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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 !