Transcript Multidisciplinary Free Material Optimization of 2D and Laminate Structures Alemseged G Weldeyesus, PhD student Mathias Stolpe, Senior Scientist Stefanie Gaile WCSMO10, Orlando,USA, May 19-24, 2013

Multidisciplinary Free Material Optimization
of 2D and Laminate Structures
Alemseged G Weldeyesus, PhD student
Mathias Stolpe, Senior Scientist
Stefanie Gaile
WCSMO10, Orlando,USA,
1
May
19-24, 2013
Free Material Optimization (FMO)
FMO is a one of the different approaches to structural optimization.
In FMO,
• the design variable is the full material tensor,
• can vary freely at each point of the design domain,
• necessary condition for physical attainability is the only requirement.
M. P. Bendsøe, J. M. Guedes, R. B. Haber, P. Pedersen, and J. E. Taylor. An
analytical model to predict optimal material properties in the context of optimal
structural design. J. Appl. Mech. Trans. ASME, 61:930–937, 1994.
M. Kočvara and M. Stingl. Free material optimization for stress constraints.
Structural and Multidisciplinary Optimization, 33:323-335, 2007.
M. Kočvara, M. Stingl, and J. Zowe. Free material optimization: Recent progress.
Optimization, 57(1):79–100, 2008.
2
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Optimal solution
• Solution to FMO yields optimal distribution of the material as well as optimal
local material properties.
• The obtained design can be considered as an ultimately best structure.
• Conceptual, since it is difficult (or actually impossible) to manufacture a
structure such that its property varies at each point of the design.
• FMO can be used to generate benchmarks and to propose novel ideas for new
design situations.
3
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
The FMO problem formulation
Mechanical assumptions
• static loads,
• linear elasticity,
• anisotropic material.
The basic minimum compliance problem(Discrete)
w
min
u l ,E
l
f lT ul
l
subject to
K ( E )u l  f l ,
l  1,  , L
  Tr ( Ei )   ,
i  1, , m
2D, 3D structures
Ei  I 0,
i  1, , m
m
 Tr ( E )  V
i 1
i
Additional structural requirements can also be included through constraints
G
• on local stresses
• on local strains
• on displacement
4

k 1
Ei Bik u
G

k 1
Bik u
2
2
 s
 s
Adding such constraints destroys
suitable problem properties, e.g.
convexity.
Cu  d
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Goals
• Propose FMO model for laminate structures.
• Extend existing robust and efficient primal dual interior point method for
nonlinear programming to FMO problems. ( second order method)
5
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
FMO for laminate structures
Based on FSDT
 11  C 11
  C
 22   12
 23    0
  
 31   0
 12  C 16
C12
C 22
0
0
C 26
0
0
C 44
C 54
45
0
C16
16    11 
 
C 26
26   22 
0   23 
 
0   31 
 
C 66
66   12 
0
0
C 45
45
C 55
55
0
C11
C  C12
C16
C12
C 22
C 21
C16 
C
C 21 , D   44
C 54
C 66 
C 45 
C 55 
Minimum compliance problem
min
u l , l , C , D
 w c(u , )
subject to
K (C , D)(u l , l )  f l ,
l  1,, L
1
2
Cik  I 0, Dik  I 0,
i  1,, m , k  1, , N
l
l
l
l
  t k Tr (Cik )  t k Tr ( Dik )   , i  1,, m , k  1,, N
 (t Tr (C
k
i ,k
6
1
)

t k Tr ( Dik ))  V
ik
2
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Optimization method
• All FMO problems are SemiDefinite Programs (SDP), an optimization
problem with many matrix inequalities.
• On the extension of primal-dual interior point methods for nonlinear
programming to FMO problems (SDP).
• FMO problems can be represented by
min
f (X,u )
n
XS,uR
subject to
g j (X,u )  0,
X i 0,
j  1,, k
( P)
i  1,, m
f : S R n  R, g : S R n  R k , sufficiently smooth.
Introducing slack variables, the associated barrier problem is
min
XS,uRn ,s,Rk
subject to
f (X,u )    ln(det(X i ))    ln(s j )
i
g j (X,u )  s j  0,
j
j  1,, k .
We solve problem (BP) for a sequence of barrier parameter k0.
7
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
( BP)
Numerical results
Without stress constraints
• 28800 design variables,
• ~9600 nonlinear constraints,
• 4800 matrix inequalities
Design domain,
bc, forces
Optimal density
distribution
Optimal strain norms
Optimal stress norms
• 43 iterations
• Higher stress concentration
around the re-entrant corner
8
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Numerical results …
Optimal density
distribution
With stress constraints (max stress is
decreased by 30%)
• 28800 design variables
• ~9600 nonlinear constraints
• 4800 matrix inequalities
• 4800 additional nonlinear constraints
(stress constraints)
Optimal strain norms
Optimal stress norms
• 72 iterations
• Higher stress concentrations
are distributed to negibhour
regions
M. Kočvara and M. Stingl. Free material optimization for stress constraints.
Structural and Multidisciplinary Optimization, 33:323-335, 2007.
9
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Numerical results …
Optimal density distribution
4 layers
Optimal density distribution, Layer=1
Optimal density distribution, Layer=2
Optimal density distribution, Layer=3
Optimal density distribution, Layer=4
Design domain, bc and forces
• No distinction between layers.
• Similar to 2D results.
• No out plane deformation.
• No material for D,
min(trace( C))/max(trace(D))=2.2413
10
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Numerical results …
Optimal density distribution
4 layers
• 129600 design variables
• 28800 matrix inequalities
Optimal density distribution, Layer=1
Optimal density distribution, Layer=2
Optimal density distribution, Layer=3
Optimal density distribution, Layer=4
Design domain, bc and forces
• 47 iterations
• Symmetric laminate
• High material distribution on top and bottom layers
11
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Numerical results …
In both cases, we have
• single layer,
• 90000 design variables,
• 20000 matrix inequalities. Design domain, bc and forces
Optimal density
distribution
44 iterations
Optimal density distribution, Layer=1
47 iterations
12
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Conclusion
• The extended primal dual interior point method solves FMO problems in a
reasonable number of iterations.
• Number of iterations is almost independent to problem size.
• FMO has been extended to optimal design of laminate structures.
13
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013
Thank you for your attention !
14
DTU Wind Energy, Technical University of Denmark
WCSMO10, Orlando,USA,
May 19-24, 2013