new optimization algorithm for topology optimization

The Center of Innovative Design Optimization Technology Applied Mechanics and Optimal Design Lab.1 Applied Mechanics and Optimal De-sign Lab.

The Center of Innovative Design Optimization Technology

A Filtered Sequential Approximate Optimization Algo-rithm

Based on Dual Subproblems Using an Enhanced Two-point

Diagonal Quadratic Approximation for Structure Optimization

Seonho Park1, Seung Hyun Jeong2,

Gil Ho Yoon3, Albert A. Groenwold4, and Dong-Hoon Choi3

1 Graduate School of Mechanical Engineering, Hanyang University, KoreaMIDAS IT Co. Ltd, Korea

2 Graduate School of Mechanical Engineering, Hanyang university, Korea3 Department of Mechanical Engineering, Hanyang university, Korea

4 Department of Mechanical Engineering, University of Stellenbosch, South Africa

Multidisciplinary Analysis & Optimization Conference 2012

The Center of Innovative Design Optimization Technology Applied Mechanics and Optimal Design Lab.2

Outline

1

2

4

Introduction

Diagonal Quadratic Approximation (DQA)

Numerical Examples

5Conclusions

3Filtered Diagonal Quadratic Approximation (FDQA)


Outline

1

2

4

Introduction


Numerical Examples

5Conclusions



Problem Statement

● Considered Problem

0

, ,

minimize ( )

subject to ( ) 0, 1, ,

where , , 1, ,j

i i L i U

f

f j m

x x x i n

xx

x

Nonlinear and continuously differentiable functions are considered. The number of design variables is much larger than that of constraints. (n>>m) Computational cost for the analysis is very high.

To solve this kind of problems, Sequential Approximate Optimization (SAO) with the dual method has been developed in the last two decades.


SAO with the dual method

yes

no

Compute 0 0,j jf fx x

0, , .j m

2

Construct a dual subproblem

3

Solve the dual sub-problem4

Compute

0, , .j m

Set k=k+18

END6

7

Optimization parameter setting

1

Move to the next point5

( 1) x kjf ( 1) ,k

jf x

( 1) ( )* x xk k

What is the dual subproblem?

Converged?


SAO with the dual method

Dual Subproblem

Primal SubproblemAt each k th iteration point

0

, ,

minimize ( )

subject to ( ) 0, 1, ,

where , 1, ,

k

kj

k ki i L i U

f

f j m

x x x i n

xx

x

( )0

1

maximize ( ), ( ) ( )

subject to 0, 1, , .

mk kk

j jj

j

L f f

j m

λx λ λ x λ x λ

( )kx

If n >> m the dual method is more efficient than the primal method.

The primal variable is explicit function of the dual variable.

Primal variable

Dual variable


Convex Separable Approximation

Diagonal Quadratic Approximation (DQA) ( )

( ) ( ) ( ) ( ) ( ) 2,

1 1

1( ) ( ) ( ) ( ) 0, , .2

kn njk k k k k

j i i i j i ii ii

ff f x x h x x j m

x

x x

• DQA is expressed by the first order Taylor’s expansion and the quadratic term.

What is the Convex Separable Approximation?

1. Separable : off-diagonal Hessian terms of the approximate functions are all zero

2. Convex : diagonal Hessian terms of the approximate functions are non-negative

To effectively construct the dual subproblem,convex separable approximation should be used for the approximating func-tions.

• Good approximation method of true diagonal Hessian terms is very important.


Previous Studies of the DQA

( )

( ) ( ) ( ) ( ) ( ) 2,

1 1

1( ) ( ) ( ) ( ) 0, , .2

kn njk k k k k

j i i i j i ii ii

ff f x x h x x j m

x

x x

Previous methods approximating the diagonal hessian terms

Exponential approximation MMA approximation CONLIN approximation TANA approximation

Groenwold, A. A., Etman, L. F. P., and Wood, D. W., "Approximated approximations for SAO," Structural and Multidisciplinary Optimization, Vol. 41, No. 1, 2010, pp. 39-56.Groenwold, A. A., Etman, L. F. P., Snyman, J. A., and Rooda, J. E., “Incomplete series expansion for func-tion approximation”, Structural and Multidisciplinary Optimization, Vol. 34, No. 1, 2007, pp. 21-40.

2 ( )

,( )k

ki j

i jxfh

x

x One point approximation

Two point approximation



Previous methods to enhance the convergence property of the SAO

5Fletcher, R., Leyffer, S., and Toint, P. L., "On the global convergence of a filter SQP algorithm," Siam Journal on Optimization, Vol. 13, No. 1, 2002, pp. 44-59.4Svanberg, K., "A class of globally convergent optimization methods based on conservative convex sepa-rable approximations," Siam Journal on Optimization, Vol. 12, No. 2, 2001, pp. 555-573.7Groenwold, A. A., and Etman, L. F. P., "On the conditional acceptance of iterates in SAO algorithms based on convex separable approximations," Structural and Multidisciplinary Optimization, Vol. 42, No. 2, 2010, pp. 165-178.

NLP filter for the SQP5

GCMMA4

NLP filter for the DQA7

Filter tests if is accept-able.

If not acceptable, the inner iteration is conducted.

In the inner iteration,

( )*kx

( ), ,

kki j i jh h

• Adjust move limit.• Enforce conservatism.

User defined parameter


Research Objectives1. Propose a DQA with highly accurate diagonal

hessian termsemploying a gradient based two-point approximation method.

2. Propose a Nonlinear Programming(NLP) filter appropriate to the proposed DQA to improve convergence property.Two-Point Approximation

f x 0f x

0fx

x 1f x

1fx

x

°( )f xy

n


0, , .j m

2

Construct dual subproblem

3

Solve dual subproblem4

Compute0, , .j m

5

Compute0, , .j m

Set k=k+113

ENDTestConvergence

11

12


1 Move to the next point

Initialize

10

0ρ ρ

( )*x kjf

( 1) x kjf

( 1) ( ) x xk kx

( 1) ,kjf x

( 1) ( )* x xk k

NLP filter

NLP filter


Outline

1

2

4

Introduction


Numerical Examples

5Conclusions



Approximate Diagonal Hessian Term using eTDQA

( )

( ) ( ) ( ) ( ) ( ) 2,

1 1

1( ) ( ) ( ) ( ) 0, , .2

kn njk k k k k

j i i i j i ii ii

ff f x x h x x j m

x

x x

*Kim, J. R., and Choi, D. H., “Enhanced two-point diagonal quadratic approximation methods for design optimization,” Computer methods in applied mechanics and engineering, Vol. 197,

2008, pp. 846-856

2 ( ) ( )( )eTDQA k

iji j

hx

fx

x Ⅱ

( ) 2( )

( ) ( ) ( ) ( ) 2 1

( 1) 2 ( ) 21 1

1 1

( )1 1( ) ( ) ( ) ( )2 2 ( ) ( )

nk

k e i i in neTDQA k k k i

i i i i i n nk ki ii

i i i i i ii i

H y yff f y y G y yy H y y H y y

x x Ⅱ

ipi i iy x c where

For the diagonal hessian terms, we use the second or-der derivative of eTDQA.

Determination of these parameters is provided in the previous research.


( )

( )

2( 1)( )2

( ) ( 1)

1

( 1) ( )

if

0 if

i

l l

ki

kii i

pke iij i i i in p pk k

l l i l il

p fxx c

Hh G p x c i jH x c x c

i j

x

Approximate Diagonal Hessian Term using eTDQA

Analytically derive hessian terms of eTDQA

Not convex!

Separable!

( ) 2( )

( ) ( ) ( ) ( ) 2 1

( 1) 2 ( ) 21 1

1 1

( )1 1( ) ( ) ( ) ( )2 2 ( ) ( )

nk



i i i i i ii i


x x Ⅱ

where, ipi i iy x c


Convexifying the DQA (1/3)

● Convexifying Operation of Part 1

( ), 1 2

2 ( ( ) ( ) 2( 1)( )2 ( ) 2

( ) ( 1)

1

( ) ( 1) ( ) i

l l

ki j

eTDQA k k pj ki e ii i i ink p pk ki ii i

l l i l il

h P P

f p Hf G p x cx xx c H x c x c

x x Ⅱ )

Part 1

( )

( )

if / 0 1

if / 0 1

ki i

ki i

f x p

f x p

1then 0P

*S. Park, D. Choi (2011) A new convex separable approximation based on two-point diagonal quadratic approximation for large-scale structural design optimization, WCSMO 9



ip ( )( ) / 0k

if x x ( )( ) / 0kif x x

( ) ( 1)k ki ix x ( ) ( 1)k k

i ix x ( ) ( 1)k ki ix x ( ) ( 1)k k

i ix x ( )

( 1)

( / ) 1( / )

ki

ki

f xf x

( )

( 1)

( / )0 1( / )

ki

ki

f xf x

( )

( 1)

( / ) 0( / )

ki

ki

f xf x

● Determination of Exponents Term pi

†

Current point Previous point

( 1) ( )

( 1) ( )1 ln lnk k

k ki i i i i

i i

f fp x c x cx x

1 1 1 1

-1

-13

3

Eqn.†

Eqn.† Eqn.†

Eqn.†

*S. Park, D. Choi (2011) A new convex separable approximation based on two-point diagonal quadratic approximation for large-scale structural design optimization, WCSMO 9



● Convexifying Operation of part 2

2 ( ( ) ( ) 2( 1)( )2 ( ) 2

( ) ( 1)

1

( ) ( 1) ( ) i

l l

eTDQA k k pj ki e ii i i ink p pk ki ii i

l l i l il

f p Hf G p x cx xx c H x c x c

x x Ⅱ )

Part 2

2( 1)( )

2 2( ) ( 1)

1

max , i

l l

pke ii i in p pk k

l l ll

HP G p xH x x

After convexifying procedure, all diagonal terms of hessian matrix become positive.


Outline

1

2

4

Introduction


Numerical Examples

5Conclusions



Numerical procedure of DQA

k: outer iteration numberl: inner iteration number

yes

no


0, , .j m

2

Construct a dual subproblem

3

Solve the dual subproblem4Compute

0, , .j m

Set k=k+18

END6

7


1Move to the next point

5

( 1) x kjf ( 1) ,k

jf x

( 1) ( )* x xk k

Converged?


Numerical procedure of FDQA

k: outer iteration numberl: inner iteration number

yes

no


0, , .j m

2

Construct dual subproblem

3

Solve dual subproblem4

Compute0, , .j m

5

Compute0, , .j m

Set k=k+113

END11

12


1 Move to the next point

Initialize

10

0ρ ρ

( )*x kjf

( 1) x kjf ( 1) ,k

jf x

( 1) ( )* x xk k

Converged?NLP filter is adopted

to improve convergence

property


Numerical procedure of Filtered SAO

is acceptableto the current filter?

Slanting Envelope

Test

Updating filter

( )*x k

1l l

Slanting envelope test using NLP Filter

0f f max 0, , 1, , .jh f j m and ,h f ( )*kx

For the brevity, At the kth iteration, the pair is obtained at . For the filter, slanting envelop is used to prove con-

vergence.f

h (1) (1),h f (2) (2),h f

(3) (3),h f

(4) (4),h f

,h f

( ) ( )andj jf h f h Reh ject

( ) ( ) orj jf h f h h Acceptable (5) (5),h f

( )

( (

)

) )

(

,j j

j j

Domin

f h f and h h

ate f h

Acceptable

(6) (6),h f

,h f

,h f (4) (4),h f

yesno

yesno

Inner iteration

Slanting envelope

*R. Fletcher, S. Leyffer, P. Toint (2002)On the global convergence of a filter-SQP algorithm

Reduce move limit Conservative approximate

1l l

Inner iteration Reduce move limit Conservative approximation

no

Satisfy sufficient re-duction criterion?


Inner iteration

is acceptableto the current filter

and 0f q q

Updating filter

( )*kx Reduce the move limit

yn

ny

1l l


Conservative approximation

Solve the dual subproblem

ix

The move limit is decreased by one half in this study.

,,k l

i Lx ,,k l

i Ux , 1,k l

i Lx , 1,k l

i Ux ,k lix

Move limit

( ) ( ), , 0, , .k k

i j j i jh h j m

The approximate functions can be easily conservative by increasing hessian terms.

max 1.1,j j ( ) ( )( ) ( )k k

j jf fx xj is determined to match

where

( , ) ( )* ( )*( ) ( )k l k kj jf fx x

The approximate function is conservative if


Sufficient Reduction Criterion

is acceptableto the current filter?

( )*kx

0.1

( ) ( )*0 0

k kq f f x x

( ) ( )*0 0

k kf f f x x

0 and q f q

yesno

yesno

1l l

Inner iteration Reduce move limit Conservative approximate

Updating filter

noyes

Updating filter

1l l


Next it-eration

Satisfy sufficient re-duction criterion?

Test if the reduction of real objective func-tion is smaller than we expected.

Let,

If

reduction of objective function is not sufficient go to inner itera-tion

Otherwise, update filter and go to the next iteration


Outline

1

2

4

Introduction


Numerical Examples

5Conclusions



Numerical Examples

No. Op-tion Problem n m

1a

Vanderplaats' cantilever beam problem720 21

b 200 201

2 　 Svanberg's 5-variate cantilever beam problem9 5 1

3a MBB beam topology optimization problem (p=1, 75

by 25)91875 1

b MBB beam topology optimization problem (p=3, 75 by 25)

1875 1

7Groenwold, A. A., and Etman, L. F. P., "On the conditional acceptance of iterates in SAO algorithms based on convex separable approximations," Structural and Multidisciplinary Optimization, Vol. 42, No. 2, 2010, pp. 165-178.9Groenwold, A. A., Etman, L. F. P., and Wood, D. W., "Approximated approximations for SAO," Structural and Multidisciplinary Optimization, Vol. 41, No. 1, 2010, pp. 39-56.

Performance of the proposed algorithm is compared with those of previous studies.7,9

Initial condition and convergence criteria are same as those of previous studies.7,9

Convergence criterion is . For the 3b problem, performance of the proposed algorithm is compared with the

MMA and GCMMA.

( 1) ( )k kx

x x


Numerical Examples

01

2 1

Minimize ( , )

subject to ( , ) 1 0 1,..., ,

( , ) 20 0 1,..., ,

( , ) 1 0,

1.0 100,5.0 100.

p

i i ii

ij

p j i i

pp

i

i

f b h l

f j p

f h b j p

yf

ybh

xb h

b h

b h

b h

Vanderplaats’ cantilever beam problem No. Method

1ap=10

SAO-A 64244.83 1.68E-06

SAO-B 64244.83 1.11E-06

SAO-C 64244.83 3.21E-06

SAO-D 64244.83 2.41E-05

DQA 64244.83 5.35E-06

FDQA 64244.83 5.35E-06

*k *l *0f max jf

*k*l: The number of outer iterations: The number of inner iterations

DQA and FDQA obtain appropriate optimum point.

DQA and FDQA show better effi-ciency compared to other meth-ods.

1bp=10

0

SAO-A 63678.1 1.78E-06SAO-B 63678.1 4.90E-06SAO-C 63678.1 2.98E-07SAO-D 63678.1 4.06E-05DQA 63678.1 8.66E-06

FDQA 63678.1 1.66E-06

38 -101 26140 9429 210 -10 034 -457 253530 2529 111 -10 1


Numerical Examples

Svanberg’s 5-variate cantilever beam

0 1 1 2 3 4 5

3 3 31 1 2 3

3 34 5 2

1 2

(0)

Minimize ( ) ( )

subject to ( ) 61/ 37 / 19 /

7 / 1/ 0,0.001 10 1, 2,3,4,5

where 0.0624, 1.0,

5.0, 5.0, 5.0, 5.0, 5.0

i

T

f c x x x x x

f x x x

x x cx i

c c

xx

x

x

MethodT2:R 1.33995

6 -

T2:E 1.339956 -

T2:MMA 1.339956 -

T2:TANA-3 1.339956 -

GCMMA 1.339956 -

DQA 1.339957 -1.26E-06

FDQA 1.339957 -1.26E-06

*k *l *0f max jf

DQA and FDQA obtain appropriate optimum point.

DQA and FDQA show good efficiency.

10 813 720 1510 419 208 -8 0


Numerical Examples

MBB beam topology optimization

01

1 01

Minimize ( )

subject to ( ) 0,

0.001 1

nT p T

e e e ee

n

ee

e

f x

f x fV

x

x u Ku u k u

x

Ku f

1, 0.3, 1E P

P

No. Method

3a

R 165.8839

-1.11E-13

T2:R 165.8839

-1.11E-13

E 165.8839

-5.60E-12

T2:E 165.8838

-5.46E-16

MMA 165.8839 5.84E-11

T2:MMA

165.8838

-4.53E-11

T2:R 165.8839

-1.11E-13

T2:E 165.8838

-5.46E-16

T2:MMA

165.8838

-6.06E-16

GCMMA

165.9624 1.90E-08

DQA 165.4939

-1.02E-11

FDQA 165.4936 3.18E-11

*k *l *0f max jf

All of methods obtain similar ob-jective function.

For 3a problem, the E method is better than the proposed FDQA.

FDQA is more efficient than DQA.

59 -58 -33 -35 -68 -51 -58 035 043 1137 10

356 -39 2


Numerical Examples

MBB beam topology optimization

Method

10-3

DQA 205.6923 6.87E-11MMA 205.784 -5.22E-05FDQA 203.9638 -1.78E-07

GCMMA 315.4972 -1.29E-06

*k *l *0f max jfx

For 3b problem, the proposed FDQA shows best performance.

Optimization is not converged ex-cept for the proposed FDQA when εx=10-4.

DQA MMA

Optimized layouts

FDQA GCMMA

The penalization parameter is set to 3.

10-4

DQA - -MMA - -FDQA 203.963

8-1.78E-

07GCMM

A - -

956 -301 -72 74347 144

4not con-verged -not con-verged -

72 74not con-verged -


5

Outline

1

2

4

Introduction


Numerical Examples

Conclusions



Conclusions

Propose an SAO algorithm with highly accurate hessian terms by using the eTDQA.

Propose a filtered SAO algorithm appropriate to the pro-posed DQA.

Investigate the efficiency and accuracy of the proposed algo-rithm by solving the numerical examples.

Improve convergence property without worsening the effi-ciency through the proposed algorithm.



Thank you!



Back Data



Previous methods related with the accuracy of approximate method for the SAO

Method Year Keyword AuthorLinear or reciprocal

approximation1986 CONLIN Fluery, C.1987 MMA Svanberg, K.

Exponential approximation 1990 TPEA Fadel, G. M., Riley, M. F., Barthelemy, J. M.

Diagonal quadratic

approximation

1995 Quasi-Newton update Duysinx, P. Z., Zhang, W. H., Fluery, C.

2002 Dynamic-Q Snyman, J. A., Hay, A. M.

2007 Incomplete series expansion

Groenwold, A. A., Etman, L. F. P., Snyman, J. A., Rooda, J. E.

20109 Approximated approximations9 Groenwold, A. A., Etman, L. F. P., Wood, D. W.

9Groenwold, A. A., Etman, L. F. P., and Wood, D. W., "Approximated approximations for SAO," Structural and Multidisciplinary Optimization, Vol. 41, No. 1, 2010, pp. 39-56.

Several SAO with the dual methods are compared.


Previous Studies of SAO with the Dual MethodPrevious methods related with the convergence property of the SAO

Method Year Author

Trust-region like framework1998 Alexandrow, N. M., Dennis, J. E., Lewis, R. M.,

Torczon, V.2000 Gonn, A. R., Gould, N. I. M., Toint, P. L.

Nonlinear acceptance filter for SQP

1998 Fletcher, R., Leyffer, S., Toint, P. L.

2002 Fletcher, R., Gould, N. I. M., Leyffer, S., Toint, P. L., Wächter, A.

Globally convergent version of MMA 2002 Svanberg, K.

Filter for the dual SAO 2009 Groenwold, A. A., Wood, D. W., Etman, L. F. P., Tosserams, S.

Filtered conservatism7 2010 Groenwold, A. A., Etman, L. F. P.

7Groenwold, A. A., and Etman, L. F. P., "On the conditional acceptance of iterates in SAO algorithms based on convex separable approximations," Structural and Multidisciplinary Optimization, Vol. 42, No. 2, 2010, pp. 165-178.

According to the filter option, SAO-A, SAO-B, SAO-C, and SAO-D are compared.


Enhanced Two-point Diagonal Quadratic Approximation (1/2)

Enhanced Two-point Diagonal Quadratic Approximation (eTDQA)*

( ) 2( )

( ) ( ) ( ) ( ) 2 1

( 1) 2 ( ) 21 1

1 1

( )1 1( ) ( ) ( ) ( )2 2 ( ) ( )

nk



i i i i i ii i


x x Ⅱ

1. Intervening variable with shifting constant ci

1

f

ix

to define an intervening variable whenthe design variable value is near zeroor negative

ipiii cxy

0otherwise,1,if

i

Liii

cxcx

*Kim, J. R., and Choi, D. H., “Enhanced two-point diagonal qua-dratic approximation methods for design optimization,” Com-

puter methods in applied mechanics and engineering, Vol. 197, 2008, pp. 846-856

i i i i ec p G H Each parameter is calculated sequentially

ipi i iy x c

where


Enhanced Two-point Diagonal Quadratic Approximation (2/2)

3. Gi is determined to match when pi calculation fails, otherwise, Gi is set to 0.

( ) ° ( )( 1) ( 1)k kf fy y- -Ñ =Ñ

2. pi is determined to match .

5. e is a correction factor to match .

ipi i iy x c

where( ) 2

( )

( ) ( ) ( ) ( ) 2 1

( 1) 2 ( ) 21 1

1 1

( )1 1( ) ( ) ( ) ( )2 2 ( ) ( )

nk



i i i i i ii i


x x Ⅱ

( ) ° ( )( 1) ( 1)k kf fy y- -=

( ) ° ( )( 1) ( 1)k kf fy y- -Ñ =Ñ

Calculation of parameters

4. .( )( ) ( )( )1if / / 01 otherwise

k ki i i

iG f x f xH

-ìï ¶ ¶ ¶ ¶ £ï=íïïîg


Two-Point Approximation

f x 0f x

0fx

x 1f x

1fx

x

°( )f x

What is the Two Point Approximation?

Current Design Point

Previous Design Point

Design Information

Conventional SAO with dual method are usually utilize only the current point information to construct approximate function.

Two-point approximation methods use function values and first order deriva-tive values of two design points.

( 1) ( 1) ( 1) ( 1)1 2, ,...,

Tk k k knx x x x

( 1)

( 1) ,k

jkj

i

ff

x

x

( ) ( ) ( ) ( )1 2, ,...,

Tk k k knx x x x

( )

( ) ,k

jkj

i

ff

x

x


Enhance conservatism of FDQA

K. Svanberg (2002), A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Ap-proximations

• If the condition described below is satisfied, the approximate function is said to be conservative

• If the approximate functions are not conservative, convergence property cannot be guaranteed.*

• If the hessian terms are increased, it is possible to obtain optimum point near to the current point.

• In the proposed method, the approximate functions can be easily conserva-tive by increasing hessian terms.

FDQA

( , ) ( )* ( )*( ) ( )k l k kj jf fx x

Example) Let . 0 1, ' 0 2f f

Then, 2 21 10 ' 0 0 0 1 22 2

f x f f x h x x hx

8h

2h