Institute for Statics und Dynamics of Structures

Fuzzy Structural Design

Bernd Möller

Slide 2

Institute for Statics and Dynamics of Structures

Overview

1 Conceptual idea

2 Cluster methods

3 Fuzzy cluster design

4 Applications

Slide 3

Institute for Statics and Dynamics of StructuresConceptual idea (1)

( ) ( )1 2 m 1 2 n

x zz , z , ..., z f x , x , ..., x

→

=

fuzzy input value x1 fuzzy input value x2~~

mappingoperator

0.01.0 ακ 0.0 1.0ακ

µ(x1) µ(x2)α-level optimization

1.0ακ

0.0

µ(zj)

fuzzy result value zj~

zj

1x

k1,x α

k1,x α ,l

k1,x α ,r

2 2, kx x α ∈

kj j,z z α∈

1 1, kx x α ∈

2x

k2,x α ,l

k2,x α ,r

k2,x α

kj,z α

kj, lz α kj, rz α

Slide 4

Institute for Statics and Dynamics of Structures

fuzzy input values x1 and x2

z

F(z)

"

1

perm_z

fuzzy result value z

F(x1,x2)

x1

x2 "

α

0

1

1

~

z = f(x1,x2)

~

~

points from - level optimization

permissible points

non-permissible points

Conceptual idea (2)

α

( ) ( )1i i i iz f x x f z−= ↔ =

design contraint

Slide 5

Institute for Statics and Dynamics of Structures

support x1,"=0

support subspace X"=0

supportsubspace Z"=0

xi xi,2 = di,2

xi,1 = di,1

zi

zi,2 = ri,2

zi,1 = ri,1

xi = f -1(zi)

support z1,"=0

Conceptual idea (3)

( ) ( )1i i i iz f x x f z−= ↔ =

supp

ortx

2,a=

0

supp

ortz

2,a=

0

Slide 6

Institute for Statics and Dynamics of Structures

6

clusters with permissiblepoints(permissible clusters)

x1

x2

support x1,"=0

xi

xi,1 = di,1

clusters with non-permissiblepoints(non-permissible clusters)

" = 1

designconstraint

{z1,"=0 } = {ri,1}z1

non-permissiblepermissible

{z1,"=0} = {ri,1}_

membership function ofthe fuzzy result value z1

z1,"=0

X X X XXX X

µ(z1)

Conceptual idea (4)

supp

ortx

2,a=

0

x i,2

= d i

,2

Slide 7

Institute for Statics and Dynamics of Structures

Overview

1 Conceptual idea

2 Cluster methods

3 Fuzzy cluster design

4 Applications

Slide 8

Institute for Statics and Dynamics of Structures

+

+

++

+++

++++

+++++++

10

x2

00

20 x110

representativecluster object

representativecluster object

k-medoid cluster method

Application of cluster methods (1)

• k predefined clusters

• selecting of k representative objects and clustering the remaining objects

• improving the set of representative objects andhence clustering

• assessing the quality ofclustering by numericalcriterions

Slide 9

Institute for Statics and Dynamics of Structures

9

0.2

0.10.3

0.3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.2

0.2

10

x2

00

20 x110

0.6

0.5

0.5 0.4

0.3

0.2

0.9

Fuzzy cluster method

Application of cluster methods (2)

• k predefine cluster

• initializing membership values

• iterative improvingthe membership valuesof the objects

• non-linear optimisation problem with constraints

• assessing the quality ofclustering by numerical criterions

Slide 10

Institute for Statics and Dynamics of Structures

Application of cluster methods (3)

Numerical criterions to assess the quality of clustering

k-medoid method fuzzy cluster method

2 4 6 8 10 12 14 16 18 20

averageseparation

silhouette coefficientSCASP

number ofclusters

2 4 6 8 10

PCN,SD

number ofclusters

sepration degree SD

normalized partion coefficient (PCN)

vmki i

v 1 i 1v i i

a b1 1SCk m max[a ,b ]= =

−= ∑ ∑

k n2

ivv 1 i 1

k 1PCN 1 11 k n = =

⎛ ⎞= − − µ⎜ ⎟− ⎝ ⎠

∑ ∑

Slide 11

Institute for Statics and Dynamics of Structures

Overview

1 Conceptual idea

2 Cluster methods

3 Fuzzy cluster design

4 Applications

Slide 12

Institute for Statics and Dynamics of Structures

Composition of fuzzy cluster design (1)

Space of design parameters Dn

x2

7.00

5.94

5.21

4.47

3.74

15.608.40 10.80 13.20 18.00

3.00

6.00 x1

6.68

permissible design parameters di

non-permissibledesign parameters di

{ }

{ }

{ }

1 1, 0 u n , 0

n1 2 n

d 1 i nd

d 1 i nd

x d d

D x , , D x

D D D D

M d , ,d , ,d

M d , ,d , ,d

M M ,M

α= α== =

= ×

=

=

=

…

…

… …

… …

D2

= x 2,

a=0

D1 = x1,a=0 D2

Slide 13

Institute for Statics and Dynamics of Structures

Composition of fuzzy cluster design (2)

Space of the restricted parameters Rm

{ }

{ }

{ }

1 1, 0 m m , 0

m1 2 m

r 1 i mr

r 1 i mr

z r r

R z , , R z

R R R R

M r , , r , , r

M r , , r , , r

M M ,M

α= α== =

= ×

=

=

=

…

…

… …

… …

1.00

µ($)

$2.498 4.666 6.6890.00

3.8

design constraint

non-restrictedparameters ri

× ××××

non-restrictedparameters ri

R1

Slide 14

Institute for Statics and Dynamics of Structures

Composition of fuzzy cluster design (3)

Algorithmic procedure

Step I : Initialization of Dn and Rm

Step II: Evaluation of the points in Rm by the design constraintsStep III: Determining the permissible points di and non-permissible points di

in Dn Md and Md

Step IV: Clustering the sets Md and Md; result: k1 permissible cluster

Step VII: Testing, whether all points of the R1, ... , Rm fulfil the constraints[v][v]

Step VI: Verification of the design variants by -level optimization R1, ... , Rmα[v][v]

Step V: Constructing of the modified sets D1, ... ,Dn from the permissible cluster;result alternative structural design variants

[v][v]

Slide 15

Institute for Statics and Dynamics of Structures

Composition of fuzzy cluster design (4)

Assessment of the alternative permissible design variants by criterions

Criterion I: constraint distance(measure for distance)

Criterion II: robustness(measure for sensitivity of the result variables )

[ ]( )[ ]( )v

m n n j

v vj 1 h 1 n x

H zB

H x= =

= ∑ ∑

design constraint

:(z)

z1[v]

design constraint distance

defuzzifiedvalue

Slide 16

Institute for Statics and Dynamics of Structures

Overview

1 Conceptual idea

2 Cluster methods

3 Fuzzy cluster design

4 Applications

Slide 17

Institute for Statics and Dynamics of Structures

design parameters:

nodal mass

distributed mass

global mass (full interaction)

distance

Example 1: Steel girder (1)

l1P(t)

M(1)~

µ~v(1,t)

n(1,t)

2 1 3

~

~

~

12 m

P(t) = P1@cos ( S1 @t) + P2@cos ( S2@t)P1 = P2 = 10 kN S1 = 44 s-1 S2 = 66 s-1

Young´s modulus E = 2.1@108 kN/m2

moment of inertia I = 1.5@10-3 m4

rotational masses are neglected

steel girder

dynamic load

µ~

Design for time dependent load

, t10M (1) = 2, 63

t,m

µ 1 5 = , 13 9

, tµ 12m⋅M = M (1) + = 6, 10 18

1l

Slide 18

Institute for Statics and Dynamics of Structures

Example 1: Steel girder (2)

µ(v)

max_v [10-3]1.97

1.00

0.50

0.003.01 max_v = 5

non-permissible points ripermissible points ri

R1 = max_v"=0

design constraint

× × ××× ×

Restricted parameter: displacement norm ϕ+

2 2

2 2

v(1,t) (1,t)v(t) =1 r

Slide 19

Institute for Statics and Dynamics of Structures

Example 1: Steel girder (3)

Cluster configuration in the space D2 based on k-medoid methodx1

7.00

5.94

5.21

4.47

3.74

15.608.40 10.80 13.20 18.00

3.00

6.00 x2

permissible clustersnon-permissible clusters

intersections

D1 = x1,"=0

6.68

permissible points dinon-permissible points di

D2

k1 = 12 permissible Cluster

k2 = 6 non-permissibleCluster

Slide 20

Institute for Statics and Dynamics of Structures

Example 1: Steel girder (4)

Clusters selected for alternative design variants

modified sets =alternative design variants

C7

C10

C5

C4

8.90 13.50 18.006.00

D1(10)

8.00 11.50

6.68

6.20

4.204.10

3.00

D1(7) D1

(4,5)

5.50

5.00

D1(5)

C4,5

7.00

cluster C5 : D1[5], D2

[5]

cluster C7 : D1[7], D2

[7]

cluster C10 : D1[10], D2

[10]

cluster C4,5 : D1[4,5], D2

[4,5]

Slide 21

Institute for Statics and Dynamics of Structures

Example 1: Steel girder (5)

Verification of structural design alternatives yields modified fuzzy results

1.0

0.01.97 5.00 15.62

µ(max_v[7])

max_v[7] [10-3]

permissiblepoints

design comstraint

non-permissiblepoints

1.0

0.02.56 3.22 4.51

µ(max_v[5])

max_v[5] [10-3]

1.0

0.02.63 2.81 4.76

µ(max_v[10])

max_v[10] [10-3]

1.0

0.02.41 2.62 4.56

µ(max_v[4,5])

max_v[4,5] [10-3]

Slide 22

Institute for Statics and Dynamics of Structures

Example 1: Steel girder (6)

Computed design variants

Design variant I (obtained from cluster 5)

Design variant II(obtained from cluster 10)

Design variant III(obtained from cluster 4,5)

M[5] = [10.8, 13.5] t

l1 [5] = [4.20, 4.80] m

M[10] = [8.00, 11.5] t

l1 [10] = [5.00, 5.50] m

M[4,5] = [8.90, 13.5] t

l1 [4,5] = [4.20, 6.20] m

Slide 23

Institute for Statics and Dynamics of Structures

Example 1: Steel girder (7)

Assessment of the design variants

0.00137 (min)0.004580.03933 (max)robustness measure B

2.97 (min)3.103.30 (max)level rank method

4.304.60 (max)4.00 (min)Chen method

3.11 (min)3.39 (max)3.37centroid method

max_vo[4,5]max_vo

[10]max_v0[5]

defuzzification results

Slide 24

Institute for Statics and Dynamics of Structures

Example 1: Steel girder (8)

Cluster configuration in the space D2 based on fuzzy cluster method

18.00

C2

C3

C4

x2

x1

permissible clusters intersectionsnon permissible clusters

D1 = x1,"=0

D 1

[2,3,4]

D2

C2,3,4

13.45 15.606.00 8.40 13.20

6.68

5.94

4.47

3.74

3.00

5.21

10.8010.00

3.25

6.687.00

Slide 25

Institute for Statics and Dynamics of StructuresRecent research results about non-classical methods in uncertainty modeling

http://www.uncertainty-in-engineering.net

and

Fuzzy Randomness

B. Möller, M. Beer, Springer 2004

Slide 26

Institute for Statics and Dynamics of Structures

Thank you !