an efficient second-order sqp method for structural

An e�cient second-order SQP method for structural topology

optimization

Susana Rojas Labanda, PhD student

Mathias Stolpe, Professor

WCSMO-11, June 2015, Sydney

What is Topology Optimization?

• Find optimal distribution of material ina prescribed design domain givenboundary conditions and external loads.

• Optimize the design of a structure byminimizing an objective function andsatisfying certain constraints.

• The design domain is discretized byfinite elements

Bendsøe, M. P and Sigmund, O. Topology optimization: Theory, methods and applications Springer 2003.


Minimum compliance problem

• NESTED formulation

minimizet2Rn

uT(t)K(t)u(t) (u(t) = K(t)�1f)

subject to vTt V

0 t 1.

• f 2 Rd the force vector.

• v 2 Rn the relative volume of the elements.

•V > 0 is the upper volume fraction.

• Use the SIMP material interpolation to penalize intermediate densities

t

i

= t

p

i

p > 1

• Use density filter to avoid checker-boards and mesh-dependency issues.

• Assume linear elasticityK(t) = Â E(t

e

)Ke


Optimization methods

Topologyoptimizationproblem

+non linearproblem

• OC: Optimality criteria method.

• MMA: The Method of Moving Asymptotes.

• GCMMA: Global convergence MMA.

• Interior point solvers

• Sequential quadratic programming methods (SQP)

Rozvany G.I.N and Zhou M. The COC algorithm, part I: Cross-section optimization or sizing. Computer Methods in AppliedMechanics and Engineering, 89(1-3): 281–308, 1991.

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.


Sequential Quadratic Programming for topology optimization

• 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

• IPQ = convex problem• EQP = linear systems

• SQP+ ensures progress with a line search combined with a reduction in a meritfunctions

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.


SQP+ Algorithm

Require: Define x0, l0 and the optimality tolerance w.

repeat1- Define an approximation of the Hessian of the Lagrange function, B

k

� 0such as B

k

⇡ r2L(x

k

, lk

).

2- Solve IQP sub problem.

3- Determine the working set of constraints.

4- Solve EQP sub problem (using active constraints).

5- Compute the contraction parameter b 2 (0, 1] such as the linearized

contraints of the sub problem are feasible at the iterate point x

k

+ d

iq

k

+ bd

eq

k

.

6- Acceptance/rejection step.

7- Update the primal and dual iterates.

until convergencereturn


Finding an approximate positive definite Hessian

• Hessian of the compliance

r2f (t) = 2FT(t)K�1(t)F(t)�Q(t)

• Use information of the exact Hessian

Hk

= 2FT(tk

)K�1(tk

)F(tk

) ⌫ 0

r2f (t) ⇠ H

k


potentially non-convex part

convex part

IQP sub problem

• IQP sub problem:

minimized2Rn

�r f (xk

)Td+ 12d

T(2FT

k

K�1k

Fk

)d

subject to Ak

d bk

.

• Define a new variable z = Fk

d.

minimized2Rn ,z2Rd

�r f (xk

)Td+ zTK�1k

z

subject to Ak

d bk

,z = F

k

d.

• Dual IQ sub problem

minimizea2R2n+1,b2Rd

14 bTK

k

b + aTbk

subject to �r f (xk

) +AT

k

a � FT

k

b = 0,a � 0.

• The strong duality property holds ) dual and primal solutions are equivalent.


DENSE AND EXPENSIVE!

SPARSE

EQP sub problem

• EQP sub problem

minimized

(r f (xk

) +Hk

diq

k

)Td+ 12d

T(2FT

k

K�1k

Fk

)d

subject to Ai

d = 0 i 2 W .

✓2FT

k

K�1k

Fk

AWAT

W 0

◆✓d

eq

k

leq

k

◆= �

r f (x

k

) +Hk

diq

k

0

!

• Define new variable v = 2K�1k

Fk

d.

FT

k

v = �(r f (xk

) +Hk

diq), and 1/2Kk

vk

� Fk

deq = 0.

• Expanded KKT system

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


DENSE ANDEXPENSIVE!

SPARSE

Benefits of SQP+

Characteristics

• Use of exact information of the Hessian

• No need of storage the dense Hessian

• Two phases to promote fast convergence


Implementation

• SQP+ written in MATLAB.

• IQP solver: Gurobi Optimizer Software (barrier algorithm).

• EQP solved using direct methods.


Test set of problems

• Michell, Cantilever and MBB domains,respectively.

• Di↵erent length ratios, volumeconstraints and discretizations.

• Size of the set: 400-40000 finiteelements.

• Distribution of instances respect to thenumber of elements.

[0:4] [4:8] [8:12] [12:16] [16:20] [20:24] [24:28] [28:32] [32:36] [36:40]0

10

20

30

40

50

60

70

80

number of elements ×103

numberofproblem

instances


Numerical Experiments

• How? Using performance profiles.

• Evaluate the cumulative ratio for a performance metric.

• Represent for each solver, the percentage of instances that achieve a criterion fordi↵erent ratio values.

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.


Robustness

Not as good solver

Good Solver

Solution close to the best one

Best Solution

Numerical Experiments II

• Criteria:

• Objective function value• Number of iterations• Number of sti↵ness matrix assemblies ⌘ Number of function evaluations• Computational time

• Solvers:

• SQP +• IPOPT-N (interior point software that solves thenested formulation (BFGS approximation of theHessian))

• IPOPT-S (interior point software that solves theSAND formulation (Exact Hessian))

• SNOPT (SQP software that solves the nestedformulation (BFGS approximation of the Hessian))

• GCMMA (first-order solver, nested formulation)0 0.5 1 1.5 2

0

10

20

30

40

50

60

70

80

90

100

τ (log10)

%problems

SQP+IPOPT NIPOPT SSNOPTGCMMA


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


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


Performance profiles in a reduce test set of 194 instances.

Penalization of problems with KKT error higher than w = 1e � 4.


First-order solver

BFGS approximation

Exact Hessian


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


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






Number of sti↵ness matrix assemblies

0 0.5 1 1.5 20

10

20

30

40

50

60

70

80

90

100

τ (log10)

%problems


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





Conclusions

SQP+ is a good choice for solving topology optimization problems:

• The use of the exact Hessian (IPOPT-S and SQP+) produces betterdesigns thant using BFGS approximations (IPOPT-N and SNOPT).

• SQP+ produces better designs than any solver in the nested formulation.

• SQP+ requires very few iterations (and function evaluations).

• Need to improve the computational time spent in SQP+.


THANK YOU !!!

This research is funded by the Villum Foundation through the research project Topology

Optimization – the Next Generation (NextTop).WCSMO-11, June 2015, Sydney

an efficient second-order sqp method for structural

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