Engineering Design Handbook - Computer Aided Design of Mechanical Systems

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AMCP 706-192

mC PAMPHLET

oENGT

ERlJh~,LNG DESIGN H11ANDBOOK

I COPUTRAl DED DESIG'N OF M EC-H A".N ICAL, SYSTEMS

JAUQUARYERS, U.S. AP'lY MATERIEL COMMAND

JULY 1973

R-,.v.od bw

N'ATIONAL TEC.WX"AL 114FORMATiON StRVICE f,41

VA

-

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HEADQUARTERS US ARMY HAIEBIEL COM10ND 5001 EISENHOWER AVENUE ALEXANDRIA, VA 22304 15 July 1973

AHC PAIU'HLET NO, 706-192

ENGINEERING DESIGN HANDBOOK COMPUTER AIDED DESIGN OF O ECHANICAL SYSTEMS

TABLE OF CONTENTS

Paragraph LIST OF ILLUSTRATIONS ................. LIST OF TABLES ......................... PREFACE . ..............................

Page ix xiii

xv

CHAPTER I ELEMENTS OF COMPUTER AIDED DESIGN S-1 --2 1-3 1-4 !-5 1-5.1 1-5.2 1-5.3

Synthesis vs Analysis in Engineering Design .... I- I The Philosophy of System Engineering ......... 1-2 Computer Aided Design in the Mechanical Sciences ....................... --4 Mathematical Preliminaries ................... 1-5 Illustrative Military Computer Aided Design Problems .. ........ ... .......... ! -8 Optimal Design of Structures ............ ... .1-8 Application of the Steepest Descent Method in !nteractive Computer Aided Design .......... Dc,ign of Artillery Recoil Mechanisms ......... Rc'.erences ................ ............

-- 13 I - 15 1 - 17

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TABLE OF CONTENTS (Can't.) Page

TRamgraph CHAPTER 2. FINITE DIMENSIONAL UNCONSTRAINED OPTIMIZATION 2-1!

2-

Introduction........ ......................

Necessary Conditions for Extrema ............... One-dimensional Minimization .................. ... Quadratic Interpolation .................. Fibonacci Search (or Golden Se~ction Search) . . The Method of Steepest Descent (or Gradient) ... .. A Generalized Newton Method ............ Methods of Conjugate Directions ..............The Conjugate Gradient Method .......... The Method of Fletcher and Powell . ... A Conjugate Direction Method Without Derivatives ................ 2-7 Comparison of the Various Methods 2-7.1 Method of Steepest Descent ............ 2-7.1.1 Cost Function f (X).............. 2-7.1.2 Cost Function f2 "X) ................ 2-7.1.3 Cost Function f 3 (Y) ....... . Generalized Newton Method . .. 2-7 2

2-2 2--3 2--3.1 2-3 2 2.4 2-5 2-6 2-6.1 2-6.2 2-6.3

2 -7.2. 1 2-7.2.2

Cost Function f,(W)....... Cost Function f(x) ...............

..

2-7.2.3 2-7.3 2-7.3.1 2-7.3.2

Cost Functionf 3 (x) ........ Conjugate Gradient Method .. ................ x... .... Cost Functionf, W( Cost Function f 2(X)......................

2--7.3.3 2-7.4

Cost Function f 3 (r) Fletcher-Po'vcll Method ..

2-7.4.1 2-7.4.2 2-7.4.3 2-7.5 2-7.5.1 2- 7.5.2 -7.5.3 28

2-2 2-6 2-6 2-7 2 'I 12 1 2-16 2-17 2-18 2 - 18 2-18

19

.. 2 2- 1

....................

.

2-21

2-21 Cost Functionfj(x) ...................... 2-22 Cost Function f 2(x) ..... ................. 2-221 Cost Functionf 3 (X)...................... 2- 22 Conjuigate Directions Without Deriva'ives ........ 2--22 W ..... .............. Cost Functionf1 (x... .... 2-23 Cost Function f 2(W)............... 23 Cost Functionf 3 (x)............2An Application of Ur~constiained Optriization to Stru,-ural Anadysis.........2- 24 2 25 ...... .... References....

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AFMCP 706-192

TABLE OF CONTENTS WCon't..) Paragaph?

Faie CHAPTER 3. LINEAR PROGRAMMING

3-1 12 3-3 3- 3.1 3-3.2 3--3.3 3-4 .3-5

lntrodv~ction .............................. Properties of Lineat Programs .................. The Simplex Algorithm .......... ............ Deterziination of a Basic Feasible Point .......... Solution of LP........... ................ 1Iihe Degenerate Cae ...................... Mininum Weight Truss Design ................. An Applicat~on of Linear Programming to Analysis .......................... ... References..............................

3-1 3-2 3-8 3-8 3-9 3-10 3-12 3-i4 3-17

CHAPTER 4. NONLINEAR PROGRAMMING AND FINITE DIMENSIONAL OPTIMAL DESIGN

t.-1

Introduction to tlie Theory of Nonlinear Programming (NLP) .......

.........

412 Global Theory ............... ............. Theory of Finite Dimensional Optimsl Design . 421 Finite Dimensional Optimal Design Problems................ ..............

42

4-23.2 4-3.1 4- 3. 4--.3

4 4.3

4-4$ 48 4-8

Loa ITheory to..........................4-10 -3.2 ecxtierio Method.........................414 Iii1In tirEterior Method.................. 4 H1 FxVterinatMonho an.I.t..or..oin....... ...... 4 -18 ted estDescrent eriosMetor NL.... .. ........16

4 -4.1 4- 4.2 4 -4.2, 1 4-.42-~2 '7'4-4.2.3

...

4-1

The Dircetion of Steepest Descent .............. Step Siz~e Determination................... .4Rosen's Method for Linear Constrints ......... Fixed Step With Variable Weigtihig .... Steepest D)escent With Constraint Tolerances ..... ................... ... A Steepest Desect Alethod With Constraint E'.ror Compensation. .. .. .. ....

4-20 23 4 -23 4 -24 4 25 4 2

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AMCF 706-192

TABLE OF CONTENTS (Con't.) Paragraph Steepest Descent Solution of the Finite 4-5 Dimensional Optimal Design Problem ......... 4-5.1 An Approximation of the Problem OD ........ 4-5.2 Solution of the Approximate Problem ......... 4-5.3 Steepest Desceit A gorithm ................. 4-5.4 Use of the Computat'onal Algorithm .......... References ...............................

Page 4-27 4-28 4-30 4-33 4-34 4-35

CHAPTER 5. FINITE DIMENSIONAL OPTIMAL STRUCTURAL DESIGN 5- 1 5-1.1 5- 1.2 5-1.2.1 5-1.2.2 5-1.2.., 5-1.2.4 5-I .3 5-2 5-2.1 5-2.2 5-2.3 5-2.4 5-3 5-3.1 5-3.2 5-3.3 5-4 5-4.) 5-4.2 5- 4.3 .5-4.4

Introduction ......... ................... Lightweight vs Structural Performance Trade-offs ....... ................ Weapon Development P A. "', ociated With Lightweight Require,.%. .......... Aircraft Armament ......... ............ Gun Barrel Design .................. Towed Artillery ......................... Other Weapon Problems ................... Plan for Technique Development ............. Elements of the Elastic Structural Design Problem ... ...................... The Optimality Criterion ....................

[

....

5-2

.....

5-2 5-3

Stcepest Descent Algorithnm foi Optimal Structural Design. ................. Computation. Consideration, . .. ......... Optimization of Special Purpose Structures ...... A Minimum Weight Column ................. A Minimum Weight Vibratirg Bean- ........ ... A Minimum Wei2.t Pcrtal Frame With a Natural 'requeicy ('c.:istral .t / Minimum Weight Frame With Multiple .

..

.5

. 5-2

...

Stress and Displacement Due to Static Loading .......................... Natural Frequency and Buckling ............. Method of Solution ....................... Steepest Descent Programming for Optimal Structural Design ...... ... ....... Linearized Cost and Constraint Functions .....

Failure Criteria iv

5-1

5-3 5-4 5-4 5-4 5- S 5 -5 5-6 5-7 5-7 5-8 5-10 5- 10 5- 1I 5-- I! 5-- 14 5 -16 18

k

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AMCP 706-192 TABLE OF CONTENTS (Con't.) Page

Paragraph

~-4.5

A Minimum Weight Plate With Frequency 5-20 Constiaints............................. General Treatment of Truss Design..............5-22 5-5 5-22 Special Problem Formulation ................. 5-5.1 5-24 5 -5.1. 1 Frequency Constraints..................... 5-5.1.2 Stress Constraints ........................ 5-25 5-25, 5-5. 1.3 Buckling Constraints...................... 5-5.1.4 Displactment Consiraints................... 5-263 5-5.1.5 Buund.,.on Design Vari ihles.................5-26 5-5.2 Multiple Loading Condtions.................. 5-28 S-5.3 Example Problems ............... ......... 5-28 5-6 5-6.1 5 --6.2 5-0.3 5-7 5--7.1 5-7.2 5 -7.3 5-7.4

A General Treatment of Plane Frame Design ... Proble.m Formulation .................. . Stress Constraint Calcu~ations ................. Example Problems ........................ Interactive Computing in Structural Optimization ............................ The Interactive Approach.................... Interactive Structural Design Using Sensitivity Data ......................... Example Problems ................. ....... Interactive Computing Conclusions ............. References ... ...........................

5-36 55-39 5-46 5-51 5-51 5-52 5-54 5-61 5-62

CHAPTER 6. THE CALCULUS OF VARIATIONS AND OPTIMAL PROCESS THEORY 6-1 6-2

Introduction.......................... ... The Fundamental Pcoblein of the Calculus of Variations............................. 6-2.1I Ntcessary Conditions for the Fundamental Problem..................... .......... 6- 2.2 Special Ca~ses and Exampies ................... f- 2.3 Vaiiat~on,: Notation and Second-order Conditions ............................ 6'-2.4 Di~rect Methods ... .............. ........ 6 -2.4.1 The Ritz Method ......... ............. 6- 2 4.2, Method of Finite Differences. .............6--3 A Problem of Bolza..............6-16 Statement of the Problem..................66 3I .. ... A Multiplim Rale.. ............ 4-3.2

6-I 6-4 6-5 6-8 6-10 6-13 6-14 15 16 6- 17

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MCP 706-192 TABLE OF CONTENTS (Con't.) Paragraph o-3.3 6- 3.4 6-4 6-4.1 5-4.2 6-4.3 6-5 6-5,1I 6-5.2

page2

Ntvcessary Conditions for the Bolza Problem.............................. Application of the Boiza Problem .............. Fhoblems of Optimal Design and Control .......... Design Variable Inequality Constrants ........... State V'ariable Inequality Constraints ............ Application of the Theory of Optimal Design............................... Methods of Satisfying Necessary Coin(titions ....... Initial Value Methods (or Shooting Teclh-niques) ............................ A Generalized Newton Method ................ References .... ..........................

6-18 6-22 6-27 6-28 6-31 6-33 6-40 6-,*0 6-42 6-45

CHAPTER 7. OPTIMAL STRUCTURAL DESIGN BY THE INDIRECT METrHOD 7-1I 7- 1.1 7-1.2 7-1.3 7-2 7-3 7-3.1I 7--3.2 7- .2.1 7-3.2.2

7-3.3 7--3.3.1 7 -3.3.2

7 3.4 7 -4

Introduction .............................. The Class of Problems Considered ............. H-istorical Development .................. ... Methods Employed .................... A Minimum Weight Column ................ A Minimum Weight Structure With Angular Deflection Requirements .............. Statement of the Problem .......... ......... Tower With One Design Variable ............... Method 1. Tower With Base Rigidly Fastened to the Earth ............. ..... Method 2. Trower With Case Pinned to Earth and With Top Supported by Guy L~ines ............................ Tower With f~wo Design Variables.....- ... .. Method 1. Trower With Base Rigidly Fastened to the Earth ............ ...... Method 2. Tower With Base Pinned to Earth arid With Top SuiM,.irted by Guy Lines .. .. .. .. .. .. .. .. .. .. .. .. ... Discussion of Results ....... .......... ..... Minimum Weight Design of IReans With Inequality ('onstrmints onl Stress and Deflection

vi

7-1 7-1 7-I 2 7- 5 7-S 7- -7

7

7T7-3 7-14

7- 14 7-15

7 16

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F,'Paragraph AMCP 706-192

TABLE OF CONT ENTS (Con't.)

Page

7-4.1I 7-4.2

7-4.3 7-4.4

Statement of the Problem ......... .......... Necessary Conditions for the Beam Des~gn Problem ......................... Statically Indeterminate Problems ... .......... Solutions of the Equations of Theorem 7-1I..

Beams With Rectangular Cross Section of Variable Depth ........................ 7-4.5.1 A Problem Which Can Be Solved Analytically ............ ........ ....... Positive With Beam Supported Simpiy 7-4.5.2 Distributed Load........................ 7-4.5.3 A Problem of a More General Typ-. .... ....... 7 -4.5.4 Conclusions ............................ References ......................... .....

7-16

7-19 7-23 7-24

7-4.5

t

7-25 7-29 7-32 7-38 7-43 7-44

FOR OPTIMAL DESIGN PROBLEMS 8-1 3-2

R

Introduction ............................. 8-I A Steepest Descent Method for the Basic ~Optimal Design Problem..................... 8-2 8-211 The Problem Considered..................... 8-2 t5- 2. 2 Effects of Small Changes in Design Variables and Parameters ........ ........... 8-2 8-- 2..3 A Steepcst Descent Approach .............. 8-5 8-3 A Stcepest Descent Method for a General Optimal D~esign Problem ......... ...... .... 8-15 8-3.1I The Problem Considered....................8-15 8-3.2 The Effect of Small Changes in Design Variables and Parameters ............... ...8--3.3 A Steepest Descent Computational rth................... --.......... 8-23 8 -4 Steepest Descent Programming to, a Class Systemn Described by Faitial Diffezwna~al iEq''rns.......8-25 8-4.1 The Class oi Pioblems Considered ............... 8-25 n 4.2 Effeut of Silall Changes in Design ........ 8-27 Variables and idaer......... 8 -4.3 A Steepest Descent Comnputational Akor~R-

10

v!I

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AMOP 706-192

TABLE OF CONTENTS (Can't.) Paragraph

Page

Optimal Desigii of an Ariller R"coi Mecharism ..................... ........ Formnulation of the Problem.................. 8-.1 8-5.2 Equations of Motion for the XM 164 Howitzer............................. 8-5.3 Steepest Descent Formulation................ 8-5.3. 1 Determination of the Adjoint Equations .... 8-5.3.2 Determination of the B'undar Conditions for the Adjoint Equations................. 8-5.3.3 Computation of Design Improvemjents .......... 8-5.4 Results and Conclusions .................... References.............................. 8-5

8-35 8-37 8-39 8-43 8-44 8-45 8-47 3-48 8-49

CHAPTER 9. APPLICATION OF STEEPEST DESCENT METHODS TO OPTIMAL STRUCTURAL DESIGN 9-1 9.2 9-3 9-4 9-5 9-6 9-7

Vil

Introduction .............................. Steepest Descent Method for Optimal Structural Design.......................... A Minimum Weight Column................... A Minimum Weight Vibrating Beam .............. A Minimum Weight Vibrating Frame ............. A Minimum Weight Frame With Multiple Failure Criteria........................... A Minimum Weight Vibrating Plate .............. References .................. ..... .......

9-14 9-16 9- 18

APPENDIX A CONVEXITY .......

A-I

...........

9-1 9-2 9-8 9-9 9-10

APPENDIX B ANALYSIS OF I3EAM-T\ PE STRUCTURES ........ .. ......

B-1

INDEX........................

I1-I

....

.....

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AMCP 706-192

LIST OF ILLUSTRATIONS Fig. No.

A System Engineering Model ................. Structural Requirement .................... Conceptual Designs ....... ................ Uniform Initial Design ...................... Direction of Steepest Descent .................

1-6

Tower With Base Rigidly Fitcned to the Earth ......... ................... Tower With Base Simply Supported and Top Supported With Guy Lines .............. Sensitivity to Design Variations ............... Sensitivity to Two Performance Indicators .......

1-8 1-9 1-10 1-11 1-12 1-13

Howitzer, Towed, 105 mmi, XM164 ............. Traditional Recoil Design Goal ............... Recoil Distribution in Time .................. Sensitivity to Gun Hop ......................

1-14 2-1 2-2 2-3 2-4

Optimum Reioil Ctuve ...................... f(x) = (x - 2)2 ........................... A Cost Function ......................... Function of Single Variable .................. Interval Partition ...................

2-5 3-1 3-2 3-3 3-4 3-5

j

Page

1-.1 1-2 1-3 1-4 1-5

1-7

>1

Title

4-1 4--2 4-3 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5- 9

...... Descent Steps ............................ Graphical Solution of Example 3-1 ........... Polyhedral Constraint Set ...... ............. Admissible Joints for Bridge Truss ............. Optimum Bridge Trus;es ............ ...... Boundary Conditi.n for Example 3-4 ..........

1-2 1-8 1-9 1-11 1-12

.

1- 12 .1

13 1-14 1-14

1-16 1-16 .1 -16 1-i7 1-17 2-2 2-3 2-7 2-7

. .

2-10 3-2 3-5 3-14 3-14 3-16

Graphical Solution of Example 4-1 ............ First-order Con~strairt Qualification ............ Penalty Function . ......................... Column ................................. Column Eleament .................. ....... Profiles ot Optimal Columns ................. Stepped Beam ............. ..............

4-3 4-6 4-11

Tyrical Elem-nt .......................... Profile of Optimum Beam ................... Portal Frame ... . .. ............... .... Typical Elements .............. .......... Optimum Portal Frame for = 3000

5 5 5 5

rad/sec

...................

5- 12 5--12 5- 14 5-14 15 -16 -16 16

5 18 Ix

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AMCP 706-192

LIST OF I L'.USTRATIONS (Con't.) _

Fig.No.

Page

5-10

Frame With Side Loading ....................

5-18

5-11

Typical Elements ..........

5-19

5-12

Profile of Optimal Frame With Multiple Criteria (q = 25 lb/in,) ........ ............ Rectangular Plate .......................... Collocation Points .......................... Optimal Design Variable h(x, y) for Vibrating Plate ........................ Description of a Truss ...................... A Truss Element ........................... Four-bar Truss (Example 5-I ) ................ Iteration vs Weight Curves for Example 5-1, Four-bar Truss .............. ............ Transmission Tower (Example 5-2) ............ Iteration vs Weight Curves for Example 5-2, Transmission Tower ....................... 47-Bar P!ane Truss (Example 5-3) ............. Iteiation vs Weight Curves for Example 5-3, 47-Bar Plane Truss ........................ Description of a Frame ...................... A Frame Element .......................... Simple Portal Frame (Example 5-4) ............ Iteration vs Weight Curves for Example 5-4, Simple Portal Frame ....................... One-bay, Two-story Frame (Example 5-5) ....... Iteration vs Weight Curves for Example 5-5; One-bay, Two-story Frame ................. Two-bay, Six-story Frame (Example 5-6) ........ Iteration vs Weight Curves for Example 5-6; Two-bay, Six-story Frame, With Stress Constraint3 Only ....................... . Iteration vs Weight Curves for Example 5-6; Two-bay, Six-story Frame, Witt All Constraints ... .......................... Vector Change in Design Space ............ Three-bar Truss .......... ........... Display of Design Sensitivity Data .......... Local Optima ........................

5-13 5-14 5-15 5-16 5-17 5-18 5-19 5-20 5-21 5-22 5-23 5-24 5-25 5-26 5-27 5-28 5--29 5 -30 5-3J

5-32

5-33 5-34 5-35 5-36 5- 37 x

Title

Trusses (Examplc 5-7)

...........

.......

5--20 5-20 5-21 5-22 5-23 5-23 5-29 5-29 5-31 5-33 5-34 5-36 5--38 5-38 5-46 5-47 5-48 5-49 5-49

5-51

5 -5I 5- 52 5- 53 5 -53 5- 54 5 -55

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AMCP 706-192

r

K/l5 K.

-39

'

is Weight Curves for Example 5-7. ~~Three-bar Trusts, With Alles Constraints.....55j

-iteration

O-Tanmnl Toc Eape53.........................

5-41 5-41

6-1

P6-4

r6-9

iteration vs Weight Curves for Example 5-7,I Treanrso Twr, With Strs Constraints..... 56 O54 rnmissio......(Exampe...................5-58 Iteration vs Weight Curves for Example 5-8, Transmission Tower, With Alles Constraints 56

Shortest Path.................. ........... 6-- Crv-- orMinimum T-me.....................

63 EapeofContinuous Functions .............. 6-5 6-6 6-7 6-8 6-10 6-1l 6-12 6-13 6-14 6-15 7- i 7-2 7-3 7-4 7-5 7-6 7-7 7-8 7--9 710 7- 11

5-55

6-1 6-2

6-3I

A Neighborhood of.~t)...................... Perturbation from Optimum ................... Graphical Proof of Lemma 6-1. ................ Minimizing Seq, ence .............. .......... Particle in Motion .........................

6-4 6-5 6-6 6-9 6-22

Orbit Transfer

6-25

...

......

I........

Thrust Program ........................... Ground Vehicle ........................... Extremal Arcs .... ........................ Extremal Acs WVith Straight Section ............. Bounded Brachistochrone................. Bounded Brachistochrone Solution .............. Column Under Consideration ................... Profiles of Optimal Columns ................... Tower Considered.......................... Loading of Tower .......................... Tower With Base Rigidly Fastened to the Earth ... ................... ........... Tower With Guy Lines .................. .... Tower With Base Simply Supported and Top Supported With Guy Lines ................ Beam Loaded in a General Wa ................. Rectangular Cross Section .................... Simple Cantilever Beam ....... ............... Cantilever Beam of Minium W-emght .............

6-25 6-33 6-36 6--36 03 6-40 7-2 7-6 7-6 7-8 7-12 7 12 7-13 7- 17 7 -25 7-29 7 -32

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AMCP 706-192

-

Fig. No. 7-12 7-13 7-14. 7-15 7-16 7-17 7-18 8-1 8-2 8-3 8-4 8-5 8-6 8-7

LIST OF ILLU.STRATIONS (Con't.) Title

Page

Simply Supported Beam With Positive Distributed Load ......................... Subdivision of the Beam With Distributed Load ................................... Beam 1"ith an Inflection Point ................ Subdivision of the Beam ..................... Volume vs Deflection Requirement ............ Profile of Optimal Beam for A = 0.16 ........... Profile of Optimal Beam for A = 0. 15 .......... Howitzer, Towed, 105 mm, XM164 ........... Recoil Force for a Rigia Mount ................ Time Intervals ............................ Schematic of XM 164 105 mm Towed Howitzer - Dynamic Model ................. Recoil Time Interval ....................... Optimal Rod Force ........................ Optimal Control Pod Design .................

7-32 7-33 7-38 7-39 7-43 7-43 7-43 8-36 8-37 8-37 8-39 8-45 8-48 8-48

Simply Supported Beam .....................

9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-9 9-10 A-I A-2 B-1

xii

Simply Supported i;brating Beam ............. Profile of Optimal Beam ..................... Portal Frame ............................. Profile of Optimum Frame ................... Laterally Loaded Frame ..................... Free Bodies .............................. Profile of Minimum Weight Frame ............. Simply Supported Plate ..................... C mtours ot'Optimum P'.aie ................. Examples, Convex Cse and Nonconvex Case ..... Graph off(x) = x2 in R . . . . . . . . . . . . . . . . . . . . Basic Beam Element .......................

.

9-2

9-9 9-11 9-Il 9-13 9-14 9-14 9-17 9-17 9-18 A-1 A-I B-I

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-~~~~~AC

A-t-~--~--'."

-

-

6

''- 70-6A-192

6

..

>S.-

-

ite

Tal No

Metho

6'L

Pg

ItrtveDt

2-

StcesDscn

2-4

Gteeraedewont Method - Iterative Data for Cost Function f, (x) ..................... SGeeaest ewnM'od - Iterative ata -for Cost Function ft2 (x) ..................... SGeeaest een Method - Iterative Data for Cost Function f3 (x) ....... ............. GeneraieGdet Method - Iterative Data

-

-

for CoghsotFnonwr.........................2-139-

2-2 2-3

12-4 S ffor

Cost Function f, (X)..................... 2-8

Ifor j

2-6 2-70 2-81

2l

Fletcher-Powell Method - Iterative Dama for Cost Functionf (X) . .. .. .. .. .. .. .. .. .... FleCnjtce reoll Metho t erativest

2-2 Fece-elMethod - Iterative DatafrCs fo tFunctionf 3 (x......................... 2-13 Conjugate Directions Without Derivatives Method - Iterative Data for Cost Functionf (X)........................... 2-14 Conjugate Directions Without Derivatives Method - Iterative Data for Cost Functionf 2 (X).......... ................ 5-_

-

2 5- 3 5-4 5 5

2-29 2-19

2-21

GeneuaieGdet Method - Iterative DJata for Cost Function f 2 (X).....................2-21 GeneraieGdet Method - Iterative Data Cost Function f 3(x) ..................... 2-21 Conutceradiwen Method - Iterative Data for Cost Function f, (x) ....... ............. 2-22 Conutceraien Method - Iterative Data for Cost Function f, (x) ............... ...... 2-22

2-10

65-

2-19

Comparison of Uniform and uptinuil Columns . .. Cross-sectional Areas of Optimum Colinuns ... Comparison of' Optimum 5cams ................ Material Properties for Aluminum. . . . ..... Comparison of' Uiform and Optimal Framnes for Aluminum .............

-

2-22

2-23

2-23

2-23 13 5 14 5 lb( 5 l17 18 xiii

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, --.-

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-,.

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A_

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AMP 7.0*192

,o

ALIST

-t

OF TA IkUS (Const.)

Table No.

nile

5-6

Optimal Design Variable b, for

5-7

Optimal Design Variable b for

Vibrating Frame . ............ :Static

Frame ..

Rage

...........

. . . . .. . . .. . .

5-18 5-20

5-8 5-9 5-10 5-I1 5-12 5-13 5-14

Vnlume of Optimum Frame ................... 5-2G Four-bar Truss (Example 5-!) .............. 5-30 Transmission Tower (Example 5-2) ............ -" 47-bar Plane Truss (Example 5-3) .............. 5- 35 Simple Portal Frame (Example 5-4) ............ .5-47 One-bay, Two-story Frame (Example 5-5) ........ 5-48 Two-bay, Six-story Frame (E.xample 5-6) ........ 5-50 5-15 Optimum Three-member Trusses (Example 5-7) .. 5-57 5-16 Derign Information for Transmission Tower 3--I (Example 5-8) ........................... 5-59 5-17 Optimum Transmission Towers With Stress Constraints Only (Example 5-8) .............. 5-60 5 18 Optimum Transmission Towers With All Constraints (Example 5-8) .................. 5-61 7-I Results for Column Problem ................. 7-5 7-2 Constants ........... .................... 7-7 7-3(A) Weights of Simply Supported Tower'% One Design Variable ............ ......... 7-11 7-3(B) Weights of Guy-line Supported Towers, One Design Variable ...................... 7-II 7-3(C) Weights of Towers ......................... 7-11 7 -4 Results for Simply Supported Beam With q(t) = . .... ..

7-5 9- I 9-2 9-3

xIv

..... ...................

7 -37

Results for Beam With S-shaped Deflection Cvrve .................................. Co-,iparison of Optimal Beams ................ Weight of Optimum Fr':mes ................ Volume of Optimum l-ame .................

7 -42 9-1 9-13 9-16

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Fr.EFACE The Engineering Design Handbooks of the U. S. Army Materiel Command are a coordinated series of handbooks containing basic information and fundamental data useful in the design and development of Army materiel and systems.

I I

This text treats a broad class of optimal design problems through use of a consistent set of computational techniques ideally suited for computer application to mechanical design problems. No attempt has been made to be exhaustive in the treatment of optimization techniques or the full range of mechanica! applications. Rather, the class of problems treated is concisely formulated (in Chapters 4 and following) in terms of design and state variabls that occur in mechanical design. A steepest-descent approach which has served as a workhorse, reliable technique in fields such as aerodynamic system design, control theory, and nonlinear programming - is developed here for mechanical system design. Extensive application of design optimization techniques is made in the field of structural design, as well as in a limited number of specific weapon design problems. The examples are presented in considerable detail, as they are encountered in practice, to provide the practicing engineer w;th insight into use of the methods for his class of problems. A consistent design philosophy is maintained throughout the text to ass*- t the designer in extrapolating the methods to classes of problems that are only similar mathematically to the examples treated here. The text is structured so that it can be understood and used by practicing engineers with a good background in calculus and matrix theory. Computational algorithnis are ,tated in considerable detail so that they can be effectively implemented by junior engineers, with only problem formulation and general supervision provided by a senior project engimneer. As with virtually all comouter aided design techniques, some computing art is required for effctive implementation of these te,.hniques. The detailed treatment of structural applications in Chapters 5, , and 9 should provide insight into this onmiputational art. References are given to more advanced liteiature fo' proofs ol theorems and extensions methods to other classes , rProblems. x\

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AMC60 d,192

F,

The Handbook was wiitten, by Dr. Edward-J. Haug, Jr. of the U. S.Army Weapons Comiiid, -It is, based on -Iecfure hiaterials -used 'by -hiffi in-a t~-eetrF0.t~qec on "Optimization of Strutura['Systems", taught at the University of Iowa since 1968'. Fxamples treated in the text are derived, primarily from Di. Haug's-rcsearch, Dr. .Jasbir Arona's University of Iowa disseriation, and the work of Messrs. Tonm Streeter and Stephen Newell of the U.S. Army Weapons Command.

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The Engineering Design Handbooks fall into two basic categories, those approved for iekase and sale, and those classified for secutity reasons. The Army Materiel Command policy is to release these Engineering Design Handbooks to other DOD activities and their contractors and other Government agencies in accordance with current Army Regulation 70-31, dated 9 September 1966. It will tic noted that the majority of these Handbookcs can be obtained froni the National Technical Information Service (NTIS). Procedures for acquiring those Handbooks follow:

4,

a. Activities wilhin AMC, DOD agencies and Government agencies other than DOD having raeed for the Handbooks should direct their iCquest on an official formn to: Commaander

-

Letterkenny Army Depott ATTN, AMXLE-ATD Chambersburg, PA 17201 b. Contractors and universities must forward their rcquests to: National Technical Information Service Department of Commerce (Requests for classified documents must be sent, with appropriate "Need to Kno" ustfiatintoLetterkenny Army Depot.) Comments and suggcsiions on this Handbook are welcome and should be addressed to:

V

US Army Materiel Command ATTN: AMCRD-TV 5001 Eisenhower Avenue Alexandria, VA 22304 D)A Formis 2028 (Rzo.nmended Changes to Publications), which are available through normial pub..cations -1py dhannels, may be used for cun) nen IS/suggest ions. Xvi

-

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44

:"

AMCO 706-1 92

0.M.P:T"ERAIDE CH APTER 1

i i

'

i-ELEMENTS OF.COMPUTERAIDED DES

1-1 'SYNTESIS, VS,'ANALYSIS INENGI

NEERi"46 Ei GN Engineering is defined (Ref. I) as "the art or -science of making practical application of the knowlede -of pure sciences such as -physics, chenistry, biology, etc.". Although broad, this definition implies that the job of -engineeringisto synthesize, or put together, useful systems'by applying knowledge and methods derived from the "pure" sciences. * -

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sig changes are made and the structure is re.

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analyzed. This -process continues until the designer is satisfied with his design, Thi. has been the principal use of the computer in the design process. In general, then; before the designer can assure himself that he has the best system, he riust be capable of analyzing all candidates. lithe pa half century, outstanding advances in engineering analysis have been made. The

The meaning of "practical" in the given

digital computer has-allowed the engineef to[

definition should be interpreted as best, or optimal; i.e., the job of engineering design is to- develop the best possible system for the given application, consistent with tit.e resources allocated to the development phase. The purpose of this handbook is to present a class of methods that allow for efficient use of the computer in the design process.

quantitatively analyze the behavior of systems that were examined only qualitatively in thtpast. The mechanical sciences, particularly, have benefited from this boom in analysis capability. Structural analysis, stress analysis, analysis of mechanisms, and heat transfer analysis, just to name a few, have made spectacular advances in the past twenty years.

Since the computer can be viewed simply as a device to handle large quantities oi' data and perform simple algebraic opeiations and logic rapidly, it is important to look first into the role of calculation in design. The usuai approach to design is to conceive of a candidate system and then test it to see if it works. Great strides have been made with digital computers in the past two decades to allow for numerical analysis as a test of the idea, or concept, rather than previous cut-andtry techniques. For example, in structural design one chooses the configuration and member sizes, and tien, tests tihe tructure by analyzing its response to given loads If the structure does not behave as desired, then de-

Until the early 1960's, and even to the present day to a lesser extent, the attention of engineering research has been focused primadly on developing analysis capability. During this ptriod of emphasis on analysis, inadequate attention was paid to de eioping a synthesis, or design, capability that is able to efficienrtly use the newly developed analysis methods. In sonic of the mechanical sciences, this problem is paiticularly .cute. In structural mechanics, for example, it is possible to analyze a structure under a given loading to obtain accurate values for stress, displacenient, and even nitural frequency It is not clear, however, how a ',tructure should be laid out and proportioned to efficiently utilize I-I

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AMCP 70S-92 material in order to meet strength requiremints. A more difficult probIem is 4he proportioning of a structure so as to efficiently limit displacement, and meet constraints on natural frequency ard buckling. 'For a review of 'he state of optimal structural design through 1967,.seeRef. 2.

2&

I

It appears that the analysis capability needed for computer aided design is available. The next problem- to be addressed, then, is the matter of what is meant by best, or optimum. The idea• of best enters very naturally into engineering design efiorts. In profit-motivated industries as well n in Government laboratories, the objective is to maximize some return function while satisfying constraints such as resource allocation, performance requirements, and human limitations. 9nce some return function or measure of value is chosen and constraints are identified, the system designer would like to have some optimal design methodology that is capable of aiding him in the determination of the best, or practically best, system. It must be e'nphasized at this point that the search is not for an automatic optimization technique that can solve any design problem fed to it. Rather, the need is for an optimal design methodology that can aid the engineer in the implementation of his concepts and guide him in a direction which, if continued indefinitely, would yield a mathematical optimum.

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A key challenge to developers of practical computer aids to designers is to take maximum advantage of human ,udgment in the design process. The potential of interactive computation and. design information display is only now in a developing stage and holds promise for significant improvement of the value of the computer in design.,

I [

1-2 THE PHILOSOPHY OF SYSTEM ENGINEERING -.

In the middle 1950's a formalized approach to the development of large-scale, man-made systems began to appear in the literature, see Refs. 3, 4, 5. This approach, which has features common to most problem solving processes, was given the name "system engineering" and is the very essence of computer aided design. A feature which sets system engineering and computer aided design off from nost of the logical problem solving schemes is the explicit inclusien of key considerations peculiar to engineering design of systems. A second important feature of system engineering is the attention paid to quantitative description of the system and its behavior. The basi, idea in system engineering is to begin with a statement of system requirements and objectives, and move in an organized way toward an optimum system. A process which illustrates the approach is shown .n Fig. I-1.

Analysi

Figure I 1. 4 System Engneering Model I-2

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AMCP 706-102

Thle purpose fihisiexi.;s notto0 iea :detailed tree cient :ofsystem engineering, but ihiie to -p efit,asp ects-of the -theory of zcomputer- aided desi !twitli eiphasis on iplied- model, of-a, _stem-engineering process shows-that- optis -malt desJis azpart if, systemengiineering, but, indee, by nomeais thedominiant Part. The puiposeof this paragraph is to-discui-'s,the6 interface of optim.al design ,th the rernainilig essential elements of system engineering.

to describe analytically. Conceptual design, as its name implies, is the identification of the various concepts or basic system configurations that might meet the syst n objectives. It is desirable in this step to leave the concepts as general as possible so as not to eliminate candidate syskems that might be very eff tive. For example, if the function to be performed is to propel a vehicle over the surface of the earth, conceptual designs might include wheels, tracks, legs, air cushion, etc.

System engineering begins with the identification of a need by a potential user of the system to be: developed. It is often- the case that the user knows that he needs a system to do a job, but he~nay have difficulty in stating his needs and objectives quantitatively. It then becomes the joint responsibility of the system engineer and user to quantify system objectives so that a meaningful set of objectives may be establisbed for the development to follow.

It is important at this time to identify ranges of values of parameters describing the system so that, for any parameter in, this range of values, the system will p,,rform the functions identified in the previous step, i.e., the set nf parameters describing admissible systems is identified. It is at this time that the experienced designer can be extremely valuable in reflecting state-of-the-art capabilities o,° technologies involved in the system development.

Once the needs and objectives for a system are identified, it is necessary to define functions that must be performed by the system and any subsystems that are required. This

The optimal design step has as its objective the choice of the undetermined parameters identified in the previous step. These parameters must be in the ranges defined by

purpose is to pick out functions or operations

tions. The critezion for choosing system

that must be perfoned in order to accomplish the mission required of the system being developed. These functions then become lower levei objectives for the development of subsystems. Identification of functions tends to be qualitative in nature. However, once a function or operation iF identified, it must be described in quantitative terms, if at all possible. For example, if a function t'ast occur quickly, the tame allowed should be specified.

parameters is maximization of system worth or value. It should be emphasized that a mathematically precise optimum may be impossible to attain and must therefore serve only as a goal. Methods for choosing system parameters should, however, have the property that if an optimum does exist, then given erough patience and computer time, that optimum should be approached as a limit.

The next step shown :n Fig. I-I is one ot the most difficult functions in system engineerirng and certainly the most difficult step

What appears to be the final step in the system engineering model of Fig. I-I Description, is, in reality, probably just an intermedi;te step. Unless the sysicm (sign procedu-e has bern unusually effective, the sys1-3

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AMCP 706192

I -

tern decided upon will probably not satisfy the user. More likely, it will probably not satisfy the system engineering team. Having the results of one pass through the system engineering process, the user can probably remember sonie constraints which he forgot to specify and which the optimum system violates. The designer probably also will see concepts that he did not see before. Much as the user, he too will remember technological constraints which he for,'.t to specify and which the optimum .ytem violates. Finally, the sponsoring activity will undoubtedly decide that it will be all right to decrease the measure of system value a small amount if it will save some money, The next step in the procedure is for each member of the team to take a deep breath, sigh, and go back to work, armed with his hard earned new knowledge. It is for this purpose that all the feedback paths in the iteiativc model of Fig. I-1 are shown. T "s procedure is then continued uiti! the sponsoring activity decides that the syst.em developed is what it really needs. This will probably be another human decision, rather than a programmed mathematical one. The remaining chapters will be devoted to the problem of c,,;puter aided and optimal design. If the (esign. ,,thcds presented later arc : be of maxiznui value to the reader, lie :?-nst h'ie a clcat picture of how dese methods fit into thc lan;r -robiem of system i :,;in-.n-mg. For fui'hc.- hterature on the basic ideas involved in system engineering, see Refs. 3, 4, and 5. 1-3 COMPUTER AIDED DESIGN IN THE MNICHANICAL SCIENCES The theory of competer aidtd and optimal d."iga is developed insubsequent chapters as 1-4

it applies to the mechanical sciences. There are peculiarities of mechanical design, as opposed to classical control system design, which require specialized trcatmvnt. Further, the mathematics involved in mechanical systern design is quite different from the mathematics of control theory. These distinctions ar highlighted throughout the text. hithe chapters that follow, optimal control theory is interpreted as treating feedback controllers; i.e., an optimal control system h.,,s active elements that sense errors in output, due to fluctuations in inputs, and adjust system coairols so as to maximize some measure of system performance. Optimal design, on the other hand, is taken as the problem of choosing system elements or parameters describing these elemen!s, which are fixed for the life of the elements, so that the system is optimum in some sense. In control literature this is called open loop control. The principal difference in the two problems is that the variables chosen in the optimal design problem are fixed for the life of the system, whereas variables in a feedback control device are to be adjusted according to inputs as the system operates. Mathematically, the difference in the two results is that the control law describes how the system variables should he adjusted as a function of the state of the system, whereas aa optimum design is simply a set of paramcter describing system elements and will not be changed during the life of'the system. This distinction is not unifcm in the control literature but is used here to identify the class of problems treated. In most literature on control problems, sequential systems are treated, i.e., operations of the system progress one after another as if they were occurring intine in a pre-arranged order. Many optimal design problems are not

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AMCP

2.

1

of this kind. For example, in designing a struzlure c= mu tb concerned with stresses due to applied loads. These stresses are interpreted as the state of the structural system. They are determined by a boul 'aryvalue problem that cannot be interpreted as a sequential process (initial-value problem). !n some design problems it is vossib1-c to define auxiliary variables so t-at the governing equaan initial-vaiue problem with additons fotmr tional constraints. This procedure, however, generally complicates the problem unncccssarily. For this reason the problems in succeeding chapters are formulated as boundary as opposed to initial-value problems.

,.

In order to illustrate the use of the methods presented, applications are made primarily in optimal structural design. Applications are chosen to ilustrate the use of the methods on problems having a number of design variables which might be found in engineering applications. Further, since many of the methods are relatively new, it is anticipated that improvements in computational efficiency may be realized in specific problems if advartage is taken of special features of the class of problems treated. It is appropriate to highlight a significant computational distinction between two classes of design problem. The reader may note that Chapters 2 through 5 of this text deal with problems in which system design and performance are specified by a finite number of parameters (real numbers). Chapters 6 through 9, on the other hand, deal with systems that are desciibed by functions on some given space or time domain. Mathematically, these problems are called finite and infinite dimensional, respectively. Optimization theory for these two classes of problems can be put in the same form, but there are very real differences in the computational

70-9

_

techniques available for design opifzation. Since the subject of this handbook is com-

puter Bid to design, the practical distinction Is made here. For a unifying matihviat-i! M treatment, the rcoder is referred to Ref. 7. riysally, it is important lo re;aize that engineering design optmnization and engineering analysis are fundamentally different in nature. In -nalvyis. one is generally assured that a solution exists and numerical methods are generally stable. in optimal design, on the other hand, existence of even a nominal desian satisfving objectives is not assured, much less existence of an optimal design. Moreover, even when an optimum exists, numerical methods for its solution are often quite sensitive to initial estimates and require much computation.l art for iterative convergence. These properties will be observed over and over in this handbook when exampl problems are treated. It is important that the reader take a mathematical outlook when doing computer aided design and optimization. A purely intuitive approach can lead to erroneous results that may not be apparent until someone happens onto a nominal desipi which is vastly superior to a "sure'" " optimum design. 1-4 MATHEMATICAL PRELIMl"AARIES The level of nathematical background required for an understanding of the mnc.hods of optimal design presented in the follov. ing chapter-,, is a course in advanced calculus and th:e abihty to use matrix notation. Since engineers often require results of rather deep mathematical analyses to -Alve real-world problems, several results have been acc, pted with references given to proofs. The purr )se of this paragrvph s to present notation and 1-5

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AJVCP-06-19

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chapters is the idea of the derivative ot' a vector funiction with respect to its vector

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AMCP 706-192

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An important theorem in the analysis cf functions appearing in optimal ,esign problems is Taylor's Theorem, Taylor's Theorem: Let the real valued function fix) have k + I continuou's derivatives in Rn. Then for ..El?", there is a point ux~ + (I with 0 < a < 1, such that -a)y

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where by Eq. 14l1 the error in Eq. t- 12 is at most a constant times 1[y - x11 2 if fix) has bounded second order derivatives. The left side of Eq. 1-12 is generally denoted 'y Sf(x), where y - x is denoted 8x. in this notation, SA)=df S.(-3

Eq. 1-13 may be thought of as a total differential. Even for vector fumiltions g(x). Eq. 1-13 holds for each component so if

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15 ILLUS' RATIVE

light as possible for purposes of transportation and erection on the battlefield, or per

MILITARY

COM

military problems two are illustrative thisdesign paragraph formulated, and

haps mounting on a helicopter. A basic design sca the strtre shat the have an top than shall 0not mounted device angular hoptimal deflectiononofthemore radians, in

":omputer aided design technique,, are out14,,d for their solution. The treatm,.nt here is

order to hit the re.eiver or tarqet. The loading that is to be considered isiwn odo up

for the purpose of introducing concepts. onese examples are treated inmor depth in Chapters 7 and 8. oThe 1d5. 1oTIMAL .DESIGN OF STRUCTURES

to a given wlocity, which w..)ld cause angular deflection of the top of the tower.

T1-,. optimizatin technique I scribed in this pa ragrapih wa, ir'tially devieooped for application to minlmswaveight structural u probi:qs. adesignFor this re,-son, and to give

Further, the requirebent that the tower support the device with only a given allowable angular deflection s the only basic function required of the tower; thus the function

engineeriiig ;.'lfor 1I applic.'ion the technique, the sethod be p.ratedofalong

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such as the earth. A schematic of problem is shown in Fig. 1-2, The basic-'

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needs and objectives in this design problem are well established, so no addihional inputs need be considered at the present time.

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Four different conceptual designs are shown in Fig. I-3. The flrst two concepts,

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ent area and spacing as desired. Three are shown for convenience in the figure.

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area of the main structural members as a function of sieight. The second set of concepts, Figs. 1-3(C) and (D), involves towers that are pinncd at their base to the supporting

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structure and that are supported by guy wires at the top of the structure Likewise, in both

variable spacing of ti:e members of the tower. In two of the concepts, Figs. I-3(B1) and (D)),

~of ~main ~second ~length

In each of tlh conceptual towers of Fig. I-3,. the variables b1 through b3 de'_ nbe the

these concepts, variable spacing of the vertical members is allowed. In the

b4 through b6 specify the variable areas in the construction of the main vertical member.

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conlcept, variation of arca along the of the tower is also allowtd, h should be noted that the conceptual desigihs in Fig.

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I1-3 can have as many sutbsections with differ-

These variaIWes serve as design parameters, in that the designer can choosc these variables and completely specify the design of the tower. 1-9

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In addition Ato the desigrn vaiibles, a main part of thedesignprobleni is~the behavior of the. structure under wind,-load, since one of the major constraints on behavior of the structure is that the angular deflection of the top of the tower not exceed an angle 0. For this reason, the angular deflection of each of

ture, it is required that the design variables be bounded uniformly away from zero. This is given formally by the inequality

the joints must be determined, along with

problem is that the angular deflection at the

lateral deflection due to lateral wind loading. This is a relativ?,y routine analysis problem

top of the tower not exceed the angle 0. This is expressed analytically by the inequality

bi > blo > O,i = I,..., m.

(1-19)

The fundamental constraint in the present

when one uses the techniques of finite ele-

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structural analysis. Not shown in Fig. 1-3, but required in the construction, are cross members which maintain spacing of the design problemninathematically, first

The final step in formulation of an optimal tion to be minimized. In the present case, the

vectors of design variables b, and state

cost function is structural weight J and is given by an expression of the form

z,

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(1-17)

the dependence of stiffness on the variables is explicitly shown. Using this the structural response is given by the

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IY~ c b/

(1-21I)

wherey3 is material density and c are weighting factors3 rep;esenting lengths of structural elements and weight requir¢eme.nts for lateral stiffners. We now have an optimal st-,uctural design problem that is well formulated from a nrithematical point of view. The objective is

i.j.where

a is the wind loading matrix.

;Now Sdesign

that the relationship between the variables and the structural response is

o find the design variables bt through b m that satisfy constraint Eqs. 1-19 and 1-20, a~id which minimize the structural weight as given tDy Eq. 1-2 1. The technique used to solve this problem, and in fact a large class of' optimal system design problews, is based on a very simple idea of engineering design. The idea of the technique is to allow small variations in

specified by Eq. 1-18, the next step in formulating an optimal design problem is the identification of constraints. (n order to prevent dimensions or structural areas from going to zero, resulting in an unstable struc-

some nominal design, and analyze the effect of :he'. variations on the .'quations of the problem and the -.o,t function associated with the problem- Ab a result of allowing only small design changes, certain approximatio~is

• A (h)z = q

(0 18)

,

'

~L

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,

may tbezmade that allow the best change in desighaariables to be. determinfied'in order to decre ase, the cost function of the problem as much as possible,-while still not violating constraints of tho-deign problem. For example, one might, choose as an initial estimate of the optimal design a uniform tower as shown in Fig. 1-4. The estimated design variable in this case is denoted by MO ).

6bi > bo -b.

(1-24)

Or, if the angular deflection constraint is violated, for example, >0

(1-25)

then, to correct the constraint error it is required that &It< 0 - z1.

(1-26)

Finally, the change in structural weight due to the change in design Sh is given by m

6J

X c=,br

I"

//////// //7 /determine

Let 5b be a small change in the design variable b(° ). Any change in the design variable will result in a change in the structural response, denoted by 6:. The nature of the structural analysis problem guarantees that small 6b yields small Sz. Further, a Taylor series approximation of terms appearing in Eq. 1-18 yields A (b(O))Sz +,

(A(b)zb

(o))6b

=

0. (1-22)

If an inequality constraint is violated, such

bi < bio

(1-23)

then in oidet to correct the cons.raint error it is required that

(1-2')

The object of the new ptoblem is to 6b so as to minimize the linearized cost function of Eq. 1-27, subject to constraint Eqs. 1-24 and 1-26. Due to the special nature of this problem, the optimum change 6b can be determined in closed form. For a detailed derivation of this optimum perturbation, the reader is referred to Chapter 5. For discussion here, the results of this calculation will be denoted by 6b = 7B + C

(1-28)

where the vectors B and C depend on MO ), constraint errors, and equations of the problem. The parameter i? is an undetertained parameter that plays the role of a step size, when viewed in the geometry of design variable space. For example, if there are only two design parameters b , and b2 ,the dire,tion of the desired change is shown by B in Fig. 1-5, and 17 is a step size along that direction. In the terminology of optimization theory, B is known as thv direction ot steepest descent. It 1, analogous to tr. direction one would go downhill in ordei to reduce i-I I

"

, '"

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,

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-

"

.. .

, __ -. ___ ='_ __ _____________,___ L:

AMCP 706-92

b2

I ,

I

S

-the

... ' b

___________________b_

Figure 1-5. Direction of Steepest Descent his altitude as rapidly as possible. It is clear that on normal hills, as in most design cost functions, the direction of steepest descent change!, dependine on the location on that hill. For this reason, the direction of steepestdescent does not generally pass through the optimum point as shown in Fig. 1-5. There are many techniques for choosing the step size 7. The one used in the steepest

descent method if; based on requesting a certain reduction in the cost function due to changed 8b. This request, then, determines the step size 71and one can calculate 6b from Eq. 1-28. This 6b is the best change in This best the estimated dcsign variable b change is then added to the initial estimate to obtain a new est'nate tOat corresponds to a structure of less weight and that still satisfies the constraints of the problem, i.e., b

=O

( ) + 6b

This process is repeated as many times as required to obtain convergence to the minimum weight structore. The optimum towers for each of the four basic configurations chosen are shown in Figs. 1-6 and 1-7, with a table ot results being given in fable 1-1. These results were obtained

47

(A) One Design Variable

(8)Two Design Variable

Figure 1.6. Tower With Base k Puly Fastened to the Earth 1-12

(1-29)

.

,

. -

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7N

-7,

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AMCP 707-192

a finite element model with) appr 'niifiately forty, elements so that. th resv Linfg

I'sing

tY WASi bltobfmaterial, and, spacing., he sown in, Table. 'I1 crespondinw to no design variables aesimply the wegl,,s of the toershavig uniform me, ibe'.s andoptmur

no variation in spacing. -,Note that there is aI significant reduction- in structur& weight for the tapered optimum towers r trer uniform towers. Extensive examples of this kind are presented in Chapters 5, 7, and 9.

2,

1-5.2 APPLICATION OF TH1 SThEPEST DESCENT METV'OD IN INTERACTIVE CO1MPUTF a AID)ED DESIGN

(A) One Design (8) I'wo Design Variable Variable Figure 1-7 Tower With Base Simply Supported and Top Supported With Guy Lines

Very often in de,,ign rroblems, it is not practical to specify Asunique cost ffunction to be minirriized, hence tbe formal optimization problen described ii, par. 1-5.1 does not apply directly. The fact that the vector B in Eq. 1-28 is a direction of steepest descent, however, is extremely valuable information to a designer. The ipplization of this information to a structural design problem, using Iinteractive graphics, is a technique which shows considerable. promise in design

Consider, for example, the proleia treated in par. 1-5.1. The initial astimrite of tha. optimum tower was taken w~, a uniform tow,r. The components of the vector 6b can br, p-ojected on a cathode raiy tube, along with a picture of the structure as shown in Fig. 1,-8. The algebraic sign of .he components of bib, corresponding to eacli of the design variables, is an indication of the effect ,' change in that design variable will have on the cost function

)

TABLE 1-1 WEIGHTS OF TOWERS

Cantilevered

Cantilevered

Cantilevered

Cuv.ine Supported

Guy-line Supported

Guy-line Supporter.

hiumbfr of Design~ Variables

0

1

2

P~est Weight height

W 2440.61lb h=63.7 in.

W -2111.4 hmax = 91.4

W"= 1827.9 h =ax80,2

W= 1563.99 W =1356.6 h 46 hmax = 46.5

W -1265,71 ',a 36.55

A , 6 97

Ama

A 3.84

Ame

Cross-sectional area of member A 7.961lb

0

10 03

1

A =4,434

2

4.95

1-13

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information he can in altering the 'he otherthat ihand, the use d&signer Nas treWd

,bo

1

I Sb 4

ii

3

-able

0.

.

/7

6b6

7 7

1-8. Sansitivity to Design Variations

4Figure

of For example, if Ob t were positive, thisinterest. would indicate that an increase in the

Sweight.

dimension b, wil decrease the structural On the other hand if the algebraic sign of 6b, were negative, then an increase in 6b, would inctease the structural weight. Likewise, the algebraic signs of 6b through 5b6 indicte the effect that a change in these element areas will have on structural weight. These data give the designer valuable information, according to which he should change his rominal design to improve the structure, while still satisfying all the essential constraints. Traditionally, in structural design by graph ics, the designer puts areas and dimensions imo a structural analysis routine and then requests a stress calculation, the results of which are shown on the screen of a cathode ray tube. This technique has been used by Lock heed-Georgia in the design of the CSA. While this technique has been quite useful in structural design, it is extremely difficult i'or the designer with only stress information to determi 0Iould tochange overall reducejust one the he sttucture el ement ieinhow struc:ural weight. The difficulty con,es in the interplay between structural constiaints. if, 1-14

distribution of material in a structure, he can better use his experierice in making design inprovements. This capvbility can be invaluable to large-scale structu.al designers. It includes the effect of individual design varichanges on overall structural value, while taking into account the effect of that design change on all design constraints. In real-world structural design problems, the designer must design his structure for more than simply light weight. He icut -he concerned with structural vibration and buckling characteristics, since these are major sources of structural failure. Often, as ip par. 1-5.1. it is possible to determine design perturbations that have a desirable effect on such structuial properties as natural frequency and weight simultaneously. Both of these factors can then be displayed on a cathode iay tube as shown in Fig. 1-9. In this case bb I indicates the diretion in which the

be changed should variable design stru,tural weight, and Wb indicates tothereduce direcw tion in which the variable should be changed to increase natural frequency. This information can then be used by experienced design

6b

0

S

6 2

6h 2

7/

-

4

b2

"

11

Fiqure 1-9. Sensitivity to Two Performance Indicators

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AMCP 706-192

-persone! in making design changes that will have desirable effects on overall dii'caft structural PrOperties, fdr example. Y1,s is extremely important in large-scale sfructzral-design duo to the difficulty in determining the effect S,f changes in an individual design parameter on several different structural properties. ComputaCon of these data and interactive aspects of- ihe technique are discussed in Chapter 5.

-generate

This design technique is feasible from a computational point of view in that very little additional computer time is 'equired to sensitivity information from stress and vibration analyses that are required. While most structural optimization work has been done in the batch mode, it is shown in Chapter 5 that utilization of the steepestdescent technique with interactive graphics is a much more practical way to design structures, particularly in cases where several measures of structural performance are important. Development and display of sensitivity information in design is a form of information transfer to design personnel. This technique depends on the availability of interactive graphics software and nardware, which are currently being developed, 1-5.3 DESIGN OF ART!LLERV RECOIL MECHANISMS

tires. A photograph of the first prototype of this wcapon is shown in Fig. 1-10. V Tho recoil mechanism for this weapon was designed according to traditional recoil mechanism design goals. Namely, the objective in the design was for a constant -retarding force which is transmitted by the recoil mechanism to the undercarriage, as shown in Fig. 1-11. A recoil mechanism was designed which delivered approximately this recoil force R(t) as a function of time. When the weapon was built and fired, a nearly constant recoil occurred, as desired; but, at high angles of fire, the weapon exhibited unacceptable dynamic response. During firing, the tires of the weapon compressed and after firing and the subseque.it release of the recoil forces, the weppon rebounded off the ground approximately 6 hi. This unacceptable behavior required a redesign cycle for the recoil mechanism with a design goal of minimizing the dynamic response, or hop, of the .eapoti after firing. It was determined that the peak recoil force could be allowed to reach 22,000 lb without damaging the support structurm. The optimization problem is then to determine the recoil force R(t) as a f.inction time such that R(t)7 minmu at 3F i--D--f

2

f

(AMCPf7W(2-1)

for all x-An D. The junct~on g(x) has an absolute iwimumi at x if - A~x) has a~i absolute nminimumn there. The minimum iscalled- strict If only strict inequalifies hold in3 Eq. 2-1 for x 3.

01

1 Figure 2-2 A Cost Function

Note that fix) can have a strict absolute minimum at only one point in D whereas it

fi)Wihumangseasmposast

nimm t sveal coud aveanabslue distinct points in D provided it has the same valupoits.a atall hes

the regularity of fix) it is difficult to verify threuediqulte.Cndrtecaeo function fix) of the real variable x which

Defiitin Afuntio 22: fix deine on a subset D of R1 has a relative minimum (maximum) at Y if there exists an e > 0 s0o that fix) has an qihsolute minimum (maximum) in a subset of D whose points satisfy

has two continuous derivatives. The Taylor fruai ~ f)

Defiitinn22: frmua funtioi&)defned +

Example 2-2: Locate all relative and abso lute maxima and minima of qix) on 0 < x~ < 3, where &i) is giveni graphically in M~g. 2-2.

-and

The function fi) has a strict absolute maximum at x = , absolute minimna (not strict) at x =0 and 2, relative maxima at x-I 3, and relative mininia at x 0Oand 2. In Definitions 2-1 and 2-2 no continuity or differentiability requirements were placedl on

(-2

0 f Ix t ) + k~sl. It is then known that (k - 1)5 -. a 4 (k + 1)5 contains the minimum point and a more +

accurate result, if required, may be obtained by reducing 6 and repeating the process from

.. 6 2.6186

.. 5.2326

"" , i

9.4666

Figure 2.3. Function of Single Variable

Once the minimum point is restricted to some interval, this interval is broken up into three subintervals by inserting points located a distance of 0.382 times the length of the interval from each end. A test is then performed to see which subinterval the minimum point lies in.For a given subinterval the partitioning is shown in Fig. 2-4.

K ir

0.382 (a-ae)

aa

0,382 (%-

ab

0

Figure 2-4. Interval Partition 2-7

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AMCP 70IS-12

The search process is terminated when the

winimum, point is isolated in a sufficiently smaii LubintervaJ.

Nt

hta a

Z k 0

6

I

6 8

ks

,asl is kncwn.

The Fibonacci search method lias the property of being best in a certain sense among all

-Step 3. Compae x' aS and x1 + cbio and go to Step 4, 5, or 6.

search techniques which isolate a in an

as

such technique is the ratio of the length of ihe largest interval in which a may lie after n steps to the length of the original interval which contained ac. It is shown in R-f. 4, pageCiipt 253, that if f rx') + as) has a unique relative minimum as a function of at, then Fibonacci search minimizes the number of interval

fpartitions.

then ot, < P() < as,. By the choice of a,, and ab, the new points 04, a. and tv,,

no where ao Go to Step 7.

omputational

ii

P)+II'6(161)

S2 (l

k= b (l.6l8)1.

Z'.

=e Q+

0.018 (a

W. b

is suitably small, put +) aI stop..18k delete the primecs on cl

a , ae' and C and return to Step 3 DE-

The simplest and probably thle best known

Step 2. ('onputef 100 + Cbsl, wheic

ab

pula asad ' pu i'=a c,,un Return to Step 2.

2-4 THE METHOD OF STEEPEST SCENT (OR GRADIENT)

6 18 )k

(y, + 0. 382 (a

v).

T S 6(l61)ks)Otherwise,

Ther upper and lower bounds on cet ') are

k-o

+as

0.382 (a.'

Go to Step 7.

Step 7. If a',-,

/-I fa'I(a +~

Qu

+

have ~r=a,'

f[x'

aind l Soae tluat a0 abCo-'1pu(te fIX")'.v a' SI whlere a1b

Step 1. First an upper bound must be found for a, %~.It is clear that 0 isStp6Iff[)+CS lower bound, a,. For a chosen Ste small step size 6 in a, let i be the smallest integer such that f

a

Ster 5. Iff Ix") +a sI ) fix(0 + absl1, a 0 < CYl : (Y. Siiiiihir to the pr-nedure in St-ep 4, put a'

Th bmythen

be given in the form of algorithm:

Ix1

)cost ').

of the (direct miethods of minimization is the Mlethod of Steepest Descent (or Gradient). This mnethod is based on thc fact that if the surface ISsmooth0, then~ Its tnUll)Llt pl1an1e is a goodl approximation to the st'rface near the point of taIngency. The phiIoS Phy of' thle

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AMCP 708.192

V,

Method of Steepest Descent is apparent in its titl. 9e hane wshesto ~ y a inremn-dx in such a way that f(x), dx, is reduced as much as possible for a given length of increment. The directioa of the increment d:: is called the direction of steepest descent. The direction of steepest descen~t is giyvenY by Theorem 2-4.

Theorem 2-4: Let Aix) be differentiable in, W.The direction of steepest descent at a point£ is(where dx

=

VfT(-')

(Z-4)

as a better estimate of the minimum point and the process is continved until Vf [x(PJj 0o xi ~'iciently T~=~+ small This method may :,e given in compact ",)nn as the steepest dsetagrtm Step 1. Malke the best engineering estimate x(0 ) of the minimum point. Step 2. Compute Vf fx ] and define 2 nornialji.ed gradlent s

v 7~ 1

-~n'n1 Find o. '()wi minimizes f [xi') + (Ys i is the number of itea~tions completed). If Vf [x(") = 0, terinate the process end P) is D

where ai> 0 is a scalar !astc,.rlt.emiiu.pit The proof of Theorem 2-1 illustrates clearly that the direction of steepest ascent -is dX = VfT(X)

(2-5)

for & > 0. The reader should note carefully that Eqs. 2-4 and 2-5 give only the direction in the design parameter space R" which yields the maximum rate of change of &~). Since thle factor a is not determined explicitly, thle size of step is not specified. In order to start the steepest descent iterative technique, the designer makes the best estimate of the design parameter mvailabl, x(O. The gradient Vfijx(()) is then computed at P~ and a new point P~ ) is

dtermined by

-whcre a0 ;. 0 is chosen using methods of paf 2-3 so thait f IxO a~fT(~.(O)) I iSa,[ minmu a ~ fic~onof~. f /fI( 0 )1 0 then f Jxl Ij < f (0) 1 orl is taiken

Step 3. Put P-l = X(; UMS. If I ') and 11V f Jx" '] II 1 are less than predetet mined limits, terminate thc process and let P~4 ') be the approximation to the minimum point. Otherwise refurn to Step 2. It is interesting to note that successive. direct ions of steepest descent are orthogonal to one-another in this algorithm -- i.e., VfAx (j+ I1,V/fT IXt" = 0]. To see this, recall that &(') is chosen so that ffx(P) - as] is a mm'amum in a. Trhe necessary condition of I iieoremn 2-1 then requires I n Da 1-fW )1 ax'i~ a IxI = 11Vf X III

whtich was to be sho'%n. 1

In the case where X 2 JFig. 2-5 is a v-ew of the design variable space. Tile closed 2-9

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J)kMCP Y(I0-2

2

(3) If Y is thc only point in S for which Vfl) = 0, then x( converges to Y.

N .(1)

j

Ssay r

77 X

Figure 2-5. Descent Steps curves in this figure -re lines of constant f(x), A relatively general convergence theorem pertaining to this algorithm will now be stated. The proof of this theorem may be found in Ref. 5, page 80. Theorem 2-5: Let f(x) be a continuous function defined on R' and let x (° ) be any point such that the closed set '

X I AX) < fiX(0 ) I

S

}f

is bounded, and f(x) is twice continuously differentiable on S. Let the mateix of second deri-- off'x), f ,x)-

Several things which Theorem 2-5 does nrt are worthy of note. First, the theoem does not guarantee that the sequer,,e of x 0 g,nerated by the Meihod of -points Steepest Descent will converge. Further, unless the assumpticn of (3) holds, the sequence need not c,,nv,:rge lo an absolhte minimum; it may converge to a relative r~inimum. The choice of th ;nivial estimate x( ° ) can have a great deal tr/ do with the limit point of the algorithm if it does converge. If it is not known befcrrnand that a Lnique relative minimum e.ists, It is general practice to start the iterative process at several initial estimates. if the sequence P) converges to the same point Y each time, then one is led to believe that he has indeed found an absolute minimum. Logic .,;ch as this can cause sleepless nights, however, particularly if a decision involving considerable resources and perhaps even one's job depends on the outcome. For .this reason, the importance of at least making a serious attempt to apply theorems svch as those of par. 2-2 cannot be overemphasized Theorem 2-3, for example, if properly applied, may prevent many anxious moments.

?

satisfy the condition

In spite of the simplicity of the Method of Steepest Descent, it has several severe restrictions which are discussed in Ref. 5, page 159.

;yTHy I< MyTy

These are: for so'ae M, every., in R", and every xinS. Then for the sequenice [xP I generated by the steepest descent algorithm:

(I) A subsequence P"r point

)

converges to a in S for which Vfl.V) n 0.

(2) f lx(tin)l decreases monotonically to

I. Even though convergence may be guaranteed by Theorem 2-5, an infinite number of iterations may be required for the minimiztion of even a positive definite quadratic form. 2. Each iteration is calculakd indepen dently of the others so that no information is

2-10

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AMCP 706-192

stored which might be used to accelerate convergence

matrix is required H(x):: V 2f(X)

3. The rate of convergence depends strongly on properties of the cost function. If the ratio of the.largest and smallest eigenvalues of the matrix of second derivatives is large, the steepest descent algorithm generates short zig-zagging moves. Convergence is, therefore, very slow. For an bxtensive treatment of modifications of the steepest descent method, which prevents certain of these difficulties, see Ref. 4, Chapter 7. Several methods, pireented -in the next three paragraphs, doot suffer so severely from the problemisjust described. 2.5 A GENERALIZED NEWTON METHOD In the Steepest Descent Method of par. 2-4, only first-order derivatives that determine the tangent plane of the cost surface are used to represent the behavior of this surface. One would expect that if second derivativs of the cost function were avilable, then a quadratic approxifunction could be constructed as ant mation to the surface. The quadratic approxiroation should allow for much betty:approximation of the minimum point of the cost function. !"

afx nxn

Note that it is implicitly assumed here that fix) has two derivatives. By Taylor's formula, f[xt 0 ) + ax] -fixo)] + Vfix()]bx 1 (2-6) + -AxTH[x(°)]Ax 2 ( ) In case f(x) is where Ax isa change in x locally convex - convex in a neighborhood of ) is .0)- Theorem A-3 shows thatH [ I] posiive semi-definite. If, in addition, -

H [x(0 )] is posie definite,~then it has an 0 -h° +A'. in Eq. 2-6 is ,[x( -u inverse. convex in,-xso Tiini2-3 insures the existence 6fa unqi-;e niihium point of the quadratic,- funtion'in Eq. 2-6. By Theorem this unique riinimua point is determined 2:-1, by VfT xt0)J +H[x(0 ) Ax= 0 or 'Ax

HI

ix") IvfT IX(0 )1

(2-7)

a id the new estimate of the minimum point is AU) =x(O) +AX.

The idea of this m.,thod is to first use a second-degree Taylor for.ula as an approximation to fix). If f(%) is cz.ivex, rjust convex near a minimum point then the minimum point of the quadratic should be near the minimum point of fix). The minimum point of the quadratic approximation is then determined analytically and is taken as a good approximation of the minimum point of fix).

( ) Since Eq. 2-6 is just an approxi:ation, x l will probably not be the precise minimum poYo of fix). Realizing that evaluation of l1(x) requires computation of n (n + 1)/2 second derivatives of f(x), one might be tempted to improve the estimate for the minimum point before recalculating all these derivatives.

In order to utilize Taylor's fornwla inci. iie , ond degree terms, the following

An easy way of improving the estimate of the minimum point is to change the length of

-

2-11

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AM4CP 706-192

the sep Ax without altering its direction. The scalar of _ I will be determined by methods of par. 2-3 so as to minimize f!x(0) + aAxI. This procedure may now be put down in the form of a computational algorithm called Generalized Newton Method:

2. In nonconvex problems an iteration does not necessarily decrease f[xP' when the current iterate P) is not near th, minimur ,oint.

Step 1. Make an engineering • .. estimate x(O) of the n"imm point of f(x),

po m Hx) a engineering egei 3..Fr Frt many problems, Hix) will be extremely messy if not im-

Step 2. Compute xY+',

=P

-

a=

[Xt1 ) VfT[XV) )iconwc

f x(i)_-aH- I[x(iI VfT 'x( I as a function of a. Here, the index i is the nuber of iterations conpleted. Step 3. If 1Kf [.c(" 1ll and IIx + ) -- xU)ll are suffciently small, terminate the process and take x" ' 0 as the minimun point off(x). Otheiwise, return to Step 2. rtr Se2 The Generalized Newton Method presented in this paragraph has been called the best for minimizing convex cost functions when second derivaives are available (see Ref. 5, page 162). Even in the case in which the cost funtion is nonconvex. if the starting point x(° ) is near enough to a relative ininimium point so thai the cost function is convex at Sx() , then good convergence may still be expected. In spite of the advantages of this method, it still has several shctcomings. 1, Even if f(x) is conve(

2-12

ai inverse cf

j

I.

Even in nonconvex minimization problems the Generalized Newton Method may be used in conjunction with a Steepest Descent Method to form an extremely effective tool. The Steepest Descent Method has the property of making good progress even though only a poor estimate of the minimum point is available. As a relative minimum is approached, however, the rate of convergence of the Steepest Descent Method decreases. At this point, however, the cost function should be convex since a minimum point is nearby. Therefore, the Generalized Newton Method may be employed for rapid convergence to the relative minimum point. 2-6 METHODS OF CONJUGATE DIRECTIONS In par. 2.4 it is pointed out that the Method of Steepest l)escent had rather poor convergence prof erties in many problems because it uses only first-order approxiiations (involving only first-order derivatives). Further, the Steepest Descent Method is not a learning process in that it does not store information from past iterations. The first deficiency is corrected in par. 2-5 where a Generalized Newton Me liod employing second derivatives is presented This method, while haing outslfaning convergence properties, requires the comnputatin of n(n + 1),2

-

1'

,,-4 "1l

I

possible to compute efficiently.

d) where -minimizes

s

H(x) may not exist unless H(x) is strictly positive definite.

,o,,

I

'

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s

AMCP 706-192

Sin

S

~

4

second-order derivatives at each iteration (.x is R") lit most engineering desi_2P problems this is an extremely tedifous;-if not impossible, task. Further,-' he Generalized Newton Meth0d-is not a learning process.

vectors S! are linearly independent. To see that this is true, form the linear combination

The methods presented in this paragraph require the computation of inly first derivatives. However, by makinp Lise of information obtained from prev-,us derivatives, conas the minimum is tipvergence is o f proached. In fact, as one of the methods pi ogresses, it develops an approximation to the matrix of second derivatives. In this respect the methods here have the desirable features of both the Method of Steepest Descent and the Generalized Newton Method.

where the a, are scalars. Multiplying this sum othletby S/TAyed

All Methods of Conjugate Directions are based on the philosophy "if a method works well in minimizing all positive definite qua-2 dratic forms, then t ought to work pretty well on any smiooth cost function." To be more specific, Conjugate Gradient Methods are guaranteed to minimize any positive definite quadratic form in n iterations (tile design parameter is in RFZ). Although this Wdeal behavior will not carry over to general cost functions, since a convex cost function often looks very much like a positive definite quadratic form, similar behavior could .be expoctedl. Experience has shown that this is tile case.

-

ii0

n1ST :1PaSPAS!

mari

the conveA function Ax)

=

BTx + .L Ax

hoet.16LtSO..,"Ibnnzr Tveorm in Let which . bronue nonzer vesecto ile whitc arfie natei wit Choose scalar, X = ".i =0, ..., n 1,r sticcessively which~ minimize 1.00

+

is po\*%t

A- 0

x1 o (2-8)

definite, thle

Q

~

'~'I%

Since A

1+(XS,0)

where f(x) isgiven in Eq

-and

-

(2-9)

where x is in R", B is an nzx I mnatrix and A is a synimetric positive definite, nix n matrix. The central idea of all methods based on coniuga',c directions is contained in Theorem 2

1, ... n c nonzero vectors inR". are called .oi jugate with resp~ect co A if !

0

Consider now the problem of minimizing

Definilun .2%5:Let A be a symmetric

r1A

AS1

ansicS/A 0 ,al .Sneiws abtar,01 ,j0,I n-,adthsi jutt.efiio flnaidpnec.

In order to be more precise, one inaI~es posiiod~nit -5.

aI S

conjugat(

abo.

te

Iit

~

any poil n III R, p.l..t

two ileihlods t:idI

011

is tile

'/WmR

follow m .irv

i~

2

Downloaded from http://www.everyspec.com

AMCP 706,192

based on different weys of generating conb~~naf,. jugate directions. There are an unlimited

5 k+1

k A S' S+1 r _____ -"' r-o STr AS

=

number of ways to generate conjugate directions. Several ways are discussed in Ref. 6. 2-6.1 THE CONJUGATE METHOD

S':

Many sets of vectors v, could be chosen to

GRADIENT

Given any set of n linearly independent vectors and a positive -'SOdefinite n x m matrix A a set of conjugate directions with respect to A

generate conjugate directions. A natural choice, the set of gradient vectors of fix), however, g'= Vff is x(), where P) are defined inTheorem 2-6. Define =

_0 g

r0

-.

can be generated by a Gram-Schmidt ortho-

kg+TS

gonalization technique. Let v° , ... . be linearly independent vectors and define SO = Vo Now put

,--'.

:7S

gP + E

-

Si

-

(2-12)

S.

SITA

1. 0

: ;

Alternatively, gk+l =_,s*+

+ k

SOTASI =0 Sgl

=

STA1v

°

+aIoS

)

Sincef(x)

and alo

(2-13)

g+,TASs"

o STA S'

For A-conjugacy, it is required that

2'IXTAx +BTx,

= Vf[x ( k ) ] =Ax ( ) + B, or from the proof of Theorem 2-6,

-SOTAS

=_vTASO ASOS

S*

i

=l

+

.

I + ak+I.o'r + ..+ O'k+ I.,-k

For A-conjugacy itis required that Sk+ITA S==PT

i

=g1 +1 +A L

AssLningS',.....S are A-conjugate, put

+Ck+,rST A S"

(21,4)

.

Now, T.,T,,

,&+

i

=0,i< k due to A-conjugacy of the S'and

(2-15)

where the second equality holds by S-con-

jugacy, so otk+,r

Vflx(k Vk+I TA Sr r S ,r

)ISk = 0,k = 0... ,n - I. (2-16)

!

From Eqs. 2-13 and 2-14 ..k

-i i

gk Tg, = gk T By induction, the resulting direct'ons are A-conjugate and

2-14

S + E

I = 0,1 k

S

I

giT A S1 i

2

Si S

Si'AS (2-17)

Downloaded from http://www.everyspec.com

AMCP 706-192 Thus

Thus the g',i o, 09 .. ,-lr indepindent and the S =10, 1,

~ >I

kIT

ierygI+TS n..-nIlare

S

k19tI gTI ITAS

l

A-conjugate.

...

By Eq. 2-17, for 1 < k, the right side of the The Conjugate Direction Method of ThecabveqtinszroFr rem 2-6 may now fie applied using the conjugate gradients A' The result is called the gkIAS k+1Tg+ Conjugate Gradient Method. In orderto applySkAkTk this- Method to nonquadratic problems, it is g g first necessary 1o eliminate explicit depenSubstituting this result into Eq. 2- i 2yields dence of the algorithnm on the form of 1(x). By definition,Sk1-g+

gi' 1

Axl' I +B =A Wx + ?U)SI)

+sB

or

Eq. 2-20 now gives an algorithm for determining the conjugate directions, even without knowledge of the matrix A. For a general function .1(x),

(2-1)

By Eq 2-16unconstrained By Eq 2-16Conjugate

g

VfT[X(i)]

and thc following algorithm for finding the minimum of ftx) is called the Gradient Method:

/l+ITsI=0 =gI SI + ?XU) SITA

Step 1. Make an engineering estimate x(O) the minimum point and Loinpute

Thusof SSo

Subsituing or 1i fom E. 212 ad uingStep

7vfT [X(o)

Subtittin S fro Eqfo 2-2 ad u~gminimizesf

2. For i

=

0, 1,

Eq. 2-15, this is gITS! XU)

..., find a = a~i) which [P + aSil.

Step 3. Compute =

9(2-19)

From Eqs. 2-18 and 2-19

Si-I

.5/

VfT1I

1)1 + 3'Si

where. gg No~v,

=vf[x"+

I) IVf T v('+ I) flX(L)iVfT IxX'il 2-15

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'25

AMCP 706.9

Step 4.

Terminate the process if 11 11 III+' and IJx(+ 0 - xU) are sufficiently small. Otherwise, return to Step 2.

little insight into use of the method. For a direct proof of convergence, etc., the reader is referred to Ref. 7. The computational algorithm is:

When this ilgozithm is applied to problems in wh'z!h ff.-) is nZA of the form of Eq. 2-9, will not occur in it steps. Fletcher and Reeves recommend that after it steps the algorithm should be "restarted", i.e., X(n+i) should be treated as x(0 ) in the algorithm. In a sense, the first few iterations of the algotithmn build up information about the curvature of the cost surface. After n

Jconvergence

Step 1. Make an engineering estimate x(O of the minimum point and choose a symmetric positive definite matrix H1(O). Step 2. For 1 0,

.. ,compute

~) 0

S(=

VJT [P() 1.

iterations, this information is disci'rded and a new estimate of curvature is built up during the next n iterations. This method then does

Step 3. Compute a f[()+ aS~') 1.

not accumulate information about curvature; of the cost surface over the entire iteratv',e

Step 4. Compute

which minimizes

V

process. 2-6.2 THE METHOD OF FLETCHER N2 POWELL

X+ t

of conjugate directions was suggestel by Davidon (Ref. 8) and modified by Fletcner and Powell] (Ref. 9). This method is repo-ted to be onec of the most p*., .-ful known methods for general functions fix), (Ref. 10). A major reason for the success of this me hsod is its capability to accumulate informa ion about the curvature of the cost surface dutin the entire iteralive process, even thiouL:_m ( nly

first ordler der~vatives of the cost func (on need to be coi% ited. The directions SO generated by the i1g riftin that follows, are coniugate if ,(x) is of the farm of Eq. 2-9. This proof is given n Ixefs. 7 and 9. In Ref. 6 it is shown that t ie mnethod of Fletcher and Powell fits naturd v into a large class of conjuigate directioni mnethods The derivatinr is tedious and lends

)

U)+

'

t, +t)C

If") + A (1) 01I

I/('

Asecond method based on a differer iset

where y0

V

A

aI

J~

(+I V ( fl~~ j-VTXi

a (1, T Y U

0I)'

.-

1

()IykJti0

Y

Step 5. If 11Vfix""')111 and lixti+ x''ill are sufficiently smnall, terminate the process. Othierwise return to Step 2. -

Fletcher and Powell (Ref. 9) prove that this l the following propert ies algorithm h&

Downloaded from http://www.everyspec.com

AMCP 706-192

I. The matrix HO is positive definite for all, f.This implies the method will always converge to a stationary point since

n, which I. ,k minimize f!x(k ' 1) + ask I

Step 2 Find a -

d fix(0 +aes(,,) 1 !0

d =-

ffx")IU )vfT[x(i)

< 0

provided Vf[x t1 ) ] 0. This means that f[x (0 I may be decreased by choosing a > 0 ifffx( ) ] *. G.s

.

. ,.,-

-

Step 1. Make an engineering estimate of the minimum point x(° ) of fix). in. .n, Choose vectors s" the coordinate directions of Rn.

2. When this method is applied to the positive definite quadratic from Eq. 2-9, ( G Iconverges to A "'. This method might be called a learning process in that only first derivatives are ever as the Agorithm progresses an compute I, ",ut of the matrix of second derivaapproxi rated. Many experienced retives is searchers a. the area of optimization methods laud this method as one of the best available

""

where Y k =k-I +aok sk k

k

.,Io'.

"'0

n,

and i is the number of iterations which have been completed. Note that in the one dimensional minimization for ak, it is possible for a< 0. "% Step 3. Find the integer m, I < m< n

-

which fAy"- 1)

fy",)

is the largest and define 2.6.3 A CONJUGATh DIRECTION METHOD WITHOUT DERIVATIVES Occasionally in applications, one is faced

with a problem in which computation of derivatives of the cost function is impossible or at least prohibitive from a computational point of view. There are many techniques for solving this sort of problem given in Ref. 4. An efficient technique, not presented in common texts, was developed by Powell (Ref. II) using conjugate directions. A computational algorithm is presented here without proof. For a proof that the algorithm generates conjugate directions the reader is referred to Ref. I1.The computanis' tional algorithm

Afym

1

3

) _f(y, ).

Step 4. Define ft =/0,0), f2

f3 =fl2yl

ftyn), and

0).

Step 5. Iff 3 > f, or (f - 2/ +f ) x (V, -f2

A)

A 2 put .

) yn

Y l'ermnmate the process if !I 11 Otheris sufficiently "Imnat 2-17

Downloaded from http://www.everyspec.com

AMCP 706-492


0 .

where the matrices A, B, and C are the same as in LP, and are called the dual of LP. The results of Theorem 3-1 relating LP and LPD are proved in Ref. 3, page 41, and Ref. 4, page 118.

~Then

I

I

Theorem 3-1: Let x and y be in the coistraint sets of LP and LPD, respectively,

I. Cry , BTx.

(3-10)

(3-11) 2. If Cry = BTX thenxandy are, the solutions of LP aid LPD, respectively. 3. If LP (LPD) is unboupded, then LPD (L.P) is infeasible.

4, If LP (L'D) is feasible and LPD (LF) is infeasible, then LP (LPD) in unbounued,

feasible. Then both have olutions 3 and Y., respectively, and B Ty = CT'. The proof of Theorema 3-2 is involved and does not yield a method of constructing solutions. It may be found in Ref. 3, page 44, or Ref. 4, page 118. Since the solution of LP must lie on a vertex of the polyhedral constraint set, i suffices to check at most a finite number of points for the miniknum, This procedure is followed in an organized way by beginning at any vertex of the constraint set. If the cost function cannot be decreased by moving along an edge of the polyhedron that intersects this vertex, then this vertex is the solution. If, however, the cost function decreases by moving along some edge, this policy is followed until a second vertex is reached and the cost function has been reduced. Since there are only a finite number of vertices and it is impossible to return to a previously occupied vertex, the process must terminate at the minimum over the constraint set. In order to illustrate the argument presented in the preceding paragraph, consider Example3-2. Example 3-2: By moving along edges of constrain! set, solve the LP

te

2x I -

minimize f(x 1 , x 2

subject to

These results are useful in constructing solutions of linear programming problems.

-X, >

Tw'y are also used in providing Tl-eorem 3 2 that i central to linear program.

3-4

,g theory.

--

2x1

-

2x

-

3

X

:::.S 'A

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AMCP 708-192

x ,X2

........ ..

> 0.

Solution: The polyhedral constraint set is shown in Fig. 3-2.

The only move admissible is toward (1/2, 1). A unit move in this direction is obtained from

x2

A121 X [~

2

which causes a change in f,

->2

., : Figure 3-2. Polyhadral Constraint Set

df

Vf(l,l/2)dx

+ VT-

-

->0. 2

moving Therefore, f may not be decreased in .. Tai

The vector

from the vertex (1, 1/2) so this point is the

The vector

solution of the problem.

whose direction as shown in Fig 3-2 is the direction of steepest descent of f(x). Starting

vertex to vertex is of moving from ideavisualization Thefor good but is poor for higher dimensional problems. The same idea, however, can be implemented algebraically. In order to obtain relations which will be required for solution of LP, define slack vari-

at (0,0) a unit movement along the x, -axis

ablesu 1 ..... u m so that

-otVfr(x

,x)= O

[2

:

yields a change (3-12)

Ax-C=u00. Vf(O,O)dx

S-df

-

2

and a unit movement along the x2 -axis yields a change

The cost function of Eq. 3-5 will be denoted by the variable (3-13)

, ,,w=Brx.

df

Vf(0,0)dx-

I

so both moves yield a decrease in fix). Choose

The problem LP now takes the form Ax - C - u

the xI-axis and move to the first vertex (1,0). The only movement rossible is in the

x> 0

-direction from (1,0). A unit move in this direction yields

u> 0

0

LP'.

+ X2

df

Vf(l,O)d

=

I

which decreases f. Move in this direction to the first vertex (1, 1/2).

-

R T Y=

I

minimum

The solution of LP' is the same as the solution of LP. 3-5

Downloaded from http://www.everyspec.com

.......

o AMCP 708.91f

-

The information contained in Eqs. 3-12 and 3-13 is contained in the following matrix equation (called the simplex tableau): all 1112

ali

...

1 -CI

X1_

Theorem 3-3: If in Eq. 3-15 b' 0, i 1, .n, and - C1, 0,1 =I m,then the solu~tion of LP is

IIIl

I a21 a22 ... a2,

C

X2

U2

(3-14) a,, I

ar2

...

a,,n

b,~

b2

...

bI0

Cm

using a method which is based largely on Thorm3.

X1"

Urn

I

W

s.c, W

,., . '

It is clear from this theorem that any Eq. 3-14 may be viewed as mn+ Iequations invovin thevarable x1 ...,x,, u1 . rns w. At present Eq. 3-14 may be interpreted as

method of choosing the vaiables Y, and r which will terminate with non-negative entries in the last row and column, except perhapsj fr6 ilsrea ehdo ovn P Before developing such a mcthod, several definitions will be helpful.

deteminng u.Ur~ an w xpliitl in terms of x1 ,... x, It might be desirable to determine some other combination~ of in + I of intems he ariblef te rmaiingn.Definition3-5: In Eq. 3-15, the variables Except in singular cases, this is possible. sPj 1.. in, are called basik variables, while Assuit ip tflat in + I of the variables sj. smrnW' anc fiI determined explicitly in terms of the iing n variables rl,..., r,n Eq. 3-14 wi!1 tie e ihe forma

the variables r1, I vrals

1__n are called nonbasic

.',e

ra'

a-, II22 .. Z

a1

a2

..

,..

a1

~0,

.

,

1]

2

'

2

L Sjpoint

1r

(3~15)

a.1 I

an2

b'Ib

'7 a

b

C', 6

"

S

LJI

where pinies denote coefficients obtained when the original set of equations is solved for rI . _Sm , and w. The solution of LP will be constructed 3-6

Definition 3-6: The set of variabless. S I rl, . r,, will be called a basic point. If c, 1_.., it, in Eq. 3-15, thern the basic will be called a basic feasible point. A certain geometric interpretation may

now be given for the nonbasic variables. In LP' it is clear that the boundary of the constraint set of LIP is obtained by setting

various combinations of the vat 'bles x1,i ,

it and u,./ Iit, equal to zero. !n the space R" of the design variable x, ., vertex of the polyhedral constraint set is obtained by having it equality constraints among the xi. i = 1,.n i, enforced. By the discussion, this occurs wheun r, = 0, i1 -. An edgeof this polyhedron is a line in R" obtained by setting r,0 for n I indices i. From Def. 3-6 and -

Downloaded from http://www.everyspec.com

AMCP 7C48.192

*

Eq. 3-15, it is clear that a basic feasible point corresponds to a vertex of thd Polyhedral set. This is true since setting the.-nofibasic van-[ ables of the basic feasible point equal to zero 2yields adiissible basic Variables. Further, two. verices lieojithe:samre edg6 of the constraint' -set- if they' have iin I of ti-.eir nonbasic6

The prociess fat interchanging the roles of-a basi an a.nonbsicvarablethu beom~ the or entalethos tol bsed n Teo~ :rern 3-3. Suppose it is desired to make ara nonbasic variable and ra basiF, variable. If a; * 0 then the At equation from Eq. 3-15, + +a',Ir 1

cis C-

ar,

+.

a'~~

k akn-

i

L

'

c4 -k

Sk.

a",

is thus clear how the coefficients inEq

varabls i~ cnixon.It

ar

[

+...

3-15 change as, the roles of a pair of Variables are interchanged. This process may be described concisely in the language of Definition

Definition 3-7: The entiy a;,, * 0, prodeding Eq. 3-16, is called the pivot of the transformation. The trarsformation itself is called a pivot St p.

may be solved for rlto obtain a; a,.iThe

c'~

j~)

ct a,

effect of the pivot step on the coefficient matrix of Eq. 315 niay be illustrated

'-

easily by the diagram a .

-*SI

+

rl+

(3-16) a 1 , ;,

I1

or ,, r ma be Usin ths exresion eliminated from the left sides of the remaining equations in Eq. 3-13. For k * this yields

O

p~~iF -~1(3-18) The diagram shown by Eq. 3.18 simply relates ta ntecefcetmti fE.31 h floigcagsocr h io srpae follrowin as t

LAI +

+aki

k

+[a a,, Ik

d~]negative

a,,. Iak, a j

-

a,,+,a,, .

Ii (3-17)

-

ge

are urhe pivotmutilepbyated

inverse of the pivot. All other elements in the same column as the pivot are multiplied by tlw. inverse of' the pivot. All other elements in the matrix a.e decreased by thle product of the element in their colun and the tow of the pivot, the element in their row and the column of the pivot and the inverse of the pivot. 3

Downloaded from http://www.everyspec.com

: ';

AMCP 706-192

Example 3-3: Giv:

'

-:

I'..

j'!l5

13

given stage of the solution process and the primes of Eq.3-1 5 are dropped, i.e., =

2ra.!I

Givea 1~np-4C-m-

,.

iioi

2.

interchange the role of rI and S2

_

--

F"2/3

3

1/3

13

-'

",

2

.applied

1/3 -2 -,i£"_ _"..'5/3

1K

14/3

:-'

r2

11 !3..J

=r

Lw.W.l

The pivoting transformation is an organized tool which allows oue to interchange basic and nonbasic variables. It remains only to obtain an algorithm which uses this tool and Theorem 3-3 to construct the solution of LIP. . 3 THE SIMPLEX ALGORITHM As was shown in par. 3-2, !he solution of the linear programming problem may be reduced to the choice of pivot points, The algorithm presented here will have two phases. The first phase will co~asist of an algorithm for obtaining a basic feasible point. The s,.cond phase will operate only with basic feasible points and will successively reduce cost function until the hypotheses of ,,rem 3-3 are satisfie6. For

j

1

now be used to denote the

coefficients that result from a pivot step to Fq. 3-19. These new coefficients

are determinui by applying Eq. 3-18. 3-31 DETERMINATION OF A BASIC FEAPOINT If some elements in the right-hand column of the matrix of Eq. 3-19 (other than 6)are negative, then the present choice of variables is not a basic feasible point. Let - ck be the negative entry nearest the bottom of the column (again eyduding 6). Since when r= 0, / = 1...n. sk < 0, if there are admissible points in tlhe constraint set of LP, then it must be possible to increase sk by increasing some r,frcm zero; i.e., there must be some positive akj Choose Jo so that a 1 > 0. This fixes the column index of the pivot. To find an admissible row index io,consider first that after the pivot step ,, C o CIo = a ° /o It is clear then that candidates for the pivot U must be limited to indices i for which

onvenic;ce in the discussion which

follows, it is assumed that the choice of basic and nonbasic variables has beea made a, a 3-8

Primes will

-:SIBLE

It is shown in Ref. 3, page 53, -tat this pivoting transformation preserves the dual linear programmhing problem.

6

bn

.

Solution: The new matrix relation is

( -9

-

a,1o

0.

(3-20)

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---

r

-

AMCIP76-92

T

With this restriction in mind, consider the t values of c after the pivot step with i 0 o. These are

The nrocess describe may be given quite simply as the iterative Algorithm LP-A: I. Choose - ck as the lowest negative entry (with the exception of 6) in

!Step

(321)the

,+

right-hand column of the coefficient matrix of Eq. 3-19.

a 10 1i ;

In order to insure -c,

> 0, i > k, it is

Step 2. Choose any positive element ak/°

that c +

4

ir

-

S

> 0, i > k, ii.

(3-22)

If all ;k 0 this clearly holds. If a11o < 0, however, the -equirement, Fq. 3-22, may be

row of the matrix of Eq.

Step 3 rhoose 1i as in Eq. 3-25. Step

Perform the pivot step with pivot

-ewritten as c....(

i > al, a

1

o

, i > k, ii

0

.

323 (3-23)

Further, for i =k.

since akl

°

has been found and the process may be terminated.

+c>3-24 a

c

3-3.2 SOLUTION OF LP In par. 3-3.1 an algorithm is given for

> 0.

Inequalities, Eqs. 3-23 and 3-24, show that if i 0 is chosen so that

a, 0 =

0, 1= 1, ..., k, choose Step 5. If an- that one with largest index i and return to Step 1. If- c, > 0, i = 1, m, then a basic feasible solutio',

_. ,

(3-25)

010

finding a basic feasible point. Once this has been accomplished, the object is to find a second algorithm which successively reduces

Since by Eq. 3-19, IV = b r +... + b r + 1 6, it is clear that if bo < 0 for some J J0 thvn IV may be reduced by increasin, rio from

- ck . If - ck then - c, ;t 0, 1 > k and -is still negative, the proceis m.,y be repeated. Otherwise choose the next entry above -- ck which is negative and repeat the process.

If all the c,, i > k are ionzero, only a finitu number ,I' basic points are possible since the process is monotone (nonrepeating). If there exists a point with - , > 0, i = , ... ti, this process must find it. The legenerate case in which some c= 0, 1 k is dis:ussed later.

zero. If a pivot step is performed which makes rio a basic variable then iv will be decreased The choice of the basic variable s, which is to he made nonbasic must be made in such a way that the point obtained after the pivot step is still a basic feasible point, i.e., so that in. However, this is - ci > 0, i = , restriction which led to the 0,e precisely choice of i o in p,.n 3-3.1. Therefore, th- ;..ne procedure for choosiop l may "c employed here. 3-9

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AMCP 706-192

Since w' = w - cioblolaoo, the pivot step determined here guarantees w' < w provided all - c, > 0, i = , ..., m. InI this case, therefore, only a finite number of pivot steps may be made, and the process n Tst terminate at the sokItion of the linear programming problem. Termination occurs when b; 0,1 Theorem.n. 3-3 shows that this is the solution of the linear programming problem. The degenerate case where some c,= 0 will be discussed pa7. 3-3.3.

Viewed geometrically, the difficulty occurs because the path which successive basic points follow on the polygonal constraint boundary may form a closed loop. To prevent this behavior with only a small error in the final solution an entry, - c, which is zero, is replaced by an arbitrarily small parameter e > 0.The problem is not degenerate any longer and cycling cannot occur. Therefore, the altered probiem will proceed toward the solution.

This orocess is given explicitly in Algorithm LP-B

Example 3-4. Use the simplex algorithm to solve the LP

Step I Choose any negativi entry (except 5) blo in the bottom row of the coefficient matrix of Eq. 3-19. Step 2. Choose io according to Eq. 3-25 with k = I.

minimize 2xI + 9x 2 + %3 subject to xi +4x 2 +2x 3 > 5 3xI +x 2 +2x 3 > 4

Step 3.Perform the pivot step with pivot a.ol o"

x 1 -0

Step4. If any bl < 0. = i.n, choose one b, < 0 and return to Step 1. Ifbl , 0, ] = 1... n,then the solution of LP has beer found.

X2 > 0 X3 > 0.

First, LP' is: 3-3.3 THE DEGENERATE CASE minimize w where In both pars. 3-3.1 and 3-3.2 the computatioral algorithms could have problems if some c,= 0. This situation is called degenerate since when n constraints are made equalities by putting r, = 0, j =

I , one has s,= c,

which means that still another constraint is an equality. The degeneracy arises from the fact that ii, Li' the a dimensional design variable x = (xI ,...,xn) satisfies n +1 linear equalities. Therefore, the a + I equations are not linearly independent.

I

4 2 -5

3I

2

2 0 1

1

l

Li

4

0, Li J

subj -ct to x310 0, i=1,2,3,

3-10

Ff1

0,,l2

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For the first pivot step in algorithm LP-A,

pivot step leads to

largeond row. = 2 is the only choice available in Eq. 3-25 and c24

4> 0.

L17/6 -7/3

The pivot is a2 2 = 1. This pivot step interchanges u2 and .x,.The result is -

9

-

=

=--11/4,

-1/2 -2 Note that this basic point is already a basic feasible point so that the process now transfers to algorithm LP-B. Since b', is most negative, choose -1. Now, C1 all

I

-1/3

5/

-1/2 3

2

7

1

3/2 5/2J

is the pivot. A pivot u

xX3

x2

U2

x

v

T

Since this is a basic feasible point and the

three elements in the third ro are n the ti medae the then the solution is immediate. The nonbasic variables are zero, pst

L3/11 4/11 - 6/11

425/11

= 1,

4

r2 a21

i/1i

-

L1/2

so .rst step is to interchange xI and result of a pivotinthebasc u Ths rsuls fesibe pintpositiv, us* This results in the basic feasible point r

L WJ.

2,

so = 2 and a2 2 50 step yields

36J

-17

xl

1

u1= x2 , :i:

~

-

-25

76 29/6j

1/3

x3

__J Put 1o

-U Fuul

11

F 11 4 -6

u1

2/3

1/3-113

C

1

2/3 - 11/6 11/6

1o 1/6

,9

7

-] U

-

1/11 -4/1l

-

1/1l -37/Ill I

I

u2

=0

and the basic variables take on the value

2

x

=

'

x. =

5/2 ,

11

=

1,andw

=

5/2.

Therefore, the solution to the original 11 is X, =0

Choose/o -. 3. CI -

a

so io

X2=

C2 -

11/6,-.-

a

11/4,

. The pivot Is a,2

0

x3 = 5/ 2 . 6/11 and a

The minimum value of(x) attained is5/2. 3-11

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AMCP 706-192

3-4 MINIMUM WEIGHT TRUSS DESIGN" As, will become apparent in subsequent chfters, most ortimal design problems are nonlinear. Even the problems considered in this paragraph appear at first glance to be nonlinear. However, it is shown that the problem can actually be solved as a linee program. This will not be the casv in general. The class of problems and their solutions tha, are discussed in this paragraph are takenfrom o ,, an outstanding paper by Dorm, GoniyAfdGreenberg (Ref. 5). Similtr results h_.&bbvp reported mo-e recently (Ref. 6). The problem treated here is minimum weight design of plane :russes with constraints on stress. The initial restrictions on the truss include only the location of joints in the by the truss truss. Th" loads to be s, purted 1 are applied at jints. A member with nonnegative cross-sectional area is allowed to cormect each pair of joints. If there are p ;aints, there may be p(A - 1)/2 members in the truss. Ira general, then, statically indeterminate trusses are allowed. = I... n, denote the crossLet A, sectional area of jth membe" and S, the load in that member due to the external loads applied to the truss; S > 0 denotes tension. If In = 2u, then equilibrium of tbe joints of the nuss is specified by the equations

2;- aS = F., i

in

(3-26)

linearly independent equations in Eq. 3-26. If a is the maximum allowable stress (both tensile and compressive) f3r the material from which the truss is constructed, then stress constraints am. I S/I( oA 1 .

(3-27]

Further, if ma is the weight density of the structural material, the total weight W of the truss which is to be minimized is A

W =P

(328)

A1Q1

where R,is the length of the jth member. The problem of minimizing W of Eq. 3-28 subject to khe constraints of Eqs. 3-26 and 3-27 is rot the complete truss design problem. In addition to the equilirium conditions of Eq. 3-26, a set of c-mpatibility conditions between displacements cf the joints must be satisfied. These compatibility conditions will be nonlinear in the variables S, und A. In its coi ,lete formulation, then, the truss design pr 'em is not a linear programming prob'm. It will be shown, however, that if the conpatibility conditions are ignored and te problem described by Eqs. 3-26. 3-27, and 3-28 is solved, its solution satisfies the compatibility conditions and is, therefore, the solution of the truss design problem. Recalling that compatibility relati,iis are being ignored, it is required that

whe. e Iiare compcments of applied forces at the joints, and a are direction cosines of the elements of the structure intersecting the /th joint. All a are zero if the ith element does not iitersect the point of application of F In order to satisfy three equilibrium equations for the applied loads (including reactions at supports), it is assumed there are m* = in --3 3-12

I

oAi / -.

n.

(3-29)

This is true since if I SiI < oA for some j, then A, could be reduced with an accompanying reduction in IV. The constraint, Eq. 3-27, is therefore replaced by Eq. 3-21. The reader should note that this irgument would not be

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AMCP 706-102

validif compatibility conditions were being enforced- since4 ardductiofi in some A, may result in a vidli'idn. of a constraint not. involving A, explicitly.

.P', namely, find x to minimize BtX

(3-30)

subject to Since by Eq. 3-29, A= 1 I 1,the optimization problem is now to minimize P-

Ax-C=0

(3-31)

X 0.

(3-32)

Or i =l

subject to Eq. 3-26. In order to treat this problem as a linear programming problem, define SJ, if S1 , 0,It 0if S,:jt

S =/

.

I

( 0, if SJl>0

=

i - 3o, ifS < 0

Now,

and S+ r8

SI Denote

)

IS

A

(a,,

-

a,,)m x 2n

and BT

R

.

As pointed out in Ref. 5, page 32, there will be m* posibility nonzero components of x (basic variables) in the solution, corresponding to linealy independent columns of the matrix A; i e., only m* of the, will poEibly be nonzero. According to Eq. 3-27, thern, only m* of the areas may be nonzero. Further, fkrces the member of A is m*, The the rankdetermined. since are uniquely resulting truss is, therefore, statically determinate and hence is the solution of the original truss design problem.

,r ))...

cT = (F

Thisicar programming problem may now be solvedby- he simplex method. Before the solution o4he linear programming problem can be takE as the solution of the truss design proble:,W, owever, it must be shown that it satisfit "E mpatibility conditio s is clear thT ifftruss specified by 07 con"ditions tati sati "f linear programmifigrWpO0blem is staticaul~ie determinate, it satisfies the compatibility conditions trivially (i.e., there are no compatibility conditions). For the analysis here, statically determinate is taken to mean that the member forces S/ are uniquely determined by the given loads and the equilibrium conditions of Eq. 3-26.

1I .' ,")"

In this notation, the problem is of the foin

It is pointeJ out (Ref. 5) that the simplex method for solving many member truss lesign problems is relatively time-consuming. It is proposed that the method be refined for this lass of problems to obtain a practical mcthod 3-13

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AMCP 7M6192

of solving engineering design problems. Several examples are solved in considerable detail in Ref. 5; the results of one of these problems will be discussed here.

subinterval, the member-sizes are different. A plot of W vs a aifd'the forms of optimal trusses are shown in Fig. 3-4.

w

"

A bridge truss is to be designed to span two points, I and 13 o; Fig. 3-3. Three vertical levels of joints are allowed with five horizontal se, a total of 15 points, as shown in Fig. 3-3. In the general case there could be 15(14)/2 = 105 members in the truss. Loads on th! floor of the truss are shown in Fig. 3-3.

2422 20-

16 14

1 T

L

H+ 2

1i

3

40

H"

vinterested

H,+V Tt

Figure 3-3. Admissible Joints for Bridge Truss

In the solution presented in Ref. 5, it is assumed that the truss is symmetric about the line of joints 7-8-9. This assumption reduces the number of variables to 57. Further, due to the assumed symmetry, there are only 14 independent equilibrium conditions. Therefore, there will be only 14 members which can be nonzero in the optimum truss. In the solution presented in Ref. 5 the problem is made nondimensiona! by defining cx = ht/ and 0 = l/V where h and 2 are the vertical and horizontal spacing, respectively, and I1 and V are applied loads shown in Fig. 3-3. The solution presented in Ref. 5, page 45, for a fixed value of P(3 = I) shows that there are three subintervals of values of a on each of which the truss has a constant geometrical form. For different values of a within a given 3-14

2

3

4

Figure 3-4. Optimum Bridge Trusses

The discussion here only touches on the highlights of the very complete treatment of the truss design problem in Ref. 5. The reader is encouraged to study this outstanding article in detail. Before leaving the truss design problem, a point of interest in the present results and in the results obtained in future chapters may be noted. In Fig. 3-4 it is clear that at two values of u the form of the optimal truss changes form drastically Still, even though the topology of the structure is not continuous in a, the weight apparently is a continuous function of oa. The same sort of behavior occurs in a oeam design problem with constraints on deflection which is discussed in par. 7-4. These problems might lead one to suspect that there is some basic mathematical structure of the optimal structural design problem that has not been uncovered. 3.5 AN APPLICATION OF LINEAR PROGRAMMING TO ANALYSIS • A major application of lincar programming in engineering design is, oddly enough, in

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AMCP 706-1 92

nonlinear- programming , It -is seldom- that a realistic engineering design problem can, be formulated as an LP. Realist,. problems are 'rge ally nonlinear When considered as a

,j;-'

1

function of both state and design varibles. Several techniques tif solution of nonlinear progranming preblems are based on approximation of the nonlinear probiem by a linear one, at least: locally. These methods then that the approximating IP be solved. This subject will be deterred until a discussion of the general theory of nonlinear programming -::L~z]=has !"eengiven.

Ki .require

I

A second application oi linear programming which is of concern to the engineer is in the solution of linear boundary-value problems that arise in such fields as continuum mechanics, It should be emphasized here that this application is not of an optinral design nature, but rather tall in the field of engineering analysis. One of tie important methods of solving linear boundary-value problems is to approximate the solution by a linear combination of known functions. The question arises, "How should the coefficients be chosen so as to obtain the 'best' approximation to the true solution?" "Best' may b. defined in many ways. A relatively new concept of "best" will be discussed i' this pa-agraph. The general linear boundary-value problem may be stated in operator notation as

tial equations on x 4 x m

Lfz] l

a(x)

X

(3-35)

-

and the bou-.ary operatnr is B[z]

Az(x 1 ) Bz(x 2 ).

=

(3-36)

In the case of partial differential equations,

E

l Ilm

a~(x a,(x)

aI

z

a

a x" Ix ... n Inax

~

n

(3-37) and the boundary cperator is B[z] = A (x)z (x), x on r.

(3-38)

The method to be d;-cussed treats both the partial and ordinary differential equations in the same way. Let 0P(X),: 1,.k satisfy the homogeneous differential equation L[ 1

=

0, in a2.

(3-39)

Further, let ¢0 (x) be found such that L[ 0 ] = Q(x), in2.

(3-40)

Since the operator L is linear, the new function k

x in P

(3-33)

B (zl =q(x), xonF

(3-34)

L [zJ

=Q(x),

where , is th, domain of the independent variable xcR n and 1' is its boundary. The dependent variable is a vector function of x. z(x) in Rm. In the case of ordinary differen-

Z =-00 +

, C.(x) -

(3-41)

satisfies the differential Eq. 3-3. regardless of the value of the constantF c, The object is now to find these constants so that 7 satisfies the boundary roditions of Eq.3-34 as closely as possible. 3.15

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AMCP 706-192 Define

Examl1e 3-4: Obtain an approximate solution of

a2Z

a2Z

In this notation, Y will be the solution of the boundary-value problcmn if and only if

Az-+ --

ax

I

aZ

2

a

--

nl={x,

z-

for all points xon r.

x

2

=1

(3-48)

~1 1 I

The method to be treated here attempts to

minimize the error in Eq. 3-43 at q large number of points x', R?=!..

L,

max

Y

Ii [Wx) I - q(xt) 11

(3-44)

The object now is to choose the constants as to minimize -y.To see that this is a linear programming problem, note that Itq. 3-44 is equivalent to B, ['i(x2 ) I -q(x') 0 and C 0 such that

The domain fl and its boundary I' are shown in Fig. 3-5. Partial derivatives wiih reSpeOt to the interior normal are shown. 23Z 2o

I,

+,. 0, vg(-)y < 0 for g,(x-) = and v, = 0, and Vh(iny -O,it is true that yTVIL(Vv w)y > 0

The' rem 4-6 states additional conditions which are required to hold if the functions .ppearing in NLP' have two derivatives. Theorem 4.6: (Second-order Necessary Conditions): Let fix), g(x), and h(x) have two continuous derivatives at a point Y in D'. Further, let the vectors vg,(x-), for all i with g1(-) = 0, and h(-) be linearly independent, If 7 yields a relative minimum for NLP', then it is necessary that there exist v and w satisfying Eqs. 4-6, 4-7, and 4-8. Further, for every yeRn such that Vg(x-)y = 0 when g1(3) = 0, and Vh(x)y = 0, it is necessary that yVIL(7, v,w)y > 0

(4-10)

For proof of this thorem, see Ref. I, page 25. Note that the existence of v and iv satisfying Eqs. 4-6, 4-7, and 4-9 is a consequence of Theorem 4-5. Even though this theorem involves second-order conditions, it sti!l gives only necessary conditions, A theorem which gives conditions which, if satisfied at some point, are su ficient to guarantee that this point yields a relative minimum for NLP' will now be stated. For proof of this theorem, see Ref. I, page 30. Theorem 4-7: (Second-ordzr Sufficient Conditions): Let f(x), g(x), and h(x) be twice differentiable functions at a point X If for xeD' there exist v and iv .atisfying ""~~ ",g,(Px)

O 0)

,

(4-11)

then Y yields an isolated relative minimum for NLP'. It should be noted that there is a gap between he sufficient conditions of Theorem 4-7 and the necessary conditions of Theorem 4-6. Strict inequality is required in Eq. 4-11 for a larger set of vectors y that may yield only equality in Eq. 4-10. It is doubtful that a single, tractable set of conditions exist that are Uoth neee,.;ary and sufficient for the generd problem NLP'. There is one class of nonlinear programming problems in which conditions may be given that are both necessary and sufficient for ai. abselute -xtremum. This class is the convex programming problem. Theo,'em 4-6: Let flx) and g,(x). i = I, m, be continiously differentiahle and convex, then necessary and sufficient conditions for Y to be an absolu,.e minimum point of NLP are that there exists ik:R" such that

g(-)

0

v, > , i =1 ... ' in and T(.

~ ,i ~t g(T) 0 .

iV The tuchnical presentation of par. 4-1 ends

VLt-r.w)

=0

with this satiffying result Several -,;nn,ents 4-7

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AMCP 706-192

are, however, appropriate at this point. The analytic necessary and suficient conditions of

from the theorems stated in the preceding paragraph.

par. 4-1 could be used to construct solutions of NLP by solving systems of nonlinear equations. This is particularly truL of the results of Theorem 4-8. If one reads the .current; literature, however, he is led to the dliiilt conclusion that iterative methods bia, d"on successive improvements are too effective to bypass in favor of metho s that rzquire solution of complicated, nonlinear, algebraic equations. Even if the results of par. 4-! are never used by the designer to cons'-t%:t solutions of nonlinear programming problems, they are still very powerful toois. Verification of the hypotheses of one of the theorems may mean the difft. nce between going onto the conputer with the comforting knowledige that a unique solution exist's As opposed to the frustrating experience of having computer print-out which may be meaningless.

4-2 THEORY OF FINITE DIMENSIONAL OPTIMAL DESIGN The nonlinear programming problems of par. 4-I are quite general and may be app!';ed to a variety of optimization problems. As is frequently the case with very general formulations of problems, special features of some problems within the class being studied are not exploited. This appears to be the case when general nonlinear programming theory is applied to solve optimal design problems. lnt,,, etation of certain of the variables a;id constraints in the problem NLP'. w the context of optimal design, yields very rfective computatior,a! methods of soluton This pai!igraph wiyl be devoted to stat.ng the finite dimensionn! optimal design prc.h!er, drawing an analogy with NLP'. and stalng necessary and sufficient coralitions .nat follow awiectll 4-it

4-2.1 FINITE DIMENSIONAL DESIGN PROBLEMS

OPTIMAL

The class of problems to be treated inthis paragraph is, in a sense, a special case of the nonlinear programming problem NLP'. However, by developing a theory ior the new class of problems which takes advantage of its special features, a more efficient solution algorithm may be obtained. The general optimal design problem must have several of the features of NLP'. Namely, it is required to have a cost (return) function which is to be minimized (maximized) and a set of constraints that describe the performance demanded of the object being designed. It is in the representation of constraints that the optimal design problem differs from NLP'.

In most problems of design in the realworkl, the object being d-signed is required to beha ie according to some law of physics. This behav-ior is described anaiytically by a set of variables called state variabks. Further, there is a second set of variableb that describe the object itself rather than its bchivior. These var;ables are called design variables since they zrc to be chosen by the designer so hat the object being designed perfo;ms its required function. It generally Lappens that the lws of physics that determine the state varhbles depend on the design variables so tht two sets of variables are related. To illustrate the differen.e between state and design varables. consider the following design p,-oblens I ltird [he Loeffcicent of damping in dn

.

- -

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AMCP 706-192

automobile shock absorber so tt'peak acceleration in the passenger compartment due to !-o.-4 conditions is as small as possible.

optimal design problem (OD) is a problem of determining bceRk to

~ydnmiif~z~) ~(4-12) The coefficient -of dampIng is the design variable since it describes the object being designed, and its magnitude is to be fixed by the designer.- Acceleration on the other hand is a state -variable since it describes theJ behavior of the object being designed. Further, this state variable may be determined by Newton's laws of motion. Note that the desigater has no direct control over the state voxiable. He may effect it only indirectly by adjusting the design vzriable. This is typical of state and design variables.

subject to OD It(z~b)

(.3

0

O(z,b) 4 0

(4-14)

where h1 (z~b) ,(zb)

,

( h,,(z~b)_J L

2. Determine the size of beams to be used

0

in a structure so that whei. a given set of loads(zb

I

limits, the deflection of certain points on the structure is within given limits, and theLrn structure is as.jght in weight as possible. Beam sizes are the design variables in this problem since they- describe the structure being lesio ed andi !!e, must be chosen by the designer. Stress and deflection, however, are state variables that are deterinndt by equiiibrium and force deflection relations. Again, the designer bas no direct couitrol over stress and deflcction. HeI may effect these quantities only 1)y varying the size or beams ;n the structure.)h417 In most real-world desilgw problems the state and design variables are J:early idcntilied. In what follows, the s!tile varajblu will be an pt-vector, zrR and the desigli %ariablewill be a k-vector, b&-Rl. The basic elements of the optimal design problem arc (lesntled by IDeinition 4-6 O') imion 4-6

The

finite diensional

1(z

,(zb

bJ

and all the functions of the probliemt are required to have first-order deriv~atives. Furthler, it is required that the (r, + 0~ vectors (4-16) 3

b]

arc linearly inderendent for ah i with 01(z~b) =0 and that thew atrix

is tionsingular. Teasipinta n nti Thc aZupinta n arx~ nonsingflar gtiaraiitees. hy the implicit itunction tlteorein (Ref 4. page 181). that for give,, / thler, -, a %iiq~ olut ion of Lhq4-1 1 for, inrther ihL. stale vanal,"le determined

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AMCP 706-1932 from Eq. 4-13 as a function of b, is differentiable with respect to b. This fact will be needed later when constructive methods are developed.

linearly, independent at F. Then there exist" multipliers .eRn 'and peR', with g > 0 such that for Hfz~b +XThz~z,b) + pTO(zb)

4

(4-21)

4-7. -LOCAL THEORY Since it is very seldom that the state ~±wations (Eq. 4-13) are linear in both z and b, convexity of the constraint set and hence the problem will be radye. For this rmason, no global results based on convexity wvill be discurssed. In case Eq. 4-13 is linear, however, global results mbay be obtain-ed by applying the Th eorewrs 4-3 and 4-8. It is clear that ;f a new variable xeR" I k iSThe defincii as

X

(4- 8)

Lbi(-8

then OD he poble ~y h l~r form NIVP. According to Theorem first-order constraint qualification satifie wit rotOD xeR + as dent variabbl) if the row vectors

i~1 ti~In 4Z, the will be ndeen-

[ahi al,

Bz a

;jot

.b

.... n(4!

(4-23)

and 9(b)=0,

j=

.inl.

(4-24)

proof of this thecrein may be constructed by simply writing Jown the necessary conditions of T.'eorem 4-5 in terms of x and then separating the componentr of x

as in Eq. 4-18.

exactly the same way thc second-order necessary and sufficient conditic-is of Thres46ad -,rspcily my be stated for the problem 01). No essential simplification of the statements of those theorems o-zurs, however, so the theorems Theorem 4-9, just as Theortem 4-5, is difficult to use in constructing solutions of

/ (OD01 3~b j

for/ with 0,(z~b) =0

(4-20)

are. linearly inde.Penden:. Theorem 4-5 may now bc appiied to the problem 0OD. ,:,nrevn 4-9 (Firstordei Vere.xery ConLet al, the functions appearing ini

~..015

OD) be differen~tiable at a point Z. 9 whith s.M.sfies EqIs 4-13, 4-14, aid .1-1 F urt her let Ihz, vec;w'i, HIqs 4- 19 -20 and 4-21 ) be 4-10

aH a

arc not restated hecre.

1

-

Liz

(4-22)

0

a

Considerable difficuilty arises because one doea not know wich of the ineqtu4lities in 0OD is an equality. For problems with a small number of inequality corstrl.ints this may not be a difficult ob.;tacle, pr-ticularly if the designer has a good intuitive idea of which constraints will be qualilies. If, on the offher hand, there! art a large numnber of in-,quahity corn raimls. thei' the number of i01ombina ~toiis of -.onstrain1t, which, may Ie equalities. is large It m~ therefore, difficuilt t0 deternline just Mokah Combination,. wil; bc equaitie, An;

.

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AMCP 706-192

araly tic Aolution is extremely difficult in this case.ical Rather than attempt to use the necessary, cor'ditions to construct candidate solutions, a more direct approqach will be followed. The remai, der of this chapter will be devoted to direct methods of~~bI hg NLP, NLP', and OD. 4M3NSEQUETIALL UCNUSRAINED MINIIZAION ECHIQUE (SMT) faorit f slvin dificut mehod A A mthodof faorie olvng dffiult problems, particularly among mathematicians, is to reduce a difficult problem to a sequence of easy problems. Each of t!,,e easy prob?ems is solved and if the method is any good, the sequence of solutions or easy problems will converge to flt: solution of the 0,fficult problem-. As the title might imply, SUMT follows just this pattern. It should be clear that a central vaii of this method must be results which guarantee convergence, at leastI

of Fiacco and McCormick (Ref. 1). Theoretresults guaranteeing convergence are., presented here to indicate the -level of the known theory of SLIMT, rather than as a complete treatment of the subject. 4-3.1 INTERIOR METHOD The interior SUMT is based on the idea of using the constraint functions to erect a barrier at the boundary of the constraint set D of NLP by adding a penalty function to 'x) which approaches infinity as the boundary of D is approached from thz -interior. Once the solution of the augmented problem is obtained, the penalty function is altered so as to effct f(x) less in the interior of D. This behavior is ilfustrated in Fig. 4-3.

I

~

in cases where solutions are known to exist. Thelt

mehdprsnedhrxesnialie

duces NLP and NLP' to a sequence' of auxilhjry problems which may be solved by the methods of Chapter 2. The cost rwiction of NLP or NLP' is augmented by a function called a peniblty function. The penalty function is formed from the constraint functions in suchn a way that as a paramieter approaches zero for perhaps infinity) the unco,:st'ained miinimum of the augmecnted cost ftiw~tior. converges to ic -nhieinn of NLP1 or NLP'. Two basically diWferent way,. of doing this are prestnted here Each has its comiputational andl theoretical advaitage , andl disadvantag,_ tha: will be described Waei. DueC tu the large bod) or theory concerning SUMT. restlits will he pr-,sented in flfls parr

graph witlioui proof. '1he reader v referred SLIM

I

i) +Penalty Funciia(1)

to the .olnpett and well-svrittell tCe3

Pnlyfx 1 Fnto _____

X1~v (A) fjx; + Penalty Fuction (2)

Penaly(2

Fnto 2 (2

X2 F~qg.qe 4.3. Penaotl Fi,nctions 4-1l

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the minimum of the first augmented cost

I. l(x) is continuous and non-negative on the interior of the constraint set D and if' xk ) is any sequence of points in 12converging to . where g,(.;) = 0 for some/, then k"-. I(xk) = + cc

function P 1.The idea, of course, is that the sequence of points P)1 generated inthis way converges to x.

2. S(r) is continuous and if rI > r2 > 0, then S(r 1 )> S(r2 ) 0andif risasequence

As illustrated in Fig. 4-3, when the penalty function is decreased on the interior of D, the minimum of the second augmented cost function x(2) is closer to the solution Wthan

of numbers converging to zero, tifl

It should be clear why this approach is discussed only for NLP and not NLP'. The constraint set of NLP' can have no interior due to the equality constraints. It is possible that NLP has no ilerior and in this case the interior SUMT is not applicable. In what

=

m S(r)

0.

Probably the most common penalty functions (x) and f(r) are l(x)

1

-

(4-26)

/II g'(x)

follows, it is assumed that the -cnstraInt set D

of NLP has an interior. The sequence of points x(i) which is to converge to the minimum point is generated by minimizing f(x) + S(r:) 1(x)

(4-25)

without regard to constraints, where S(r,) l(x) D and in the interior of for x ay is continuousS~r, = isuchtha V.0= 0 - orJ~) )(+forI < any in.Aisuc tatfoecg 1 (in for any I < I n. t is clcar that if one begins an iterative minimization techmque of ChapD, then a ter 2 at a point in the interior of relative minimum point will he found which must lie in the interior of D. Otherwise, the minimizing sequence would have had to cli:nb over a portion of the auxiliary cost surface .hal is infinitely high and none of the will (d0this. -,n,,his

and (4-27'

S(r) = r.

Any pair of functions satisfying properties I and isNo. 2 ass.ciated No. the suitable. It maywithbeEq.to 4-25. however, e o e choos t sdatage ers dei dsigner's advantage to choose another form for any particular problem. For other suitable choices of penalty fuctions, see Ref. I, tage

Tile rlgoritho, tor solving NLP by the interior point techuque is given in )efinmtion

ldDejtion 4-7. The .iterior point sequenIn order to obtain the sequence of points , the parameter r is allow cd to approach X(11 zero. To insure that the ,,equtt.ne 0 1) coll-

inhniurn point, the verges to a rel:ativ" functions /(x and S(r) are required to have the following properte

4-12

14illy UnLonstramcd minimization algoritlim is given by the tollow:,g

Step I. l)efinc the function '(xr) =

) + (r) ,(v.

'4-28)

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AMCP 706.192

where S(r) and I(x) satisfy properties No. I and No. 2. Choose r. > 0 and x(° ) in the interior of the constrain" set D. Step 2. Beginning at x(° ) minimize U(x,r0 ) without regard to constraints to obtain P ). Any of the methods of Chapter 2 may be employed for this purpose. > 0Fr ,chooser, Step 3. For i= 0,1,2,... such thatr + < r,. Beginning at () minimize U(xj + t)without regard to constraints to obtain ( 1), where I is the iteration index. Step A, As ri -.o,if x(i+ t)- P) IIand 1f xP + 0) - f [xP)l I are sufficiently small, terminat, the process and take x + 1 as the solution of NLP. Otherwise return to Step 3.

S

In order to be sure that this algorithm will lead to a solution of NLP, one would like to have a result that as rk - 0. a solution is appioached. Such a result is contained in Theorem 4-I10. h Theorem 4-1: In the interior point algorithm jast given let: 1W[). g,(x ), - - ',g,n (x) be coin(4-29) tinuous on the constraint set D, S(r) lnd tr)

satisfy

-operties No. (4 30)

and No. 2

(4-31D

!'he interior of D be nonei)pt ,

where x"is not an isolated point of (4-32) D, {rt} be a strictly decreasing sequence which converges to zero. (4-33) Then for x( 0 ) sufficiently near x and r, sufficiently small, lirx

(4-34)

=

Further,, 0

tim

0

SJ ln fxP) I

m UUIxP , I I,.'

f(') (4-36)

(fIx t ) I I is monotone decreasing (4-37) and {I[x(11 l

is monotone increasing. (4-38)

For pioof of this theorem see R,.,. 1. page 47. It has been noted throt,0hout the previous development that if NLP ii convex - i.e., then g,,, (x) are convex Aix). g,(X). "nice" things happen. One of these "nice" tmiings is given in Thcorem 4-I I. T:%'orem 4-!1: If NLP is convex with a m. unioue minimum point . glx), = I.= ... re twice contiluoLly Jiffvrentiablc, and if E.s 4-29 tlhough 4-33 hold. then '0) gcncrated by the given .igorithn will converge ,o point. i III- inimum

rlh-re be -. reltine 1imU;nu

Ioini x- f(vt f-)r all x thh InI) / 1 l ood of ;k'd III S s w ae 11t .

It

hould he not-d that Sw", i o

aOgi ri'hii ()ef 4-") ;cquired a p ;',

whe i(O)I

4-1 3

.1

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AMCP 706-192

the interior of the constraint set but no method -of dbtainhig such a point was given. This question will be addressed later in this paragraph. Exaple4-3 SlvetheLPthe f~x , 22) 1 +x 91XI 2 )

iniumfor

4-3.2 EXTERIOR METHOD

Unlike the interiof method, starting points for the exterior SUMT are not required to be tecntraint set of NLP. The basic idea in 1, exterio, method is to add to the cost function a penalty function that is positive points outsidie the constraint set and zero

cost function from being too far

X1augmented

.x 2 )

-

~from

SMT.constraint th intrio usinpoit usinpoit th intrio SMT.may SoluioE:on

U(X'r) = '+

x2 -

I

r

7.L-X

Th ucin()

2 ]

Thefuntios fx),g,(x), and g, (x) are

convex and by Theorem A-5, Appendix A,so are - I g, (x) and 11g 2 (X). Since r > 0, U(x.r) is convex and thus has a uniqueA minimum. To find it, put

the constraint set if the original pst function fix) is "well behaved" outsi(!e set. It is clear that this approach not be taken if fix) is inmdefined or takes negatively infinite "alues outside the constraint set. One very appealing aspect of the exterior method is that it handles equaiity as well as inequality constraints without difficulty, so that it can be used on NLP'.

th

The penalty function employed for te4

etrc

ehdwl

aetefr 4-)

()

where P(t) and E(x) are required to satisfy the conditions:

(xI

ax1

au

=0

12.E(x) =0 if x is in the constraint set, and EWx)> 0 if x is outside the, constraint set. r

1

2.j P"(t) is cotnou

-X)

'(

then

n if t2 "1 > 0.~ Further, if

> 0, +c

N'~ t 1) = +c

Probably the most common choicc for P(z) X2 rPWl)

As r-

C',x,

'C anid X2

(4-40) 0 so tile solution

and

or Eixample 4-3 is

~ (,0)Ex)= 4-14

Z !g1(X)+ Ig1(V)II

(4-41)

-

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AMCP 706-11.2

obtain an unconstrained minimuin point of

and

x~)

whredenoted

The basic idea for the exterior method was given by R. Courant in 1943 (Ref. 5). Hesufcety argued that if

Step 4. As ti -)oo, if 11[P)' P~ 4 jI and IffP1' - f[xP'')] I are matrineth process and take P) as the solution of NLP. Otherwise, return to

(4-42)Stp3

7T(x, ) 'f(x) + P(1) E(x)

wereminmizd wthot rgar tocontrantsVery much ab in the case of the interior wthou tocertainnt were regadinimzed method, Theorem 4 -guaranteesaceti using tj and 12 with t2 > tl, then since the mesrofucs. augmented cost function is pinmaiized more when 12 is used than when 11 is used, the minimum point corresponding to 12 should be closer to the conitraint set and hence,closer to the mlnimum point of &i) on the cntanse.for

Thcieren 4-12: In the exterior point algorithm let: f)g)W ... g(xbecnius all x.

An explicit algorithm for solving NLP orEx)aiPt)sifycntosN. NLP' by this method is given in Definition EIx ad o. 4-8.ladN2ofE.43.(4)

(4-43) saify coniton No..(444

There be a relative minimum point 3F in that admissible domain D such that fti < &i) for all x 0 "i in akean ellii-zte ~o)some neighborhood of i, where Y is Step1.ngieerig nta sltdpito!.(-5 of the solution of NLIP or NLP;.

Definition 4-8: rheexterior point sequentially unconstrained minimization algorithm is given by the following:

Step 2. Choose t, > 0 and beginiting at x(O find an unconstrained minimum point of

denoted x~i Step 3 C.ontilnue WilliIh I

I

is strictly in(4-46)

T'hen for x(0 ) sufficiently close to

f(x) + P(t, EWx

'x,: t

The sequence ( tj cIeas!ig to + -.

and t

sufficiently la~gt., (4-47)

lilt)

2.

,

by Choosing

and %artig from r(1-1)

l'(Q

)I1

-

0

(4-48)

4-15

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AMCP 706-192 Urn f l

xt 1)

f()

(4-49)

4x2 + 2t(3x2

-

1)= 0

~andT~x')

1121

(4-50)

T[x(. t] =f(x)

t

fxP)]

is monotone decreasing

(4-51)

E[xP)]} is monotone decreasing. (4-52)

As

For proof of this theorem, see Ref. 1,page 57.

,x

Very much as in the interior method, if the NLP or NLP' is convex, then convergence is guaranteed by Theorem 4-13.

The solution is then /12\ (x,x'2 )

Theorem 4-13- If NLP or NLP' is convex with a unique minimum point, and if Eqs. 4-43, 4-44, and 4-46 hold, then regardless of the estimates x( ,) and tp the sequence P) generated by the algorithm given by Theorem ,-12 will converge to the minimum point.

.

3

and x

3.nd

MIXED INTERlOVEXTERIOR METHOD

Both theinterior and the exterior methods presenmed in pars. 4-3,1 atid 4-3.2 are not applicable in certain kinds of problems. In

Example 4-4: Solve

particular, the interior method cannot be used if the interior of the constraint set is empty,

f(x,,x 2 ) =x1 + 2x2 = minimum

such as in the case with equality constraints. The exterior method connot be used if some constraint function is not defined or is illbehaved otitside the constvaint. A combination of the two methods will now be given which allows the treatment of probles which may have both these undesirable features and thus could not be treated by either pure interior or exterior methods,

h(x,x)

-I =0 __

by the exterior point SUMT. 2

+ IV2 + t 2

a7"-=2x ,+2t(xi t

aT -

=

+x

+X2-1)=0

For convenience, consider NLP'

4x2 + 2t(xI + x, -- )= 0

4x 2 - 2x

4-! A

minim~zef(A)

(4-53)

subject to

Subtracting,

"rl,,

-1)2

O .or x = 2x2 .

g(x)

,i

h (x) 0,1

I.

m

I.... p

(4-54) (4-55)

k -

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AMCP 706-192

-

where the set of all points which satisfy the m inequalities Eq. 4-54, has an inttrior. As might be expected, the constraints, Eq. 4-54, will be dealt with using an interior point penalty function and the constraints, Eq. 4455, will be dealt with using an exterior point penalty function.

denoted x5'4) Step 4. Ac r, 4A X(t-1Ia

cojf 1jjxU ]~ 1 J-fx'' r mll"emae h process and take xfQ as the"6& tion of NLP'.Otews46A rl*

The penalty function used here will be

S As might be expe~1ii oriisuyoth two mtostr the mixed indtid; a conrF61~ euti

whr SA ~tEq.

Wsaif cni

No.I ndNo 2of q.4-9.Itis unesodTheorem

ta

x)iafucinoonythe constrairt

given by Theorem 4-14.

gorithm let:.'~

4-14: J.ln;4he mixed point al-

ingforNLPis lgoith ow gvenir. Defintion4-9.straint

the nonempty interior of their conset and A~x), h, (x),.h,,(x) bo continuous frlx. (4-5 8)

seqentall unonsraied iniizaionalgorihm y te sfolowng:conditions gien

s(r), AA KO) ,jafid E(x) satisfy No. I and No. 2 preced-

ing Eq. 4-26, and Na. I and No. 21 of the solution of iNLP'.ofE.43.(-)___ Step 2. Choose r, > 0 and tj > 0 and obtain an unconstrained minimnum of V(x,rl,tz) f(x) +S(r, ) (x) + P(t ) E(x),

Thers exist a relative minimum point 3F in the admissible domain D' of Eqs. 4-S4 And 4-55 combined, such that flu) < f (x) for all x :0 in some n'~ghborhood of Y, where 3F is not an isolated p~ji,' of D'. (4.60)

(4-56)

The sequence I r, be strictly dedenoted P ).

Step

3. Continue with i = 2, ... by choosing r, < r11i and ti> t,,and starting from P'1i) finding an unconstrained minimum point of V(x,r 1,t,) =f(x) -f S(r,) 1(x) +

P~) ~x)n"'S(r,)

y!fcenl

creasing to 0 and increasing to +

tjIbe strictly (4-61)

Then for x(O suziciently close to XV, sufficiently small, and t~sufficiently large, I [P)I = 0

r,

(4-62) 4-17

Downloaded from http://www.everyspec.com

"7-7-1

'-r

77-77

Since (XIX) HITV[x('I,

A.-For

:J=~)46)

,

-~0

Satisfiedatllims

1. Taking the limit as r

>

proof, see Ref. 1Ipage 60.

< 0is

then,

-*0,

2

1

As I -+ it is necessary that x, +x2 -I or Eq. 4-66 will be violated. Therefore, In

Example 4-5: Solve fAX 1 ,X 2 )-X1 g(x 1,

2)

-n

+ X2

£x

2
0, then a re,3ti~ve midnimum has been bypassed. To 1a; ate this relative nimum, do a ont.,dir, nsional search in th~e direction V.'star',."j 4*xif ) to obtain xU

ROSEN'S METHOD FOR LINEAR CONSTRAINTS

• - .4-4.2.1 . .If

)

P

;P+-)

presented in.the

"Thei'e,are~ntany~techniques

i: i 3be •

O

A

This process may be summ-tid 40,,os,, -'s Algorithm:

the constraint functions are linear, thenStp1Co is of stecpest descent direction the lnte the :=found, itmay be followed without-7min leaving gr(x)elativd Tonstraint a ti constraint fun 0

This algorithm,

i not in for C,

therefore, can lead to rather long step sizes. Constraints here are restricted to the torm

ue t '

'fo

c ha

Aoxr):. Step 2. Comput af

b < 0, 1 = 1i,.

GTx

if

where G, is an i x I matrix of constants. The step size is to be determined so that k is as small as possible and still SGT(x(,) + k81

-bi

= 0

for some iLA [x" 1.Only those i need to be considered for which GT&. > 0, since otherwise this constraint can never go from strict inequality to equality. The step size k, therefore, is chosen as

[-b -G rxU/)

mrin

i:

k =_ iA 4 [xL

Gil,GiT6 bx > 0 )

0, since otherwise this constraint can never go from strict inequality to equality. The step size k,therefore, is chosen as

Ix> )

k

....m

O,1 ,

fhe where G, is an n x I matrix of constants. step size is to be determined so that k is as small as possible and still G;T[x(I) +

Step I. Compute mC

x(I

'

x ( ) + k8.

and go to Step 4. Step 3. If

0,

GTx(I) G

) therefore is given by

then find A so as to minimize flx~' [X, )

] 4-23

Downloaded from http://www.everyspec.com

!-

-.

.,, . ... _i

-ut

XU Step 4.

=

.

I-

. and go;to

r, x

and goAo

Step 4. IfIfl[xUl+t01-f [xUI Iandll~xU* ) Stp4

ax

:steepest-descent algcri2hm.]

axx/

ax

and

22

-

8x =-E-/.-"r¢:

44.2.2 FIXED STEP WITH VARIABLE WEIGHTINS3L

calculation then, rather than asking for a direction U. satisfying Eq. 4-73, the designer might request a change 6x in xU) which satisfies Sxlt6x = R2

tx ax_

.,,o:,,

(4-82)

tr

Tfg2a

When the auxiliary problem, lqs. 4-7) through 4-73, was formulated, it would have been possible to ask for the step size directly rather than just the direction of steepest descent. many eases, the oehavior of the solution isInmuch more sensitive to changes in one variable than another. For stability of

-

-1

-ix\x

~L

x'11ll are sufficiently small, termirate and take xUl) as the solution of NLP. Otherwise, return to Step 2 of the constrained

r

P

7

1 j

/

'5ax

ax

1/2 afr

";: atTf\

j (4-83)

N

t

h

e

i

m

l

then considerable progress may be made toward the minimum point. However, since the constraint functions are nonlinear, violations may occur at any iteration. After a new i point func ) has been computed, the constraint runctions should be checked. If any constraints are violated in excess of fixed tolerance, the method of par. 4-3 may be used to move xU+ ) back info the constraint set.

...

(4-80)

The computational method is then Jewhere TV is a positive definite matrix (usually scribed in Algorithm for Steepest De".:nt diagonal) and R is a predetermined constant. With Fixed Step Size: The elements of IVare often chosen so that expected changes in various components of x, Step I. Using the method of par. 4-3, 8x, wiil contribute approximately the same interior testimate toilematrx magntud T I5X. The magnitude to 6xTIVSx. matrix T, tW, thre-obtain thereslto an fNP ~ hc of sithe solution of NLP, x (° ) , which is in fore, is chosen based on the designer's exthe constraint set. Further, choose perience. the weighting matrix It' and step size Qin Eq. 4-80. The analysis performed in obtaining the direction of steepest descent follows with Step 2. Let j denote the number of the only minor chantes. The only changes of present iteration. Computeg,[xU) Compute . and form the set A [i)on interest, computationally, are

I afT =_(.agV-1 ag)-, ag If/-a' x ax! \ ax (4-81) 4-24

-

3fax[Px)j and ieA [xU.

a>/axtx')j for

Step 3. Compute ,Xin Eq. 4-81. For all X,

h

Downloaded from http://www.everyspec.com

:7, -s-

,

,S

,

-

-

,-

.

,-,-

V.

,

F e,, or until a minimum of f[xU) + kA.x1

4-4.3 A Ss..EPEST DESCENT METHOD WITH CONSTRAINT ERROR COMPENSATION In previous subparagraphs, steepes descent npeiu uprgahseps ecn methods were given which at boundary points generated steps parallel to a constraint boundaty in a direction which decreased the cost function as rapidly as possible. Due to nonlinearity of the const.aint functions, and the finite step size, however, seine constraints u ill invariably be violated. It is the object in thiparagraph to present a new method motivated by !he article (Ref. 6) which automatically corrects for violation in constraints.

ror each 1=A[xU) I and e,ch step in k. The

Let A[x/)l ( il g1[x(1) ] > 0 1 be the indices of constraint functions which are zero or are violated. As in the preceding development of this paragraph first-order Taylor approximations will be used to approximate functions appearing in NLP. The linearized version of NLP at an approximation to the

'ultiplier k is increased monotonically pro-

solution, xQ ) , is

A uniform step size in k may be chosen and step, fraken, checking gtx

(

) + k6x]

4-2c

Downloaded from http://www.everyspec.com

V4

'minimize

-Cx

6f

-.

t1

(484

-

ninear'ad'cab

subject to l

iWhether all c

vot4-

ostait a

J

ccssively, the full violation may be eorfected; i.e. ieA[x t 1 )l.

(4-87)

Warx =R

where 2 is small. Assuming Eq. 4-85 is an equality, necessary conditions for the linearized ,problem are obtained by using Theorem 4-5. From r a L= f

Premultivlying Eq.-488 by a§lax and using Eq. 4-85 yields

a0- X+ 2Saf

+

ax b0

ax ax

it is assumed, as usual, that the gradients of ali constraint functions which are zro or violated are linearly independent. Therefore, the coefficient matrix of X is noncingular and

(..2 i

T YgT ax ax/ a T + 2Pa g.

Lax

g

ax +

(-89)

J3~

Substituting Eq. 4-89 into Eq. 4-88 yieldt

+ PIaxT8X

and Theorem 4-5, it is nccessary that Xi

and

aa aax -+

hn~h

(4-86)

In order that step size is not excessive, it is required that

a-x-

i~

violated and it may be conclixidthkLthe kth copintoEq485sudhaebe allowed to be a strict inequality. Tile index k is then deleted from A.

--

ax

* .+ 2PA-e

0

0,

(4-88)

6x

-

1=

_I r

( -_ x',a

?grT /

x ax/ agr

-

1

'x/a7

'

ax.

and

X, g-.S\(iax

Atg,)

0o,le A.

This set of equations is nor ,near in X and 4-26

rit R

porefits are- no-negati e. If

any carnapbniiis neAtic

Whee g L*~ nd ~ s tkenas he changie in ~,i.e., the total change -desired taken at the designer's discretion. Usually, so long a4 the constraints are not violated ex-

g.l--x(I)],

,4-88nn~q. q-88 h

I' and Eq. 4-85 asanequhi. a -o -is- i, e Ts-takenahan thn behdeteri'inied and .tcheck nide-to see-

-

.

te eichct'il x.13 redue o * necessaryqond4t-8s 5a~uli~te

(4-90) This expression for 6x could now be substi. tuted into Eq. 4-87 to find 0. To be more general, ioweve., put I/(2p) =y3> 0 and define

(

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AMCP706-192

summarized, in. ,the, qteepqsi-ideicent! Atstralnt~Error

gb2rlthmi'llth, L-

-~~~

~~X~iOX (4912 and' ox2

=-

(49) A. axa /af/ax

-

Using this new~notation-Eq. 4-90 becomes .Step Ox x2=76x

+

This representation of Ox has important properties gie yTerm41.Calculate Theorem 4-15: Ox1 and 6X2 of Eqs. 4-91co -92 atisy-te andcoditi'~szero, 8XIfrom

. x1 2 =1.0 2..-

2 = Ogx

ax

4

6OX 1.

tp. 1 Make an, engine~iing estimate of solution of NLR:I.zK Step 2. LUt-the'iteration-number be ~0~ 1~~ anJ or Coih~ute A[x~l) ] And I .' ad /axi] Step 3. Compute [x(l) I and choose the desired change Aj in g.

I-

4. Comfpute 6x and Wxin Eqs. 4-91 -und 4-92. Step 5. Choose -f by a suitable scheme. X in Eq. 4-89. If any onts2aelsthnzr for gj x0l)] which are close to remove these components g and return to Step 3. If all

xi-a 0, proceed. Step 6. Form

g -

3. L- Oxl

Compensdtion.

0

0

A method o" choosing 7j still has not lb en given. This parameter is interpreted as a step-size and may be determined by one dimensional search or any other scheme Chosen by the designer. In differcat applications, different methods have proved effective. No single scheme has been found that seems best. rhe choic.e of -f at this time constitutes an art as much as a science. The u~se of this method may now be

5x

'y8x t + Ox 2

and X('+ 1)

A

X(/) +

Ox.

Step 7. If If~xt1 ')I - f(x 1 ) I Iand 11Ox 11 are sufficiently small, terminate the procesb. Otherwise return to Step 2. 4-5 S~tbEPr"ST DESCENT SOLUTION OF THE FiITE DIMENSIONAL OPTIMAL DECIGN PROBLEM In thic naragraph a steepest-descent method of solution of the problem OD is developed. 4-27

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AMCP798-192

-

Inmre y ways, the:method'of thispataph is similar to,-the method of par. 4-4. Herye however; a'distinction is made biteen design --c- and state-'.variables, :and the -two 'types of f

used to solve Eq. 4.5for z.The object-now is-to deteinr,'ie b,change in b(° ), denoted 6b, such.that b(1) =:b.( 0 ) + b:

ariables arerentl

Spa.

The-problemt-tobe solved here is,just s in -2:, Choose bdRk and.zeR"!o n,..,amize (4-94)

f(z,'b)

-

- "ah/az[z(°),b(

subject to h(z, b) = 0

(4-95)

O OD

¢(z, b) < 0

I

(4-96)

where h(z,b) = [h1(z,b),.... h,(z,b)JT, and O(z,b) = [01(z,b), ... ,Om(z,b)]T. The state equations, Eq. 4-95, are put into vector form here in order to take advantage of tle compact matrix calculus notation. The steepest descent algorithm for OD is developed here by first approximating the nonlinear elements of OD by linear expressions in the various variables. The difference between the method presented lre and that of par. 4-4 lies in the treatment of the state variable. In a sense, the state variable is a nuisance since it does not really describe the system being designed. The algorithm presented here is obtained by first eliminating the state variable from the linearized problem and then solving an explicit problem for 'a optimum improvement in the design varz3b .b Very much as in par. 4-4, an engineering estinatc u; ,o opth.iui design ib madu. Its denoted by b(° ). Then the state equations, Eq. 4-95, are so!ved for the corresponding state z(° ). Any method of analysis may be 4-28

(

--

wilbe an "improved" design. The meaning of "improyed" will be made clear as the analysis pfogresses. If the new -design vaHoble b( 1 ) were substituted into 'Eq. 4-95, this equation could 'be solved for the corresponding new state variable z(1). Since the matrix 0 )]' ibnonsingula;-, the implicit guaranfunction theorem, Ref. 4, page 181, tees that if 118b IIis small, then z(1) - z(O) will be small. The change in z is denoted 5z so that

and

(

-

t

U

()+

Sa.

[-f(0), b(OiZ a(4-99)

+af [ h[z(o)" bO)

:

-

(4-98)

4-5.1 AN APPROXIMATION OF THE PROBLEM OD The basic idea in the approach to OD presented here is to constuct an approxiniation of OD which can be solved to obtain an improvement 8b in b(° ).The approximate problem is obtaineJ by making linear approxiinations to nonlinear functions in OD. Linear approximations to the changes in f(z,b). h(z.b), and 0 (z,b) due to the small changes 8b in b(° ) and 5z in z(° ) are, by Taylor's Formula, Ref. 7, pag Sr Uf(0) b(O)l

,

,(O)

r',o,. ,,(o)]

z

(4-100) +L r (" b(O)],bj

Bb

'

t

' -

Downloaded from http://www.everyspec.com

AmCp 7064i92.

zo), b(O)] '[o) b(O)AU

(4401)

The set %(possibly-empty) simply, contains all- the iidi6es. ,'Af constraints that wil be r-uired to-satisfy Eq 4103. To--iake -- XHi use of vector calculus notation, definthe columinmatric

~(z~b)3:L In the- ddvlopnefit:.tfiat follbws, .the-arguments [ O(),b(°).]. -of a1lAfuntions-will be understood unless otherwise explicitly noted. The-4mbol:S in front of a quantity simply denotes the total differential of that quantity. I fSinceh[z(° ), b(0)]= 0 and z(O) + 6z is to

IsA

(4-105)

If the set A is empty, then is defined as zero; i.e., all the constraint functions, whose indices are in A are placed in a column matrix, In ihis way, the conditions, Eq.

_8 z+ ~ 4 AO, _simply z Sb ,

(416 (4-106)

where the column matrix 60 is defined as

ah

A107)

Eq. 4-102 isviewed as determining 6z as a | functionof 8b. It Isclear -that Eq. 4-102 canr

be olv, for8 since ah/az bben osnua.A ~ ~ the matrix ~ ~ has~asue

I

Inequlity constraints, Eq. 4-96, will be treated in an approximate manner. The mrethodployed i pp sato r ir. Th method employed here is to require that iff 0l[((°),b (0)] 0,then

The object of the following analysi$ will be to choose Sb so that f[z (° ) + Sz,b(° ) + 8b] i3 as small as possible. If this nonlinear function of 6z and 8b i- replaced by its Taylor approximation, the problem is to choose 8z and 6b to minimize

(4-103)

60/ < AO/,

where AO, is the required change in the value ( ) of , dut to the changes 6z and 6b in z O and

isempty, 40 is defined to be zero.

Sf=-=f6z + f6b. Sf=-Sz+-Sb.(4-108) ab aZ

b(O).

For convenience of notatoi,, define the set of indices *

A

j

. 1[z(o),

b(O)] ,

.

(4-104)

2

4-103, may now be written.

satisfy the 'qihtion.h[z(O) +8z,b)8b] -is 0o' the linearized version of this condition .is

ah

I

The entire argument up to this point has been based on the fact that II6b II will be small. In order to insure that this is the case, it wll be required that ,;b

(4-'09) 4-29

.....

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j~o,. :

-

Sr

,'-C.

I7

AMCP 706(192 for e small and Wia-positivecdefinite matrix. The. matrix _ will. be. uicd_:in---,.rticu!ar__ problelms to assign 'weights to the various components of 6b. This is often necessary when the components of b represent different physical quantities that may be of different ord!sof-magriitude. Usually W is diagonal. To summarize the -approximate problem, 8b and 5z are to be chosen to minimize

af -8Z az

.f

+2 Jb ab

(4-110)

Define the column matrix NJ as the solution of

subject to the constraints

ar

S6Z + 5b =O, b

8b < A - z+ z ab -

dence on 8z is to solve Eq. 4-111 for 6z-as a .function-.of 8b. This, however, requires the inversion of the matrix ahlaz. The preceding approach of applying- necessary conditions was-scuttled for just this reason, so another method of eliminating 8z must be four Np.te - 'that if the terms (af/Oz) 8z an. (a 13z) z could be found in terms of 8b, then ,dependenco on 8z would be eliminated. This is th approach that will be taken here and also in a later chapter on infinite dimensional problems.

(4-111)

(4-112)

ah az

7,j

brlV6b

t

.

(4-113)

(4-114)

and the matrix X as the solution of and=a Oz

and

fT 3z

. O)z

(4.115)

Note that V6is a matrix whose columns are ser&tions of

4.5.2 SOLUTION OF THE APPROXIMATE PROBLEM Necessary conditions of Theorem 4-9 could now be applied directly to the approximate problem, Eqs. 4-110 through 4-113. If this course of action is followed, however, an explicit inverse of 3h/8z must be computed. Since the dimension n of this matrix isoften quite high, this operation would be very costly. Instead of applying necessary conditions immediately, Eq. 4-111 will he used to eliminate the dependence of the remaining iunctions of the problem on 6z. Necessary conditions may then be easily applied for the determination of 8b. The obvious method of eliminating depen4-JU

"

h(4-116) az 0 for leA. Note that Eqs. 4-114 and 4-115 require tihe repeated solution of equations with the same matrix on the left and different right-hand sides. There are efficient computation codes which can construct all the solutions simultaneously. To see how the~e newly defined matrices are helpful, compute the transpose of both sides of Eqs. 4-114 and 4-115 and multiply through on the right by 8z to obtain

X

T h -3 z

Oa z

(4-17)

-

-

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TanaTh ax'z

j

Z2

z.

(4-118)

,Note~that the terms on the right side of these be which are to urones exactly equations are elimnate 4HO frmad4-12. Eq. the eliminated from Ecis. 8z4-110 4-11 inL-.both Fur appears that and Oh/az) ther; the-term te thefd tier (ah ) Sz taapersind ot.

left-hand sides can be obtained from Eq. 4-111 as all

]

-'l

T

af

r T

z

b 7

h T-Sb

E'"z. 3Z

Substituting these relations into Eqs. 4-110 and 4-112, the approximate problem becomes: Sb is to be chosen to minimize S(only (4119)

Sb subject to the constraints T

r5b -c SbTWIb

(4-120) Z2

(4-121)

where

Q" =-~fT ab and

t should be noted that if the limitation, o th if th liitisn Eq. 4i21, on the size of 116b 11 is not Eq. sho21, Wn efretnthpobmEs4-119 enforced, then the problem, Eqs. 4-1 and 4-120, is just a linear programming problem a beesle eh ta well-established techsolved byywl-salse that may niqses of linear programming. This technique

' -fr aT -h ab

the literature. Thv necessary conditions of ,'heorem 4-9 may now be applied ,othis reduced problem. order to apply the theorem and in later In z:I-N1"tions, it is required that the matrix Qr have full row i.e., linealy that the ro-w; (columns of rank; 0) are Id nd nt.

and i

:

-

is similar to that used in Zoutendijk's method of feasible directions (Ref. 8). For a discussion of this method the reader is referred to

Sb.

Using this relation, Eqs. 4-117 and 4-118 become

WTall

A is empty no £ = 00, ifA , isifempty.

(4-122) (412

Further, for use of the theorem ,t is required hat the column vector WSb be linearly independent of the columns of 0. It may be noted that these assumptions require that there can be no more than k - I constraint functiuns which are zero or positive at any iteration. This is true since the matrix 0 has k rows and since its columns must be linearly independent of 2$, there can be at most k - I remaining linearly independent columns. These assumptions are reasonable from a physical point of view. If 0 had rank k then the equation 26= would uniquely determine 6b, and there would be no optirnizatiin problem. The constraints, Eqs. 4-120 and 4-12 1, will be treated differently, so different multiplier 4-31

°

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AMCP 706.192

notation In Theorem 4-9 will be used for each. First, define

4-121 and the problem is again solved with the reduced number of constraints. In any method of solution of the approxi-

t = T£1 iLb + Q

6b +v6bT R6b.

mate problem, no information is gained if v = 0. Therefore, in the following v > 0 will be assumej.

Theorem 4-9 requires that t=0 al

=

Rjr T + Ar67"++2v~brl

where A, . 0 and 2(,tQ6b

V

6b

0

A¢I)= 0, IA

(4-125)

-

(£ ( + 0 A).

2

(4-127)

l is now assumed (to be checked later) that Eq. 4-120 is an equality. Substituting for Sb from E'1 . 4-127 into the equality Eq. 4-120,

and ,(6bT V6b

Eq. 4-124 for 6b,

(Solving (4-124)

t) = 0.

(4-126)

At this point, a computational difficulty arises. It is difficult to determine 6b from I qs. 4-124,4-125, and 4-126 since it is not

L

Rewriting this equation.

I''

known which of the c-j,,.,;znts, Eqs. 4-120

and 4-1"21. will be eualities and which will be strict inequalities. The question is, "Which of the inequalities. Eq. 4-120 or Eq. 4-121, will become stict inequalitics"' This can be interpreted geometrically as a ,uestion of leaving the boundary znd goilg into the interior of the coastraint set defined by Eqs. 4-120 and 4-121 It has been the experiene with this technique that once a constraint. say 0i(z. b), becomes zero, then for several small steps 6b it will remain zero. This observation has led to the following computational procedure. fir' all constraints. Eqs 4-120 and 4-121, will be assumed equalities and 6b is determined using Eqs 4-124, 4-120. and 4-121. [hen the algebraic %inj,of the A, ,nd P are checked It

+ 2% p

2v

/ = -.

Ji,

2vA .

Since OT is required to have full row rank and IV' is nonsingular, the matrix I , if A is empty = 14-128) T0It' Q6, if 4 is not empty

el (

It,-'

+ 2vA).

(4-12')

Note that in the unconstrained ase .,hien 4 is empty, p = Oiice v T 0 and A = 0. Subtituting from Fq 4-12) into I'q. 4-127

they are alinon-negative, then this i%the

desired solution of thle problem 11.onl thle other hand, ,ome j,or v art negative, then the constraints corresporiding to these multipher are remove'd roni Eq 4-120 or Eq 4-32

6

1i, 2P It'

0r

tI

(4-1"0

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AMCP 706-192

This expression for Bb could now be substituted into bbT W6b = t2 to solve for v. However, in practice it seems just as realistic to choose v > 0 in an iterative process as to 'hoose t. Once v > 0 has been chosen A may be evaluated in Eq. 4-129. If any components are negative, the corresponding elements of are removed and 6b is calculated using the matrix. new k,' inte4130forprbeine mdesigner Taiin in Eq. 4-130 for 5b, define 8b' = l1- 1

-M;'

R"Tw-')2J (4-131)

and M;0 A4

SO = W-1

(4132)

1

- -B

bl + b2 .

Theorem 416" The vectors 6b' and 8b2 of the following Eqs. 4-131 and 4-132 have

An obvious check on convergence is to monitor 6b and the associated reduction in f &f.When small 6b occur and essentially no improvement is made in f, the process is terminated. This test, however, leaves a great deal to be desired since the choice of v can lead theis falselyprocess steps yield veryto small iterative that8btheand believe converging. A much better test is to monitor the constrained gradient 8b. Since in an unconstrained problem the gradient must approach zero at a minimum, one might expect that once A = 0, the constrained gradient Sb should approach zero. The real quantity

Theorem 4.17: Let f(z. b), h(z,b), and (z., b)be continuously differentiable functions. If the sequences (0 )] and [P( )) generated by the above algorithm converge to the solution. z b of the problem OD and if = 0 for all sufficiently large /. then it is necessary that 6bI approaches zero as/ approaches o.

STEEPEST DESCENT ALGORITHM

in this developed The iterative as follows: may beprocedure summarized paragraph Step I. Make an engineering estimate of

properties:

2. Q1r 6b' =0

4. -RJ T < 0.

(4-133)

The vector 6b' may be interpreted as a constrained gradient with I/2v taken as a step size. The matrix which multiplies V in Eq. 4-131 e.entially projects the gradient V of the cost function onto a tangent plane to the constraint set. The term b 2 serves to drive any errors in constraint functions to zero. are supported by TheoThese interpretations rem .16.4-5.3 rem 4-16.

!.6b TllV6b

7'bb 2

1 1 could then serve as a convergence 11b60 cIek. Theorem 4-17 makes these ideas more rigoroas.

In this notation, b=

3.

0

the optimufi design variable, b(0 ). Step 2 In theith iteration,ji, 0, soive Eq. 4-95 for z(1) corresponding to 0 ). 4-33

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AMCP 70&1i92

Step 3. Form the vector of constraint functions -in Eq. 4-105 and solve Eqs. 4-114 and 4-115 for W' and

Step 4. Compute R. and £ in Eqs. 4-122 and 4-123.

Step 6. ComputeM¢

in Eq. 4-108.

S . eb and b Step1 7. Cmu 4132tion 4-131 and Eq. 4-132.

in Eqs.

Step 8. Choose v > 0 and evaluate fi in Eq. 4-129. If any components of fi arc negative, take the corresponding elements out of q and return to Step 3. Step 9. Compute

2v Step 10. If lf[x1+1)] - f[Px)] I and IISb' IIare sufficiently small, terminate. Otherwise, return to Step 2. 4-5.4 USE OF THE COMPUTATIONAL ALGORITHM The algorithm presented in par. 4-5.3 will certainly not solve all optimization problems. It is presented primarily to guide the designer to the proper equations developed in par. 4-5 while he is solving a problem. A.most surely a complicated real-world optimal design problem will hame some feature which is not explicitly contained in the general fornmulation OD. In order to utilize a steepest-d',scent philosophy similar to tile one developed taere, 4-34

the designer should method of obtainirng this way, problems often can be treated algorithm slightly.

be familiar '4h the ftle given algorithm. In with peculiar features by altering the general

There are two steps in the algorithm of par. 4-S.3' which are not complete. They are Steps 8 and 10. In Step 8, a p2xameter v is to be chosen, but no analytical method of choosing it is given. This is the classical difficulty with steepest-descent methods. They give a direcbut, unfortunately, they do not allow analytical determination of a step size (l /(2v) in this case). A simple technique for choosing v which has worked well in a number of problems is givea here as a candidate scheme. Since it is the 8b' component of 6b which tends to reduce f, the step size determination will he based on 6bI. The basic idea is to choose v in order to obtain a certain percentage reduction in f Let 4f (a negative quantity) be the desired reduction in f for a single iteration (perhaps a 5% to 10% reduction). Since for A - 0, 7I

-

6b.

(4-134)

v is chosen as QJT~b,

P

2A(

(4-134)

In many problems v has been chosen according to Eq. 4-134 on the first iteration and held constant throughout the iterative process. In other problems convergence propert-es were improved it v is changed during the iterative process.

A

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AMCP 706-192

1. ANV. Fincco aid G.P. McCormick, NonSolution of Problems of Equilibrium and linear _Progranming. Sequential UnconVibrations", Bull. Amer. Mat/h. Soc., 49, strairnedM lMnh~flztlon Techniques, John193p.12. Wile- -&-Sons, New-York, 1968. 6. 1.0. Melts, "Nonlinear Programming Methods for Optimizing Dynamical Sys2. J. Abadie, Ed,, Nonlinear -11rogramming, tems -inFunction Space", Automation and John Wiley &Sons, New York, 1967. Remote Control, No. 1, Jdnuary 1968, pp. 68-73. 3. D.J. Wilde and '2.S. lBeightler, Foundations of Optimization, Prentice-Hall, Englewood 7. C.Goffman, Calculus of Several Variables, Cliffs, New Jersey, 1967. Harper and Row, New York, 1965. 4. W. Rudin, Principles of Mathematical 8. G. Zoutendijk, Methods of Feasible Direc. Analysis, McGraw-Hill, New York, 1953. tions, Elsevier Publishing Company, Amsterdam, 1960. 5. R. Courant, "Variational Mexhods for the

4-35

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.

ACP 706-192

CHAPTER 5

FINITE DIMENSIONAL OPTIMAL STRUCTURAL DESIGN

5.1 INTRODUCTION" Throughout this handbook, structural optimization problems are chosen to illustrate the use of the design methods developed, There are two principal reasons for using structural problems for illustratibn. First, there has been great emphasis on helicopter and man portability of materiel, which places a premium on strutstural weight. Illustrative of Army concern with lightweight structures is the theme of the 1970 Army Mechanics Conference, "Lightweight Structures" (Ref. 33). A second key reason for highlighting structural optimization is its advanced state of development, relative to other areas of the mechanical engineerig sciences such as dynamics of machinery and mechanisms. A few examples in these related areas are treated in this handbook, but development of coin putational techniques remains to be done. It is felt that if the reader develops a thorough understanding of structural optimization and computational techniques, he will be ,n a good position to address problems outside the realm of structures. The fact that the mathematicb of structural analysis parallels that of related mechanical disciplines strengthens this feeling. A cursory review of Army materiel needs convinces one that light weight is a requirement for a majority of weapon systems being develop-, by the Army. The high priority placed on air mobility as well as lightweilght

infantry equipment has presented weapon system designers with a major challenge. In the case of air mobility, minimum equipment weight is a necessary condition for maximum helicopter payload. In infantry applications equipment weight limits the soldier's firepower and mobility. In seeking lightweight designs, one is tempted to simply use lightweight materials and lower safety factors. It becomes apparent, lioweer, that structural weight reduction can significantly degrade system performance. For example, when the weight of an artillery piece is reduced by 30%, dynamic response due to firing the weapon becomes much more severe. In infantry weapons, the requirement of reduced weight has led designers to lighter weight operating metlanisms for inlividual weapons. In lightweight rifles, for example, bolts are much lighter than in previous weapons and hence arc more sensitive to c.hanges in friction due to dust and externa! pari.cs than were the more massive bolts in tl'e M14 and MI Rifles. There are many eamples, some of which will be discus.ed later in this handbook, of instances in which sintply reducing weight of subsystens causes problems which did not occur in heavier designs. The lightweight objective, then, requires that the developer take in overall system view and consider the interation between weapoa %%eight and perforiare of the weapon systern As is true in virtually evwry design problem in %.hicli the limits oi technology are approathed, the liglhtweight weapon de'ag:n 5-i

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(

AMCP 706-192

problem must be considered simultaneously with elI other aspects of system design. It is not practical to expect, therefore, that one will find lightweight structural design specialists operating independently of designers concerned with other aspects of the weapon development. A technology is needed which will allow trade-offs concerning weaponweight to be integrated into the overall weapon design process. The objective in this treatment has been to formula'e the minimum weight structural design problem with constraints realistically reflecing the performance requirements of the weapon system. A detailed formulation and solution of this structural design problem is presented in this Chapter as well as Chapters 7 and 9. 5-1.1 LIGHTWEIGHT VS STRUCTURAL PERFOPMANCE TRADE-OFFS Normal!y, achieving a lightweight structure

result, the minimum weight design problem is often stated with explicit constraints on structural deflection, natural frequency, buckling load, and strength. A central part of the design problem, then, is representation of weapon system performance requirements that have an impact on structural design. It is often required that in doing structural design, dynamic weapon performance must be analyzed to assure that the proper constraints are included in the structural design problem.

5-1.2 WEAPON DEVELOPMENT PROBLEMS ASSOCIATED WITH LIGHTWEIGHT REQUIREMENTS To further explore some of the trade-ofs, betw:een lightweight and Weapon system pcrformance, several typical problems encountered in weapon development will be discussion in this paragraph. discussed here is presented to highlightThe some typical problems, not necessarily to identify all light-

Nornial'., a lighweight acltievng requires a reduction in the amount of material used. The consequence is an increase in structural flexibility that causes increased deflections, deLreased natural frequencies, and decreased buckling loads. Conseq uity, failure modes that were not previously critical, may now become limiting factor, in i gu suporingstrcdesgn.Forexaple design. For example, in gun supporting struetures, tncreased deflection often reduces effectiveness of tie weapon system by increasing dispersion. There are many ways in which such changes in structural performance can have an impact on overall system behavior,

structural heeispe entdtsighlgh sromed yia veight structural design problems faced in weapon development.

The only effective approach to nuimum weight structural design is to formulate tire structural design problem to include constraints on performance which are dictated by functional use of ,lIe weapon system. As a

generates a very difficult class of minimum weight strudtural design problems. The weapon developer's interest in structural design for aircraft armament he3 primarily in the area of weapon and weapon support structures.

5-1.2.1 AIRCRAFT ARMAMENT Some of the most critical lightweight trictural development problems in weaponry today are in the field of aircraft armament. This is due to the very high priority placed by the Army on improved air mobility and the need for minimum weigait weapon systems to be carried by helicopters. The combination of lightweight structural requirements and the extreme environment under which the structure must perform in helicopter application.

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AMCP7Q641P2

The key structurAl'-requirementi ih this =/applidatioifis'aecurate aimillj of ani automatic

gun 'barrel development, particularly,With for inifaftryautomatic _weapon application.

weapo n duri fn the weaponi support,,structure due to inputs from the weippgh, and from the airframe, S which vibrates duie to aerodynamic inputs, "i .nust be.conside redr in the design problem.

a great deal of emphasis being placed on lightweight infantry weapons, the barrel is a natural component in which to beck weight reduction. This-is -particularly true for rapid fire weapons Inwhich heavy barrels llie

The mbst difficult. feature of the minimum weight structural design problem for aircraft weapon applications, is-the variety of peiformance and failure constraints which must be treateO in the design process. Constraints must generally be placed'on stresses arising in the structure, anguiar deflection of the structure at the gun mount, and natural frequency of the supporting structure. These constraints generally appear in the form of inequalities, For example, stress is required to be less than or equal to the ailowable stress for the material. This kind of constraint is very realistic, from an engineering point of view, but makes the solution of the optimal design problem rather difficult.

traditionally been used to alleviate temperatare problems. For a particular buanti co.* figuration, decrea;cd mass tendK to cause elevated temperatures and strosses, To co|i. plicete matters, material strengths a0,hi8ghly temperature dependent, making stress co straints difficult to handle, Another polentlal problem, as one tends toward optlinality in barrel design, Is the possibility that material yield properties will become critically depen dent upon strain rates and require their explicit inclusion In the design process.

)

Inaddition to altering the geometry and distribution of material in the structure to obtain desirable performance, it is also possible to induce damping into the s'icture and to use active feedback control devices to reduce response. These two mothods of reducing dynamic response will require additional weight on board the helicopter. There is a trade-off between design of the structure and design of other means of obtaining improved weapon system performance. These trade-offs, then, require that we treat the aircraft weapon design problem as a system problem, explicitly accounting for the interaction between structural behavior, damping, and active feedback control. 5.1.2.2 GUN BARREL DESIGN A second area in which lightweight structural design is of critical importance is that of

, %" ,"

Another problem, which can arise in reduced weight design, is harrel deflection with resusting reduction in weapon accuracy. Deflection constraints must, therefore, be considered. The objective of the barrel design problem is to choose barrel dimensions and structural material to minimize barrel weight in the presence of constraints on dollar cost, temperature, stres%, and perhaps strain rate. The optimal design problem must then include equations of state of stress and temperature as a function of time, both depending on the barrel design features. 5.1.2.3 TOWED ARTII.LERY 1 lie principal objective in towed artillery design is to provide support for a large-caliber tube that will, upon firing, transmit momentuin to the earth without doing damage to the support structure and without undue dynamic response. The fundamentals of the design

5-3

I

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AMCP 706-162

problem then lie in the field of mechanics and, in particular, are highly dependent upon the weight distribution within the artillery piece. In -traditional artillery design, the support structure is flexible but has been quite heavy and stiff in the past so that the flexibility of the structure was a higher order effect. Also, heavier carriages reduced the severity of the dynamic responseopirblem due to their higher mass. Recent ,d'evelopments, such as the M102, 105 mm Howitzer, have resulted in a weapon that weighs approximately 3200 lb, as compared to the older M0 which weighed 4500 lb. As a result of the reduced weight, problems have arisen in providing a firm support for the artillery piece on soil. More recent design efforts, including the XM164, 105 mm Howitzer, and XM198, 155 mm Howitzer, have resulted in weapons which are considerably lighter than their predecessors. As a result of the reduced str'ictural weight of the weapon, dynamic response in both of these weapons became ,ritical and had to be treated as a key desig" constraint in development of the recoil mechanism. For a discussion of a particular problem, the reader is referred to the artillery design example of par. 8-5. Although these are primarily mechanical system design problems, they have arisen due to the lightweight design criterion. For this reason, when one considers lightweight struL tural design he must be wilhng to fit his structural design problem into a larger system design program and clearly understand the interfaces arising between structural and other system performance characteristics. 5-1.2.4 OTHER WEAPON PROBLEMS The example problems cited in par. 5-1.2 are meant to illustrate the essenti 0 f.,atures of 5-4

some of the more ccmplex-lightweight'structural desigitproblems faced intweapon, development. They are simplifications of the real problems but are difficult'enough to illustrate the need for research in development of design methods. In view of the current emphasis within the Army on air mobility knd lightweight systems, new design methods are required which are capable of solving these and many more lightweight design problemrs. 5-1.3 PLAN FOR TECHNIQUE DEVELOP. MENT The remainder of this chapter will be devoted to formulation and application of a method of structural optimization. As noted at the beginning of par. 5-1, an in-depth treatment of lightweight structural design provides insight into application of the general methods of Chapter 4. For a comprehensive review of structural optimization through 1967, the reader is referred to Refs. I and 2. Several of the major classds of optimal structural design problems are outlined in Ref. 2. Some of the key papers which have appeared in the literature since 1961 ,,re listed in Refs. 3 through 18.

5.2 ELEMENTS OF THE ELASTIC STRUCTURAL DESIGN PROBLEM A class of optimal structural design rroblems in which the structure must remain elastit. is treated in this paragraph. The objecfive of this ,'aragraph is to show how the optimization methods of Chapter 4 can he used to solve realistic optimal design problems. No attempt is made here to present a complete thcor) of optimal structural design that is capable of solving all problems [he reader shuuld note that, e en for the

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..

-

-,,--

.

b

-

-%.Ax -M....

"r

-:

,.'.°.A

t

AMCP 706-1g2

classof problems, conidered, herd, it js not

pog ible-to'blindly applythe, techniques of Ch pker.4.A- certairamo int ofknowledge of structural aqr,..;ty-la is equired before a reasonable stalenient of the-design problem and a mdthod of- ioluion can be obtained. Even more importa ., tle structural deigner.needs to 'hive a-ihorotW *_-knowledge -of the opti-, mization metiiods 'f hapter 4 and their development. Ag-wilfibe seeni in-some cases it is required that:pats of the -design problem be interpreted -in lkht of the derivation of the optimization method. Inthis way the nicthod may be_ adapted for solution of a particular class of pbilnis.

5.2.1 THE OPTIMALITY CRITERION The meaning of optimal or best must be clearly estabJished in each problem of interest. In order 'o have a problem which may be solved by th; previously developed ,'-timization methods, a real valued measure of the cost of the struct-re (value of the structure) must be chosen; Such measures as dollar cost of the structure, weight of the structure, or dynamic response of the structure may be chosen. XVoig with the choice of a cost function, the parameters, or dcsign variables, that represant all design alternatives must be chosen. These parameters will ofter be dimensions of structural members, area of member cross sections, or locations of joints in the btrufture. In keeping with the notation of the preceding chap.er, these design variables will be denoted as bl,. I = I ...., m. For convenience of notation, thae variables will be put in the vectnr form b = [b . ,, b,, IT.

Invariably, the behavior of the structure under load will have to bu considered in the

ment; buckling loads, andnaturalfreqffenCy_

The,:.olltion of allvai'abfes requrc .to: describe-this respons'e due to aoplied 16ad will be denoted by the statt variable vector z. The, manner in which-z is related to the-dc~ignvaiables and applied loads will be discussed in some detail later inthis paragraph. 'The cost of the structure must now be described as a -eal valued function of the design and behavior variables. In keeping with te preceding notation this function will be denoted as J = f(z, b, t).

(5-1)

where t is -ne or more eigenvalues -such :o bunkling load and natural frequency, Befere a meaningful discussion of treatment of the stru(tural design- problert. may be given, the behavior of the structare due to loads and constraints on that behavior mwst be analyzed.

5-2.2 STRESS AND DISPLACEMENT DUE TO STATIC LOADING It is assumed for now that the structure of interest is either made up of a finite number of distinct interconnected members or that large continuous members in tht. structure have been approximated by a finite number of elements as in finite element techniques. Further, it is assumed that the entire structure is described by a vector design variable b. Let stresses at critical points in the structure be denoted Zi, . . ., z. and displacements required for the analysis and design of the structure be denoted zr+ , . ", zn. The

behavior of the structure due to any given load may then be specified by the vector state

design problem. The response of the struLture

variable z =

may irclude quantities suLh as stress, displace-

be restricted here and in the remainder of this

1: 1......, z. IT.

Attention will

-

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AMCP 706-192-

C5by

chapt~r to 9tiuctures whiich dbey Hook'law, i.6., At~essand,,aispiaceifient are~ determined linear 'eq~tiois,. It is-dea ,t:owever,-that the--disighi iariables-play a 1ar& -part- irivtf e respons e of th structure to loads.- The depen-dence -on2 thd design variibles enters ~th~se linear -equations through the coefficients. The equations for z.will be denoted A (b)z -P-

These. constraints ca -generally be wvrittenjan the form O(z, b. t) 40

(5-4)

Where O(z,;= [0~~~~ . . ,zbt The inequality constraints Eq. 5-4, arc -required to besatisfied for~eachiof the states Zi due to different applied, loads A1.

-52

clear that the Eqs. 5-2 and constraints, Eqs. 5-4, fit into the formulat ion of the finite dimensional -optimal design problem of par. 4-5. r-eqtment of the restrictions imposed by Eq. 5-4, however, must be delayed untilsimilar restrictions due to other behavior constraints are accounted for. The entire -Itis

where P isa matrix o~f loads and A (b) -(a,1(b)]I

,,

(5-3)

-wfosdpen ohtim is a m~tri elcidns -is -mitri -woseelemnts diendon-thd design variables, In this formulation of the prol~lem, x and may be- generalized state and baad variables. Eq. 5-2 may 'be obtained through direct application of equilibrium and compatibility conditions or through appILcation of -wiational criterion for equilibrium. In today's structural analysis technology, Eqs. 5-2 are very likely to be obtained by finite element methods (Refs. 19, 20). If the structural analysis problem is properly formulated, the matrix A(M is nom2 .gular and z may be obtained by solving Eq. 5-2. It is assumed that the alementb-of the matrix A4(b) are diffarentiable wah respea to b.

problem will be treated in par. 5-3. 5-2.3 NATURAL BUCKLING

Conistraints on behavior of the structure due to each of the applied loads P ,j include bounds on stresses and uisplacements. 5-6

AND

As pointed out in par. 5-1, the desire to obtain lightweight structures has led to rsnnepolm nlkwsbcln problems. It is necessary, then, that a meaningful optimal design methodology be capable neinvue ofeorigcntats associated with the system response. The sort of constr. 'nt considered here is

ofefrigcntansoKievhe >

In most- ra-rld structural design problems t, strucluri o r~zquired to carry a whole £ indly of loads that ucc-ur at different times in the life of the structure. The treatment here wvill be limitea to a finite number of loads, denoted P1. 1 = 1, . ., s. Associated with each !oad is a state ?t determined by Eq. 5-2.

FREQUENCY

(5-5)

where Is buckling load or natural frequency and 'Is a lower bound on that eigenvalue. More general restrictions than those of Eq. 5-5 are Included In the general constraint, Er+ 5-4.

Much as in Eq. 5-2, the equations of vibration or buckling may be written in the fornm K (b)y

~(b)), Al

(5-6)

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,-7.

is an eignvectoi wherey= [yi,. " whi6lplays the roic of-asi~itevaf-iable,. K(b) - -[k,1(b)] ,

(-7

isgenerally, symmetric positive definite, matrix, and

5.2.4 METHO. OF SOLUTION In the preceding formulatiin 'fthe opti-

(5-8)

mal design probiem, the cost functions and constraints Associated with stress and displace-

is generally also a symmetric positive definite matrix. Eq. 5-6 is often obtained through

ment can be put into the format of the problem treated in par. 4-5. The constraints

a finite element formulation of the structural analysis problem (Refs. 19, ~20).

associated with natural-frequency and buckling, however, are not of exactly the same fOrm. One difficulty is that the coefficient for the ,cigenvector y. K(b) - $M(b), be singular at the solution. This clearly contradicts the assumption in par. 4-5 that

=

J

of this- quotient is the'*smallest e!genyaIu,..Adirect, .-rethod: of 'minimiiing, the Rayeigh, quotient is discussed in par. 2-8..

.There ar many methodsor omatrix gmust f t eigenvalue and associated- eigenvector in Eq. 5-6. The first method requires that the inverse of K(b) be computed. Multiplying through Eq. 5-6 by K- I (b), K- I(b)M(b)y

I

(5-9)

This problem is now in standard form and the largest eigenvalue of K- I (b)M(b) is sought. The power method of obtaining this eigenvalue is quite effective (Ref. 21). Itis particularly effective when a good estimate of the eigenvector is rvailable. In the iterative design technique, a good estimate is generally available from the prcvious iteration. The power method is, therefore, well suited for use in iterative techniques. This method does have the severe disadiantage that K-i(b) must be computed for each new b. A different method of finding the smallest eigenvalue and associated eigenvector of Eq. 5-6 without computing K- I(b) is based on the Rayleigh quotient us discussed in par. 2-8 and Ref. 23. The smallest eigenvalue of Eq. 5-6 is obtained by choosing a normalized vector y which minimizes the quotient yTK(b)y/lyTM(b)y]. The minimum value

state variable. This situation is a direct result of Murphy's law "if anything can go wrong itwill". Actually, it is not realistic to expect that a mathematical fonulation of the kind presented in par. 4-5 should contain all realworld design problems. Already, an important problem has been encounter'J which requires an understanding of .! ..evelopment of par. 4-5 in order to include ,he new problem in the steepest-descent algorithm. The eigenvalue problem, fortunately, can oe treated very nicely by the steepest-descent technique. Developmert of the method will be done in par. 5-3. 5-3 STEEPEST DESCENT PROGRAMMING FOR OPTIMAL STRUCTURAL DE. SIGN !n order to obtain a steepest-descent algorithm for the design problem with constraints on r'3envalues, it is nucessary to go back into the derivation of the algorithm of par. 4-5. The major effort required here will 5-7

:

-

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'be theiin.aration of'the siuctural design,. .probl 61edtaj..;an'appiq,

W~t:rilbl

'2',

0f

.

ifhe ,kind ddksibed, by 134s. 44119 -through 4-121.for t,.1 LINEA111ZED cosT AND .sT : NCTI0N i!N S

CON.

depend. 'b,. - the first orderb Sinc on. tbe zi cost ainand d coi stit :functions be rturbation

AN"!

=" :

(5-I15)

7b

-- " ;

each 0/, 0. All this follows since t,(z.b) in . formalation is simply A(b)z in the the-general present piiobleif so = -"-

[(b)z I:

these ofunctions due to es' small

chunges Sz, 6b, aad 8r'An z, b, and ' is += +=

-a

(-(5) wit A symmetric so A T = A.

and

-

Thus, the explicitydependence of Eqs. 5-10 5-11 o eand can lz be easily eliminated. It az

ar

M-

6',

(-1

remain', 'q le~ermine 8r"in terms of Sb. This probl3mhas been addressed ina completely

The problem of writingexplicitly the perturbed and cons intermscost :f

rigorcus manner by Kato (Ref. 23). Explicit expreisions are given there under quite restric-

6b now reduces to obtaining explicit expresslons for the terms involving 6z and St.

tire hyootheses. A formal development will be givai here which obtains the same result.

From Eqs. 4-117 and 4-118, and the perturbed state equation we obtain, just as

Itisassumed that the eigenvaes and eigenvectors of

Eq. 4-119,

K(b)y

6z aznd

T a [A(b)z

b

where

5-8

(5-16)

depend continuously on b and further, that to first order, the following perturbation equation isaccurate =

d

-

[A(bzreb

(-13)

az ab and X are determined by

AXJ h)-z and

M(b)y

(5-12)

and •-

=

K(b)6y +l- IK(b)y ISb = ab(5 + 2-[ add(b)y 6b +

(5-14)

b.Ti(b)y

-17) c(b)ly

where y and t satisfy Eq. 5-16. IfK(b)

cr

ot(b) is not symmetrice itis

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AMCP 706.192

-

necessary to solve the adjoint eigenvalue problem,

tion-thatEq. 5-f7holds is highly questionable from an operator the itid point of view. Under reasonable assumptions on- the fitite

(5-18)

dimensiona! eigenvalue problem treated here however, Eq. 5-19 is shown to hold (Ref. 23); i.e., even though the jqstificat.on given here is not mathematically rigorous, the result, Eq. 5-19, holds for a-large class of problems,

KT(b)7

-

=

MT(b)F

that has the same eigenvalue t as Eq. 5-16 but a different eigenvector 37. Rearranging and premultiplying by y T this is "

;37'[K(b) - tM(b)]8y + 7T _

[K(b)y] 8b

"

3.

Defining

Sr [M(b)y} 5b br[.

.

3b

[A(b)z] ,b

+ I ~L/ 3 7 T(b)y1ij

=3TtAf.(b)y

--

1K

x(

Since the first term is a scalar, tf jK(b) 3F.218

3-A[K(b)y]

T.

[M(b)y I I . 7

-

(5-20)

and

Sincej7 is an eigenvector of Eq. 5-18, MT(b) 7

[KT(b)-

=

BT

Aumn

[K(b)y

b

3b

+ (

[K(b)y]

0

and this equation becomes -yT 6 aM(b)y)

6b{1

T

:=-'{' I [M(b)y 1

T

y 7

( (S-21)

= j7T Af(b ) y . Assuming be the case,

T

A(b)y

*

,

0 which will generally

x Ib/M(b)Yl, or 0, if is empty. 5-10 and 5-11 become

r"bKbEqs. .'f" [_(b)y]

6b

(5-19) l'TrM(b)yl.

6J =

I

6b

(5-22)

6b.

(5-23)

and Derivation of the perturbation formula, Eq. 5-19, has been strictly formal. The assump-

=

5-9

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AMCP 706-192

The lineafized problem'is -now to minimize &f,,Eq. 5-22, subject tb-constraints T

where A is the desired correct'on in constraint error and

-M; (0 T | 1 .+ 2v,). If any component of IA is negative, remove the corresponding row from and return to Step 3. Step8.bCmput Step 8. Compute

6b

(- Ra'

=

T

5bTIVlb < and where IV is positive defirite and is small. This is precisely the same pr(,blem in par. 4-5, Eqs. 4-119 through 4-12 1, so the theoretical results and steepest-desceht algorithm of that paragraph apply with proper interpretation. 5-3.2 STEEPEST DESCENT ALGORITHM FOR OPTIMAL STRUCTURAL DESIGN Step 1.Make an engineering estimate of the optimum design variable b(° ). Step 2. For 0, 1..., solve Eq. 5-2 for q), z Eq. 5-6 for yU) and W, and Eq. 5-9 for 37 (if k(b) or M(b) is not symmetric) with b = b).

5b2

= jj'l R£Af;0A&

and form 5b

I --2 bl + 60.

Step 9. Compute b(T+t) = bU) + 6b. Step 10. If all constraints are satisfied and Sb is sufficientiy small, terminate. Otherwise, return to Step 2 and continue the process.

Step 4. Compute V and £2 in Eqs. 5-20 and 5-21.

All te properties of 3b' and Sb derived ir par. 4-5,2 hold in tJ'*.. case. Further, the discussion of that par igraph regarding suh things as choosing v also hold. The reader should refer to that paragraph for detailed discussions.

Step 5. Choose Af as the desired reduction in corstraint error.

5-3.3 COMPUTATIONAL CONSIDERATIONS

S -p( Co.-,-.i

,sewhere

Several comments on the computational art used in solution of these problems are in order. First. if a feasible design was chosen initially, large steps could be taken unil one or more constraints were violated, d, which

w.te ip

time the step size was reduced. Second, it was noted that a- the optimum was approached,

Step 3. Form as in Eq. 4-105. Solve Eqs. 5-14 and 5-15 for NJ and X .

(

1, 1.k is empty Tl

S

Step 7. Choose v > (,.. _ 5-10

,

It

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--.

=

-

________________

"j

a

AMCP 706-192

oscillation occurred. By monitoring the dot O

product, 6bU)T 5b - 1), oscillations were sensed when negative values of the dot product occurred. Thus, step size, 1/(2v), was divided by two when negative values of the dot product occurred on tw%oaccessive iterations. Finally, the most effective method of adjusting step size was to monitor successive reductions in cost function after feasibility had occurred. Once insignificant reductions occurred, the step size was reduced to obtain finer convergence, The Power method used to compute the smallest eigenvalue performs quite well. At every iteration, the starting value for the eigenvector is taken from the previcus iteration which manifested a very rapid rate of eonvergence. An accuracy of 0.1% in eac! component of the elgenvector was used to compute the new eigenvector. The stiffness matrir for the structure was inverted by the Gauss-Jordan .. // elimination procedure.

number of constraints with positive components of p was always less than or equal to the number of design variables of the problem. This procedure of adjusting the constraint set has worked very well and has minimized the possibility of divergence of the algorithm. The method piesented is relatively automatic in the sense that, for the computer program developed, the input data given is the only pertinent design information required for solution of the problem. All the necessary matrices and their derivatives are automatically generated in the computer. Any person with a reasonable knowledge of FORTRAN language should be able to handle the programming without any difficulty. The method is developed to meet simultaneously displacement, strength, and frequency requirements on the structure. The technique, therefore, can be made user oriented.

5-4 OPTIMIZATION OF SPECIAL PURPOSE STRUCTURES

•

Another comment that is appropriate here concerns the sign check on the Lagrange multiplier vector called for in Step 7 of theoptimizamultplir pcaled vctoor n Sep ofthe tion problems are solved in this paragraph on, tonpblmarslvdithsaagpho computational algorithm (par. 5-3.2). The comutalgeraic sigoehm cpnen of. The an ad-hoc basis to illustrate the method of

algerai ofeachcomonen sig

ofthe

par. 5-3. Subsequent paragraphs will treat

vector p was checked at Langrainge multiple each iteration.eachiteatin. iff sme some oof th the coponnts components were negative, then the matrix 0 and the vector A were adjusted accoidingly. This procedure is particularly useful whenever there were redundant constraint violations. in some cases, thL number of constraints violated is more than the number of design variables of the problem, yielding a singular matrix coefficient of p. In such c- es numerical noise yielded a solution such that some of the components of the vector p were always negative, indicating that the corresponding constraints would be strictly satisfied in the next iteration. In numerical examples, the

l treat s par. s-a. Subeq large scale problems in a more unified man-

nr

5-4.1 A MINIMUM WEiGHT COLUMN A column is to be constructed by making its cross section piecewise uniform as shown in Fig. 5-1. The objective of the design problem is to choose the element areas so that the column will support a vertical load P0 without buckling or yielding under compressive load. For the purpose of the present problem te geometric shape of each column elerrent is tixed and symmetric about two 3-11I

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Q

AMCP 706-192

1o

PO Yr-.

where ax

is the allowable stress of the

column material in compression.

fx X

Sb,

In order to apply the optimization method of par. 5-3, the equations which determine the buckling load in terms ofb = bl,.... bkIT must be obtained. Using the generalized coordinates shown in Fig. 5-2 and Eq. B-4, Appendix B, the potential energy of the ith element under the buckling load P is

Figure 6.1. Column

PE, JuT K(b)u

orthogonal axes so that the cross-sectional area b, of the ith element comple.ely specifies the element. With this assumption, if a is the second moment of the cross section of )mit area, then

41 01,hi2

(5-24)

where II = (Ul, II,u. u t

k4

2; biL,

r

is as shown in Fig. 5-2. The matrices K(b) and D(b) are from Eqs. B-4 and B-8, Appendix B

In this problem, weight of the column is to be minimized so that the cost function is J=

Pu DI(b)uI (5-28)

-

u1

Q

UI

(5-25)

(at

where y is material density and L, is the length of the Ith element of the column. There are two basic constraints that must be satisfied in this desigi: problem. Fh'st, to insure that the buckling load P is not less than th ipplied load Po, it is required that

Figure 5-2 Column Element

.(5-26) Second, in order to insure that the column material does rot yield under the applied load Po, it is necessary that matria des iotyild Oj - (Po/b0)-

12 -6., i -12 &b K(b)

0,

5-12

and

-6LI

- 6L,

4L'

6L,

2L.'

2

6L,

12

2L,2

6L

2'

2L

4,L

L3

52 9)

1

(5-27)

S .k

"

I

1.

4L

.29

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AMCP 706-192

3

1

SL,

20

3

1

5L1

20

T

L

1

15

20

(5-33)

3

1

3

1

5L,

20

5i,

20[

1

Li

I 11

L(

60 (5-30)

D(b) = 2

76 20

O.

I=,., k

1 20

2;0

.

_

ab

since

0, does rot depend on P, 1= 1,..., k and

ekl

T5t

~kI=

-

Summing the total potential energies of all the elements from Eq. 5-28 and defining a new variable

i

ab (1T/JT [D(b)y]

) Ty)

+ > 0

{0], i f Ok+I < 0.

Y = [YI , Y2 ... Y2k I T

(5-34)

The computations required in Eq. 5-34 are 2T u2, Ut1 t, u4., 1[ut energy PE of the column the total potential may be written

messy but they can be programmed for automatic computation. All expressions required for direct application of the steepest descent algorithm of par. 5-3 are now available. Numerical results and profiles of optimum columns are shown in Tables 5-1 and 5-2, and Fig. 5-3. Numerical

PEJ Y T K(b)y-P YTD(b)y 2 2

where K(b) and D(b) are made up ofelements of KI(b) and Dr(b) and are symmetric. Applying the theorem of minimum total potential energy given in Appendix B, the governing equations of buckling are

data for the example problems are E = 3.0 x 107 psi, a = 0.079577, oma x = 20,000 psi, and L = 10.0 in. Computation in each case required approximately 0.1 sec per iteration

TABLE 5.1

(5-31)

K (b)y - P D (b)y.

COMPARISON OF UNIFORM AND OPTIMAL COLUMNS Eq. 5-31 is now in the form of Eq. 5-6, with proper interpretation of notation. In order to implement the computational

aigorithm of par. 5-3, the following vectors are required:

= =

1.

yL 2 . . ... jLk

(5-32)

P, lb

Volume of Volume of Unifor:* Optimal Column, In? Column, In?

500 1Wo 1500 2000 4000

0.806 1.143 1.411 1.640 2.412

0.923 1.300 1.600 1.840 2.600

Material Savings, % 12.7 12.1 11.8 10.9 7.2

'Lighrest uniform column which will support load P.

5-13

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"Y•

AMCP 7C66192

TABLE 5-2 CR03S-SECTIONAL AREAS OF OPTIMUM COLUMNS Element No. I

P - 500 b

P - 1000 lb

P - 1500 lb

P - 20001 b

P 4000 lb

1 2 3 4 5 6 7 8

-0.1070 '0.155 0.1035 0,l0o0 0.0960 0.0831 0.0738 0.0623

0.1499 0.1480 0.1442 0.1383 '0.1303 0.198 0.1064 0.0892

0.1833 0.1809 0.1763 0.1691 0.1593 0.1464 0.1299 0.1088

0.2106 0.2076 0.2023 0.1942 0,A831 0.1683 0.1493 0.12F0

0.2947 0.2875 0.7789 0.2683 0.2505 0.2302 0.2020 0.2000

9 10

0.0477 0.0267

0.0668 0.0500

0.0812 0.0750

0.1000 0.1000

0.2000 0.2'000

together uniforn sections of beams as shown in Fig. 5-4. The objective is to choose the sections so that the beam is as light in weight as possible and still satisfies constraints on strength and natural frequency. Due to dynamic inputs to the beam, it is required thit the natural -frequency of the beam be above a given limit woo to prevent oscillation probP=500

P= lt40

F - 1500

F"ctulems.

As in the preceding column design problem, the cross-sectional geometry is chosen, but all dimensions of the cross section may be varied in the same proportion:. Thus, if b denotes thu area of the ith section, then the second moment of the cross-sectional area is = ab2

7

P 2000

P 4000

P-6974

Figure 5.3. Profiles of Optimal Columns

(5-35)

where a is a constant of proportionality depending on the geometry of the cross section. The problem at hand is to minimize

and 15 iterations to converge on an IBM 360-65.

b

b2

b3

5.4.2 A MINIMUM WEIGHT VIBRATING BEAM

A beam is to be designed by piech.g 5-14

Y

Figure 5.4. Stepped Beam

(

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AMCP 706-102

[

weight, so the-cost function is .kX=biLt

(S-36)

j

_ 6/L 1 . _ 12/

_64L"

=E-

.1 1 '

i.1di)

6J -.1

the is-material density and L, -is,

wheic

121

q6%1 L1:'

4Il 2 614L !, '2,/L 1

length'of theithsection.

-(540)

As atrength constraifi, it iseq'jiared that i

-j

O,

-

.(5-37)

=-1..., c

Wiere:b4 0 0 is chosen so that theban 1"support a -lateral- load. The coustraint oh natural frequency can be written as

'Forming as 6gl.vector y,thati contain:. all displacemets:androtations for-the beam, the total kinetic and potehtia, efiergies are ij~'(b~;/2 and y-K,(b)yI2, respectively. Lagrange',e uations,.Eq. B-i 7, are then 1

+ K(b)y(t) =0

-[08M(b)(t)

By neglecting compression of the beani, deformation of a typical element is sY own in Fig. S-5. By Appendix B, the kineti( eAergy

(5-41)

For harmonic motion of the structure, y(t) y sin Cot, where y is just a constant vector, tis time, and w is natural frequency. Substituting 2 = w , the into Eq. 5-41 and defining eigenvalue equation is

& AI

u

2

Figure 55. Typical Eleme it

of an element is,VlM'(bs '/2, wlire, from Eq. B-6 156 -

Mf(b

)

pbLi 420

- 22LI

22L2 4L

13LI

54 -

13L,

-

3L

(5-42)

The problem of minimizing J of Eq. 5-36, subject to the constraints of Eqs. 5-37 and 5-38, and with state Eq. 5-42, is in the form of the general proolem of par. 5-3. The steepest-descent computational algorithm of

that paragraph can be applied dh'ectly to this problem.

As a numerical example, the beam problem

13L1 156 22 13LS- 3Li 22LI

K(b) y = rM(b)y

22L1 4L

4I (5-39)

Likewise, the potential erergy of the ith element is UITKI(b)uI/2, where, from Eq. B-4

was solved with the data E = 3x101 psi, L = 10 in., a = 1.0. and p = 0.00208 lb-sec / Win?. The computational algorithm required about 0.6 sec per iteration on an IBM 360-65 system and approximately IS iterations to converge. Results for a range of natural frequencies are given in Table 5-3 ard the profile of an optimum beam is shown in Fig. 5-6. 5-15

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AMP706.192

TABLE 5.3-* COMPARISON OF OPTIMUM BEAMSA 'Volu-me of Frequency, Unifbim radlec arn,lin?

Optinm V6 u-me. 16? 1.062 0.897

Material 5 Ssvlng's,% 8.0

1.3'97 1.663 1.9512.263' 2.598

1.269 1.481 1.727 1.993 2.283

9.74 10.94 11;4813 11.93 12.12

7.217

6.330

12.29

51.6 0 0,935

3130 40 D 46

'4800 5200 600 6000

10000

2 Y

4.06

1f

L

1

t

*Uniform beam of lowest volume having required

5.7. Portal Frame

natura freqency.Figure

IIIment stiffness matrbt fzm Appendix Bis Figure 5&6. Profile of Optimum Beam

1 E0 2 1 Rb

-6L

1

.. 1.2

Neglecting strain energy due to axial deformation of the horizontal member, the ele5-1 6

6LI

6L,

12

-L 2L2 61.,3

2~ 6

5-4.3 A MINIMUM WEIGHT PORTAL FRAME WITH A NATURAL FREQUENCY CONSTRAINT(53 A portal frame as shown in Fig. 5-7 is to be proportioned so that it weighs as little as possible and has its fundamental frequency at least as large as a specified frequency o. Each member of the planar frame is formed from several un~iform sections whose areas are to be determined as design variables. As in the.)U preceding problems, the cross-sectional geometry is taken as fixed and all dimensions of cross sections varied proportiona~y. The second moment of the cross-sectional area about a centroidal axis is 1,=a2weeb s the cross-sectional area of the ith element.

6,-2 4L2

-

6L

IL

I

where L, is the length of the lth member and the element deformation v riabies are shown in Fig. 5-8. The potential energy PEof the 1th element is

3 U1

U3

U

AlU2

1

Figure5&8. Typical Eiemerts

U4

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AMCP 7O6192,

PE, ~

'1* 1 1 T' K(i)u. 54)4 -2,~rbtatfion,andP

"".M(b)'ae4formed-

*h~rey~s the ect&i f l4 dipjacemenits and: caTheluaticesK'1(b) and from elemfent-stiffiiess afid' lhtieas0:1id in'Appendik B.

whaw

miss

'Likei, fr o Appendix ;B 'the kietic enehj 'E f h t', i al66 .i ii0rs

and-.the '4iatrikx..'or ,'tlis pofblem ~issimply wveight of thi'stiucir,whicii k

'4 -~

uM~bu~

m

(5.48)

4ZbiL1

wher'z~dentes irie~eivatve'f uandWhere-p i-.defisity,'oftthe .stnictura1 material' 1lS6

212Lt5,

13h

The -constrhiifiiPbse-d~oh athe: probl~em include Iowei limits on o--iitibnal'irea

420

54

- 13LI

156

1L3122L

22LI 4

2

(5.45)

Takng ntoaco.un th laera riid ody motion-of Member 2, th3 total kinetic energy of the structure is 1 KE Z 2" i'M(b)z4'

T

MH

(546

wher mssMs fteMeber2 ad ~ is the horizontal velocity of point A. Rquiingharoni moionwit frquecy w, the displacement vector yQt) made up of all isvibratiort dsplaemens E

q~b-

1

.54)

where b0 > 0-and a. lower limiit on natural frequency

~

o-

(550)

0

where ,is the lowest allowable eigenvalue of Eq. 5-47, to =w The steepest-i lescent algorithm may now be applied directly. Data for the specific problems solved arc given !nTable 5-4. The results for an aluminum portal frame are given in Tbe - n -,wt yia rfl shown in Fig. 5-9. The design variable b, shows the distribution of material for a minimum wteight frame whose frequency of must be greater than or equal to a

de~i)diii wt~scli ~si

TABLE -

where y is a constant vector. Applying Lagrange's equations and eliminating time depedene yeld

K(b)y = ft(b)y

~,.,.

0

(5-47)

MATERIAL PROPERTIES FOR ALUMINUM

a, dimensionless .42.61&Wi0 rlb/in.

P b-SO/i 2

10, n

L, in.

0.07058 10.3x'10'

.40.009825

10.0

5-17

-C

Downloaded from http://www.everyspec.com

IA'

Ft~kiv'..6

-.

'

COMMAI*ISON OF uNiFoRM'.AD OPIMAL

" E',F(fi RA AO~iNM -:. - ...... 4tABLo ttbf Frequency, rad/w-, "______..... 2000' 3000, 4000 5000

' .resits

"f~rm...Opthl

Fre,

Weight lb Fame lb- Reduction,%

___, ________________

3.748' 8.434 1.4.994 23.428

-1729 2.562' 350-4.6A8

'53.9 69.6 76.1 80.0

TABLE 56, OPTIMALROES IG ARIABLE FOR VIBRATnG FRAME +,-~'r-,,l/ 2000 3000. '4000 5000

OPTfIAL 6ESIGdNIVARIAB'LE bi-

bi

-

b, 1.577 1.964 b2 0.883 1.604 b3 0.552 1.416 b4 0.374 0,866 b50.350 0.360 bE 0.350 0350

2.907 2.484 1.912 1.290 0.671 0.350

4,020 3.321 2.622 1.725 0.836 0.350

the same for Members I arnd 3; and Member 2 converges~to the lower bound Io,so only the for r the dare reposg va ed. or ae frequencies the values for theb, for Member2 are equal to 0.350. 5-4.4 A-MINIMUM WEiGHT AMEWITH MULTIPLE FAI -URECRITERIA To illustrate the applicability of the stecpest-descent method for the minimum weight design of-structureswvith'-stres, buck ling,, and displacement cofistraints, an example of a statically, loaded, frame problem-is presented. Fig. 5-10 -shows the, geometrical

B

34

tdgure & 10 Frame With Side Loading

Figure 5-9. Optimum Portal Frame for w =3The specified value. It can be seen from Table ,-5 that a significant material saving is possible in comparison to the portal frame with members of constant cross section. io18

( .

configuration of the frame that is considered. All members are assumed to be of the same length L. Member I is subjected to a lateral loading q(1). Member 3 has a uniform crosssectional area which is prescribed and will not be allowed to vary. The connections at points A and B are frictionless pins. finite element method is used to obtain the elastic response of the system for a given set of design variables, i.e., the crosssectional areas of the elemenis. As in the preceding problem, the geometry of each cross section is the same with all dimensions

-

Downloaded from http://www.everyspec.com

'AMCP-7O6.i2

--.-..

of'Cross section varying proporionaly. Th,

I,-cib" where b1 ii -th e bss, dtkiiinar~aOf,

where the

trix, P isderived.-fronmiohe

shortening of Member 2 as in par-. 54; Vand,.#s.

thelth lement. -The stiffness-niix Kbl)of

in :the pieoi(us, problem, K(b),is,a-.stiffriess

a typical elen~in, Fig. 5,-! i;cn'e vrtter'as-

matrix..

in pa.- '5-4.3.with geneialzed&dispkcemehts defineAby

'

1

and 2which'is "141, si ply

IT

'

The cost functib n to: mizeU'in this . -pro5lem is:the structural weight ofMenbef9s

4

U

where, .,yis the weight.density of the traterial. "

2

1-

u

-V

I. Stress constraints at the Ith node-of.' Member 1:

.,.

ea E l Figure5-I. Typical Elemnts From the fundamental beam theory, ifR is the horizontal force transmitted from the Member I to 3, and assuming that-Member'2 remains straight without buckling, then neglecting compression of 3Member 2, the deflection at A is uA = RL /(3EI 3). From the equilibrium conditions on the transverse forces and moments at the nodes of Member 1,the generalized displacement z, which 's made up of the element displacements u1 can be evaluated from tion the following matrix equa-

tion

A(b)z -F

(5-51)

where F is a vector load and A(b) is a symmetric matrix. In a similar manner, if y is the displacement vector containing all element deflections associated with Member 2, the buckling load P can be determined by solving the eigenvalue problem K(b)y

The weight of the.frame-is to'beminimized. subje tbtbt-fllbwlng constraints:

A

PD(b)y

(5-52)

0 1x =lc =01-sa x
0, to t to obtain (6-25)

Condition, Eq. 6-26, Is.-a second-order differenial equation in .~Q) and Is called the, ueagrani euton. Condition, Eq. 627 is called a transveriality condition. For each I or / such that 77,(P) or ii,(t) is not specified by Eq 6-4 q .7imle Fa;(P)=0 or aF/Ax, (0l) =0. The condition, Eq. 6-18, at discontinuities ,(called corners) In i'(1) is called the Weierstrass-Erdnin comer condition.

ieelecntnoss - 0,[I + @1)1 0 and x'(t)-is required to -be continuous so [1 + (x')2 1 *0 and-it, therefore, is required that -

i"

1"(t) 0

V

or i(t) at +1b,

I where a and b are constants. This implies th3t

1

0

Figure6.Z Minimizing Sequence

In this illustration, a solution of the problem exizts in the class of piecewise continuous functions but not in the class of twice continuously 'differentiable functions. This problem, therefore, should serve as a warning that not all innocent looking calculus of variations problems have solutions.

x(O)

b= 0

and

Therefore the solution of the problem is (6-34)

Eq. 6-31 is in this cas [FX,X, x = 0.

The end conditions yield

x(l) =a =.

Case 2. F depends only on x: F = F(x').

the shortest path between two points in a plane is a straight line. This shouldn't shake anyone up.

x(t) = t. Case 3. F depends only on: and x':

(6-35)

Example 6-4: Using the formulation of Example 6.1, find the shortest curve in the t-x plane which passes flitzugh the points (0,0) and (1,1).

F =FHt.x),

(6-36)

Eq. 6-26 is, in this case, d

t

6.9

Downloaded from http://www.everyspec.com

MCP 706-i 92

reduces to

:O~.-:this

Fx,(t.-

C

(6.3.

-where d is inarbitrary constant.

l{2gX [1 + (XI)2] or

F F_(.x;xJ.

(6-38)-

Eq.- 6-3 Hs, in this case,

.

Fx - F., x'-"F, *x" 0. M11'in byx. yields2

____

'whee~C'~is

-C

newv constant-.

The ~oltian bf titis differential equation is a family of pycloids in pafaMietricform t= rC 2 -

-x

2

1X

Case 4.,.x is a real.valued function, ind'Fxl dPends bnty. n-x anid-i

*

12c.

+C

- (s -sins)

.and, xF-(x' 2

x~~

x'x"Fx.x. 0.

F

o )

This isjust2 The constants C, and C2 are to be deter! mined so that the cycloid which passes through the given points Is fixed.

F. 0, so

F - x'FX. C

(6-39)

where C is an arbitrary constant, oiv-5.theiirchisochone Examle th Brchisochone Examle -5.Solv problem of Example 6-2.

The function 17 from Eq. 6.4 is F i +('2

6-2.3 VARIATIONAL

1l+x2 /2

it

J

(2gx)' =

6-10

C.

NOTATION

SECOND-ORDER CONDITIONS

Eq. 6.39 applies in this case and yields

2gX

It should be noted that each of the problems treated here reducedto the solution of a nonlinear differential eql'ation. hIilis charazteristic of problems of the calculus of variations. The reader is undoubtedly aware that it is only In the simptest cases that closed form solutions of these differential equations may be obtained. Further, questions of existence and uniqueness or solutions are by no means trivial.

(I + (X'

2

1112

For1(x)

AND

F(t,x~x')dt,

define the fi:,,, variation of 1(x) as 16(x)

-Ax +ecbx)I de

(6.40)

Downloaded from http://www.everyspec.com

7>

--

AMCP 701

_ -

'de J*

TF-a2 (tX, SXT

,

'

i[_.

+ I

T~xx, (ti~x')xx .-

6*O

r32F

'axax

SaF

(,x,.66

=m'-

.'+'+x

dtx

Jd

tx~-

-•

J() = 0,

+t.

.x'

Define

Note'tfat ai, this.d s notrequirethat-i(t) be tie solution.of the'fundamental problem. !f, however, x(t)= Ht) is-the solution-of. the, -fundafiiental.;problem, thent-iis ,c!eir,_trom Eq. 6-19 that ris necessary that (6-41)

a2F

A x

2 B- 2a "

2

)7

aid

for all 8x(t) for which i(t) + e~x(t) the end.ccnditions in the fundamentalsatisfy problem. In a way quite similar to-the definition of the first variation, the second variation may be defined as

C= aXF

With this notation, 67.J(x)

JXTx)

62 J(x) =--

92

(5xr'ASx +o.,tBbx '

=

+ eax)

+ x'TCSx') dt.

Performing the differentiation, this is

If firt,x,x9 has three derivatives, then by 62J(x)

Taylor's formula

d+ 2

d I_.

we

[

1 M + e4J(x + eax) =J(x)

8x T

d

L

ax-

e-o

d2Ji

d

e-0

aFT a 'r(t,x+ e6x,x'+ e6x,) + 6x-

+_

3Jl le-e

(6-42)

(t,x + S5x.x, + e6x')d e-o

where 0 < g < e. If we computed d3 J/de3 , It 6-11

Downloaded from http://www.everyspec.com

AMCP 706-192.

-would-Anvolve a'sum of terms each contdinLng third degree terms in ft,.and'6x'For'6x and -xi suficiihtly -small, this term may be -negJ cted to -obtain ,a second-order approxi-mationwhen e = I so that i ¢+ ax)Z + ax) n 6+ l2J. ..

,'

whichhas largn derivatives. One might, therefore,,beJd to believe that, the derivative teen in the inequality of EI. 6-43 is dominant. This would then require C io be positive. semi-definite. To show that.this is the case, assume-that'

tthere

is a point t*, t0, < t < 0,and~a 'nnzero

Ut is.clear then-that -and 6-J play -the role. of differenti-als--theo, ~o~0f functionals.

vector h~suchthat hC(t*h = -23 < 0. For any cotinuous.6x() such. that, 6x'(t,*):= h, there is an interval

Further, if (t) yields a xelative minimum for the fundamental problem, then 01¢ +6x) is a relative minimum at e 0. Itis, therefore, necessary'that

t* - ci < t < t* + a> 0, such that 8xV(tC(txt) 'C - 3 < 0 in:t,*-oc,4 t -c t*+

d2 J d[

0

"

Define h;Isin' 1* i,

This Isjust 82J(Sr) > o,

]t-e,

,,4t*+a

Sx(t) = 0, elsewhere

or Ltt

so that

6x + 6xrB6x ' (SxTA ( + T(

+ ax'rC6x')dt

h

0

(6-43)

for all xL-euch t6hat .i + x satisfy the end conditionX for the fundamental problem, In what follows it will be convenient to limit 6x(t) to those varlation3 which satisfy UX(t1) 6X(P ) = 0. If 6x"() is small for all t. then 6x(t) mu,' 0 also be small since x1i ) = 0. On the otner hand, It Is possible to choose 6x:t) which is zero at the endpoints and small for all t, but

6-12

0I

=

[OS

, t*--t

Z t+ 1

0lx'() ,

t~o

elsewhere

Now, Eq. 6-43 is 0 04

(6xTASx + +,..xTB~xI + 6x'T46x') dr

.

*a

2 2

.

.- o

r(tsina-

,*)

hT

Downloaded from http://www.everyspec.com

AMCP 706.192

+""-

sifi

c

v:

t 0

I ,,

incorrect. This-limplies TCh > Ofor allh and t heeforeund t)et postives r-deflative

Since

this result inaybe stated sTffin lor n 6-3.,

L" '

~~Theoremi

"--

6-3. A necessary oyditlon -for the fundamental probleni to have a relative

stOdO

miinimum. at M)4 is that SI.cos0

-

do

Gelftm~d and Forain tRef. 2, p. 104) indipo, Cate iatpeople to argue that condition a sufficient solutionareis,prone point of the

cos

where where:

for an extremum. They point out, however, that this is not 0, 0 q' + I1... q. to < 0q(t* t adtQ The conditions obtained through application of Theorem 6-5 to the optimal design problem may now be tated as Theorem 6-7.

dt

- -1

TV

,for t t1/

(6-121)

I(tP - 0) =0

(6-125)

(6-126)

H(t*-0)-H(t*+0)=0 - 0) - \(t* + 0) = 0 P(t)(tx,u.b) = 0

(6-127) I.

q

(6-128)

ga(btl.x') L.It x(t),uM.bl dt

+

i (il.l dt

all at

=0

r

(6-129)

fort=t

(6-130)

and -1=

/-

ab 6-30

0, for f =# 1I

(6- 122)

(6-123) t = 01t,x(t).u(t).b.X(t I " all dt=0 3G

A

a' _ H(t_O) + H(t + 0) = 0" at/

Theorem 6-7: If [x(t). u(t), b. t . x'l is a solution of the optimnal design problem o1 Def. 6-3 and if the matrix, Eq. 6-105, has full row rank, then there exist multipliers Xo > 0, 1. r, y 00, a Xt), =I...m, -. r' + I._r, po(t), = .... q.,,(t). 04,=q+ 1.... q, not all zero, and functions G and H of Eqs. 6-106 and 6-108 such that d

0

11[t,xthoU,h.X(t),,0] y,Ol f3r al ad imssiblu

1.

(6-131)

Q

Downloaded from http://www.everyspec.com

AMCP-706-192q

~ZA

It should be noted th~at, just as in Theoremn 6-5; the number of conditons here is just .-

n.

Existence o

sluinis, however, a

very difficult question that is treated in-Refs.

plicitly on U(t),_then this vquatsvii is of the pa;, tpobl-tetdi frreuedi rule of differentiation 0

-

-

2

cit 2at2

--

atax

INEQUALITY

64.2 STATE VARIABLE CONSTRAINTS

~ 10p df ' x di ~/T-.

In many meaningful design problems constraints may involve restrictions only on the state variable. This is the case when some of Eq. 6-101 depends only on t. x, and b. To study this problem, just one such constraint needs to be considered, i.e.,

(6-133)

where all the arguments are omitted. If the Fight side of Eq. C-133 depends explicitly on u(t) then this equation is of the form treated in par. 6-..1. 7Tis process continues until

0, to 4 1

00,0

tit

.

(6-132)

Let r -c t t, r - t", be an interval in which 0,of Liq. 6-132 is an equality. It is U, bo' the matrix, Eq. clear that ao,/au 6-103, has a zero row in this interval and hience cannot be of full tow rank. Theorem 6-7 cannot be applied directly, so further analsisis rquied.is necessary that

t4 t+ 00= 0 o i is te itervl In r4 0 0 i is .~ ~, In te ~iteral

0 =1P+ at dt

30 dx a dt

From Eq. 6-97, dx/dt mady be replaced by f and this relation becomes 0= d~o -

0

.L

x(bl(6-134)

*f inivolves u(r) explicitly in its right side and ut(t) c-an be dete,,niined as a function of x(t) and b, as in par. 6-4. 1. Trhe integer P~ A I defined to be the first integer for which this is true. The constraint, Eq. 6-132, is then called a P fh order state variable me 'umality constraint. From the theory of ordinary differential equations (Ref. 14). Eq. 6-134 throughout t < t+ and 00I t.x(r ),b I =0

(6-135)

O,8(tx(f),b]

+i00P.O ax

fI£xt,()bI(6-136)

If the rightz side of this equation depends ex-

are equivalent to 00= 0 throughout t 0-31

7

-

Downloaded from http://www.everyspec.com

-AMCP 708-192-

t , t'. 'Shis, cf course, requires that 0. have V piecewse continuous derivatives, and f have v_ piecewise continuous derivatives in r < t < t+. The point t- plays the role of a d1 in the problem stated earlier in this paragraph.

Theorem 6-8: If [x(t), u(t), b, t1 ,xi1 is a solution of the optimal design problem with state variable inequality constraints, then there exist multipliers m o 0, X (t), I = I .... n,"ya y, , . r, 0,' r'+ 1. r, 1, . .(t . (t) 0,j=q'+I...q, and 7-. i= 1, ..... v and 3 associated with a state

It will be assumed that when the right side of Eq. 6-134 is used in place of 0, in computing the matrix, Eq. 6-105, this matrix has full row rank. In this case Theorem 6-7 may be employed. To utilize this theorem, define

variable constraint and 0 in t dX af/1 dt -x' for t 0 t1, t-, t+ X dt

tt.t

- =O, for t

--

d

-

:X g.

dt 0

(6-143)

G- _X(t° ) = 0

where , =0 if 0. involves u explicitly, G=Xogo+

(6-142)

to

(6-137)

,

(6-141)

t~Caff

DG 36 --+

t

(6-138)

aGT aGT

/

x"" . Xt/ + ) 0 7O+X~

6-144t)

aGT

G=YZ€

y

jO r7' -

t-,x(t').b

(0-139)

where this sum on 0 is extended only over the indices associited with state variable inequality constraints. ri-are multipliers, and if= X, f - Xofo

+X(t- -0)-(t- +0)=0

---

X(( -0) - X(t + 0) =0 3G t+

.-

-

(t + 0) =0 -llt f

027 )=0

(6-140) 2

-

a

With G G 6 and I replacing G and I in Theoien 6-7, a set of necessary conditions for this problem are o' i::- .1They are easily computed and are given here as Theorem 6-8 6-32

-BG _

0fO) 0)Hl + +0)= 0 +(tI

G

) fl( -

-

It(t 0) + II(t# + 0) = 0

) 0

6-145)

j

Downloaded from http://www.everyspec.com

AMCP 706-192

-0)

3

-

if(t* + 0)

X(to - 0) - X(,*

0)

0

(6-146)

0

(6-147)

0(t)0(t.xub)O, 131 .

q

(6-148)

g.(b, t. x l )

S-

6-4.3 APPLICATION OF THE THEORY OF OPTIMAL DESIGN

L.[t,x(t),u(t),S] dt

= 1..r dt'fortt

In order to develop some f;nmiliarity with

=0.

to

dil

these subintervals is not knowr before the solution is computed. The generality of the problem makes it difficult to discuss all its intracics without resorting to special cases and examples.

(6-149) .1which

t',ol(6-150)

and

the methods of the preceding subpamgujphs, several examples will be trea.ed here. These problems wdl be idealizations of teal-world pr-blems but will illustrate the basic ideas carry over into more complicated problems. Example 6-8: Time.optimal Steerbg of a Ground Vehicle (Ref 19)

HI t,x(t). U,bAt).-y,0J H[l:x(t),u(),bX(),'O!

(6-151)

for all admissible U. The full set of necessary conditions em. bodied in this theorem is awesome from a theoremt in this issil awesom from aeil bodie computationalequaion point of view. The differential x ad Xaresubjct fo equations for x and 'A are subject too mlti multipoint boundary conditions that involve a set of undetermined multipliers. In a gross sense, Eqs. 6-147 may be viewed as determining intermediate points in t o 4 r -, t" and the associated boundary conditions on x(t) and

To illustrate the concepts presented in par. 64.2, an optimal vehicle steering problem will be solved Thib problem is chosen because of its clarity of formulation and solution. A ground vehicle (a tractor in this case) is to be steered so that it begin it a given point and is steered so that it reaches a given straight line pan th e shortest possible time. The vehicle and the line it is to reach are shown in Fig. Poini /, midway between the rcai wheels, is located by the coordinates x (i) and x (t).

X2

Use of the theorem is further coiplcated by the fact that the design ve.iable may be determined as the solution of Lq. C-142 which satisfies Eq. 6-15i. This means that u wil! be determined as a furcton of x, b and all the multipliers Thk expression foi u will.. generally take different forms in different subintervals of to % t , tn and tie spacaiig of

x2

/> " X1 Figure & 11. Ground Vehicle 6-33

4

.3,

Downloaded from http://www.everyspec.com

A

70.-92 -C

Orientation of the velticle is specified by a third variable x 3 (t). Steering of the vehicle is accomplished by choosing the angle 0(t). From physical grounds it is clear that the state of the vehicle is described by x(t) [XI(tV 2 (t),Xa(t)]T and the ve'.cle is controlled through choice of 0(t).

=$'

Vsinx3

(

(6-152)

ai

° initial x() = the VIL. I nd time t= G.°x0(0) O At = A, and X3 (0) A. The terminal time T is not determined but it is r-quied that X2(7) x, and x3 (7)= 0 since the vehicle muzt be tangent to the target line at time T.

4 -H=

aX 2

o

(6-153)

All other initial conditions can be obtained from these by reflection in Fig. 6-1I. The kroblem is now in the Io.m described in par. 6-3. For use in Theorem 6-7,

6-34

X

3

H

2 0-p =X 3 aseO

TO

+P2

(6-155) "

X (T)= 0 (' ='o.,(T)Vcosx3 (T)+.

(6-156) ) 2 (TJsirx 3 (T)

+ X.(T)a tan 0(7)

and as an idealization it is assumed that any steering angle in - 0o 4 0 ,-0o may be chosen instantaneously For a reasonable problem it is clca: that 0o < ir/2. Further, for definiteness, assume lx I < ir;z and x2 > X .

'Y

I

'-XV sinxx

-'

The steering angle is limited by

+

0

X2 V cos X3

Sx2 (0)

G--XT+y

(00-

- ax,

=

- 00 < 0 < 0

-92

"

Thu conditions of Cheorem 6-7 are

= a tan 0 •i 3

where x ,a

X3 (T)

+

, (0- 00)

-

'0(6-154)

3

X2

H = X, V COS x 3 + X2 V sin x3 + X.a tan 0

It is assumed that the rear axle of the vehicle moves with a constant velocity V. In this case, motion of the vwhicle is governed by the differential equation S= Vcosx

4 ix2 ()-

+

111(

0

- 0)

(6-157)

= 0

(6-158) 6-(0

-80)=0

and dH al6l

(6-159)

The first two equations in Eq. 6-154 yield . (1)

lx1(0)-x0° I

IY2 (0) - x 1 + y [X3 (0) - x I

and Eq. 6-156 impheq t equation in Eq 6-I54 is daen

=

0. The last

.

Downloaded from http://www.everyspec.com

AMCP 700-192 V cos x 3 .

i3= -

Since x0 < r/2, for t small, eithe- 0(t) = do or 0(t) =-00. From Fig. 6-11, it is

Using !he first equation in Eq. 6-152 to replace V cos x 3 , this is 3 =

-

reasonably clear that 0(t) = 00 and Eq. 6-152 can be integrateJ to obtain

x 1(t)=x ° +R [sin (x3 +bt)

x1

Therefore, X3 ()=

- sin x ]

-

2 x 1 () +

.

(6-160)

The behavior of 0(t) may be isolated to two different eases. The first is 19(t)l = 00. The second is 10(t)l < 00, in which case Eq. 5-158 implies jil(t) = ul(t) = 0. Eq. 6-155 then then shows that X3 (t) = 0. By Fq. = 6-1(0 =

x(t) is either a constant or t2

t 3 0. Triis

and Eq. 6-157 then implies Xo = 0 so all X,are zero. This is forbidden by Theorem 6-7, so x () is a constant when 10(t) I < 0o. But it x1 (t) is constant .;ij) = 0 and the first equation in Eq. 6-152 implies x 3 t) 0. The last equation in Eq. 6-152 implies 0(t) =0.

x 2 (t) x

- R [cos (x ° +bt)

(6-162)

- cos x3 x 3 (t)=x' +at tan 0o, where

b =a tan 0o R = V/b This path is just a circular arc with center at (xo - R sin x, xo + R cos xo) and counterclockwise motion.

It is clear then that if 10(t)l < 00 for some in termal of time, the path of the vehicle must be c straight line parallel to the x?-axis in Fig. 6-I. Siice the last two terms in /1 are zero, the only ex:icit dependence of Hl on 0 ;s through the term X3 (,)6 tan 0(1). The inequality, Eq. 6-131, states that 0(t) must maximize If It is clear then that if X3(t) -t 0, then o(t) = 00 sgn IX 3(t)),

(6-161)

where sgri q

Similarly, if O(t) should become - 0o at some time t* whe.e x, (t*) =x , (t*) = xj = and x 3 (t*) x3 then the path is described by xi(t) =x -- R [sin (x -- bt) -- ihx x3

x 2 (t) =x2 + R [cos (x. -- bt) --

(6-163)

cosxl

xdt) x* - at tan 0o. Iq! -R

Further, it is clear that 0() only when X3(t) = 0.

This nath is a circular arc with ctnter it (x* + sin 4. x* - R cos x*) and clockwise =0

is possibl

mot:'n. Since this circulai arc must be tangent to the 1-n. x2 =x,, the x 2-coordinate of the center must be x, R =* R cos 6-35

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AMoP 706-192

Nute that by Eq. 6-152, x 3%') must be R cosx). R cosA +x X continuous, so ,' and 2 are contiuous. Rcos~) 22 ar continuous. Therefore, the tangerit to th: paih of p. int A in the (x1 ,, )-plane is Further, tLe relation no:ed just below Eq. 6-163 is dx 2 X'2 -ta x t-Ra x4 -R= x osx.

dxJ

1 x1

-R

and this slope is continuous. This 'neans that segments of the optimal path where = - 00, 0, or 00 must be tangent where they iitlrsect. With this information, the soluticn of thproblem may be constructed geometrically.

x2

cs

These equations :eld ,

R

+[R

-Rcos xO)'

/2",

- 2 -R In Fig. 6-12 the initial arc, which is described by Eq. 6-162, is shown leaving (IAx). A whole family of second arc, is shown corresponding .o different values of 1.

2

x*=(x0 +x' 2

2

2

+R c -

mayinbe noted by exemining the family of Itpaths Fig. 6-1' that if x' > s=R + x + R COS A, then the first arc has been followed beyond a time f wlere x 3(0 = r/2. At the point x1 (i) = x - R sin +R, x 2() =x + R cos A it would have been possible to construct a vertical portion, of the optimal path. This construction is shown in Fig. 6-13. The extreminal paths constructed for x. > s satisfy all the conditions of the theorem so

0O

._X

1

Figure 6.12 ExtremalArcs

From the construction of Fig. 6-12 it is ckar that the point of taigency of the two c,-cies (x, x') is at 'ie middle of the line joining their centers, i.e., - (xO - R sin xO +x' + X si x3, 6Section 6-36

I,

(t) x2

()1 00

x2 ) v.x

_X'

Figure 6-13 Extremal Arcs With Straight

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AffC n 706-192

that they may be opti-num. It is clear that for < s there is only one possible solution of the probler. For x2 > s this is not the case as shown for x1 = il. Both the extremals leading to ihe path x2 = irl ',ltisfy the necessary conditions of the theorem. It i% geometrically, relatively 'ear that these are the only two possibilities so the one with the shortest time required to get to x 2 = X is to be chose,.. the test, Eq. 6-151, may. eliminate ot.e candidate. It seems clear that when the extremal with straight liae exists, it is best. x

it should be ioted that if x2 > s + 2R. it is impossible to intersect x2 = x with only two circular arcs so the extremal with a st-3ight section is required. This problem illustrates many of the basic iaeas and comrlexities involved in cptimal design and optimal control theory. Some of the features are worth noting because they will arise later:

this case, Eq. 6-122 provides no information. It is then required that the inequality, Eq. 6-131, must be used to determine the design variable. For a complete treatment of this subject, ;ee Ref. 20. The problem treated in this paragraph is not as complicated as most optimal design problems occurirng in the real-world. It does, however, illustrate some -f the features and difficulty encouwtered in most realistic optimal design preblems. This problem should convince the reader that the solution of opiim2l design problems is not simply a matter of plugging numbers into formulas. Even though analytical methods will be stressed in subseqcent work, the effective solution of this ciass of problems requires a sound understanding of :he theory of optimal design. Example 6-9: A Constrained Brachistochrone Problem.

I. Pieced extremals. The conditions of Theorem 6-7 give a set of curves or solutions that must be pieced together to form the optimal path in state space. In the vehicle steering problem, these curves or arcs are put together geometricaily. In more complex problems, this will have to be done analytical'y using the conditions of "hlleorem 6-7.

The problem considered here is similar to Example 6-4 but with a conztraint added. It is required to fi id the path through (0 0) which lies above the line x2 = I + x, tan c in the (x, .x2 )-plane and that carries a particle, without friction, to the vertical line x, = x, in the shortest possible time. The 1'roblem is shown in Fig. 6-14.

2. Multip!e hq-or. As seen in the fore, going problenh, wore tKin one c rnJz.e solution may e const-u-,ted. Cond-;;or. Z-, 6- 13 1, must then b. , sd cnoo~e the best of these candidates.

Thi- problem will be treated as an optimal d~eigin problem. On the assumption that there ai4 no discontunities in the velocity vector, conse.stion of energy yields

3. Singular arcs. It occasionally lhappens. is in the vehicle s'eefais problen, that there will extit a set oi values of the state riabres and i.wilipliers such that the fuicfh - I dtie not depend explicitly op 4. ' ariablc.

22 or (2x)2 6-37

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AMCP 7%)6-192

"1 A

1/2 x

z

=(2gx 2 )

1/

sin u - (2gx 2 )

1/2

tan acos

I

0t(6-167)

-

which does contain u explicitly. The constraint, Eq. 6-166, is, therefjre, a first-order

X2.h 11 2~

+x tance

state variable inequality constraint. In order to employ Theorem 6-8, define

X2

multipliers - Y1, r-, ?

-

such that

Figure 6.14. Bounded Brachistochrone

The equations of motion of the particle are then ' =(2x)/2 I = (29X2 ) -i2

=

(29X2)

/2

G = T+ 7yxt(O) +"12x 2 (0)

+cosu Cossinu(6-164)

sin

= r-(x- - x- tan ce - ih)

u2

where u is the angle between the x I-axis and the tatigent to the path on which the particle is to travel. This angle it specifies the curve, so it is the design variable. The location of the particle is specified by the point (xt,x,) so this is the state variable. The boundary conditions are X, (0)

X2 (0) = 0

X1 (7)

X'i

x

12

'= X,(2gx 2 )

cosu

(6-168)

X2 (2gx 2 )1 / 2 sin u 12

,,92)

x(cos u - tan o sin it).

Necessary conditions irom Theorem 6-8 are (6-165)

i 1 =0 The object is then to find u(t), x(t), and x 2 (') such that a particle starting at rest at (0,0) reaches x(T) = x1, in minimum time T. The path is required to satisfy the constraint 0

= x2 - x, tana - It c 0.

= - g(2gx 2

) i

2 [XI cosiu

(6-169) -t A2 sin u - p(cos u tan a sin u1l

(6.,66)

Since the constraint of Fq. 6-!66 does not involve tl'e design variable u explicitly, the problem contains a state variable inequality constraint. Computing € and substituting from Eq '-164 yields 6-38

2

(2gx

2 ) l2 1

Xi sin u + X 2 cos u

+ p(sin u + tan o cos uji =0 X2 (J') - 0

,6-17t) ((-71)

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AMCP 706-192

tIan a + X: (

-

the optimum curve is givep oy

- 0)

X-(t-+0)=0

7T (b-172)

-+X2(ft- -0) - X.2 (t-+ 0) =0 -

0

~-

-X 0 -----

BX 7Sg + 2;'YBa( -

to P 0,;+ 1423 7-0)-

(7-70) G =rO. 4 ,L 04 kt)+ 71,4.6 0(t) +r.s.s (t) +

.

(1) + r7

[A~~2 + P4 04,2 +P S 05,21

t)

+ o.4,r 0 4,(',) + T,.4., 04 0 + ,ro.s.

-j3-

$ ( r)

Tcorem 6-9 and Eq. 7-70 yield 'Theorem

p,=

P3 01

(7-73)

3

= 0, = .i,

,=,2. 3,

(7-74)

AS 05,2 = 0

(7-75)

and 44,2

at all but a f6,ute nuinLer of pon~s in (0,T).

7-1. Theorem 7-1 Necessary conditions fcr the minimum weight beam problem are

+

C -- ?,. 0) O) X+ i=

I,2

- 0) = O, (7-76) 7-21

'"

"

Downloaded from http://www.everyspec.com

(

ANTCP 706-192 S(t "

)- ?

+Tros

=

(t; -0)-

the boundary conditionsg. = 0, s = 1, 2, nimu.

T'o,

be satisfied along with the conditions 0(7-77)+

o s

02

ags

X2 (t+ 0) - X2 1(t.- - 0() 0)-Tl4r + 7, Sr

=

0

(7-78)

X (1t -+ 0) - "\ (t " +-056

Sl

XI(T

(7-85)

2

S=I y

ag,

1, 2;

(7-86)

-S

(7-79) and the Weierstrass condition

X2 (t6" + 0) - X? US - 0) - '46

ii(xi U, Ni, ) 4

(7-8O)

+T-sS =0 + 0) -

u, ,

f(tr

X2 (t)-T+

Tsr X' (t)

+ 7SXr

x'tX)

0

t - 0)H

7-61, and 7-62 with u replaced by U. The

(7-81)

0)-

-() + 1r6 G

+ r7S6 X' (t6) = 0 l(t. + o) - /(t

in (0,7), where U

statement of Theorem 7-1 is now complete.

0 -

)

satisfies Eqs. 7-52, 7-53, 7-54, 7-55, 7-60,

- 0) - To.. X 2 (t,)

-

,

is any function which along with x, and x 2

0

)+ y

w

-6

JH(X I

be satisfied for each

1( +-must

4

x

(

S4" Xhowever, (7-82)

- 0) = 0

If there is only a scalar control variable u(t), then the condition of Eq. 6-105 will be violated at points t* which are intersections of intervals in which Eq. 7-52, 7-53, or 7-54 is an equality. With an additional hypothesis, the conclusions of Theorem - I are still valid.

72

At a point "* co, it is assumed that $ = = 0, where , > Oand 2 > 0 are any

two of the constraints of Eqs. 7-5 2, 7-53, and 7-54. If 4 ,s defined as '= rmin (s1 ,4 2 ), then '> 0 replaces the conditions i I > 0 and 02 0. It is assumed that 301/Iu and O42 /0u

and ti(t++ 0) - i(t+- 0) = 0 foi all ot, r, 6, and r/;at each of the pointsS= t r aiid tF, either 0(S) = ,I (I) = 0

(7-83)

02 (S)= 02 (S) =0,

(7-84)

or

7-21

iI,21YS

'g)

and

0 ) - 704 6

=0

l(tr + 0) -

O

are not zero at t* The new constraint now satisfies the conditions of Eq. 6-105. If Theorem 6-8 ki applied to the newv formulation of the problem, the result is identical to Theoiem 7-1 wili the exception of Eqs. 7-73, 7-74, and 7-70. However, the new 'omditons on u are identical to those

Downloaded from http://www.everyspec.com

)

AMCP 706.192

implied by Eqs. 7-73, 7-74, and 7,76. The roles played by p, and g2 in Theorem 7-1 would simply be combined in a new variable p. This result may be stated as Corollary 7-1. Corollary 7-1: Let there be a scalar design 'ariable u(t) and assume that any two of the inequality constraints, Eqs. 7-52, 7-53, and 7-54 arc equalities at t*. If the first partial derivatives of these two constraint functions with respect to u are not zero at t*, then Theorem 7-1 holds. One further result may be easily obtained. If a jlau and 3a0/au are nonzero at t*and are of the same sign, then u is continuous at t*. To prove this, it is supposed first that alau> 0 and u(t* + 0) =u(t* -0)- e,e > 0. Taylor's theorem (Ref. 16, p. 56) implies

P1 t*.u(t* + 0)) = e

[[t*

-

0))

[t*u(t " - 0) - 0el au -treated

%vhere0 < 0 < 1. But, a, /au > 0 and e > 0,

finite number of unknown constants, or 2. The beam is supported in such a way that an infinite number of constants are required to specify the reactions (e.g., a beam on an elastic foundation). In the first case, the unknown constants appear in the expressions for M and V. By defining new state variables, x, with I > 3, to be these paiameters, the following differential equations must be satisfied: dx d

,

3.

In this way, statically indeterminate problems of the first type are reduced to variational problems to which Theorem 6-8 applies.

For statically indeterminate problems of the se,-ond type, however, morc basic changes in formulation must be made. The fourthorder differential equation, Eq. 7-50 must be since q(t) may be q[tx(t),x2 ()J. The fourth-order equation is equivalent to the first-order system

so 0=0,[t*u+0)l


1/2 max , Eqs. 7-52 and 7-53 are satisfied if and only if Eqs. 7-56 and 7-57 are satisfied. This result may be proved by expressing oP and rP as functions of d and applying methods of ordinary calculus. The T

restriction rmax ; 1/(2 a vi%) is necessary,

7-4.5 BEAMS WITH RECTANGULAR CROSS SECTION OF VARIABLE DEPTH Three examples are

onsiderd in this the beam example, each subparagraph. In cross section is rectangular with fixed width and variable depth. Also, the constraint, Eq. 7-55, is taken as - A - x(: < A where A > 0 is a constant.

since at the extreme fiber of the beam, the principal shear stress is half the principal normal stress. This relation between 7max and 0

ma is no restriction for design of metallic for condition. steel and other beams. common Yield metalsstresses satisfy this

In this case, there is only one design variable h(t); Eqs. 7-44 through 7-48 are l(t)

=

h 3(t)

(7-89)

12

(7-88) A \t)

=

bh(t)

(7-90) 7-25

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'IT

b(t)b

(7-91) 2

b 8

(t)

any iffirWa1 wheiV' Eq. 7-54' is%-in

'i

3 th

(7-92)

7h -)

and 30

h)=

C~)= t.(7-93)

2~)

M(t) and V(t) 3re assumed to be known, piecewise twice continuously differentiable functions of-t-whose discontinuities occur at points t= wo.

""

Eqs. 7-89 through 7-93 and Eq. 7-70 along with Xo = 1, yield

14 (LLO

1

(7-95)

In any interval, (t, it',where .qw,-W-i-s an equality, direct differentiation and use of Eqs. 7-71 and 7-72 yield

0

0= - 12M)

7-96)

.r(-96

-FI12M(t)]j (";/(7-94)

L Ebj

(7-94)In

The procedure outlined is now used to determine h(t). In any interval wher Eq 7-56 is an equality, 61M(t)i . Q)

Inany .nterval where Eqs. 7-56, 7-57, 7-54, and 7-88 are all strict inequalities, Eqs. 7-84 and 7-75 show that pi(t) 7-73 then is

so h~~t)= I~61M(t)i~ t b a_ L ax

'

order for Eq. 7-96 to be satisfied, it is necessary tlat M(t) = 0 (hence also V(t) = 0 q(t)) must be identically satisfied in (r. 0. If this is the case, h(t) is gaven by Eq. 7-95.

=

1 .... 5. Eq.

3Mt •h-b)

In any interval where Eq. 7-57 is an equality,

31 VQ)1

0,

b-+X(t)0 ,t

b

(7-97)

so

hWt

F

(t) 2

2bh(t) so h(t) =3(t) 2braax 7-26

It is worthwhile to note th!at in order for Eq. 7-97 to hold the product X2lt)AI(t) must be positive. That is, X2(t) and M(t) must have the same algebraic sign throughout any interval in which Eq. 7-97 holds.

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AIMCP:706-19Z

31V(t)l

"

(1/2

C

-

ifIr -ma

,\., 3E2)/ 2-

x

q3

7r-.....,, '

/3

.4b2\

ma'fol= and thefunction sgn (

q sgn(q) 2 Q, MQ)] 1

')is defified-by tht;

q

foralq.

if3X, m jfkk

1 < 0mx,,., -andp 'The Weierstrass condition shows that the largest-of the expressi6ns in Eq. 7-98,is the

propier value fh(t).

Equatimns which determine the special t t and t mayiuowebofoud.n are the problem at hand, X, (t)and X 2 (t) constant, so their derivatives are zero. Eq. 7-81 is then

fi; - O)= -; + 0)

(7-100)

Eq. 7-98 in Eq. 7-64 yields C1MQ)I VW1 t

-2

'I

-

xo) A t)

=

x

if I I IEqs.

if 01 = , Io = onx MAif

Experience has shown that on both ides of t;, Eqs. 7-56, 7-57, and 7-54 are strict inequalities. Assuming this is this is the case, 7-94 and 1-98 together wit!' Eq. 7-83 or 7-84 may be used to simplify Eq. 7-100. The resuat is

i--I(t; [o2 t

if I = 1

- --o O

(7-99)

-C3 IX 2 ()r

1

14

x IM(t)l'

x sgn [M(t)],

/4

j

if Iri

[X (t; + O)M(t; + 0)] 114

(7-101)

max,

( 1


Eq. 7-98 and 1 > 0 imply M(t;) 0 0, so if M(t) is continuous at t;, then Eq. 7-101 eo,

(7-102)

),2(t; - 0) = X2(t; + 0).

Points of discontinuity of M(t) must be checked in Eq. 7-101 as possible t;.

where 3

C3 =- ___

9E

x32br

Eq. 7-96 shows that points t+and t can 6

a

occur only in intervals where M(t) (hence also 7-27

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AMP 7O6-192

"V(t).ahd q(t)) is identically zero. Since -this situation is not commbn in p~ractical pr6b-

3-b

lems, such'intervals will not be discussed here.

IV(Q)I

C1 20(QM(QI V(Q) I3

According to Theorem 7-1, the points t;

36

are deteminedby the condition

o) h(t + 0)

11(1j

(7.103)

By direct computation it is seen that the partial derivatives mith respect to h of the left sides of

,14 the definition of C3 in this equation andUsing manipulating tha result yields,..

14

j3V(Q)

lrI-max

0

I o I - ma'.

0

14

-bx2()M(Q)I/

4

44

X2(Q)M(Q)

anI and

I/C 14-

9E

-' ' V(Q)I1] 314 ×L-,'-T''E-

LX 2QMQJ

/

64b2

I, - I< 0

.

9E

ax

=4.

(7-105) By putting

are all negative at points where M(t) :/0 : V(t). The result stated just below Corollary 7-1 then shows that points of intersection of intervals in which one or more of the above inequalities is an equality may be determined by the condition h(t

-

0) = h(t* + 0) .

r

l

V(Q)

1 114

LX2 (O)M(Q) J and (

64b'r4.

a

/4

(7-104)

Eq. 7-105 becomE. If Eq. 7-104 is used to determine t, then points wa must be checked in Eq. 7-1C3 as possible t. Let Q :0 w. be defined to be a point of intersection of two intervals such that IH= max on one side of t '-Q and Eqs. 7-56, 7-57, 7-54, and 7-88 are strict inequalities on the other. Point Q is to be determined by Eq. 7-103. Due to continuity at t Q, Eq. 7-103 may be written as 7-28

3P4

-

4CP

+

C4 =0.

The roots of this equation ir P are C. C.C(I + V/2), and C(1 - V21), where i2 = - 1. The fourth powers of the last two roots are not real, so the only real solution of Eq. 7-103 is IV(Q)14

64br

4

max

X2 (Q)M().

9E (7-106)

.

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AMCPY706-19

Siilarly, S t##:iw' ',is defiuiedtoba .;oint. of.~ite cttn.6of;tw6, interyais such siEqs. Stgnd)E1s. '4e anc. i- i ar

that',IoIf~4"'7',

3;

'iiies brijtbe -6thir..Jiistas.ab,ve,.Eq. 7-103 rcduces to .3',- 46P +4

c,anid

*eeA 0 be small enough so that

Jo0I

(121 41 6M

\b

and

2

If this is,the case thenl 1(t) > 10 for ali t. Further, since r(t) 0,- Eq. 7-98 may be simplified to 6M 3/2 6M 1 ifil Oax max maxd hQt) =0, E2

114Q)

iflo0l< Um~ax' (7-11)

Also, Eq. 7-99 becomes

j

(2bGu3

X,W

0oJ

f2Qj1)d?7dv.

1

(W

The inequality f2(t) < 0 and Eq. 7-113 implyX 2 < 0

in(W), so there can be no

points t;. The only possible point at which Eq. 7-88 is an equality is It= T. Since f 2 (t) < Eq. 7-114 implies x I() -c 0. Therefore, if Eq. 7-88 is an equality, it m~ust be x1 (T) =6 A. Further, sinre there are no 4", t+ q, or ca.X and W~2are continuous and there is only one pair of constants, tj and t2 to be determined. It is assumed fist that x (7')> - A. In this case, Eq. 7-86 implies

~=

IImax

4

and

/)l

iflol1< Umax'

(7-112) Integration of the differential equations

which in turn implies tj 2 0. Since X 1 0 throughout (0,7), the second part of Eq. 7-111 cannot occur. Therefore, the bea,ii of minimum weight is uniform. Eqs. 7-1 12 and 7-1 14 then yield 2bc1laa

7.30=

(7-114)

1/2

T EM A

(7-113)

f2 (,)d7

x2 (t)

122

Downloaded from http://www.everyspec.com

-c x is the

MC76f9

A

xl(T) =(T - t*)4

s~ulf~f-h 0lepdievvdho tina4'idcnefit'Alssu6h that

)14.

6E 2M(71)X If -the- deflection requirement A does not-4

9

satisfy Eq. 7-115, then it is ncesay that x 1,( '--A. Otherivise, thisariimffeflW6uld hold; and ihC'defiection-atT would violate Eq. 7-88, Therefore, the additional boundary condition, 9 3 =X 1 (7)+ A=0,

;mustxbe -satisfied. Tito two const~nts owb fond.have ~ mst

and

The only useful relation given by Eq. 7-86

it_)5/4

v/-34 A(T

+f

lid

I

L5' 112 2b 3 (T-*) 3ETA

3

t

The right-hand side of this equation may be simplifed by eliminating, either t, or t* osro t~u s 1 q .16 ic asmuch physlcal significance as t*, it is eliminated. The conditions x 1(7) beo s -A

is

j

This implest 2 = t r.so that (

-

-)t*)

.4T

5

(T -

(7-117)

and only ti remai*ns to be founJ. On physical grounds, it is expected that the beam should be stiffest near t =0 in order to reduce the deflection at T efficiently. Also, 0,thesecnd artof sine X islarestat Eq. 7-111 would tend to stiffen the beam there is there. It is assumed, therefore, that justonepoin t~havig Ia I 0mx onitsAs left and on Iahvn I max rgton rqirenjeitbi We- ioblenii cosid&id[here is-given in Fig. 7-16. From this

407

optimalbeams-with deflection rq -uirements gieatradest 0.156 have cohsiderably differentAfrmn. The' beam profiligs ofTigs. 7-17 nd 7-18 illustrate this difference gaP-hicallyi -As-&Ucrfss toward 0.156, the jump in h(t) a025(se.Fig. T-17) beconmes more ~~~~~proPoahced.'Ho#6ver,-o eysihWls thaji 0.156, the prs~file is continuous, ijiuch as in Fig. 7-18. 7-4.5.4 CONCLUSIONS

The examples considered in pars. 7-4.5.2 and 7-4.5.3 are of the order of complexity that might be found in actual practice. In these examples, -a-saving -of nutorial 33% is realized when nonuniform, optimal ' 4 O .I '. beams are used initead of uniform beams. For more- complex 'loading situations, the saving F/gu~e;~ &Vdl'uk4,iis Deflection Rdqc.!r~rienrmay be even greater. From an engineerhig viewpoint, such savings are significant. 10:

graph, it -appears that the volume of the optimal' beam is a continuous function of deflecticun requirement. This is a rather remarkable result in view of the fact that

In structural applications, this saving may be offset by additional cost of fabricat~on. ARldimensions InInches

1.14

0.81

0

3.2

25

40

Figure 747. Profle of Optimal Beam for A

a. 16

dimensions In Inches

_____________All

1.21

0.81

o

2.6

24. f

31.0 33.7

40

Figure 7.18. Profile of Optimal Beam for A = 0.15 7.43

-p7to

Downloaded from http://www.everyspec.com

-

---

-AM

AMCP706-192 However, for applications in which weight is a prEmiumi, such is ifi aeio~sj~c _Wdrk, abilcation of-iiium weight structural members may be quite feasible. Further, if the cost of

forming nonuniform beams is not prohibitive, such is in the iiiiizfa-uie of reiforhced concrete beams, then nonuniform otimal beams may be used to advantage.

REFERENCES 1. R.A. Ridha and R. N.Wright, "Minimum Cost D~esign of Frames", . of the Structural Division, ASCE, Vol. 93, No. ST4, pp. 165-183, August 1967. 2. J.L. Lagrange, "Sur la forces des Ressorts plies", Mern Acad., Berlin, 177 1. 3. T. Clausen. "Ulber die For architektonischcr Saulen", Bulletin physicomethemnatique de l'Academlc, T. AC., pp~. 386-379., St. Petersburg, 1851. 4. Z. Wasiutynski, and A. Brandt, "The Present State of Knowledge in the Field of Optimum Design of Structures", App. Mech. Reviews, Vol. 16, No. 5, pp. 34 1-350, May 1963. 5. C.Y. Sheu and W.Prager, "Recent Developments in Optimal Structural Design'", App. Alfec/s. Reviews, VNI. 21, No. 10, pp. 985-2, 168.Optimal ctobr

Plastic Bodies", Proc. Symposium JUTAM, pp. 139-146, Warsaw, 1958. 9. J.B.Keller, "The Shape of the Strongest Column", Archive Rati. Mech. AnaL, Vol. 5, pp. 275-285, 1960. 10. 1. Tadjbakhsh and J.B. Keller, "Strongest Columns and Isoperametric Inequaliits for Eigenvalues", J. App!. Mech., Vol. 29, No. 1, p,). 159-164, March 1962. 11. J.B. Keller and F.I. Niordson, "The Tallest Column", 1. of Math. and Mech., Vol. 16, No. 5, pp. 433-446, 1966. 12. F.I. Niordson, "The Optimal Design of a Vibration Beam", Quarterly of App!. Mat/h., Vol. 23, No. 1, pp. 47-53, April 1965. 13. W. Prager and i.E. Taylor, "Problems of Structural Design", J. App!.

March 1966.35No1,p.0206 6. D.C. Drucker and R.T. Shield, "Design for Minimum Weigh:", Proc. 9th Inter-Mac196 national Congr. App!. Mech., Brussels, 14. i.E. Taylor, "Minimum Mass Bar for 1956. AxiJl Vibration at Specified Natural Fre7. W. Prager, "Minimum-Weight Design of a

Portal Frame", Proc. ASCE Vol. 82,

qec"

M

.

o.S

o

0

p

1911-1913, October 1967.

1956.

8. D.C. Drucker, "On Minimum Weight Design and Strength of Nonhomogeneous 7-44

15. J.P. Taylor an(' C.Y. Liu, "On the Optimal D-sign of Columns", AMAA J., Vol. 6, No. 8, August 1968.

(

Downloaded from http://www.everyspec.com

AMCPOO1k -16. 1,C G6ffinin -ala&Iui of Ser~ Vari1.G.A. Bliss,, Lectures on the Calculus of V aions, -Uivrsity' Chiag Press,

1.Chicago, 1946. 1.J.F. Traub, Iterative. 'Methods .for the Solution of Oquatiohsi -Prentice-PHl, Englewo. Cliff, New jersey, 1964.

7.45

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AMCP 706-192,

CHAPTER 8

METHODS OF STEEPEST DESCENT FOR OPTIMAL PESIGN PROBLEMS

8.1 INTRODUCTION As seen 'by thie examples of Chapter 7, solution of the necessary conditions for tne general problem of optimal dsign is difficult. Even in idealized design problems numerical methods must normally be employed to construct a solution. The numerical *-,chniques for the indirect method presented ir par. 6-5 and in Chapter 7 are iterative in nature. Each of the techniques requires that an estimate of the solution be made before the iterative process may be initiated. In many cas.s, particularly in new problem areas, the designer may have only a gross notion of what to expect of the solution so his initial estimate may be poor. Convergnnce of the techniques of Chapters 6 and 7 are reported to be very poor unless good estimates of the solution are available. In fact, these iterative techniques often diverge for poor estimates of the solution. On the other hand, i" a good initial estimate is available, theze methods converge very rapidly. the need for a This discussion illustrates workhorse technique that may be used even when only poor estimates of the solution of the optimal design problem are available. The method should be capable of making steady improvement in an estimated solution and, in fact, converge to the solution. Rate of convergence could be sacrificed for dependability if required.

A second desirable property of a general method of optimal design is that ii apply routinely tq a large class of real-world optimal design problems. To be useful to the working design cmineer, the method should -pply whenever the designer has developed the capability to analyze the system to be designed. Further, the method should be explicit enouh so that a senior cngineer can set the problem up for computation and a less experienced junior engineer can program the algorithm for use on a digital computer. The methods to be developed in this chapter and applied in the next have many of these nice-to-have pruperties. The basic idea of these direct methods is to simplify the basic design problem so that it will readily yield information which allow3 the designer to make a small improvement in an estimated optimum design. After the improvement is rade, a new and better estimate of the solution of the optimal design problem is obtained. The process is repeated successively to obtain small improvements in the best available estimate of the solution until the design obtained is sufficiently near the optimum. The basic method of simplification of the design problem is to expand functions involved in the problems through use of Taylor's Formula. In this way, a simplified problem is obtained which serves as a good approximation of the original problem provided only small changes are allowed in certain variables. 8-1

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AMCP 706-192

8-2 A STEEPEST DESCENT METHOD FOR THE BASIC OPTIMAL DESIGN PROB-

and

LEM

0,#(tu; =,3P =

q',

t

a(t,u) < 0, P3 7'+ , .... q, to < t

8-2.1 THE PROBLEM CONSIDERED

lems with fixed endpoints, no discontinuities in the basic problems, and no intermediate

tt .

Just as in Chapter 6, the variables x(t), u(t), and b ;,ce vectors, x(t) = [x1 ().xn(t)J T, u"' u [(t), ..., Um(t)]JT and b

conditions on the state variable. As seen in par. 6-4, this eliminates state variable inequality constreints from direct treatment. All these features of more general optimal design problems will be treated in par. 8-3. Specifically, the problem treated here is to find u(t), to ; t c t, and b which minimize

t', ti
P 10 [ 1U(0 I)

FaL.T

It should be noted that the collection BWt of indices may change with the variable t.

AIP (t)

Define the column vector of elements 0,, with >0

=

f

1

iEAab

and a similar column vector of functions 4i,, with 0 (t) > 0

(8-22)

(8-23)

and

[Lgba r + t fc- all ciEA

r ~(:) ~()(8-18)

d

+

for all &rEA

[](8-17)

(8-20)

3B)I

L.T+ 1 !c ab 3b

d I dt (8-24)

Note that A Wi is a matrix or functions with mi rows and the same number of columns as there are indices in A The matrix 21 of

P3EB(t)]

8-5

.. .....

Kj

i

Downloaded from http://www.everyspec.com

AMCF 706-192 cc~nstantc lias k iow- an., the same number of columns'as M (t).+

G

Aj (t)bu dt

(8-25)

2~'~ 8b >

(8 26)-~

T Tp

T

6

Wbb + 'yrc ~'(8-29) + 7 obbTW

Using the rrmatit,' notation of Eqs 8-21 through 8-24 hi Eqs. 8-14, 8-16, and 8-17.suhta 8J 9J'6b+

(XA

XOA'_.yTA 0 T

-0

~~.2yfoiu TW ,

(t)

(8-30)

O

Meome O) and 5b are determined, some mechanism must K' set up fof req~uiring that th'tse variation~s are actually k-mall. For con-

.a,

dt = G

-

1

venience, P'-t

oeT+'TqT

dP2 = 6b TVb~b +

8-1

6u TwIV()Su di (8-27)

coesmlenuhthat

t)

n

bae

i

emsil to pu

Theprolemisnow reduced !o finding60

1

0=1

ovn

q.8-30

V(t

eothe trwaseilc Bolz,- problem of18-2 exis , -f=

mu~~riirs j(8-33)

ior~~83 4',

an

fr > an wtthis y-

e

>

an 8-> A"direc analytical>

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AMCP 706-192

point since Theorem 6-7 1equires

or

ly.rr64+a0)=O 7T( 6 ~+4)=

AT (t)[6(t)+ce (t)J

'

aW(t-

0

and some compor ents of 8 +a and 4(t) + c(t)will be zero. A certain amount of logic is required to fiid -y and p(t). Since only small 6u(t) and lAb are admitted, if 4 and (t) :re zero for ut °)it), b(° ) then very likely they will also be zero for u(°)(t) + 6u(t), b(°) + 6b. Following this line of reasoning, it will be assumed first that

~~A5 +a0uV and * t) + c (t) =0. Then -f and pl,) are determined by substituting bu(t)and 5b from Eqs. 8-32 and 8-33 inio these equations. The multipliers 7 and p(t) are then determined and checked for the proper sign. If y. 0 fora > r' and p(t) 0 for 3 > q', then this assumption is admissible. If, on the other hand, -y. < 0 for some a > r', then 6tps + a i -0 is incorrect and it mu~t be that 60. + ao. < 0 should occar. This is equivalent to simply iemoving 0. from 4 and recalculating. Likewise, if p(t) M< 0 for some . q', then 60,(t) + ao,(t) < 0 should occm and 01(t) should be removed from (t) and the multipliers recalculated. So much for the semi-mathematics, now to the calculations based on this arhument. If B(t) is cmpf for all r. then is not defined and p(t) neeO not be determi,ed In case B(t). not empty, is to be required that

x [(t)+AO(t)3'+

-

P(t)

+C

0

It is assumed that at poirts where B(t) is not empty, a /au has full row rank, i.e., all constraint functions which are zero or positive are independent. Since W() is nonsingular, the matrix 0 Q)

=±

rv (t)-'

-'

(8-34)

,

is nonsingular. Therefore. t

.A~)-

-

xL'C

(A + A Ty)- 2oc

1

(8-35) k" iints where B(t) is empty, put a /ou = (t) 0 and AO(t) = I. In this way, p(t) is consistently defined by Eq. 8-35 for Al t. Substituting Eq. 8-35 into Eq. 8-32 I 2(t) =- 2

+

V'

(' + A

1 IV-' -yo 4

x

u

(AJ+A y)

8-7

,.

Downloaded from http://www.everyspec.com

AMCP 70&.192

•*

or

and

'/ "

l

8u)--

':U ,

~ ~u

=

oo A"

x

a

-

au

(8-40)

au Eq. 8-37 becomes

x(A + A07 ) cWU-1

A-'.

.

(8-36)

2o (M

where I is the identity matrix.

t

Rj- +d -py _ C

t

AO+T T

-T

) ra

yr ¢'to

to

Ou xUt ( 1

-

'

(

)u

A O'-OT

(/-

(8.37)

x(A" Al"y)dt=-a . .A'RI is Defining

y=-

Tt

r IV ; I v +5

(.

AOTIV;I

X r

(I-~~~ A'

M

pLT

Xa- sT)sA^ Y1J

(838) Q

)~My d t

A dtal A'-

IV." iut

(a

Wu '

7P I~ aY 2

+

M O1

(8-41)-

Since W() is positive definite so is is';, and 'here is a nonsingular matrix s(t) such that s' (t)= sT(t)s(t). By direct multiplication, it may De verified that

Substituting bu(t) and 6b from Eqs. 8-36 and 8-33 into Eq. 8-26 and then enforcing 6f

I RpOTW_

1+Mqy)+cM¢=ai"

ISI; IA( 08-39)

,,"Itw,)A',

S- T

al

x sAy] dt>

0.

(8-2)

Downloaded from http://www.everyspec.com

~

~AMCP

Therefore, M,~ is, at least positive semi"J definite. In tha developMent that follows, itj wifi~be assumed thatWM is positive definite and, Hence, nonsingu

706192

Defining 8u1 t)W

1-5

a

A,

Eq. 841 is now solved fory~ to obtain

(8-4~3)

P)

1

.1

(IT

(8-44)

~ 1

-A()W

x AOM-A1 (ai- cM

It should be noted that if the set of indices A is empty, it does not exist so M ~ is not even defined. If, in this c&use, M . is defined as one and ' zeo, then y =0in Eqs. 8-41 and 8-36 reduces, appropriately. In thfs way, a single mathematical ana~lysis holds in all cases. Substituting Eq. 8-43 into Eqs. 8-33 and 8-36 yieldsan

x (A' -A AM

'

'

a-

SO6b=W- 1 (RQ-

A4'

@

RM- 1 M j

)

(8-45)

(-6

(46

tha xrsinsfrnd)ad baesml 6 1 (i)+ 8u(t)

-~

U .. L au

AO

-all

U

Ttheep.'rsions fo u() and b rm s.ply

and

x [T~7~

(8-49)IP

' 6,.

WbR

and

(8-48)

+(AOM

cM, )

an tetrs W and 84,o hs

27e and

6b

2

ar not multiplied48

uniisivle

n

P

b270

ae aritios 8~t) nd b fom qs.8-9

Downloaded from http://www.everyspec.com

AmCP-706-192( In fact, if and

are zero or empty, 8u(t) =

of Steepest Descent. First, Relation 1 states

"b 2 0. It appears that Su2 (t) and Sb may be interpreted as making corrections in constraint errors, or keeping constraint functions from being violated. Actually, this and more is true,

that the changes Su(t), b and Su 2(t), 0 2 are orthogonal. Relations 2 und 5 show that 6u 2 (t), 8b2 provides the requested reduction in the constraint functions. Relations 3 and 4 show, as might be expected due to the orthogonality of Relation 1, that Sul (t) SbI has no etffct on the constraint functions in and i. Finally, Relation 6, along with Eqs. 8-48 and 8-49, simply states that if =0 and ¢(t) 0 then 6u(t), 6b provides a reducion in J.

Theorem 8-1: The following identities among 6u2 (t), 8b', 6u), and 8b2 of Eqs. 8-44 through 8-47 hold: t

1. bb

6ul W~u6adu=0

ior*Before

2.

ROT 6b"

AO

+

3. V€ Tb +

t'

t T6 u 2d

stating a computational algorithm, :t is important to develop a test for convergence to tle solution of the or.ginal problem. The procedure here will be to-show, through use of the necessary conditiens of Chapter 6, that as the solution of the origiial problem is approached, it1 (t) and bb' must approach zero.

-a

T

A ' 6u'dt = 0

,II

to

4. LOul =l0

By Theorem 6-7, at the solution of the problem, Eqs. 8-1 through 8-4, there are multipliers 6:),.: = I. n, v,, t r. and (t), = I ... q such ihat for H=

f-

fo-

vr L --

f

(8-50)

and (8-51)

SGgo+vg it is required tMat

I" 6. -

Tb' -t

A' T 6u'dt

0.

By considering the case when and , are empty, it is cle.tr that in order foi bt, bb to be in the negd:i gradient direction of J (i.e., is required. 6u = - A' ad Ab = - V), -to > 0 The six relationships of Theorem 8-1 give the designer an intuitive feel for the Method 8-10

do

aitf

-dt =- --ax

8-52)

al -= 0 3G

(8-53) di

0

(8.54)

i.

4tl

A

Downloaded from http://www.everyspec.com

AMCP 706192 at

t

Further conditions from Theorem 6-7 are, from Eq. 6-124,

,,,r(-55)

,a=r

=

0t),3q'+ 1., q.

(8-56)

aGT Corresponding to the definition of 0Pand 0 i)=0(-2 in Eqs. 8.17 and 8-18, def' ie 0, Land t-as and containing only components of P,L, g, and (863 aGT corresponding to elements of V.and . In this notation - and due to Eqs. 8-55 and 8-56 -C3t) Eqs. 8-52, 8-53, and 8-54 become Multiplying Eqs. 8-62 and 8-63 by 8x0 and ax, yields (857 +fafo -T T do o T o T(o xo axag di ax ax T ~_ T.T ~ _

u

au

AL

ax0

ax 0

0

u

au

(8-64)

(8-5 8)

an

a,

ago0g

+-rI

+W7 ,(g')axl =0. abT abo - P

)'dr

0

(8-59)

(8-65)

These equntions hold for all 6x0 and 6xI. Adding Eqs. 8-64 and 8-65, 6x0 + 'g

-

ax,

Substituting from Eo. 8-1l Into Eq. 8-57-

yield +c dX'

_af

d-hY

T 5x

ax

=

Tf T XN

-

(8-60)

x

or

a

T(jo)6Xt) + WT(,)6Xl =0.

Substituting terms from Eqs. 8-13 and 8-1S into Eq. 8"6, l,

+ xJ + p(-1

)6X,

')6X

ax

(8-66)

Again., Eq. 8-66 holds for all 8oA a.. 6XI .

NJ T

+ +

a x

xo

+pT

w

jT(o)ao

_

0

+ WT(t,

)6X,

=

8-7

(8-7)

Aj

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AMCP 706.92

for all 8x9 and 6x1 satisfying the: second equation of Eq. 8-8. By collecting terms, Eq. 8-67 becomes J

[cT (. I)+XJT (tI)+pjT,

Premultiplying the transpose of Eq. 8-70 by (ai/au)- ;'Yields

u

T(t IX1a

auU

equaion 8-8 f Eq

Eqs. 8-61 and 8-i8 constiiu~'e the boundary-value problem adjoint to Eil. 8-8, where the dependent variable is (co + V~+ X~~.Due to tbme assumed well posed t.ature of Eq. 8-2, te boundary-value problem of Eq. 8-8 has a shown in Ref. 2, Chapter 4, that in this case the adjoint boundary-value problem, Eqs. 8.61 and 8"8, has a unique null solution, i.e.,

au/

872

Thcofienof nEq8-2sjutA ) of Eq. 8-34 which is nonsingular. Therefore, ~ ~ +~ -r

-A

Lau

solution for all BuQt) and 6b. It is

*unique

au

W-

+AA

0v)J

U

(8-73) Substituting Eq. 8-73 into Eq. 8-71,

8b P

5b

[+ fT(T

(8-69) Subatltuting for w(t) from Eq. 8-69 into Eqs. 8.58 ani 8-59 yields

f Qf U -~

4~T+ orX4 T) /

T Uor,

14U (8-70)

-T

all and ago +-T

ag

T+

ab

aj0 ab

in the notation of Eqs. 8-22 and 8-24, - + Q',' = 0. (8-74)

Substituting for i in Eq. 8-73 into Eq. 8-70,

-'a

-

pT ai d

3b ~Jd=

ab

T)aT

to

af

+ af0 T + air ]t=

0T

-

P+aA-P1;A

2K A

auK

To (8-71)

-

IV' A

=0. (8-75)

Downloaded from http://www.everyspec.com

-4 AMCP 706.192and

Prmultiplying Eq. 8-75 by AVTW-' integrating yields

:

o

or,

"

a•

x(AJ

(AO T W; AJ -A*W;1

-A'O M 'M.,)

=0.

(8-79)

t

Eq. 8-79 is the desired result, u (t) = 0.

x LOT AO-1 a' W'.z A' dt

au

Substituting 0 from Eq. 8-78 into Eq. 8-74 yields'

Tu

A0vm-y Wu+jAO -A@

and this implies, by Eq. 8-46, that b1 =0.

a~r

r,pW_ W

at

AO

- "

It is now possible to state a computational algorithm employing the rmults of the pre-

u

ceding analysis and discussion. L x W

u

I

t d]

0

(8-76)

£T y i j + £

Make an( ° ),engineering estimate Step l. u(°)(t), b of the optimum dc-

(8-77)

= O.

-

Computational Algorithm:

yields, Premultiplying Eq. 8-7/4 by Q TW qA i

sign function and parameter. Adding Eqs. 8-76 and 8-77 finally yields Step 2. Solve Eq. 8-2 for x(° ) corresponding to u(O)(t), b(0).

,A10 j + ,Al0 0 P 0

ft - ,

,~,i

Step 3. Check constraints and form O(t) of Eqs. 8-17 and 8-18.

(8-78) 7)Step

,

4. Solve

,'AO

71u TU

±

-

equation

Ste 5. Compute At(t), IF, AO (t), and £Q in Eqs. 8-21 through 3-24 and

IV.A

, + O- MA' au 7U

and

8-11 with h and the boundary conditions of Eqs. 8-13 and 8-15 to obtain Wc(t) and Xv *(t), respectively.

Substituting 0 from Eq. 8-78 into Eq. 8-75, AJ

the differential

'

AO(1) inEq. 8-34.

I

Stel. o. Choose the correction factors a and c0in lqs. 8-19 and 8-20.

M

8-13

..-

... ..

.. ..

. ..

.

;,

.e,

,,,

-

-

"

..

- ..

..

,

.'-

•

.

_: .-. .

Downloaded from http://www.everyspec.com

i.

i

.'u '---

-

AMCP 706-192

Step 7. Compute

M*,

M,

,

and M

in

Eqs. 8-38, 8-39, and 8-40.

Substituting these expressions into Eq. 8-25 and using Eqs. 8-44 and 8-46,

Step 8. Choose yo > 0 and compute 7 and p(t) of Eqs. 8-43 and 8-35. If any components of i with &> r', or ju(t) with 3 > q', are negative, redefine i and R(t) by deletingt corresponding terms and return to

&r =-

T .

7

A

I uI

o

x (I -

Step 9. Compute Sul(t), U2(t), 6b', and 2 8b of Eqs. 8-44 through 8-47.

V- AO-

2'oM-(i,

1(

-

Mr,

M'Al

M,

J

where u(t) u

= u°)(t) -

+ 6u2(t) b(l) = b(O) _ - i 1t

.

"_

- A' M;,M¢,, dt

x

Step 10. Compute (t )

o (M

-

Step S.

u

".'oo I 2

MJJ b t + SO .

kJxP

bQ+

(IA1V / au

i

T

IV I

Jdt

_

Step 11. If the constraints are satisfied and

Sul(t) and Sb are sufficiently small, terminate. Otherwise, return to Step 2 with u(O). b(0 ) being replaced by u(1), 0 ) . An algorithm of this kind invariably hivolves a certain amount of computational art. The critical elemert of this algorithm is the choice of the parameter "to in Step 8. Once the constraints are satisfied to acceptab. accuracy, 60(t) and b1 will be approximately zero and l/(2yo)can be viewed as a step size in the ,llrection &uP(1),

1b. In

this case the chuanc in u(t) and b is S.requires t ) 6u(t) u(i 2"re 6b

8-14

2

SO

7o

With Eq. 8-80 it is possible to request a reasonabl. magnitude for U and compute the yo which ;hould give this reduction in J. In this way, it is possible to choose a reasonable To. Experience with this method on structural design problems, of the kind discussed in the 'ollowing chapter, has indicated that a request of 2% to 10% reduction in tie cost function on the first iteration gives a valte of yo that yields convergence. Often, this value of y'o must be adjusted during the iterative process to prevent d;vergence or tc, speed convergence. This matter of choosing step size in Step 8 a great deal more attention. With a little experience one can develop a "feel" for how to adjust Vo to get good convergence, even in complex problems. A feasible autonatic method of choosing -o is desirable for

Downloaded from http://www.everyspec.com

AMCP 706-192

use on high-speed computers. No reliable

method is known to the writer at this time.

g. (b,tI,),

8-3 A STEEPEST DESCENTMETHOD FOR

[t,x(t,u(t),b dt = 0,

4t

A GENERAL OPTIMAL DESIGN PROB-

=1

LEM 8.3.1 THE PROBLEM CONSIDERED

r

(8-34)

g (b,t/, x1) =g.

.

The bas;c optimal design pro'.lem with fixed endpoints, no discontinuities, and no

L. (t,x(t),u(t),b]dt< 0,

+

intermediate constraints was treated in Lhe preceding paragraph. The problem considered here w;ll be a generalization of that problem

.

-

r'+ 1.r

to include features such s variable endpoints,

and

discontinuities, and intermediate constraints. The basic idea of the method of solution will be the same as in the preceding paragraph.

0 (t, u) = ,

Accounting for the additional features of this problem, however, introduces some com-

= € (t~u)

plexity into the derivation of equations.

The problem to be treated here, is to , t tnb, and t t

o determ ine u(t), t - t

tn which minimize

C t . I'll

q'

0, to < t< tn,

(8-85)

....

,

Note that this is just a special case of the problem of Def. 6-.. Fquality constiaints will be included in Eqs. 8-84 and 8-85 in a natural way during the development. It is assumed

J = go (b,fl, x/)

that for given u(').bt I , and x/ the boundary-

+

(8-81)

fo It,x(t) u(t),bI dt ft',

subject to the conditions

w

value problem, Eq. 8-82, has a contintous solution x(t). If constraints of the form &(t,x,u,b) e 0 occur, they ma be replaced by a const72int of the form t

dxo n

fAt,x,u,b), t < t ; 7problem 0

e

T("~)F~'

5.,rt"-01

+

as a boundery co,!ition on X(t) at to.

+ 0) -F(t0 +0) + T

0

- 0)

t

-

0)

G

+ Ful 1 +0)

0

.X) =0,

S = . ... 'in

(8-113)

In thi- cawe, Eq. 8. 112 yields 0 noT

11 ,

+r 3 (

&C+)-

811

I

agT

?.Qt

$~~O

- -x =-

This is then a tundary condition on Wi) at x

In exactly .he same way, postrnuit.p'ying

W

-

0) + F(t'

-

0)1

UZI

(8l4

Downloaded from http://www.everyspec.com

F

Eq. 8-111,othotehad is just a .co

equation with'n components which cieterminesr,.. rItgvsn tion on il )

XMI'

l0)±

0)

MPO

-

explicit ifra

+ ~~~ j(g:t0)+F(t 1

No- h oudr conditions on Xt ae0 been determined. It remains,howe'ier, to determine jump zonditions on \(t) at the intermediate points t1, Post multiplying Ea. 8-101 by fl ± 0) and adding to Eq. 8-104 yields j(tJ ± 0) + 1.(t' -xT(:I

-

0)

-

F~tl+ 0)

- 0) f(:1 ±0)

N4(;

-

0))

I5)

0)

wher thenottionof qs. -10

thrugh

8-109 ha been used. The .Thdce of limit from the right or left (the plus or minus sign, ftoefr rcspectively) in Eq. 8-15i the otrier alternative will be now. One C'r chosen for computational con-icnience.OFIt is assuioed that (fhe condition Slla'.xl) 0 determnines tl as a function of x1, so it is required thit the total e-rivative of IQJ with respect to 11, h'01 ±_ 0), not be zero. Therefore, from~ Eq. 8115,

I ±0) ±0)+FQ ~ 1

-

(1

0) Lqi 1

± 0) .

0~ + XTiI J~'±0,f(,

0))

At' ±

+W X , W+ ) 0% -Pt/'. i ) S&

F(tl + 0)

-

XT (tI - 0) If(t' :t 0) - At'

+TgIf)

AT(h + 0) [f(/±O f(I + 0)] + Y 2)(8-1

-

tUit: *his~ exression into Eq. 8-10 1,

ft±0)- fiji + 0)

--

-0)

0)]1

1 ax-.

This equation, the boundary conditions at Eqs. 8-111 and 8-114, and the differential equation, Eq. 8-93, are to determitie the adjoint variable, X(t). tO < t 4 tn. ITe boundary and intermediate conditions on NMi, were constructed so that Eq. 3-97 hoids and inurq.89beo s Og CtF/F jlSQ= aSb + \O O +

(-.3F

x

Osb+X

0 Bu(t

T t

b di bT A~k

or

6Q Fab

13

+

1---+'.-.-

OLb ]6

u(t)dt. (8-117)

This equation meets the objective of this subparagraph, nanely, determination of the dep.endence of 6Q on 6b and 6vQt) explicitly. SnceQ was any functional, tlim result can be applied to the particular fuiictionals of the present problem, l and 4'.. To obtain &Iaad 8-21

Downloaded from http://www.everyspec.com

-'"

"?"

-,."

"

AMCP 706-192

ag

') and X1c( 1 ) as the solution of Eqs. 8W3, 8-! 1,8-114, and 8-116 with 80., deii

g=go and F=fo for ?J(t)

-a-

ab

r

t

+I

D afT

(8-118)

3b

\'P(I

t (8-122)

X "(t).

(8-123)

and and g g. andF=L. for Xc"-(t).

(8-119) & ,T

= --n'{t 7+ a

In this notation, Eq. 8-117 yields

l8

Ini:his Potation

rafo

ago b

1

AJT(t)6u(t)dt

+X V (t) -jdt Sb F

abJ " 6u(t)dt

and

~

I~

a

=

+

U.0

3b +

L -b ) af]dt

+

•

b

~left tll of..

j6(t)dt,

(

For a more comnpaect nottuon define +

[a

ab 3

T Jtd 4+~af b ab ' "j (8-120)

A 9-21

(t) = If- - Taf XI (t) l l) all (

Aa (t)6u()dt (8-125)

The problem of this paragraph is now in approximately the same state as the problem of par. 8-2 was in E~qs. 8-14 and 8-16. Before proceeding to derive a steepest descent algorithm, however, several comments are in order, First, the choice of limit from the right or was not made in Eq. 8-116. This choice is generally made depending on the distribution

3L,

3.

(8-124)

and 6ka = Ru r'6b +

() W -L

+

r

(8-121)

of boundary conditions on the state variable.

If most of the boundary conditions on x(t)

are given at t0, for example, then most of the boundaiy conditions on X(t) will be givei. at 0 , Since the adioint equations ar. linea, 'werposition techniques may be used to solve the boundary-value problem. These techniques involve sevetrl integrations of Eq. 8-93 from in 'o to with different starting conditions at t". These integrations must account for the itimp condition, Eq 8-116. The

Downloaded from http://www.everyspec.com

AMCP 706-192

integration then proceeds from the right and Eq. 8-1 16 should then be used to determine - ) in terms of ),(t + 0) so that the integration may continue. For this reason, the minus sign is chosen in Eq. 9-116 so that

One might argu. that the function x(t) is close to the actual ,'ate and examine the effect 3n &2(t',x(t/)] of :altring t', i.e., tl t

X(t - 0) = Xt + 0) + a I

1___

(t/

-

x(!/)] #

S21/

-O

0)-L(:

U21x dx + Txx Tdt 61/

a( +u

V (t j t

=

)

&t1

Wt

-

0)

(8-127)

+ F(t1 - 0) - F(t + 0)

vhere the plus or minus sign is chosen

+ X(tI + 0)1f(t; - 0)

- f(t+ 0)1 3

ax'

.

(8-126)

depending on whether 6tl should be positive 1 or negative to make 12 (t/ + t!), x(t/ + t1) = 0. tien chosen and if it is not 6t is The change too large, the state equations need not be re-integrated. This -rgument corresponds to a Newton-type algorithm for the determinatio;,

Since the state equations have previously beei integrated, g(tI - 0) and 121(t' - 0) can be computed in Eqs. 8-107 and 8-109.

of ti. This proceduie should be used af-" t evcry variation in u(t) and b and subi,' integraticn of the state equations, sincr . ) will be altered with an accompanying Jtp,&

The second matter that requires discussion is the determination of t ! and its variation, j=

lion in I.

0.1,

..

r/. If the state equations form an

iiitial-value problem (all initial conditions 2jven) then one can make an estimate foiu(t) and b and integrate Eq. 8-93 from o toward tq (or t toward t' if all boundary cothditions are given at 0"). As the integration progresses, I92(tx) can be monitored and the value of t for which it is zero is called fl. The siturtion is not so easy in case the state equation, form a boundary-value problem. One method of determining t/ requires that a reasonable cstimate of tI be available, perhaps from engineering intuition or prelimmary analysis. The state equations are then integrated using the engineering estimates for u(t) and b. It is likely that for the solution x(t),

TIONAL ALGORITHM The problem o deterining 6ub' .,,5b other J and j-ist satisfy reduce which will now be solved a-, in par coi,.,raints 'As in the preceding paragraph, if s-nie o, f,(t)] or , ed. If, on is less than zero, it will b the other har'd, 0. . 0 or 01:..(t)1 . 0, then it will be required that SO,,=

-a

and 6

=

a

where 0 < a < I and 0

c< I 8-23

Downloaded from http://www.everyspec.com

AMCP 706-192 A.st as in par. 8-2, define two sets of indicesrA')1

and

J.(8-133)

All (t)

That is, the columns of 2R'and AP' (t) are 2c erd A*, (t) for those ci with i. > 0. Now,

and B('

{0 1OP 1,'00(f(t >

T6 + 0;:AJ T (6u(tdt

whier;e u0 (W and W~) are the beginning estiwntte of the design variable and design 0 parameter, resi-ixtively, and x(O (t) is the issocit-d solution of the state equations. Further. di.fine tire colimn vectors of constsaint functions

[2

(~I~u~o)I~l]and (8-129) .~

By the argument of pair. 8-2, it will be required that

6ia=

T ~ ~ 6b +

(8-1 30)

(t)5ug:)dt.

have a slightly different origin, this problew. is precisely the same as that given by Eqs. 8-19, 8.20, and 8-25 of par. 8-2. All the ana!ysis required to determine Su(t) and 6b follows Theorem 8-1 h:olds. The only difference is that r'I ; Theorem 8-1 must be interpreted ais t0 inthe present problem. The algorithm of par. 8-2 may now be given with~ references to equations of t11:s paragraph. A lgoriimn. Step 1. Make

-

(8-131)u( (-131)sign

.

Using the notation of Eqs. 8 '-2 8-123. define the matrices

~

8- 24

A

(8-135)

and

L

1

Eqs 8-130 and] 8-131. Although the symbols

and

L

and

The problem of this paragraph is now to fir.d 6u(t) and bb to minimize V, subject to

lct'l i(8-128)

-

(8-134)

=

C

I

and

1Step,

J(8-1

1sion

32)

0

an engineering estimate )(t), ht0 ) of the optimum defunction and parameter.

Step 2. Estimate tO. 11 ...t, and solve Eq.- 8-8'- for xI W 3. Adjust P~ as relumied by the diSLusbelow Eq. 8-127 and reconiJ).t VIM ii required

Downloaded from http://www.everyspec.com

AMCP 706-192

Step 4. Check constraints and form 4small, and (t)of Eqs. 8428 and 8-129. Step 5. Solve the differential equation, Eq. 8-92, with boundary and intermediate conditions of Eqs. 8-111I, 8-114, and 8-116. The solutioms corresponding to the functions, Eqs. 8-118 and 8-I119, yield X() and X00(t), respectively. Step 6. Compute 2-', A'(t), Q1 , 1,9Y (t), and AO(t), in Eqs. 8-120, 8-1219 8-132. 8-133, and 8-34, respe~.tively. Step 7. Choose the correction factors a and c in Eqs. 8-130 and 8-131L and M ,, Step 8. Compute At,,, Eqs. 8-38, 8-39, and 8-40. ,,

in

> 0 and compute y and Step 9. Chooae yVo

pqt) oi Eqs. 8-43 and 8-35. If t!ny r' or compnens o ywit a p(t), with 13 > q'are ne,,ative, redefine 4' and d) by deleting coresoningtemsand return to step 6. Step. 10. Cocnpute bul(t), 50jj(t), 6b', ar,d Wh of Eqs. 8-44 through 8-47 Step. 11. Compute U0' )(t)

= tj

(

--

terrminate. Otherwise, proceed to Step 13. Step 13. Adjost t0 , t', .. , tr, as required by the discussion below Eq. 8-127. Return to Step 2 with ut 0 ), b(0 ) being replaced by u(') and bt1 ).

Fo r an alternate development of the algorithmn in the special case of a full set of initial conditions, wee Refs. 5 and 7. S.everal example problems are solved in Ref. 5.

8-4 STEEPEST DESCENT PROGRAMMING FOR A CLASS OF SYSTEMS DESCRIBED BY PARTIAL DIFFtRENTIAL EQUATIONS 8-4.1 THF CLASS OF PROBI EMS LONSlDER~r' Thtus far, all problt;.- considered have had their stzte -ariable spe~cified by algebraic equations or boundary-value problems w, ordinary differential equations. It iNpossible, however, that the state of tl', systenm being considered is governed by a boundary-value probiemn with partial differentia' equations. In such cases, the state and design variables are functions of more than one independent variable. One may then think of the design variable as being distributed o,.,r an area, volume, or higher dimensional space Foi this reason, such priblenfis have been described as distributed parameter systems

6111(f)

A gieat deal of work Iiaz been done on

~~.

+ sill (I~

I 6bi1 + 6bj' 2y~ Step 12. If the constraints are satisfied and Sulut) and 6b' are sut,ic~ itly W bto)

- -y

.wh:cl.

h:r~~d have a

time-like variable (Refs. 12.1 3); i c.. a variable makes the gov~erning differential :,Iuation hyperbolic or par..bolic. In this paragraph, :onsideration will be limited to static problemis %ich as equilibrium of )lates, shells, 8-25

Downloaded from http://www.everyspec.com

AMCP 706-192

etc. These problems are described by linear, e' iptic, partial differential equations (Ref. I). The boundary-value problem treated will be denoted

l(X1

t= +

(8-137)

where x = (x1 ,x2 ,. xk )T is the independent variable which ranges over the domain 91 in Rk with boundary P. The vector u(x) =~ 111 , .'

+ k

f gO(x~z~vb)dr

ff fO(x,z,u,b)dn. a

+

(8-140)

is a minimumi subject to the constraints of Eqs. 8-136 and 8.137,

f ga(x~z~vb)dP

the design variable

x)Tis

01 +

The object of the problem is to de:ermine uwx, xGe2. Y(x), xer, a::d b such that J=

B(v,t-)zl =q(x,Yb),xGfT

+ "t. 10,

variable v=x over r[+(boundary (x.f~) design), T is theand"e design b. (b1 ,.. b)T is the design parameter. The state

~n

variable z(x) = IzI(x),..z. (X)IT is to beI

determined by the boundary-value problem, Eqs. 8- i36 and 8-137, which is linear once u, v. and b~ are specified. It is important to note,

aXZUbd20

1, ... , r'

(8-141) (x. z, P.b)dP

=Sg,

however, that the problem depends in a nonlinear way on u, v, and b.

I + ff L,(x~z,u,b)dn-< 0, 1

An example of the form of the differential operators L(u~b)[l and B(v,b)lzl is

I

e.

XE2(8-138)

X alz

a.___xG'-

2:

r' + 1. ... , r

r,(8-142)

J,

and

W,(X.V 'C 0,XE P, ='

bb(xv,b)

~~~A'~~4 x-

(8-39)

~

JX..3k

where k )TBefore p C, =(a,,C,,).

a ),= a~.k ~ ~,.., 8-26

=

W,(X V)0, Y F P

and ft(Y,b)Izl

a

)

~

.

'(S-143)

The method of solving this piobkmn will be t aic

AwatIis

vf p.ais. 8-2 and 8-3.

At. est-,inuret u(0 )(x). r(o)(0), and 0 ) will be niad Pil ;nanges sought whiihI reduce J, subject to tC constraints of 0 e problem, desirable changes in is0), 00), and (

may be determined, of course, their

Downloaded from http://www.everyspec.com

AMCP 706-102

S

:

effect on function in the probleni must bi examined.

for xer, whlere

.

ApL~xxu, b)[z] =Ti~~~z 8-4.2 EFf:ECT OF SMALL CHANGES IN DEA.GN VARIABLES AND PARAME"ERS It will be assumeo"In the following that the

A,, L(x,u O)[N =-b L(x,u4,b)iz J A B(x'ii'b)[z!ABx 1z= B(X'v'b)[zl)

boundary-value problein, Eqs. 8-136 and 8-1137, is well-behaved in the sense that small ° , changes Su in u0° ) , byv in V0 and 81 !,n b(° ) yield a new solution P( ) + 6z (wihere z( ) is

=B(x,v,b)[! Bxvb[z) a

the solution corresponding to the estimated Sdesign

functions and parameters), where 6z is

small. To first order, 6z must satisfy the lineanizzd boundary-value problem. L[t

( ° ), (° )

b

The functionals J and 0. o are or tN' same general form, so, for their analysis definL P

(°)

=.QB[x ut°) b(°)J

it

+-Q Ix. u( 0). b(° ) I 6b To(8-144) for xsolt and crpag

aQ

ig t

f g(x,z. vb)dF + f$ F(x.zu,b)d . r -z b[b (8-147)

Once be0-. dependence of bP on3bchanges in it, v ake asz +O and and b is determined, the result may be applied directly to i andia n

au

t0 )

Fo: convenience in the, following developmient, the arguments of L and B will always be taken as it( ' ), v(°), and W( ) .

I (Az) + A UL[u ( ° ), b ( ° ) ] z (° )) bit

+ AbL Iu(°), b()

q[ v(°),b

_

To first order terms,

tgh estiate

( (8z) + APB [v(°),hb)] [z(0)) 6v

+AibB[O)fb(O)][z(O)bb

x

+7bIx

+f (

"F bz +S

6+au b d2. b

bbq](8-148)

Vb(b) aq

In otdtr to mnake use of Eq. 8-148 in tht. to'.termiiotion ena ate explthi, deppenence 6z. This or bit. by, and b,o it isndesirable

66

(8-145)

is do L* d

through use of theujoint operator ore, by emd 8-dg7

.

Downloaded from http://www.everyspec.com

AMCP 706192

(TL(ub)[Sz] ffn :

-

SJL-(u,b)[I} d2

JA[]TC[Szjd

(8-149)

r

ag 7

w . I [X] and C&5z] are ... c,.al operatc he form of the operators A, C¢, and L* is dt tei mined 'y integrating XTL(u.b)[6zl by parts. Putting =Fr Lz

(-151

Eq. 8-148 becomes af g +bg 6v + 61. l Sz-A[XTiC[6z "iz SP=

.

bPS

be explicitly independent of 6z for all Sz satisfyir Eq. 8-145. This may he interpreted , requiring that on P certain components of Sz be determined from Eq. 8-145 in terms of Sv, Sb. and the remaining Sz. The coefficients of all components of 8z remaining in Eq. must then be set equal to zero. These equations will then yield boundary conditions for;'()

6T P=

XrL(u,b)[Szl

rIT~z ¢(8-152)

Assuming all this calculation has been completed and X(x) determined, Eq. 8-151 may be written as

dP d

ab 6b +

The objective now is to eliminate explicit dependence of SP on 6z. This may be done by requiring that

L

ffAT(x)Sud.1+ j

ut

T(x)Svd

P

I'

+ ff

+ aF 6b [ ti2 . where

ab Substituting from Eq. 8-144 for L(u.b)1tz1. this is -L XTr- .b)[z] SP a,f

A F AWx =i-

aQT -

ALlu,b)WzX +

-

X

fI(x:)= coefficient of 61in Eq. 8-151 after substitution

all jab

t

b

-

4_

8-28

Sz

T

6b I d[

AL(u.b)(z]X + Y'

X d9

ab

+X b~~Ib)

+

aJLu nr

Q6 d

+

-

6 )A (8-1i)

+ f [cofficient of 6Sb in Eq 8-151 r after substitutionldf (8-153)

Downloaded from http://www.everyspec.com

[By

AMCP 706-192

putting

'(8-158)

(8-154)

gg 0, F~f) and

where

g =g,, F =L0

(8-155)

one obtains

A =(ce4i 0).

It will bc required that C1 'P

6=-

V'

(8-159)

ffMjT(x)Su da + fl

(8-160)

f IJT(xSd&M

r

T8b .1-

(-5) 2 (-156) Sb

where C, is a constant between zero and one. The idea here is to drive 'Pa toward zero if a constrtint is violated or will be violated by a change in the design variables or parameters.

and 50

ff A"* (x)it d92

For convenience define

n + f j ,*

(x)6v dl'

9~ 6b A1'(x

respectively.

The expressions, Eqs. 8-156 and 8-157, give the desired explicit dependence of Vi ard W.on 6u, Si', and Sb. The problemn is now reduced to determination of 6ut, 6r., and 6b which give the gxea est reduction in J subject to the constraints of the pmoblen

The pro~.edure will now be to choose 6u, 6Y, and 6b so as to iminuze V1 subject to tL~e constraints Eqs. 8-141 through 8-143, just %

in pars 8-1 through~ 8-3 In order to insure

aGA)

llv W)=( 110-; cEA)

Q =

(

(8-161)

aEA

In this nlotation, 6=f~

XTIJ rif~()~S ~

+

8-4.3 A STEEPEST DESCENT COMPUTATIONAL ALGORITHM

Eq 8-14 1.define

in later de-elopmrent,

f nIV 1

(xOidV'+9

Sb. (8-162)

Likewise, det'inc (X

=

[5

L

1

f)

~'

813 t8-6

Downloaded from http://www.everyspec.com

AMCP 706-192( dP'= ff6uT VU 8uda

and

CN~X)

= 1

(8-164)

1 1

LIE=E(x)J

where DWx)

(I OP() >01

(8-165)

and UI I wl(x) , 01

E(x

(8-166),

It will be required that 6R)4-

C2 4Y),

XES2

(8-167)

and bcj(x

< -

C3c,x), xGI'

(8-168)

where

+

dr+ SbTW bb

8-VTWbv

(8-171) wi-.tre dP is "small". The choke of dP will be discus--J 'ater. T.he design variables and parameter, 5u, 6v and 6b, are now to be chosen to minimize V1 of Eq. 8-156 subject i Eqs. 8-160, 8-167, J-168, and 8-17 1. A multiplier rule of Liustkrnikl an~! Sobolev, Ref. 14, page 209, will now be applied to the vr,- -, problem. It guarantees the existerice o. nultipliers, p(c), xrI2, I1,(x) >

,a>

N' o >0, and -0 such that

5(67) =0

0 < C,2 4 I and 0 < C3 x.~.(u a~

(3-172)

for all 6u, Sv and 6b, where

'I

()

(6I7~

(x)5v(x).

(8-170)

=

ff[

TOx

-

YT*A~~

andnI 5c() W

-y--

Before determining 3u, 5P', and 6b, r device should be introduced to insure that these

quantities are small as is required in order that the preceding first order analysis is a good approximation to reality. The engineer snould ch oose positive definite weighting matricesTIV W.(xW, JV,(x), and lWb so, as to associate arelative importance to all the variables. It is then required that

~V.

'YOSt

-

.5ud2

,T (X)

+ fV xOnjT(X)

-

+

[ --

iT(X

,T

5

-- -y(flp T(,V)

~6rdF ;)

y ~q

--

Y. 6b It'~ (8-173)

8-30

Downloaded from http://www.everyspec.com

AMCP 706-192 Using 87 of Eq. 8-173 in Eq. 8-172

6(ay)

0

f[X

A

()

(x

At th':, poirt it is assumed that the problem 7 = I may be chosen. Eqs.

normal so that

=-is

8-175 through 8-177 yield

_ yTA* Tox -

27 0ouTW

.Ix

- x W

+f[- xOTW(x) - ll r6x - 2d rl'

Byx

r(x) [I-

+2 v°TT

[I l"' P(X)

-

2vosblsb] 62 b,

-

.

,x (E

(8-179)

1

(8-180)

and (8-174)

6b=L 1V0(,

2

2

for all 6 ua), xEfZ, 6 p(x), xEl', and Pb. This implies -t(x)

- AM (x) -- 2-y.o I u 610)

-.- a-

a~a

2YO

-X

(8xO18.75) r

1

a ,,_,

ar

('), .G$E2

lJ W(x) - 1 (x)y - 2'y. IV,61'(x) -

((X)r

and

=0

I|

~go -Y

' l '

It

-

l lI 'y

(

forx C', ind Xo

b

Assume for the present that Eqs. 8-167 and 8-168 are equalities. Substituting Eqs. 8-178 and 8-179 into Eqs. 8-167 and 8-168 yields

forxE.2

-

(8-178)

x GS

r

2

yo¢b :: 0.

(8-177)

-

('&

dzl1

)

Downloaded from http://www.everyspec.com

AFA

AMCP 706-192

Since W~u tan4 47! are positive definite, the SicIV.

.

IV, ,r

Y

mitrices (oiau' W-1 (ap 7/au) and (a,1/',)

v(x)

-

W'"

-

W (ac T/av) are po.itive semi-definite. It will be assumed that they are positive definite and hence nonsingular. In case q or w is empty, tli.n the terms irnltiplying p and v do not exist. In ths case simply define p P*,=) v= 0 and (Olau)IV,' (a3T/rau) (3./av) IV(a (2Y!v) .I. In any c ie.

,u(x) = AO

, 2'° +

_

[(I-

_T w

A-

IV-

)

x (-Il0& rA

-2

-

It,

-(8

(-.A '- A , )]

p "

/oC3 (5 xe

x ,.

184

(8-181)

where

!n order to determine 7, these expressions are

AO =

0substituted a 1 -the

iito Eq. 8-160. Using Eq. 8-182, resultirg equatios is

and v(x)-

Aw

[2oC3 ' Ii -

IJ -I IV

(-

I I

RioI),

1

x [ (9-182)

where

-

CoA1l, ,

08185)

-C,

ar

Stbstituting from Eqs. 8-181 and 8-,82 into E:s. 8-178 tind 8-1", yields 8"(X)

-- ;[(i.§ A° ' --

u

€

~

Where ej

=

CvC,°, W~ (I-B (,~- A A. -

-x-

/

xt''-A 'y

-,

A9 -

-~11

-y(,-.v't

/ 8 TAt+ JIPTIV-1 11- "X' [)

X

(1-

8-3 2

A d2

uI

-

v

(8-186)

Downloaded from http://www.everyspec.com

AMCP 706-192

M;= fIA' J 0I

.11-

\

LO

all

AO1

xAda2

v(x)

2,yo

(8-192)

by6 ' (x)-r 671(x)

and

au r

where

fA +-- fT I; I-AO

8-187

all

fl

(8-188)

x (AJ

62x

-

~

A~ rfJt-~

~P

~ A

(

and T

AA _' A1,,) xe -n (8-194)

x[MM-f1,

all

Czi +C 2 Atj

(..

c

It was shown in par. 8-2 tha temaces

+ 3u

3w1,

in the itgad o yf tepstv ei definite. Therefore, At.~ 0 is at least positive

serii-definite. it will be assumed in wvhat

follows thu1t J/,~ ~ is positive definite and, therefore, nonsinguiar. Solving Eq. 8-185 for -y then yields

-Y= Atf; [21 0(c1

-

'.!.j

27yo

Su(X) + 611,(X) 61-

T'~M-

A

IV

x (nJ

-

fuA',

I

()wIY'

At,,~), xEP (8-196)

C3Np,)

Substituting y fiom Eq. 8-190 into Eq 8- 180, Eqs. 8-183, and 8-184 yield 6u(x)

60 (X)

(8-191)

0 r~A;~(,~ +C 1

.,)1

C 3 IVI~A

(8-197) A--'

W2x~~

8-33

Downloaded from http://www.everyspec.com

11',-

'W

-

W

MR1 1

MR

-

-'

AMCP 70B4192

Sb '(9,

20Q AC' M,)

(8-198)2

and

+ C 3 AIfj

(8-199)

It should be noted that if there were no constraints then Sa, 5v, and 6b would reduce 2hyo 2 ndo270 respectively. In order that the change in design variables and parameters should be in the negative gradied~ direction, it is clear that -y > 0 is r,;quired. ThtL magnitude of 'yo could be determinAd by substituting Eqs. 8-191 through 8-199 into Eq. 8-171. How-, ever, a'? must be chosen so it may be just as well to simply chcose -f in E%1s. 8-191, 8-192 and 8- 193.

Just as in the prohtens oi pars. 8-2 and 8.3, the variations bu' (x), bu'(xW, 60 (x), 5v'(x), 6h'I,and b" satisfy Theorem, 8-2.mu

C3 C4on I'

6. -5"'rl

-

7. -wavl

0, on I'

8.

-

PiT bbI

f

f IIJ'r5Pdr r1

fA"6 '6'~

2 0 is given. 9-9

Downloaded from http://www.everyspec.com

AMCP "8-102

.

Termiigfaueof the problem to be freqerc Ifthebea 16hasa fundamental natural frequency of wo or higher, then this is clearly the optircm beam. On the other hand, if this beam has a natural freqjuency below wo, then there must be points along the beam for Which au2 (1)> 10 and a mneaningful designi problem exists. The inequality on natu~ral frequen'.y is WO 0(9-46) -

d2 ~d2w) dx 2 \.tu dxv 2

Ip~ I

J f

iv(0) =w(L)= 0 w"(L 0a wl#L) W$P(O

(9-41)

In ordler to put the boundary-value problem, Eq. 947, ir-to the form Eqs. 9-I and 9-2, w, y2 =Eau2 (d~yj1 dx21). and define yj pwo l The problem, Eq. 9-47, is then

Ly~

1 i

0f

(-9

=0)

The boundary-value probl-m, Eqs. 9-48 nz99 ssi-doits 7! a.92 The optimal desigr. problem Is well-deIned and the notation of par. 9-2 applies directly. From Eq. 9-23,

Ld~y YI

Al'

=j

Eu

If

uy dx .

Th. computational steepest-descent algorithm may now be implemented in a direct manner. As a numerical example, the given problem r-s solved with the data E = 3 x 10" psi, L l0 in., a 1.0, and p =0,002cS slug/in-' The finite element structural analysis program. Even though there was no attempt at making the co'nputstional routines efficient, only 7 see per iteration on an IBMI 360-65 Computer were required. For must natural freque.ncies, 10 to 15 iterations were sufficient for convergen-e to within nimrical accuracy of the computations. Results for a range of natural Areqjuencies are given in Table 9-I1. The general snapes of profles of several of the optimum beams are shown in Fig. 9-3 to illustrate the

9-5 A

&2=

U-d;-

and fromt Eq. 9-25,

optimum distribution of material. MINIMUM

WEIGHT

VIBRATING

FRAME

Y

'Tau2JThe (9-48)

9-10

Y2(0) =y2(L)

igenvalue prot!.,m was solved through u-c of

where prime denotes different iat-o n with r'spect to x.

Ix

yJ (0) --Y,(L) 0

4

There are several ways in which the natural frequ.mcy of vibration of a beam may be related to the design of the beam u(t). Thef2y relationship chosen here is the boundary-value probleni describing lateral displacement during oscillation. It isgiven in Ref. 3 ?-

Fd2 Y2

with boundary conditions

distribution of mate. al along members )f the frame shown in Fig. 94 is to be

Downloaded from http://www.everyspec.com

'hie

AMCP 706-192 TABLE 9-

W2

L

COMPIARISON 6F OPTIMAL BEAMS, I~ifrm

rodlwc

1n64hvngs

0.9353

0.8967 1.0583i

4.13

4400 4800

1.3971

1.2536 1,4740

10.27 11.35

1.1546 1.13627

1.9514 2.2631 2.6980 7.2165

5200 5600 6G00 10000

tX

23Z

Rduor

3600 Iwo0

x

137189 1.9847 2.2)05 6.3172

8.34 11.92 12.30 12.61 12.46

*Uniform boom of lowest volum, having requi. dnatural rii atnc .qcyFor

determined so that the frame is as lightweight as nd pssile as funametalnatral I'

3,600 ud/secbeam

4~00ra~scc ~

WJJ 7

.17

K L

Fiue94. Portal Fiarne

-:

frequency greater than or equal to a given wo. Further, as a form of strength requirement I(x) > 10 > 0 is required. convenience, all members have the same length and all cross sections have the sm emtybtmyb cldb atr' that varies with x. In this case, the area of cross sections u,(x), i = 1,2,3, uniquely determine the design of (lie beams when the material is chosin. Further, the second moment of the cross-sectional areas are 1(x)

a'j ul(x W

where a, is a constant depending on the Lross-sectional geometry chosen. 6,000 fa/c

Defining

Y2 =Ell 11

Y3 ..

W92Iw 2

Y6

[ l3 1V3

10,000 radisec

Figure 9-3. Profile of Ogimal Beam

9-I 1

-

Downloaded from http://www.everyspec.com

AMCP 706.192 ,~% -!g,

the differential eqi,.tions for vibration of the frame are 1YI

Y1-

0

EiY3

The perturbation from Eq. 9-51 are 6yyl(BMW 0

ay~l;)=5y$(0)

8A (0) 0

Sy' 1(L)=6y'3 (0)

0

SAW=L)8y'(0)

SY3(0)

Y3 -7 jY4

A' YS-~Y

boundary conditions

0

8y 3 (L) 0

SY2(L)5y(0 BM

U3YI

SyS(L)=0

6Y4 (L)= 8y 6 (L)

L oly' J

&()+6~o

where r pW2. Boundary' conditions are Y,(0) 0 y, (L)-y (0)

A (L)

Ay'(0)

Y 3(0)= 0

Ay(4)

Y's (0)

Y 3 (L) 0

Y 2 (L)=Y 4 (0)

ys(L) 0

Y 4 (L)=y 6 (L)

8u(xy(L

U .

Two integrations by parts and cimination of boundary terms through use of Eqs. 9-5) and

9-52 yield (91

A (L) + 0) if

(L

I

yTJy dx

YTK6y dx 6U

o(

U2JL~d] y1(9-53)

The boundary-value problem, Eqs. 9-50 and 9-51, is written in seif-adjoint form. The last bounuary condition in Eq. 9-51 is jus Newton's second law applied to horizontal motion of Member 2. This boundary cond,)n does not fit Eq. 9-6 exactly due to dependence on the design variable u2 (x). It

will have to be treated as a special case according to the comment following Eq. 9-13. 9-12

Z'

(L)O0 (9-50)

A (O)0

'~

Since the boundary value problem, Eqs. 9-50 and 9-51, is self-adjoint, 7 = y in the general formulation. The derivation of Eq. 9-21 holds and Eq. 9-53 may be substituted along with

J

f.L

Lx= TM6Y dyTmy Jof

dx

Downloaded from http://www.everyspec.com

to obtain

This isprocisely~the fom:okq.jp- 7,,andthe remainder f'hte descent a ~ihnis 'valid, It-should

(~Lstbepest

IJJbe

noted, that -thigdifivation iiforial, and rigorous verification of. 134-944 is'expectd:

(L

K

Iy

to' be-e'xtemely'4ifticult.

~""''~ U()+ 1[fL ~~ ~5 Yi(L)finite

.1 2(x~x

"'1

.10are

.

The-algorlthm of par. 9-2 was used to solve this proqbezn in a-direct mfanner. Xhl6e*eigevalue problemi was solved approximately- by a blement rehod. Data for this problem a 0.079S8, p = 2.616x10 4 Ib-sec 2/in. 4 , E = 10.3x 106 psi,!10 =.00982Sn.4, and L in. Weights of optimum frames are Eiven

In Table 9-2 for several frequency requireLi r2110.0 Eai I mnents, and the profile of an optimum frame is sj

shown in Fig. 9-5.

2Y4

TABLE9.2

Solving Cor at,

CaA,, radluc

I

Optimum lb

L 2Weigh~t. 2

2A uy

+

WEIGHT OFOPIMUM FRAMES

d

C12y

2000

3000

4000

5000

1,73

2.56

3.59

4.69

I-

++~y(~ U3Y2

Ec~ 2

fl[tY- 1813

}dx (9-54)

By making the obvious choice for M, Eq. 9-54 can be written at

Li

T

A~ 6u dx

(9-55)

JoJ

Figure 9-5. Profile of Optimum Frame

Downloaded from http://www.everyspec.com

'MP7M619V AMIIMUM. '-M6 WEIHT MULTIPLEFAILURE9RI"T2R1A4n ,

Wutotstru t4Wial

Si-

________

2

Oprbk nis, con-;'

4srt iji ijembr u -slie;, dif"in and -buckllng ar* iiOfoirccd~ilnthe rinifiim,*e kt iiatefjine 'ibwiiin Fli. 9-6. 'Area iIs

Figure 9-Z. Free Bodies

the thhd member moves - (.'L) units to the right. Taus, by elementary beam theory (Ref. 3)

YL,

2

(-6

T- 3w(L)EC13 qx)

t~~

3

The differential equation of deformation of the first member is IEui 1 I(x)w"1"=-q(x) Figure 9-6. Laterally Loaded Frame

and the boundary conditions areW W(O)

allowed to vary along the length of 1W~ ftnl. and second members but the geometrical shape of the cross section is fixed. Thus,

W'(O) iv"(L)

0 =0

(9-58) 0

11W) = aud() where G,depends on the cross-sectional geometry and u,(x) 'i the cro3s-sectional area of the Ith member at the point x. The size of the third member is fixed and all members are taken the same lengthI. Frec body diagrams of th~e members are shown in Fig. 9-7.

9-14

0~

3El0v(L)

-(Ect,

u1 ") (L)

.

To get the boundary-value problem, Eqs. 9-57 and 9-58. into the form of Eq. 9-1, define Z,=I (9-59) 2

The third member is uniform with constant moeulus El3 . Further, axial deformation of the second member is neglected so the top ot

(9-57)

'I)

The boundary-value problein, Eqs. 9-57 and 9.58, is tiien

Downloaded from http://www.everyspec.com

AMCP76-192

z2

Lz

and

2)1

llIj

W+ o 01 ,2

(9-67)

The constraint, Eq. 9-65, requires that -the -q~x)

(96~

L

axial load T in the second member be less than or equal the buckling load'. A limit Sis

(x)

placedd on horizorstal deflection ofthe top of

oj

the frame in Eq. 9-66. The constraints, Eq. 9-67, are included to insure that member cross section does not go to zero anywhere. A more realistic constraint would be on bending stress but this would require a constraint of the form of Eq. 8-6. This constraint will be included in subsequent work but will not be

and Z(

Z11(0) 2()0

__z

[3E13zl(L)

4(L)

-2

01

(L

-

treated here.

Th. constraints, Eqs. 9-65 and 9-56, do not

L0 J

fit directly into thL. basic formulation of this

(9-61) text and require special treatment. The linThe equations which determine buckling eridfomofteetntaitae load P of the second member are ,() P1A i (.8

L'

Ky _2y" =P -

PAy

(9-62)

and

L3))k:LY /

(i~t2U2

and

-

0y O

0

8z 1(L)

(9-69)

A0

l~ t remains to obtain expreszions for 8z I(L) (963)

and 6P expiiotiy in terms of 6u1 (x). Fror, .-q.

The objective in the design problem is to

I re ooti nepeso o .z() r'nsictt iia oE.91 is needed. Integration twice by parts of

1y(LiL

9-2 1, 6P' ma~y be expressed in terms of 6ui2 (x).

choose i(x) and ti2 (X) to minimize the weigl't of the first two m-nibers, j

x

y

2

f.Jo)dXTL8z dx yields

96)

0

The constraints to be enforced are

L3

zl(L) - P 40

~

(9-65)

z()-S