Transcript Multidisciplinary Free Material Optimization for Laminated Plates and Shells Alemseged G. Weldeyesus, PhD student Mathias Stolpe, Senior Researcher.

Multidisciplinary Free Material Optimization
for Laminated Plates and Shells
Alemseged G. Weldeyesus, PhD student
Mathias Stolpe, Senior Researcher
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
6 November
2015
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
6 November
2015
The FMO problem formulation (solids)
Mechanical assumptions
• static loads,
• linear elasticity,
• anisotropic material.
The basic minimum compliance problem(Discrete)
min
u l ,E
subject to
2D, 3D structures
w f
l
T
l
ul
l
A( E )u l  f l ,
l  1, , L
  Tr ( Ei )   ,
i  1,  , m
Ei  I 0,
i  1,  , m
m
 Tr ( E )  V
Constraints on local stresses
 il
•
2
:

2
 l d  s ,
i 1
i
  E
i
• Highly nonlinear involving matrix variables.
• Adding such constraints destroys suitable problem properties, e.g. convexity.
4
DTU Wind Energy, Technical University of Denmark
6 November
2015
Goals
• Propose FMO model for laminated plates and shells. (Such models are
not available today)
• Develop special purpose optimization method that can efficiently solve
FMO problems. (FMO problems lead to large-scale nonlinear SDP
problems)
5
DTU Wind Energy, Technical University of Denmark
6 November
2015
FMO for laminated plates and shells
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
0
0
C 45
45
C 55
55
0
C16
16    11 
 
C 26
26   22 
0   23 
 
0   31 
 
C 66
66   12 
C11
C  C12
C16
C12
C 22
C 21
C16 
C
C 21 , D   44
C 54
C 66 
C 45 
C 55 
• (C,D ), fixed within a layer in the thickness direction.
• Slightly violates the idea of FMO.
6
DTU Wind Energy, Technical University of Denmark
6 November
2015
FMO for laminated plates and shells contd…
Minimum compliance problem
min
u l , l , C , D
w f
subject to
A(C , D)(u l , l )  f l ,
l
T
l
(u l , l )
l
1
2
Cik  I 0, Dik  I 0,
l  1, , L
  t k Tr (Cik )  t k Tr ( Dik )   , i  1,, m , k  1,, N
 (t Tr (C
k
i ,k
7
ik
i  1,, m , k  1, , N
1
)  t k Tr ( Dik ))  V
2
DTU Wind Energy, Technical University of Denmark
6 November
2015
Stress constraints
• Linear stress variation across the thickness within a layer.
• Two stress evaluations in each layer (at the top and lower surfaces) over
each finite element to capture stress extremities.
 ika ,l
2
:
2

 na,l d  s , i  1,, m , k  1,, N

 nb,l d  s , i  1,, m , k  1,, N
i
 ikb ,l
2
:
2
i
8
DTU Wind Energy, Technical University of Denmark
6 November
2015
Optimization method
• Primal-dual interior point method (second-order method).
• Combines standard known interior point methods for nonlinear
programmings & linear SDPs.
9
DTU Wind Energy, Technical University of Denmark
6 November
2015
Optimization method contd…
FMO problems can be represented by
min
XS,uRn
subject to
f (X,u )
g j (X,u )  0,
j  1,, k
X i 0,
i  1,, m
( P)
f : S R n  R, g : S R n  R k , sufficiently smooth.
Introducing slack variables and barrier paramter , 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 .
( BP)
We solve problem (BP) for a sequence of barrier parameter k0.
10
DTU Wind Energy, Technical University of Denmark
6 November
2015
Numerical results
4 layers
Optimal density distribution
Design domain, bc and forces
• # FEs 20,000.
• No distinction between layers.
• # 160,000 matrix inequalities.
• Similar to 2D results.
• # 720,000 design variables.
• No deformation out of the midsurface.
• No material for D.
11
DTU Wind Energy, Technical University of Denmark
• # 52 iterations.
6 November
2015
Numerical results …
8 layers
Optimal density distribution
Design domain, bc and forces
• 4 load-case.
• # FEs 10,000.
• Symmetric laminate.
• # 160,000 matrix inequalities.
• High material distribution on top and
• # 720,000 design variables.
bottom layers, implies sandwich
structures. ( similar results in DMO)
12
• # 25 iterations.
DTU Wind Energy, Technical University of Denmark
6 November
2015
Numerical results contd…
• Curved surfaces
• single layer
• # FEs 80,000.
Design domain, bc and
forces
Optimal density distribution
• # 160,000 matrix inequalities.
• # 720,000 design variables.
• # 54 iterations.
• # FEs 40,000.
• # 80,000 matrix inequalities.
• # 360,000 design variables.
13
DTU Wind Energy, Technical University of Denmark
• # 51 iterations.
6 November
2015
Numerical results contd…
Stress-constrained FMO problems
Design domain, bc and forces
Optimal density Optimal stress
distribution
norms
Without stress
constraints
• # FEs 7,500.
• # 7,500 matrix inequalities.
• # 45,000 design variables.
• # 7,500 stress constraints.
14
With stress
constraints
DTU Wind Energy, Technical University of Denmark
6 November
2015
Numerical results contd…
Without stress
constraints
With stress
constraints
Optimal principal
stresses
Without stress constraints
# iterations
compliance
15
30
1.8951
DTU Wind Energy, Technical University of Denmark
With stress constraints , 60%
85
1.9281
6 November
2015
Numerical results contd…
8 layers
Design domain, bc and forces
Optimal density distribution of the top 4 layers
Without stress
constraints
With stress
constraints
• # FEs 2,500.
• # 40,000 matrix inequalities.
• # 180,000 design variables.
• # 40,000 stress constraints.
16
Optimal density distribution – no such a significant
difference
DTU Wind Energy, Technical University of Denmark
6 November
2015
Numerical results contd…
Optimal stress norms at the upper and lower surfaces of the first top four layers
Without stress constraints
With stress constraints
Upper
Lower
Without stress constraints
# iterations
compliance
17
47
5.4771
DTU Wind Energy, Technical University of Denmark
With stress constraints, 30%
115
5.5624
6 November
2015
Conclusions
• The developed optimization method solves FMO problems in a reasonable
number of iterations.
• Number of iterations is almost independent to problem size.
• More number of iterations is required for solving the stress constrained
problems (but expected).
• FMO has been extended to optimal design of laminated structures.
• The change of material properties plays main role in reducing high
stresses in FMO.
• Reduction of high stresses is achieved at a small increase in compliance.
18
DTU Wind Energy, Technical University of Denmark
6 November
2015
Thank you for your attention !
20
DTU Wind Energy, Technical University of Denmark
6 November
2015