Application of Optimization Shells to Design Problems

Application of Optimization Shells to Design Problems

H a n s G u d e n s c h w a g e r , SSW Fahr- u n d Spezialschiffbau G m b H ^

1. I n t r o d u c t i o n

Most design problems may be f o r m u l a t e d as follows: Determine a set o f design variables (e.g. m a i n dimensions and interior subdivision of ship, scantlings of a construction, or characteristic values of pipes and pumps i n a pipe net) subject to certain relations between and restrictions of these variables (e.g. by physical, technical, legal, economical laws). I f more t h a n one combina-t i o n of design variables sacombina-tisfies a l l combina-these condicombina-tions, combina-the problem may be f o r m u l a combina-t e d uniquely by d e f i n i n g a measure o f m e r i t (e.g. weight, cost, yield) and asking for t h a t c o m b i n a t i o n o f design variables w h i c h optimizes the measure of m e r i t .

Design problems can be described very generally as: f r o m given quantities ( = design specifi-cations) we have to determine (design) details using k n o w n relations. T h i s general f o r m u l a t i o n could describe almost a l l computer programs. Design problems differ f r o m most other problems i n t h a t f r o m case to case other quantities are specified or u n k n o w n , and t h a t the applicable relations may change. I n designing scantlings for example, web height and flange w i d t h may be variables to be determined or they may be given due to the necessity to continue other s t r u c t u r a l regions. There may be upper bounds due to spatial l i m i t a t i o n s , or lower bounds as o u t f i t t i n g or other s t r u c t u r a l members need to transverse the scantling below the flange. C u t -outs, v a r y i n g thickness i n l o n g i t u d i n a l direction, and other s t r u c t u r a l details create a m u l t i t u d e of alternatives w h i c h have to be handled. N a t u r a l l y , most design problems for whole ships are far more complex t h a n the sketched 'simple' design problem for scantlings.

2. D e s c r i p t i o n of t h e o p t i m i z a t i o n s h e l l

So w h i l e the s o l u t i o n o f design ( o p t i m i z a t i o n ) problems is desirable, i t is also o f t e n too tedious and complex f o r the designer. Here methods of 'machine intelligence' may help by creating a u t o m a t i c a l l y f o r each i n d i v i d u a l design problem an a l g o r i t h m . T h e designer's task is then basically reduced t o supplying:

- a list of specified quantities (design specifications)

- a list of unknowns i n c l u d i n g upper and lower bounds ^.nd desired accuracy - the applicable relations (equations and inequalities)

Setting up this 'knowledge base' is far easier and flexible i n h a n d l i n g various problems t h a n conventional procedural p r o g r a m m i n g .

2.1. C H W A R I S M I a n d D E L P H I

T h e above philosophy was implemented more t h a n 20 years ago i n C H W A R I S M I , an o p t i -mization shell denoted ' o p t i m i z a t i o n compiler' by i t s creator, Söding (1977,1983). I t accepts descriptions of design problems as inputs and converts t h e m i n t o F o r t r a n programs solving these problems. C H W A R I S M I i n its earliest forms dates back to the mid-70's and has been modified and extended several times. T h e o p t i m i z a t i o n shell has been applied t o many design problems i n the f i e l d of p r e l i m i n a r y ship design, s t r u c t u r a l o p t i m i z a t i o n , h u l l f o r m design, and even electronic circuit design. T h e concept is based on F o r t r a n w i t h some special extensions to describe the o p t i m i z a t i o n problem. These extensions introduce elements of declarative pro-g r a m m i n pro-g and are easy to learn and t o handle. T h e 'compiler' then transforms the problem thus described i n t o a pure F o r t r a n source code. D E L P H I , Gudenschwager (1988), is a direct

'Lammersieth 90, D-22305 Hamburg, Germany ^POB 101240, D-27512 Bremerhaven, Germany

successor o f C H W A R I S M I . Besides i n t r o d u c i n g a better user interface and some other improve-ments, D E L P H I uses a better core optimizer which is i n essence described below. F u n c t i o n a l i t y and f l e x i b i l i t y - the user's view - of D E L P H I are absolutely comparable t o C H W A R I S M I .

T h e relations i n our o p t i m i z a t i o n shell are either equations or inequalities. For o p t i m i z a t i o n problems (more unknowns t h a n equations), i n a d d i t i o n an o p t i m i z a t i o n c o n d i t i o n is specified. T h e shell handles only real variables. W h i l e especially integer variables would be desirable f o r some design problems, they were not yet incorporated as this would make the a l g o r i t h m considerably more d i f f i c u l t and also increase c o m p u t a t i o n a l t i m e . However, extension of the shell t o include also m i x e d problems w i t h integer unknowns appears feasible i n the f u t u r e .

T h e shell works i n two steps. T h e designer compiles i n the first step a l l relevant 'knowledge' i n the f o r m of relations. T h e shell checks i f the problem can be solved w i t h the given relations. A f t e r this process, the m o d i f i e d problem is converted into a F o r t r a n program, compiled, and linked. The second step is then the actual numerical c o m p u t a t i o n using this program.

2.2. C o r e o p t i m i z e r

T h e actual o p t i m i z a t i o n is performed by the subroutine O P T . STR (1994) gives the F o r t r a n source code. So here we l i m i t ourselves t o a short description of O P T .

O P T determines / real variables rc,-, i = such t h a t the object f u n c t i o n (measure of m e r i t ) f ( x i , x j ) is m i n i m u m w i t h i n the admissible space. The latter is defined by inequality and equality constraints:

Xmin.i <Xi< Xmax,i f o r 1 = (1) gj{xu...,xj) > 0 foi- j = 1...J (2) hk{xi,...,xr) = Oiov k = l...K (3) I f constraints (1) are lacking, one should use bounds of sufficient distance f r o m the expected

o p t i m i m u m . Inequalities of the f o r m 'less or equal' are transformed to the f o r m (2) by m u l t i -phcation w i t h - 1 . I f the object should be maximized, minimize - ƒ instead. T h e two types (1) and (2) of inequalities are introduced to s i m p l i f y the application of O P T and t o improve its efficiency.

T h e subroutine determines the o p t i m u m b y generating a sequence of linear constrained optimizations, s t a r t i n g at a given i n i t i a l guess for X j . T h e admissible space for the linear optimizations is bounded by a box o f w i d t h ±6i around the current estimate, where Si,i = are given i n i t i a l l y and decreased i n steps d u r i n g the search process. W i t h i n this box, the admissible range and the object f u n c t i o n is defined by linearizations of (2) and (3). T h e linear optimizations are p e r f o r m e d using the D A N T Z I G routine, STR (1993).

T h e subroutine D A N T Z I G solves the 'linear p r o g r a m m i n g ' ( L P ) problem N

M i n i m i z e S UnXn subject to the constraints n = l

N

Y l ^inXn = ho foT 1 = and dn <Xn< e„ for n = 1...N n-1

an, bin, dn, and e „ are given and a;„ are to be determined. Compared to the standard L P f o r m a t where variables a^n have o n l y lower bounds o f 0, here a l l variables have a r b i t r a r y lower and upper bounds, and b o t h equations and inequalities can be handled directly. I n case t h a t the actual problem has variables w h i c h have no lower and upper bounds, when a p p l y i n g this r o u t i n e one should use bounds outside the expected solution domain. However, bounds outside of the expected s o l u t i o n d o m a i n by orders o f magnitude should be avoided because this would increase r o u n d i n g errors. I f a f u n c t i o n S should be maximized, m u l t i p l y by - 1 to o b t a i n a m i n i m i z a t i o n problem. Correspondingly < symbols i n constraints are transformed to > symbols.

Many routines for solving L P problems are available. Reasons for constructing t h i s routine were:

- T w o L P routines taken f r o m widespread mathematical subroutine packages t u r n e d out t o contain errors w h i c h could not be spotted due to confused and unnecessarily complicated programming.

- I n case the problem allows no solution, many L P routines stop w i t h o u t g i v i n g reasonable Xn values, whereas D A N T Z I G gives the p o i n t a;„ w h i c h results i n m i n i m u m combined violation o f the constraints; this feature is necessary i n many applications.

- I n common routines, frequently occurring cases involving variables w i t h b o t h lower and upper bounds require the i n t r o d u c t i o n of additional constraints, thus increasing program-ming, storage requirements, and c o m p u t i n g t i m e .

T h e D A N T Z I G routine applies the simplex a l g o r i t h m to variables Xn = {Xn — dn)l{en — dn) or f „ = I — Xn- T h e coefficient m a t r i x is first set up for Xn- I f ï n = 0 contradicts t o at least one constraint, an auxihary object f u n c t i o n g i v i n g the amount of accumulated constraint violations is minimized. A d d i t i o n a l l y t o the simplex a l g o r i t h m , the c o n d i t i o n S < 1 is tested; i n case o f violation a change f r o m variable S„ to S „ is performed. I n case an admissible vector x is f o u n d , the a l g o r i t h m continues w i t h the actual instead of the auxiliary object f u n c t i o n . Source codes of D A N T Z I G and O P T w i l l be emailed o n request.

3. A p p l i c a t i o n s

3,1. B l o c k of m i n i m u m s u r f a c e

The example is seemingly simple, because the intelligent human b r a i n 'sees' i m m e d i a t e l y t h a t a cube must be the solution. However, i n f o r m a l o p t i m i z a t i o n the p r o b l e m is n o t really t r i v i a l due to the involved nonlinearities. T h e problem is f o r m u l a t e d as: C o m p u t e the m a x i m u m block volume f o r a given surface. Side lengths A , B , and C are positive. T h e p r o b l e m is f o r m u l a t e d i n an 'extended' or quasi F o r t r a n language as:

C Program CUBE computes t h e side l e n g t h s A, B, C o f a b l o c k C w i t h maximum volume f o r g i v e n s u r f a c e

PROGRAM CUBE

REAL LSTART, LDELTA, LHAX

WRITE(*,*) ' I n p u t e s t i m a t e d , s t e p , maximum v a l u e o f s i d e l e n g t h ' READ(*.*) LSTART. LDELTA, LHAX

WRITE(»,*) 'Input s u r f a c e o f b l o c k ' READ(*,*) SURFAC

C Begin o f o p t i m i z a t i o n b l o c k

UNKNOWNS A(LSTART, LDELTA. 0., LHAX), & B(LSTART, LDELTA. 0., LHAX). & C(LSTART. LDELTA. 0.. LHAX)

SURFAC+0.5 = A*B + B*C + A*C MAXIMIZE VOL = A+B+C

SOLVE

C End o f o p t i m i z a t i o n b l o c k

WRITE(+,*) ' Surface = .'. SURFAC WRITE(*.*) ' Max volume = '. VOL WRITE(*.*) • Side l e n g t h A •= '. A WRITE(*.*) • Side l e n g t h B = '. B WRITE(+.+) ' Side l e n g t h C = '. C END

T h e p a r t marked as ' o p t i m i z a t i o n block' is the actual f o r m u l a t i o n of the o p t i m i z a t i o n prob-lem. The statements before a n d after are i n p u t and o u t p u t . These parts are usually realized

by file rather t h a n interactive i n p u t and o u t p u t . They may be more or less 'cosmetically' enhanced.

T h e example shows how simple and clear the p r o g r a m m i n g of the o p t i m i z a t i o n p r o b l e m becomes. Equations may be a r b i t r a r i l y stated w i t h terms allowed also o n the r.h.s. of the equation. Thus formulas can be directly encoded f r o m a text w i t h o u t conversion t o a f o r m w i t h only one variable o n the r.h.s. R u n n i n g the p r o b l e m w i t h the i n p u t : L S T A R T = 5., L D E L T A = 1., L M A X = 100., S U R F A C = 600. yields w i t h i n fractions o f a second the result: A = 10.000, B = 9.997, C = 10.003, V O L = 1000.

The i n i t i a l step size is thus not the accuracy of the solution. Most problems i n our experience are insensitive to where y o u start. T h e search domain should not be made a r t i f i c i a l l y small by imposing narrow upper and lower bounds.

3.2. S o l v i n g a n o n l i n e a r d e s i g n p r o b l e m

Instead of o p t i m i z i n g , the same shell can be used to solve a determined p r o b l e m where the number of unknowns matches the number of equations. These problems are frequent i n design where many nonlinear formulas were solved t r a d i t i o n a l l y i n a trial-and-error loop. T h i s task can be performed r o u t i n e l y more conveniently by the shell. A n example may demonstrate the similarity i n the process. T h e task is t o determine the displacement, length and installed power for a S W A T H ship (steel, diesel engine, circular cross section o f lower h u l l ) . Based on previous designs, we have some simple design formulas, Bertram and MacGregor (1992a,b,1993), which can be used to determine a first base design. T h e power estimate requires the displacement, the displacement depends among other things on the machinery weight w h i c h i n t u r n depends on the power. Simply using existing routines and formulas allows t o let the shell solve this problem.

Again, the example is j u s t chosen t o demonstrate the approach. P r a c t i c a l design programs involve more variables and far more statements f o r subroutines and f u n c t i o n s to supply es-timates for various variables. Especially the power prediction is crude, b u t could be easily extended similarly t o the weight subroutine, e.g. using i n d i v i d u a l estimations f o r resistance components or even a wave resistance c o m p u t a t i o n by a classical method. As long as the ' b u i l d i n g blocks' t o compute these quantities are available, the f o r m u l a t i o n of the o p t i m i z a -t i o n problem is s-traigh-t-forward and quickly performed. We refer -to Schneeklu-th and Ber-tram (1998) for a discussion of development strategies f o r more sophisticated design o p t i m i z a t i o n models.

REAL DISP, LIGHTW, LDA, P, RANGE, WP, VKNOT

WRITE(•,*) ' I n p u t : p a y l o a d [ t ] , speed [ k n ] , range [nm] ' READ(*.*) WP, VKNOT. RANGE

C b e g i n o f o p t i m i z a t i o n b l o c k UNKNOWNS LOA (10.. 1 . . 0.. 1000.) UNKNOWNS DISP(WP. 10.. WP, SOOOO.) UNKNOWNS P (1000.. 10.. 0.. 100000.) LOA = 5.33 * DISP**0.3333333 C Power P - 0.077 * VKN0T**2.784 * (DISP/LOA)**0.928 C Displacement DISP = LIGHTW(DISP.P.RANGE.VKNOT) + WP SOLVE C end o f o p t i m i z a t i o n b l o c k

WRITE(•,*) 'Displacement [ t ] »'. DISP WRITE(*.*) 'Payload [ t ] =', HP WRITE(*.*) 'Length o v e r a l l [m] •='. LOA WRITE(•.•) 'Power requirement [kW] ='. P END

REAL DISP, P, PAUX, RANGE, S, WAUX, WF, WH, WO, WS, VKNOT ws = (0.425 - 0.00000175 • DISP) • DISP WH = 0.0094 * P WAUX = 0.147 * DISP**0.998 WO = 0.189 * DISP**0.881 I F (P.LT.11190) THEN S = 178 ELSE S = (230-0.004647*P) END I F PAUX = 0.2505+DISP**0.924 WF = (RANGE/VKNaT)*(P*S+PAUX*210)*1.15E-6 WSM = 0.0755 * (WS+WH+WAUX+W0)*+1.075 LIGHTW = WS + WH + WAUX + WD + WF + WSH END 3.3. P r o p e l l e r o p t i m i z a t i o n

T h e o p t i m i z a t i o n shell was used to set up a program O P A L L for propeller o p t i m i z a t i o n . T h e program is applicable t o propellers of the Wageningen B-series (two t o seven blades), controllable-pitch propellers, and ducted propellers. T h e necessary functions describing the physics were taken f r o m Oosterveld and Oosanen (1975), Holtrup and Mennen (1982), Oosa-nen (1971). O P A L L determines the o p t i m a l propeller (or t w i n propellers) f o r a given speed V , resistance R T , wake f r a c t i o n W , t h r u s t deduction f r a c t i o n T D , relative r o t a t i v e efficiency E T A R , gearing efficiency E T A G , s h a f t i n g efficiency E T A S , d r a f t D R A F T , number of blades A N Z B , and number of propellers A N Z P .

T h e problem is basically described by:

minimize the necessary installed power {— maximize the propeller efficiency) where the basic equations describe t h r u s t T and power PB

Subject to the constraints t h a t the actual blade area r a t i o is larger t h a n the m i n i m u m blade area r a t i o (to keep cavitation w i t h i n acceptable l i m i t s ) and that Va/{n • D) < P/D. T h i s last constraint is an e m p i r i c a l rule that ensures t h a t KT and r]o are positive.

T h e functions for K T and K Q and the m i n i m u m blade area ratio are rather complex, but could be taken f r o m existing programs. T h e modular approach allows t h e n t o set up the actual o p t i m i z a t i o n i n a concise and economical manner:

PROGRAM OPALL

REAL DIAH, DSCH. DSCHR, DHIN, DHAX, REV, RSCH, RSCHR, RHIN, RHAX, k AEAO, AEAOHA, PZUDO, PZUDC, PB, PSCH, PSCHR, PHIN. PHAX. & DRAFT, VA, WT. TEMP, SALIN, ANZP. ANZB. THRUST. ETA

CHARACTER*10 PROTYP, KRIT. KOOSBU EXTERNAL DICHTE.FKT.FKQ

COHHON /UNKNWN/ DIAM. DSCH, DSCHR, DMIN. DHAX. REV. RSCH. RSCHR. ft RHIN. RMAX. AEAO , AEAOHA. PZUDO. PZUDC

COHHON /VARIA/ PB. PSCH, PSCHR, PHIN, PHAX,

ft DRAFT, VA, WT, TEMP, SALIN, ANZP. ANZB, THRUST. ETA COHHON /CHAR/ PRDTYP. KRIT. KOOSBU

C read i n p u t f i l e T = p{s).n''-D^-KT{AElA^,PlD,...) PB = 2npis).n^-D^-KQ{AE/Ao,P/D,...)/mi (4) (5) CALL INPUT C b e g i n o f o p t i m i z a t i o n b l o c k

UNKNOWNS DIAH (DSCH. DSCHR. DHIN. DHAX). ; REV (RSCH. RSCHR, RHIN. RHAX).

& AEAO ( 0 . 5 , 0.05, 0.2, AEAOHA). & PZUDO ( 1 . 0 . 0.1. 0.5. 1.4 ) . & PZUDC ( 1 . 0 . 0.1. -1.4, 1.4 ) . k PB (PSCH, PSCHR. PHIN. PHAX) THRUST = DICHTE(SALIN)•REV**2*DIAH**4* & FKT(AEAO,PZUDO.PZUDC,DIAH,VA,REV.ANZB,PROTYP,CHECK)*ANZP PB = 6.2832*DICHTE(SALIN)*REV**3*DIAM**5* ft FKQ(AEAO.PZODO.PZUDC,DIAH.VA,REV,ANZB,PROTYP.CHECK)*ANZP/ETA VA/(REV*DIAH) .LE. PZUDO

AEAO .GE. AEAOHI(PROTYP,KRIT,KOOSBU,THRUST,ANZB.ANZP.DRAFT.DIAH. & PZUDO,TEHP.SALIN.VA.REV) HINIHIZE PB SOLVE C end o f o p t i m i z a t i o n b l o c k C w r i t e r e s u l t s i n o u t p u t f i l e CALL OUTPUT END 3.4 R o / r o s h i p o p t i m i z a t i o n

Gudenschwager (1988) used the D E L P H I shell to f o r m u l a t e a design model for r o / r o ships w h i c h considered the special requirements for the ship h u l l geometry of these u n i t load ships. Before the actual o p t i m i z a t i o n , basis fore and a f t ship h u l l forms were stored i n a linearized a n d normalized f o r m together w i t h their corresponding hydrostatical values. D u r i n g the o p t i -mization, the routines o f the system created a combined h u l l f o r m by linear t r a n s f o r m a t i o n and interpolated the necessary geometric and hydrostatic values f r o m the stored i n f o r m a t i o n . T h i s allowed a reasonably quick representation of the influence of h u l l geometry a n d d i s t r i b u t i o n of cargo.

T h e o p t i m i z a t i o n a l g o r i t h m is l i m i t e d t o continuous variables, b u t especially the w i d t h of a r o / r o ship should only be varied i n steps. T h i s problem was solved by i n t r o d u c i n g an addi-t i o n a l conaddi-tinuous variable. T h e o p addi-t i m i z a addi-t i o n deaddi-termined addi-then a addi-theoreaddi-tical o p addi-t i m u m parameaddi-ter combination for the continuous variables. A subsequent o p t i m i z a t i o n determined the actual o p t i m u m t h a t could be realized for the nearest discontinuous values.

T h e following m a i n groups of equations and inequalities described the design m o d e l used i n the o p t i m i z a t i o n :

equation^ inequalities

geometry and m a i n dimensions 6 16

hydrostatics 9 2

resistance, propulsion, power 8 8

weights 11 1

centers of g r a v i t y 4 0

s t a b i l i t y c r i t e r i a 5 7

cargo h a n d l i n g 1 0

t o t a l 44 34

T h e model involved about 250 specified design quantities and 57 free ( u n k n o w n ) parameters w h i c h were determined i n the o p t i m i z a t i o n . Variations o f the user-specified parameters served to investigate the influence of h u l l geometry, speed, block coefficient and f u e l price o n the mea-sures of m e r i t freight rate and b u i l d i n g cost of the ship, respectively on some o f the involved free parameters.

4. C o n c l u s i o n a n d o u t l o o k

op-t i m i z a op-t i o n problems. T h e efforop-t involved is much lower op-t h a n i f each p r o b l e m had op-to be grammed directly. I t lowers thus the threshold t o using o p t i m i z a t i o n i n practical design pro-cesses. Assuming basic knowledge of programming, a user w i l l realize the f o l l o w i n g advantages:

- the shell is portable (realization i n standard procedural language) - i t is easy to learn how to use the shell

- the shell offers great f l e x i b i l i t y EIS an a r b i t r a r y number o f F o r t r a n subroutines can be linked

- parameter variations can be realized by loops outside the actual o p t i m i z a t i o n block - the feature of external 'equation blocks' (data base of o f t e n used equations) allows a

modular structure of equations and constraints - the user can design i n d i v i d u a l l y i n p u t and o u t p u t

T h e development of large o p t i m i z a t i o n models, as described i n section 3.4, is not simple. B o t h model development and i n t e r p r e t a t i o n of results require experience b o t h i n optimiza-t i o n and i n ship design. T h e resuloptimiza-ts depend soptimiza-trongly o n optimiza-the accuracy and robusoptimiza-tness of optimiza-the underlying functions used i n f o r m u l a t i n g the o p t i m i z a t i o n model.

T h e o p t i m i z a t i o n shell is not fool proof and errors occur frequently when beginners start using the shell. N o t the least of the problems is t h a t users f o r m u l a t e problems w i t h o u t solution, e.g, t r y i n g t o f i n d the ' o p t i m a l ' propeller that delivers l O M W w i t h a m_aximum_ diam_eter of 0.2m. I f the shell yields no or only unplausible results, the reason lies o f t e n i n improper upper and lower bounds.

A more f u n d a m e n t a l problem is t h a t the target f u n c t i o n to be optimized is o f t e n not clear. I n reality, o f t e n a 'good compromise' between various concrete goals is sought. T h i s is denoted as ' m u l t i - c r i t e r i a o p t i m i z a t i o n ' . I t is - as a rule - impossible t o optimize m u l t i p l e c r i t e r i a at the same time. Instead an abstract criteria formed by a weighted sum o f the various c r i t e r i a is formed and then o p t i m i z e d . However, the weighing factors are obviously subject to debate and can only be given by some m a r g i n . A sensitivity analysis t h e n may give a class o f designs - rather t h a n one single o p t i m u m - which have to be considered as equivalent.

Ever more demanding constraints of b o t h economical and technological nature increase the motivation to employ o p t i m i z a t i o n techniques to determine p r i m a r y design parameters. A decade ago, rather complex o p t i m i z a t i o n models as described i n section 3.4 required already only one minute c o m p u t a t i o n a l t i m e on a P C . Today, the increased c o m p u t a t i o n a l power allows the integration i n interactive computer-aided ship design systems enhancing the existing capa-bilities of these systems t o meet the requirements for more sophisticated ship design methods.

O p t i m i z a t i o n shells of the f u t u r e should t r y to extend the f u n c t i o n a l i t y w i t h o u t sacrificing user-friendliness. Perhaps f u r t h e r incorporation of knowledge-based techniques, namely i n for-m u l a t i n g and i n t e r p r e t i n g results, could be the p a t h to a solution, e.g. Bertrafor-m (1998). B u t even the most 'intelligent' system w i l l not relieve the designer of the task t o t h i n k and to decide.

R e f e r e n c e s

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