ ¬ §©¨ª ª ª ¨ ¬ «


self weight

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

1


self weight


Se cumple el siguiente resultado:Theorem 0.2. Sea

� V � ® � solucion del problema

�¯ � ��

entonces existen � � � � ° � * � tales que definiendo

� � ® � �� d � ° �

se cumple que el par

� V � � �es solucion de

� � �� (el problema dediseno).


�¯ � �

es equivalente a

�± � �

�± � � � �� RL N P W � V

�� ²� I A V ³ J � � � � �� -

W I A V ³ J � � � � �� -


Se cumple el siguiente resultadoTheorem 0.3. Sea

m V solucion del problema�± � �

con vectoresmultiplicadores ´� � ´ ° � * � � entonces

V h � ® m V

� A h � ® � ´� A d ´ °A � J � � � � �� -

con ® h � ACB # � ´� A d ´ °A �resuelven

� � � �


minimization of the expected compliance as an …migucarr/cursos/ma38b... · 2004-01-20 · dome...

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