The use of second-order information in structural topology...
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The use of second-order information in structural topology optimization Susana Rojas Labanda, PhD student Mathias Stolpe, Senior researcher
Transcript of The use of second-order information in structural topology...
The use of second-order information in structural topology
optimization
Susana Rojas Labanda, PhD student
Mathias Stolpe, Senior researcher
What is Topology Optimization?
• Optimize the design of a structuregiven certain constraints, loads andsupports.
• The design domain is discretized.The variables denotes the presenceof material at each element.
• The goal is to decide which elementsshould contain material and whichones not. It is a 0-1 discrete problem.
• Model as an optimization problem
minimizex
f (x)
subject to g(x) 0
h(x) = 0.
Bendsøe, M. P and Sigmund, O. Topology optimization: Theory, methods and applications Springer 2003.
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Minimum compliance problem
• SAND formulation
minimizet,u
fTu
subject to aTt V
K(t)u � f = 00 t 1.
• NESTED formulation
minimizet
uT(t)K(t)u(t)
subject to aTt V
0 t 1.
• f 2 Rd the force vector.
• a 2 Rn the volume vector.
•V > 0 is the upper volumefraction.
• Use the SIMP material interpolation topenalize intermediate densities
t
i
= t
p
i
p > 1
• Use Density filter to avoid checkerboardsand mesh-dependency issues.
˜
t
e
=1
Âi2N
e
h
ei
Âi2N
e
h
ei
t
i
h
ei
= max{0, r
min
� dist(e, i)}
• Linear Elasticity
E(ti
) = E
v
+ (E
1
+ E
v
)˜
t
p
i
K(t) = Â E(te
)Ke
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Minimum compliance problem
• SAND formulation
minimizet,u
fTu
subject to aTt V
K(t)u � f = 00 t 1.
• NESTED formulation
minimizet
uT(t)K(t)u(t)
subject to aTt V
0 t 1.
• f 2 Rd the force vector.
• a 2 Rn the volume vector.
•V > 0 is the upper volumefraction.
• Use the SIMP material interpolation topenalize intermediate densities
t
i
= t
p
i
p > 1
• Use Density filter to avoid checkerboardsand mesh-dependency issues.
˜
t
e
=1
Âi2N
e
h
ei
Âi2N
e
h
ei
t
i
h
ei
= max{0, r
min
� dist(e, i)}
• Linear Elasticity
E(ti
) = E
v
+ (E
1
+ E
v
)˜
t
p
i
K(t) = Â E(te
)Ke
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Optimization methods
Topologyoptimization
problem
+non-linearproblem
• OC: Optimality criteria method.
• MMA: Sequential convex approximations.
• GCMMA: Global convergence MMA.
• Interior point solvers: IPOPT, FMINCON,...
• Sequential quadratic programming:SNOPT,...
Andreassen, E and Clausen, A and Schevenels, M and Lazarov, B. S and Sigmund, O. Efficient topology optimization in MATLABusing 88 lines of code. Structural and Multidisciplinary Optimization, 43(1): 1–16, 2011.
Svanberg, K. The method of moving asymptotes a new method for structural optimization. International Journal for NumericalMethods in Engineering, 24(2): 359–373. 1987.
Svanberg, K. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAMJournal on Optimization, 12(2): 555-573, 2002.
Gill, P. E and Murray, W and Saunders, M. A. SNOPT: An SQP Algorithm for Large -Scale Constrained Optimization. SIAMJournal on Optimization, 47(4):99–131, 2005.
Wachter, A and Biegler, L. T. On the implementation of an interior point filter line-search algorithm for large-scale nonlinearprogramming. Mathematical Programming, 106(1):25–57, 2006.
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Optimization methods
Topologyoptimization
problem+
non-linearproblem
• OC: Optimality criteria method.
• MMA: Sequential convex approximations.
• GCMMA: Global convergence MMA.
• Interior point solvers: IPOPT, FMINCON,...
• Sequential quadratic programming:SNOPT,...
Andreassen, E and Clausen, A and Schevenels, M and Lazarov, B. S and Sigmund, O. Efficient topology optimization in MATLABusing 88 lines of code. Structural and Multidisciplinary Optimization, 43(1): 1–16, 2011.
Svanberg, K. The method of moving asymptotes a new method for structural optimization. International Journal for NumericalMethods in Engineering, 24(2): 359–373. 1987.
Svanberg, K. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAMJournal on Optimization, 12(2): 555-573, 2002.
Gill, P. E and Murray, W and Saunders, M. A. SNOPT: An SQP Algorithm for Large -Scale Constrained Optimization. SIAMJournal on Optimization, 47(4):99–131, 2005.
Wachter, A and Biegler, L. T. On the implementation of an interior point filter line-search algorithm for large-scale nonlinearprogramming. Mathematical Programming, 106(1):25–57, 2006.
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Sequential Quadratic Programming for topologyoptimization
• SQP for minimum compliance problems in the nested formulation
• Solve a sequence of approximate sub-problems• Convex quadratic approximation of the Lagrangian function.• Linearization of the constraints.
• Implementation of SQP+ = IQP + EQP
Require: Define the starting point x
0
, the initial Lagrangian multipliers l0
and the optimality tolerance w.repeat
Define an approximation of the Hessian of the Lagrange function, B
k
� 0 such as B
k
⇡ r2
L(x
k
, lk
).Solve IQP sub-problem.Determine the working set of the inequality constraints and the boundary conditions.Solve EQP sub-problem (using active constraints).Compute the contraction parameter b 2 (0, 1] such as the linearized contraints of the sub-problem are feasible at the iteratepoint x
k
+ d
iq
k
+ bd
eq
k
.Acceptance/rejection step. Use of line search strategy in conjunction with a merit function.Update the primal and dual iterates.
until convergencereturn
Morales, J.L and Nocedal, J and Wu, Y. A sequential quadratic programming algorithm with an additional equality constrainedphase. Journal of Numerical Analysis,32:553–579, 2010.5 DTU Wind Energy 11.3.2015
Finding an approximate positive definite Hessian
• Sensitivity analysis for the minimum compliance problem
r2
f (t) = 2FT(t)K�1(t)F(t)� Q(t)
ˆHk
= 2FT(tk
)K�1(tk
)F(tk
) ⌫ 0
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convex part
potentially non-convex part
Reformulations of IQP and EQP sub-problems
• Approximate Hessian computationally expensive
• Use dual formulation for the IQP sub-problem
minimized
r f (xk
)Td + 1
2
dT(2FT
k
K�1
k
Fk
)d
subject to Ak
d bk
minimizea,b
1
4
bTKk
b + aTbk
subject to AT
k
a � FT
k
b = �r f (xk
)a � 0.
• Expansion of the EQP system
minimized
(r f (xk
) + Hk
diq
k
)Td + 1
2
dT(2FT
k
K�1
k
Fk
)d
subject to Ai
d = 0 i 2 W .
0
@0 FT
k
AWF
k
�1/2Kk
0AT
W 0 0
1
A
0
@v
k
deq
k
leq
k
1
A = �
0
@r f (x
k
) + Hk
diq
00
1
A
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DENSE!
Reformulations of IQP and EQP sub-problems
• Approximate Hessian computationally expensive
• Use dual formulation for the IQP sub-problem
minimized
r f (xk
)Td + 1
2
dT(2FT
k
K�1
k
Fk
)d
subject to Ak
d bk
minimizea,b
1
4
bTKk
b + aTbk
subject to AT
k
a � FT
k
b = �r f (xk
)a � 0.
• Expansion of the EQP system
minimized
(r f (xk
) + Hk
diq
k
)Td + 1
2
dT(2FT
k
K�1
k
Fk
)d
subject to Ai
d = 0 i 2 W .
0
@0 FT
k
AWF
k
�1/2Kk
0AT
W 0 0
1
A
0
@v
k
deq
k
leq
k
1
A = �
0
@r f (x
k
) + Hk
diq
00
1
A
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SPARSE!
SPARSE!
Benchmarking in topology optimization
• How? Using performance profiles.
• Evaluate the cumulative ratio for a performance metric.• Represent for each solver, the percentage of instances that achieve a
criterion for different ratio values.
rs
(t) = 1
n
size{p 2 P : r
p,s
t},
r
p,s
=iter
p,s
min{iterp,s
: s 2 S} .
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.20
10
20
30
40
50
60
70
80
90
100
τ (iterp,s = τ min{iterp})
%problems
Performance profile
Solver1Solver2
Dolan, E. D and More, J. J. Benchmarking optimization software with performance profiles. MathematicalProgramming,91:201–213, 2002.
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Good Solver
Best Solution
Solution close to the best one
Robustness
Not as good solver
Benchmark set of topology optimization problems
• Total Problems: 225.
• From 400 to 40, 000 number of elements. (up to 81, 002 dof).
• 3 different domains
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Performance profiles
Objective function value
1 1.1 1.2 1.3 1.4 1.5 1.6 1.70
10
20
30
40
50
60
70
80
90
100
τ
%problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Number of iterations
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
10
20
30
40
50
60
70
80
90
100
τ (log10)%
problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Performance profiles in a reduce test set of 194 instances.Penalization of problems with KKT error higher than w = 1e � 4.
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Exact Hessian
First-order solver
BFGS approximation
Performance profiles
Objective function value
1 1.1 1.2 1.3 1.4 1.5 1.6 1.70
10
20
30
40
50
60
70
80
90
100
τ
%problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Number of iterations
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
10
20
30
40
50
60
70
80
90
100
τ (log10)%
problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Performance profiles in a reduce test set of 194 instances.Penalization of problems with KKT error higher than w = 1e � 4.
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Performance profiles
Number of stiffness matrix assemblies
0 0.5 1 1.5 20
10
20
30
40
50
60
70
80
90
100
τ (log10)
%problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Computational time
0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
80
90
100
τ (log10)%
problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Performance profiles in a reduce test set of 194 instances.Penalization of problems with KKT error higher than w = 1e � 4.
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Performance profiles
Number of stiffness matrix assemblies
0 0.5 1 1.5 20
10
20
30
40
50
60
70
80
90
100
τ (log10)
%problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Computational time
0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
80
90
100
τ (log10)%
problems
SQP+IPOPT NIPOPT SSNOPTGCMMA
Performance profiles in a reduce test set of 194 instances.Penalization of problems with KKT error higher than w = 1e � 4.
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tau = 15
What can we conclude from the performance profiles?
• GCMMA tends to obtain a design with large KKT error
• IPOPT-S produces the best designs followed by SQP+
• IPOPT-S and SQP+ (exact Hessian) produce better designthan IPOPT-N and SNOPT (BFGS approximations)
• SQP+ converge in the least number of iterations and stiffnessassemblies (= function evaluations)
• SAND formulation requires a lot of computational time andmemory
• Need to improve the computational time spent in SQP+
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THANK YOU !!!
This research is funded by the Villum Foundation through the research project TopologyOptimization – the Next Generation (NextTop).15 DTU Wind Energy 11.3.2015
