GENOPT A Program that Writes User-Friendly Optimization Code David Bushnell International Journal of Solids & Structures, Vol. 26, No. 9/10, pp , 1990

SUMMARY OF TALK Purpose Properties of GENOPT Two kinds of user: (a) GENOPT user and (b) end user GENOPT commands GENOPT architecture Example 1: Minimum-weight design of a plate Example 2: Minimum-weight design of an isogrid-stiffened spherical shell Example 3: Minimum-weight design of a ring-stiffened, wavy-walled cylindrical shell with use of BIGBOSOR4 Example 4: Minimum-weight design of an isogrid-stiffened ellipsoidal shell with use of BIGBOSOR4

PURPOSES OF GENOPT Convert an analysis into a user-friendly analysis Make the step into the world of automated optimization easy

PROPERTIES OF GENOPT An analysis of a fixed design is “automatically” converted into an optimization of that design concept. GENOPT can be applied in any field. It is not limited to structural analysis. User-specified data names and one-line definitions appear throughout the output. Hence the input and output is in the jargon of the GENOPT-user’s field. GENOPT is a FORTRAN program that writes other FORTRAN programs.

ARCHITECTURE OF GENOPT The program system generated by GENOPT has the “BEGIN”, “DECIDE”, “MAINSETUP”, “OPTIMIZE”, “SUPEROPT”, “CHANGE”, “CHOOSEPLOT”, “CLEANUP” architecture typical of other software written by the author for specific applications (BOSOR4, BIGBOSOR4, BOSOR5, PANDA, PANDA2)

Structural analysis, optimization

“Window” size v. design iterations

TWO TYPES OF USER GENOPT USER: Uses GENOPT to create a user-friendly system of programs for optimizing a class of objects. END USER: Uses the user-friendly system of programs created by the GENOPT user to optimize objects in the class covered by the GENOPT USER’s program system.

ROLE OF THE GENOPT USER(1) Choose a generic class of problems for which a user- friendly analysis and/or optimization program is needed. Decide which phenomena (behaviors) may affect the design. These are called “behavioral constraints”. Examples: stress, buckling, modal vibration, displacement, clearance. Establish the objective of the optimization. Examples: minimum weight, minimum cost, minimum surface rms error, etc.

ROLE OF THE GENOPT USER(2) Organize the input data. Simple constants? Arrays?, Tabular data?, Decision variables? For each input datum choose: (a) a meaningful name, (b) a clear one-line definition, (c) supporting “help” paragraph(s). Write or “borrow” algorithms to predict various behaviors, such as buckling, modal vibration, and stress, that may affect the evolution of the design during optimization cycles. Test the new user-friendly program system. Interact with the END USER.

ROLE OF THE END USER(1) Choose a specific problem that fits within the generic class established by the GENOPT USER. Choose an initial design with appropriate loads and an allowable and a factor of safety for each behavior. Choose appropriate decision variables with appropriate lower and upper bounds. Choose linked variables and linking expressions (equality constraints), if any. (These are chosen by the END USER in the processor called “DECIDE”).

ROLE OF THE END USER(2) Choose inequality constraints, if any. (To be chosen by the END USER in “DECIDE”). During optimization use enough restarts, iterations, and “CHANGE” commands in the search for a global optimum design. (This is now done automatically by “SUPEROPT”). Interact with the GENOPT USER. Check the optimum design via general-purpose programs and/or tests.

THE GENOPT MENU OF COMMANDS(1) Command for the GENOPT USER and the END USER: GENOPTLOG (activates the GENOPT menu of commands). Commands for the GENOPT USER: GENTEXT (GENOPT USER generates a prompt file with “help” paragraphs. GENTEXT produces FORTRAN program fragments, some complete FORTRAN programs, and two “skeletal” FORTRAN subroutines to be “fleshed out” later by the GENOPT user.) GENPROGRAMS (GENOPT USER generates executable elements: BEGIN, DECIDE, MAINSETUP, OPTIMIZE, CHANGE, STORE, CHOOSEPLOT, DIPLOT). INSERT (GENOPT USER adds parameters, if necessary). CLEANGEN (GENOPT user cleans up GENeric case files).

THE GENOPT MENU OF COMMANDS(2) Commands for the END USER: BEGIN (END USER provides initial design, material properties, loads, allowables, and factors of safety). DECIDE (END USER chooses decision variables, bounds, linked variables, inequality constraints, and escape variables). MAINSETUP (END USER sets up strategy parameters for simple analysis of a fixed design or optimization). OPTIMIZE (END USER performs the analysis or optimization). SUPEROPT (END USER tries to find a “global” optimum). CHANGE (END USER changes some variables).

THE GENOPT MENU OF COMMANDS(3) CHOOSEPLOT (END USER chooses which decision variables to plot versus design iterations). DIPLOT (END USER obtains postscript plot files for margins and/or decision variables and the objective versus design iterations). CLEANSPEC (END USER cleans up SPECific case files).

SEVEN ROLES THAT VARIABLES PLAY 1. A possible decision variable for optimization, typically a dimension of a structure. 2. A constant parameter (cannot vary as the design evolves), typically a control integer or material property, but not a load, allowable, or factor of safety, which are asked for later. 3. A parameter characterizing the environment, such as a load component or a temperature. 4. A quantity that describes the response (behavior) of the structure to its environment, (e.g. maximum stress, buckling load, natural frequency, maximum displacement). 5. An allowable, such as maximum allowable stress. 6. A factor of safety. 7. The objective, for example, weight.

SEVEN ROLES OF “plate” VARIABLES

SEVEN ROLES OF “plate” VARIABLES (cont’d)

EXECUTION OF GENTEXT (continued)…

EXECUTION OF GENTEXT (continued)

EXECUTION OF GENTEXT (continued)…

EXECUTION OF GENTEXT (concluded)

FILES PRODUCED BY GENTEXT

Plate.INP file generated by GENTEXT

PART OF THE plate.DEF FILE: a glossary

THIS GLOSSARY IS GENERATED BY GENTEXT

THE PROMPTING FILE, plate.PRO, produced by GENTEXT

PROMPTING FILE CREATED BY GENTEXT (continued)…

PROMPTING FILE CREATED BY GENTEXT (concluded).

FORTRAN FILE, plate.CON, created by GENTEXT

FORTRAN FILE, plate.CON, CREATED BY GENTEXT (cont’d)..

FORTRAN FILE, plate.CON, CREATED BY GENTEXT (end)

SOME IMPORTANT NOTES, ESPECIALLY CONCERNING “BEHAVIORAL CONSTRAINT” AND “MARGIN”

PART OF THE BEHAVIOR.NEW FILE CREATED BY GENTEXT

LABELLED COMMON BLOCKS CREATED BY GENTEXT, plate.COM:

GENOPT USER WRITTEN CODE INSERTED IN SUB. BEHX1

GENOPT USER WRITTEN CODE INSERTED IN SUB. BEHX2

TABLES OF PLATE BUCKLING LOAD COEFFICIENTS CAN BE FOUND IN BOOKS SUCH AS THIS…

BUCKLING OF PLATE IN AXIAL COMPRESSION: COEFAX Called KAXL(i) in BEHX2

GENOPT USER WRITTEN CODE INSERTED IN SUB. BEHX3

GENOPT USER WRITTEN CODE INSERTED INTO SUB. BEHX4

GENOPT USER WRITTEN CODE INSERTED INTO SUBROUTINE OBJECT, WEIGHT OF THE PLATE

SOME NOTES APPENDED TO TABLE 6 ON P.1193 OF THE IJSS PAPER ON GENOPT:

RUNSTREAM FOR OBTAINING AN OPTIMUM DESIGN

GENPROGRAMS CREATES THESE EXECUTABLE FILES BEGIN(end user supplies starting design, loads, etc.) DECIDE(end user chooses decision variables, bounds, equality and inequality constraints, etc.) MAINSETUP(end user chooses analysis type, which behaviors to process, how many design iterations, etc.) CHANGE(end user can change values of variables.) AUTOCHANGE (automatic random change in decision variables; used by SUPEROPT.)

GENPROGRAMS CREATES THESE EXECUTABLE FILES (continued) CHOOSEPLOT (end user chooses what variables to plot v. design iterations.) OPTIMIZE (end user launches the mainprocessor run, either analysis of a fixed design or optimization or design sensitivity analysis.) STORE (variables, margins, objective for all design iterations are stored for display in the *.OPP file.)

EXECUTION OF THE PROCESSOR CALLED “BEGIN”

EXECUTION OF “BEGIN” (continued)…

PART OF THE FILE, plate1.BEG, GENERATED BY “BEGIN”

THE ENTIRE plate1. BEG FILE (p. 1 input data for “begin”)

THE ENTIRE plate1. BEG FILE (p. 2 input data for “begin”)

THE ENTIRE plate1. BEG FILE (p. 3 input data for “begin”)

THE plate1. OPB FILE (p. 1 output from “begin”)

THE plate1. OPB FILE (p. 2 output from “begin”)

THE plate1. OPB FILE (p. 3 output from “begin”)

EXECUTION OF THE PROCESSOR CALLED “DECIDE”

EXECUTION OF THE PROCESSOR CALLED “DECIDE” (p. 2)

EXECUTION OF THE PROCESSOR CALLED “DECIDE” (p. 3)

EXECUTION OF THE PROCESSOR CALLED “DECIDE” (p. 4)

EXECUTION OF THE PROCESSOR CALLED “DECIDE” (p. 5)

EXECUTION OF THE PROCESSOR CALLED “DECIDE” (p. 6)

EXECUTION OF THE PROCESSOR CALLED “DECIDE” (p. 7)

EXECUTION OF THE PROCESSOR CALLED “DECIDE” (p. 8)

THE FILE, plate1.DEC, GENERATED BY “DECIDE”

THE FILE, plate1.OPT, GENERATED BY “MAINSETUP”

EXECUTION OF THE PROCESSOR CALLED “OPTIMIZE”

OUTPUT FROM “OPTIMIZE”: THE plate1.OPM file (p.2)

OUTPUT FROM “OPTIMIZE”: THE plate1.OPM file (p.3)

EXECUTION OF “OPTIMIZE” (continued)… Output omitted in order to save space

EXECUTION OF “OPTIMIZE” (concluded)

OUTPUT FROM “OPTIMIZE/STORE”: THE plate1.OPP file (p. 1)

OUTPUT FROM “OPTIMIZE/STORE”: THE plate1.OPP file (p. 2)

OUTPUT FROM “OPTIMIZE/STORE”: THE plate1.OPP file (p. 3)

A PLOT OBTAINED VIA “CHOOSEPLOT” AND “DIPLOT”

RESULTS FROM THE MOST RECENT VERSION OF GENOPT

ANOTHER PLOT OBTAINED VIA “CHOOSEPLOT”/”DIPLOT”

RESULTS FROM THE MOST RECENT VERSION OF GENOPT

ANOTHER PLOT OBTAINED VIA “CHOOSEPLOT”/”DIPLOT”

RESULTS FROM THE MOST RECENT VERSION OF GENOPT

A NEW PROBLEM: MINIMUM-WEIGHT DESIGN OF AN INTERNALLY ISOGRID-STIFFENED SPHERICAL SHELL UNDER UNIFORM EXTERNAL PRESSURE

FOR DETAILS, SEE LOCKHEED REPORT F372046, Jan.1990, “SPHERE” - Program for minimum-weight design of isogrid- stiffened spherical shells under uniform external pressure

REFERENCES IN THE LOCKEED REPORT F372046, January 15, 1990, “SPHERE” - Program for…”

CURVE USED TO CREATE TABLE OF KNOCKDOWN FACTORS v. IMPERFECTION AMPLITUDE FOR GENERAL BUCKLING OF ISOGRID-STIFFENED SHELL

MODELS FOR LOCAL BUCKLING OF SHELL SKIN

CURVE WITH DISCRETE POINTS SELECTED FOR TABLE OF LOCAL BUCKLING OF IMPERFECT SPHERICAL CAP v. “SHALLOWNESS” PARAMETER, LAMBDA

GLOSSARY OF USER-NAMED and USER-DEFINED VARIABLES in the GENERIC PROBLEM CALLED “SPHERE”

GLOSSARY FOR THE GENERIC PROBLEM, “SPHERE” (continued and concluded)

PART OF THE sphere.PRO FILE CREATED BY GENTEXT

“SKELETAL” BEHAVIOR.NEW FILE and BEHXi, I=1 to 5 SUBROUTINES TO BE “FLESHED OUT” BY THE USER

RUNSTREAM FOR OPTIMIZING THE SPECIFIC CASE, WHICH IS CALLED “sphere3”

OUTPUT FROM “OPTIMIZE/STORE”: THE sphere3.OPP file (p. 1)

OUTPUT FROM “OPTIMIZE/STORE”: THE sphere3.OPP file (p. 2)

PLOT OF OBJECTIVE (WEIGHT) OF “sphere3” DURING OPTIMIZATION CYCLES

PLOTS OF “sphere3” SKIN THICKNESS AND ISOGRID STIFFENER THICKNESS v. DESIGN ITERATIONS

PLOTS OF “sphere3” HEIGHT OF ISOGRID STIFFENING AND SPACING OF ISOGRID STIFFENERS v. ITERATIONS

MARGINS v. DESIGN ITERATIONS FOR “sphere3”

MARGINS FROM LATEST VERSION OF GENOPT: “sphere3”

THIS IS AN EXAMPLE IN WHICH BIGBOSOR4 IS USED TO DO THE STRUCTURAL ANALYSIS. HENCE, THERE IS A SIZABLE DATA BASE Also see, AIAA Paper AIAA (same title as above), 41st AIAA Structures, Structural Dynamics, and Materials Meeting, April 2000

SOME CHARACTERISTICS OF THE PROBLEM

VARIOUS “WAVY-WALLED” CYLINDRICAL SHELLS, “WAVYCYL”

LOADING AND BEHAVIORS The loading is uniform external pressure combined with some axial compression and static and dynamic lateral g-loading (which causes beam-type bending of the wavy-walled cylindrical shell). BEHAVIORS: 1. Maximum axisymmetric stress in shell wall from nonlinear theory, STRMAX. 2. Buckling load factor from nonlinear theory, BUCFAC. 3. High-circumferential wave buckling from nonlinear theory, BUCHIW. 4. Maximum stress at circumferential coordinate, theta=0 degree from linear theory, STR0. 5. Maximum stress at theta = 180 degree from linear theory, STR Linear buckling load factor at theta = 0 degrees, BUC0.

BEHAVIORS (continued) 7. Linear buckling at theta = 0 deg., buckling mode is anti- symmetric at the midlength symmetry plane, B0ANTI. 8. Linear buckling at theta = 0 deg., mid-circumferential-wave range, BUC0MD. 9. Linear buckling at theta = 0 deg., high-circumferential-wave range, BUC0HI. 10. Linear buckling at 180 deg., BUC Linear buckling at theta = 180 deg., high-circumferential-wave range, B180HI. 12. Linear maximum normal displacement at theta = 0 deg., WWW0

BEHAVIORS (concluded) 13. Linear maximum normal displacement at theta = 180 deg., WWW Modal frequency corresponding to beam-bending mode between lateral supports, FREQ. 15. Maximum stress from random lateral and axial excitation, STRRAN. 16. Buckling load factor from random lateral and axial excitation, BUCRAN. 17. High-circumferential-wave buckling load factor from random lateral and axial excitation, BRANHI. 18. Maximum normal displacement from random lateral and axial excitation, WWWRAN.

DECISION VARIABLES, BRINGS, WAVLEN, AMPLIT, THICK, HWEB, TWEB, HFLANG, TFLANG

GLOSSARY OF VARIABLES USED IN “WAVYCYL”

GLOSSARY OF VARIABLES IN “WAVYCYL” (continued)…

GLOSSARY OF VARIABLES IN “WAVYCYL” (concluded)

VALUES OF BEHAVIOR AND MARGINS OF OPTIMIZED SHELL

FILES THAT CAN BE USED AS DIRECT INPUT INTO BIGBOSOR4 AND WHAT QUANTITIES ARE COMPUTED CORRESPONDING TO EACH FILE

THE PART OF SUB.BEHX1 WRITTEN BY THE GENOPT USER

BUCKLING AND VIBRATION MODES FOR OPTIMIZED WAVY- WALLED CYLINDRICAL SHELL WITH “SMEARED” WAVYNESS AND ACTUAL WAVYNESS

VIBRATION MODE WITH INCREASING MAX.PERMITTED DEGREES OF FREEDOM. “MAXDOF”, IN THE BIGBOSOR4 MODEL OF THE OPTIMIZED WAVY-WALLED CYLINDER

Externally pressurized imperfect ellipsoidal shell STAGS model of amplitude of axisymmetric buckling modal imperfection

EXTERNALLY PRESSURIZED, AXISYMMETRICALLY IMPERFECT ISOGRID-STIFFENED ELLIPSOIDAL SHELLS The shell is modeled as an “equivalent” ellipsoidal shell by matching the ellipsoidal profile as closely as possible by a number of shell segments each of which is toroidal. That is, each of which has a constant meridional curvature. This is done to avoid finite element “lockup” in BIGBOSOR4, which leads to seriously unconservative predictions.

EQUIVALENT ELLIPSOIDAL SHELL UNDER EXTERNAL PRESSURE

DISCRETIZED MODEL, SHOWING 12 SEGMENT EQUIVALENT ELLIPSOIDAL SHELL. EACH OF THE 12 SEGMENTS IS TOROIDAL (constant meridional curvature). THE OVERALL PROFILE MATCHES AN ELLIPSOID. BIGBOSOR4 model

x (radial) coordinates at the ends of each of the 12 segments.

DECISION VARIABLES The thickness of the shell wall and the height of the isogrid stiffeners varies along the meridian. The spacing of the isogrid members and their thickness is uniform along the meridian. The shell is elastic. The decision variables are: 1. The height of the isogrid members at 13 “call-out” points along the shell meridian. 2. The thickness of the shell wall at the same 13 “call-out” points along the shell meridian. 3. The spacing of the isogrid members (constant). 4. The thickness of the isogrid members (constant).

REASON TO TRY TO SOLVE THIS OPTIMIZATION PROBLEM The behavior of externally pressurized, axisymmetrically imperfect ellipsoidal shells that are “flatter” than a spherical shell is quite nonlinear, even if the material remains elastic. The nonlinear behavior presents a challenge when automated optimization is attempted because early nonlinear axisymmetric collapse may occur as the behavior of various non-optimum designs are processed during optimization cycles. Reliable strategies must be introduced to prevent numerical “crashing” during optimization cycles.

TYPES OF ANALYSIS 1. Linear axisymmetric bifurcation buckling (to obtain imperfection shapes). 2. Nonlinear axisymmetric stress analysis of axisymmetrically imperfect shell (to obtain the maximum stress in skin and stiffeners in two meridional regions and to obtain local buckling of the skin and stiffeners in two meridional regions. 3. Nonlinear axisymmetric collapse analysis of axisymetrically imperfect shell (to find the maximum load-bearing capacity of the imperfect shell neglecting non-axisymmetric bifurcation buckling) 4. Nonlinear bifurcation buckling of the axisymmetrically imperfect shell (to find the maximum load-bearing capacity of the axisymmetrically imperfect shell in the presence of nonlinear non- axisymmetric bifurcation buckling.

WORK THAT IS DONE 1. The GENOPT user establishes variable names and definitions. 2. The GENOPT user decides which “behaviors” are to be considered (linear buckling, nonlinear stress and local buckling, general collapse, and nonlinear general buckling). 3. The GENOPT user writes FORTRAN routines that call the BIGBOSOR4 processors to obtain the “behaviors”. (In this case only SUBROUTINE STRUCT is “fleshed out”; BEHAVIOR.NEW is produced entirely by GENOPT.) 4. The GENOPT user compiles and checks and re-checks the software he/she has introduced. 5. The END user sets up cases for stiffened and unstiffened “equivalent” ellipsoidal shells and optimizes them. 6. The END user compares predictions from STAGS and BIGBOSOR4 for the optimized shells.

GLOSSARY OF VARIABLES IN THE “EQUIVELLIPSE” GENERIC CASE

“MODE 1” AXISYMMETRIC IMPERFECTION SHAPE BIGBOSOR4 model

“MODE 2” AXISYMMETRIC IMPERFECTION SHAPE BIGBOSOR4 model

VARIOUS BEHAVIORS CORRESPONDING TO LOAD SET 1: + MODE 1 AND + MODE 2 IMPERFECTION SHAPES

MARGINS CORRESPONDING TO THE BEHAVIORS LISTED ON THE PREVIOUS SLIDE

NOMENCLATURE IN SOME OF THE DESIGN MARGINS

MORE NOMENCLATURE IN SOME OF THE MARGINS

VARIOUS BEHAVIORS CORRESPONDING TO LOAD SET 2: -MODE 1 AND -MODE 2 IMPERFECTION SHAPES

MARGINS CORRESPONDING TO THE BEHAVIORS LISTED ON THE PREVIOUS SLIDE

ANALYSIS NO. 1: LINEAR AXISYMMETRIC BUCKLING, USED IN ORDER TO GET BUCKLING MODAL IMPERFECTION SHAPES: MODE 1 and MODE 2

ANALYSIS NO. 2, NONLINEAR AXISYMMETRIC STRESS ANALYSIS WITH +MODE 1 IMPERFECTION SHAPE AT THE DESIGN PRESSURE: p = 460 psi

ANALYSIS NO. 2, NONLINEAR AXISYMMETRIC STRESS ANALYSIS WITH +MODE 1 IMPERFECTION SHAPE (concluded)

ANALYSIS NO. 3, NONLINEAR AXISYMMETRIC STRESS ANALYSIS WITH +MODE 2 IMPERFECTION SHAPE AT THE DESIGN PRESSURE: p = 460 psi

ANALYSIS NO. 4, NONLINEAR AXISYMMETRIC COLLAPSE ANALYSIS WITH +MODE 1 IMPERFECTION SHAPE

ANALYSIS NO. 5, NONLINEAR AXISYMMETRIC COLLAPSE ANALYSIS WITH +MODE 2 IMPERFECTION SHAPE

ANALYSIS NO. 6, NONLINEAR NON-AXISYMMETRIC BIFURCATION BUCKLING ANALYSIS WITH THE +MODE 1 IMPERFECTION SHAPE

ANALYSIS NO. 7, NONLINEAR NON-AXISYMMETRIC BIFURCATION BUCKLING ANALYSIS WITH THE +MODE 2 IMPERFECTION SHAPE Next, do it all again with the minus Mode 1 and minus Mode 2 imperfections.

DESIGN ITERATIONS DURING A “SUPEROPT” RUN

OPTIMUM DESIGN OF STIFFENED EQUIVALENT ELLIPSOIDAL SHELL WITH MERIDIONALLY VARYING SKIN THICKNESS AND HEIGHT OF ISOGRID

LOAD-DEFLECTION CURVES FROM BOSOR4 AND STAGS

OBJECTIVE (WEIGHT) DURING THE 1ST “SUPEROPT” RUN FOR AN UNSTIFFENED EQUIVALENT ELLIPSOIDAL SHELL

OBJECTIVE (WEIGHT) DURING THE 2ND “SUPEROPT” RUN

OBJECTIVE (WEIGHT) DURING THE 3RD “SUPEROPT” RUN

OBJECTIVE (WEIGHT) DURING THE 4TH “SUPEROPT” RUN

OBJECTIVE (WEIGHT) DURING THE 5TH “SUPEROPT” RUN

OPTIMUM DESIGN OF UNSTIFFENED EQUIVALENT ELLIPSOIDAL SHELL WITH MERIDIONALLY VARYING SKIN THICKNESS

LINEAR BUCKLING AXISYMMETRIC MODE 1 FROM BIGBOSOR4 FOR THE OPTIMIZED UNSTIFFENED EQUIV. ELLIPSOIDAL SHELL BIGBOSOR4 model

LINEAR AXISYMMETRIC BUCKLING MODE 1 FOR THE OPTIMIZED SHELL FROM STAGS STAGS gets pcr=651 psi BIGBOSOR4 gets 642 psi

LINEAR AXISYMMETRIC MODE 2 BUCKLING FROM BIGBOSOR4 BIGBOSOR4 model

LINEAR AXISYMMETRIC BUCKLING MODE 2 FROM STAGS STAGS gets pcr = 684 psi, BIGBOSOR4 gets 667 psi.

PREBUCKLING AXISYMMETRIC DEFORMATION IN THE PRESENCE OF A -MODE 1 INITIAL IMPERFECTION SHAPE BIGBOSOR4 model

NONLINEAR NON-AXISYMMETRIC BUCKLING, -MODE 1 INITIAL IMPERFECTION SHAPE BIGBOSOR4 model

NONLINEAR NON-AXISYMMETRIC BUCKLING MODE FROM STAGS FOR THE SHELL WITH THE -MODE 1 IMPERFECTION SHAPE STAGS gets pcr=407 psi, BIGBOSOR4 gets 478 psi

COMPARISON OF STAGS AND BIGBOSOR4 PREDICTIONS FOR OPTIMIZED UNSTIFFENED SHELL WITH -MODE 1 Wimp

A VERY EARLY USE OF GENOPT. MY SON, BILL, HELPED A FRIEND OF HIS WITH SOME PHD RESEARCH THE FRIEND WAS DOING AT U. C. BERKELEY

ANOTHER APPLICATION OF GENOPT FOR PHD RESEARCH IN THE APPLIED MECHANICS DEPARTMENT AT STANFORD