12 gbm code optimum design
TRANSCRIPT
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Center for Advanced eCenter for Advanced e--System Integration TechnologySystem Integration Technology
Introduction to Optimum Design Programand ApplicationsIntroduction to Optimum Design Programand Applications
[Contents]
Selection of Good Optimization Algorithm
DOT/VisualDOCOther Optimum Design Codes
Optimum Design Examples using Optimum
Design program
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Center for Advanced eCenter for Advanced e--System Integration TechnologySystem Integration Technology
1. Selection of Good Optimization Algorithm. Selection of Good Optimization Algorithm
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Attributes of a Good Optimization AlgorithmAttributes of a Good Optimization Algorithm
Aspects need to be considered Robustness, efficiency, generality, ease to use
Reliability or Robustness
Reliabil ity of an algorithm is guaranteed if it is theoretically proven to
converge
Guarantees same optim ization results starting from different initial estimates.
Has higher prior ity over efficiency
Generality
The algorithm should be able to treat equality as well as inequality constraints.
Ease of use
Ease to use by the experienced as well as inexperienced designer
Efficiency Faster rate of convergence to the minimum point
Least number of calculations w ithin one design iteration
Efficiency w ithin an iteration implies the minimum of calculations for the
search direction and the step size
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QuestionsBefore Selecting an Optimization Algorithm
QuestionsBefore Selecting an Optimization Algorithm
Does the algorithms have proof of convergence? Is it theoretically guaranteed to converge to an optimum point starting
from any initial design estimate?
Can the starting design be infeasible?
Can the algorithm solve a general optimization problem without any
restrictions on the constraint functions?
Can it treat equality as well as inequality constraints?
Is the algorithm ease to use?
Does it require tuning for each problem?
Does the algorithm incorporate a potential constraint strategy?
Does it provide well defined GUI?
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2. DOT/VisualDOC. DOT/VisualDOC
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DOT (Design Optimization Tools)DOT (Design Optimization Tools)
Description
A general purpose numerical optim ization software (Fortran-based
program) package created by Vanderplaats Research &
Development(VR&D)
DOT can be used to solve a w ide variety of nonlinear optimization
problems.
The user provides a main program for calling DOT, and an analysis
program to evaluate the necessary function. Algorithms
For constrained optimization :
Modified Method of Feasible Direction (MMFD) : option 1
Sequential Linear Programming(SLP) with adjustable move limits : option 2
Sequential Quadratic Programming (SQP) : option 3
For Unconstrained optimization :
Broydon-Fletcher-Goldfarb-Shanno (BFGS) algorithm : option 1
Fletcher-Reeves (FR) Algorithm : option 2
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VisualDOC : DescriptionVisualDOC : Description
VisualDOC is a software system created by VR&D
It simplifies the addition of optimization to almost any design task.
VisualDOC is uses a graphical user interface(GUI) along with optimization
algorithms to setup and solve design problem
VisualDOCs optimization library is based on the general purpose optimization
software Design Optimization Control(DOC) and Design Optimization Tools(DOT).
Option of VisualDOC
Direct optimization by the VisualDOC optimizer
Optimization through the use of response surface techniques
(approximation technique) .
To calculate responses user can either supply a computer routine written in C/C++
or Fortran, or user can couple an existing program with VisualDOC. VisualDOC solves design problems by iteratively calling the optimizer to modify
the design variables and then calculating the resulting responses.
VisualDOC provides API and third party interface.
More options and capabilities included in VisualDOC 2.0
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VisualDOC : OptimizerVisualDOC : Optimizer
For constrained optimization :
Modified Method of Feasible Direction (MMFD)
Sequential Linear Programming(SLP) w ith adjustable move l imits
Sequential Quadratic Programming(SQP)
For Unconstrained optimization :
Broydon-Fletcher-Goldfarb-Shanno (BFGS) algorithm
Fletcher-Reeves (FR) Algorithm
For problems of fewer than 10 design variables, user can use response
surfaces (approximation technique).
There are no implicit limits on problem size.
Design variables may be continuous, discrete, or any combination. Linear or nonlinear design variable linking is supported.
An arbitrary number of constraints may be specified, and multiple
optimization is supported.
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VisualDOC : ComponentsVisualDOC : Components
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VisualDOC : Graphical User InterfaceVisualDOC : Graphical User Interface
Using the graphical user interface (GUI), user define a catalog ofvariables and responses.
The graphical user interface lets user enter and modify all optimization
components of a design problem.
User can supply initial values and limits for the variables.
User can also supply limits on some responses, which are called
constraints, and specify one or more responses as design objectives.
The design objectives can be minimized, maximized, or driven to target
values by a simple click of the mouse.
Different windows organize users design data, providing a clear,
concise description of users design problems.
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GUI Example(Windows) in VisualDOCGUI Example(Windows) in VisualDOC
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How VisualDOC Interface Works?How VisualDOC Interface Works?
The primary function of the VisualDOCinterface is to set up the optimization problem.
The data is passed between these modules
using a global data structure
Optimizer adjusts the values of theindependent design variables.
The modified design variable values are sent
back to the analysis routine.
The analysis code returns the direct response
values to the optimizer.
Values of objective and constraint functions
are calculated.
The objective and constraint values are
passed back to the optimizer, which then
adjusts the design variables once again.
This process continues in the same fashion
until optimum has been found.
VisualDOCGraphical User Interface
VisualDOC
Optimizer
AnalysisRoutine
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3. Other Optimum Design Codes. Other Optimum Design Codes
GBM d liGBM d li t
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GBM code listsGBM code lists
Several GBM codes for general design optimization purpose CFSQP - nonl inear and minmax optimizat ion.
CONOPT - nonlinear programming.DOT - Design Optimization Tools.
FSQP - nonl inear and minmax optimizat ion.GINO - nonl inear programming.GRG2 - nonl inear programming.LANCELOT - large-scale problems.LSGRG2 - nonlinear programming.
MINOS - l inear programming and nonlinear optimization.NLPQL - nonl inear programming.NLPQLB - nonlinear programming with constraints.NLPSPR - nonl inear programming.
NPSOL - nonl inear programming.OPSYC - OPt imisation de SYstmes Creux.OPTIMA Library - optimization and sensitivity analysis.OPTPACK- constrained and unconstrained optimization.SQP - nonl inear programming.
MINOS 5 1MINOS 5 1
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MINOS 5.1MINOS 5.1
MINOS is a Fortran-based computer system designed to solve large-
scale optimization problems.
Algorithms
Linear problem : the primal Simplex method
Nonlinear objective : a linearly constrained nonlinear program.
a reduced gradient algorithm in conjunction with a quasi-Newton algorithm
(BFGS)
Nonlinear Constraints :
a projected augmented Lagrangian algorithm
MINOS is designed to find solutions that are locally optimum.
No GUI and APIs are available (Consol Application)
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GENESIS-Structural Optimization SoftwareENESIS-Structural Optimization Software
Genesis GENESIS is a fully integrated structural analysis and design optimization
software. Analysis is based on the finite element analysis (FEA) for static,
normal modes, direct and modal analysis, heat transfer calculations, and
buckling analysis.
Design optimization is based on the advanced approximation concepts
approach to find an optimum design efficiently and reliably. Actual
optimization is performed by the well established DOT and BIGDOT
(excess of 2 mill ion design variables) optim izers, also from VR&D.
GENESIS performs both sizing (member dimension) and shape
(geometry) optimization. Typical problem sizes involve well over 100
design variables and many thousands of constraints. Almost any
calculated response (or nonlinear function of variables and responses)
can be chosen as the objective or may be constrained. These include mass,volume, eigenvalues, stresses, and deflections.
Pre- and post-processing is done by the PDA/ PATRAN and SDRC/ IDEAS
programs.
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Excel Solverxcel Solver
Excel Solver The Solver option in EXCEL 2000 (and earlier versions) may be
used to solve linear and nonlinear optimization problems.
Integer restrictions may be placed on the decision variables.
Solver may be used to solve problems w ith up to 200 decision
variables (design variables), 100 explicit constraints and 400
simple constraints (lower and upper bounds and/ or integer
restrictions on the decision variables).
Algorithms
Newtons method
Conjugate gradient method
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4. Optimum Design Examples usingOptimum Design program4. Optimum Design Examples usingOptimum Design program
Space Launch Vehicle DesignSpace Launch Vehicle Design
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Space Launch Vehicle DesignSpace Launch Vehicle Design
Design problem
Minimize CD : Drag Coefficient
Subject to Ch a : Surface heat t ransfer rateNose Pairing Volume bFineness ratio = c, where a, b, c are constant
Number of design variables : Xi, i= 1, 4
Linearization of objective function and design constraints
Sensitivity Analysis
Update the design
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DOT Main program (main for)DOT Main program (main for)
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DOT Main program (main.for)DOT Main program (main.for)
DIMENSION X(100),XL(100),XU(100),G(100),
*WK(800),IWK(200),RPRM(20),IPRM(20)
C DEFINE NRWK, NRIWK.
NRWK=800
NRIWK=200
C ZERO RPRM AND IPRM.DO 10 I=1,20
RPRM(I)=0.0
10 IPRM(I)=0
c
RPRM(1)=-0.0005
RPRM(2)=0.00005
C DEFINE METHOD,NDV,NCON.METHOD=2
NDV=4
NCON=2
C DEFINE BOUNDS AND INITIALDESIGN .DO 20 I=1,NDV
X(I) = 0.0
XL(I)=-0.2
20 XU(I)= 0.2
100 CALL DOT (INFO,METHOD,IPRINT,NDV,NCON,X,XL,XU,
*OBJ,MINMAX,G,RPRM,IPRM,WK,NRWK,IWK,NRIWK)
C FINISHED?
IF(INFO.EQ.0) STOP
C EVALUATE OBJECTIVE AND CONSTRAINT.
CALL EVAL(OBJ,X,G)
C GO CONTINUE WITH OPTIMIZATION.
GO TO 100
END
c-----OBJECTIVE & CONSTRAINTS------------------------------------------
SUBROUTINE EVAL (OBJ,X,G)
implicit real*8 (a-h, o-z)
IMENSION X(*),G(*), d(5), h(5),aold(4)
OBJ = d1 + d2* X(1) + d3* X(2) + d4* X(3) + d5* X(4)
G(1)=h1 + h2* X(1) + h3*X(2) + h4* X(3) + h5*X(4)-
0.012453
G(2)= (-1.)* vol + 0.198919
RETURN
END
Optimization method : SLP
No. of design variables : 4
No. of constraints : 2
OBJ:Objectivefunction
G(1),G(2): Constraintfunction
X(i):DesignvariableSide Constraints
2-Bar Truss Design : Structural Optimization2-Bar Truss Design : Structural Optimization
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2 Bar Truss Design : Structural Optimization2 Bar Truss Design : Structural Optimization
Description The symm etric 2-bar truss design shown in below has been studied by several
researchers
Ball ing and Clark(1992), Schmit, (1981), Sobieszczanski-Sobieski et al(1982)
The objective of this optimum problem is to min imize the weight of truss systemsubject to behavioral constraints
Related parameters
- B = 30 in
- t = 0.1 in
- = 0.3 lbs/ in3- y = 60,000 psi- E = 30E6 ps i
2-Bar Truss Design : Structural Optimization2-Bar Truss Design : Structural Optimization
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2 Bar Truss Design : Structural Optimization2 Bar Truss Design : Structural Optimization
Formulation for Optimization Minimize W(X)=2Dt(B2+H2)1/ 2
Minimize the weight of truss system
Subject to
g1(X) = e - 0 the first constraint prevents failure due to Euler buckling
g2(X) = y - 0 the second constraint prevents failure due to yield stress
Where,0.5D5.0 (in) X(1) : mean tube diameter5.0H50.0 (in) X(2) : height of the truss
The resulting optimum value from (Schmit, 1981) for W(x) is 19.8lbs.
- W * = 19 .8 lbs (at D* = 2.47 in , H* = 30.15 in )
2-Bar Truss Design : Structural Optimization2-Bar Truss Design : Structural Optimization
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2 Bar Truss Design : Structural Optimization2 Bar Truss Design : Structural Optimization
Comparisons with Algorithm and initial Valueinitial values X(1) =0. , X(2)= 0.
DOTProgram
Constrained Unconstrained
Algorithm MMFD SLP SQP BFGS F-R -X1 (D) 2.481 2.476 2.476 4.558 4.558 2.47
X2 (H) 29.870 29.992 30.00 15.031 15.031 30.15
Opt. val. (W) 19.800 19.800 19.800 28.828 28.828 19.8
Num. of Iter. 9 20 11 3 3 -
Num. of fun. eval. 77 69 45 24 24 -
true
initial values X(1) =1. , X(2)= 5.
-2626406626Num. of fun. eval.
-3310205Num. of Iter.
19.823.60123.60119.80019.80022.690Opt. val. (W)
30.1517.82017.82030.04129.99017.588X2 (H)
2.473.5883.5882.4742.4803.462X1 (D)
-F-RBFGSSQPSLPMMFDAlgorithm
UnconstrainedConstrainedtrue
DOTProgram