Discrete Variable Optimizationusing

MSC.Nastran SOL200

Dr. Hans SippelDirector MSC.Nastran Europe

……, 2002

Generic CommentsSince 1971 (!!) MSC has been developing, maintaining and marketing its proprietary version

Numerics/Performance

Dynamics

Optimization

Discrete Variable OptimizationDiscrete Variable Optimization

Design of Experiments

Discrete Design

Discrete Design Verification by

FEA

Improved Continuous

Design

Finite Element Analysis (FEA)

Initial design

Approximate Model

Continuous Optimization

Algorithm

Continuous Optimization

Sensitivity Analysis

Discrete Variable OptimizationDiscrete Variable Optimization

Conventional optimization uses Mathematical Programming that yields continuous design variablesThese design variables are not immediately usable ( i.e. you cannot make a plate of thickness 0.37582 cm ).With discrete optimization, the user can specify thicknesses available: according to standard gaugesThree methods1. Conservative Discrete Design (CDD)2. Design of Experiments (DOE)3. Rounding up/offImplemented as a postprocessing step to a continuous solution, this means 1 additional FE – analysis

Discrete Variable OptimizationDiscrete Variable Optimization

UB ContinuousOptimizationResults

DV1 ……. DV5 ……. DV9

LB

Dis

cret

e V

alue

s

* CDD: 2*9 evaluations

* DOE: 2**9 evaluations

* Round up/off

The DOE method - StepsThe DOE method - Steps

Continuous Optimization (Objective function, constraints)Build of a 2 level list consisting of the next smaller and larger discrete value for each discrete design variableSelection of a subset of that list ( orthogonal array, OA ) and calculation of the objective function and the constraints for all rows of the OA using the Approximate Model

Within MSC.Nastran for Number of Design Variables (NDV) less or equal 16 an exhaustive search is performed. If NDV>16 2**16 possible solutions are evaluated.Selection of the best solution

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0)(~ xxxxfxfxf −

∂∂

+=

The DOE method - ExampleThe DOE method - ExampleOptimization Problem:min. f(x) = a*b discrete values: a = [3,4,5,6,7,8]with g(x) = a+b-16.3≥0 b = [11,12,13,14,15,16]

1. Continuous Optimum:a = 3b = 13.3

2. Objective Function and restrictions in the continuous optimum:f(x0) = 39.9g(x0) = 0

3. Build of a two-level List with closest discrete values:a = [3,4]b = [13,14]

4. Build and analysis of the full factorial design (4 rows = 4 trials)Analysis a b f g1 3 13 39 -0.32 4 13 52 0.73 3 14 42 0.74 4 14 56 1.7

5. Selection of the best solution: a=3; b = 14

b

lines f=const.

gf ascending

discrete Optimum

cContinuousoptimum a

The DOE Method - SOL 200 Input DeckThe DOE Method - SOL 200 Input Deck

The DOE Method - SOL 200 OutputThe DOE Method - SOL 200 Output