Tutorial 3, Part 1: Optimization of a linear truss structure

Tutorial 3, Part 1: Optimization of a

linear truss structure

2 Tutorial 3, Part 1: Optimization

Truss structure

• Optimization goal is to minimize the mass of the structure (initial mass is 8393 lbs)

• Maximum stress is 25000 psi (limit stress=40166/global safety factor=1.6)• Maximum displacement is 20 in• Design variables: cross section areas of trusses 0.1 in² ≤ Xi=ai ≤ 20 in²• Load parameters: F=100000 lbf• Material parameters: E=107 psi, =0.2, =0.1 lbs/in³• Element length: L=360 in• Responses from linear finite element analysis are mass, displacements at

loading points and stresses for each element


Task description

• Parameterization of the problem• Definition and evaluation of DOE scheme• Definition and evaluation of MOP

• Single objective, constraint optimization• Gradient based optimization• Adaptive response surface method• Evolutionary algorithm

• Multi objective optimization• Pareto optimization with

evolutionary algorithm


Project manager

1. Open the project manager2. Define project name3. Create a new project directory4. Copy optiSLang Examples/10bar_truss into project directory

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2.3.


Parameterization of the problem

1. Start a new parametrize workflow2. Define workflow name3. Create a new problem specification4. Enter problem file name5. Start parametrize editor

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1. Click “open file” icon in parametrize editor2. Browse for the SLang input file 10bar_truss.s 3. Choose file type as INPUT

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1. Mark value of a1 in the input file2. Define a1 as input parameter3. Define parameter name

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1. Open parameter in parameter tree2. Enter lower and upper bounds

(0.1 … 20.0)3. Set settings as default for

other parameters

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• Repeat the procedure for the other cross section areas• Lower and upper bounds are automatically set to 0.1 … 20.0



1. Open output file of solver 10bar_response.out2. Mark output value in editor3. Define mass as output parameter4. Repeat for displacements and stress values (take care of sign)

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1. Mark final string2. Define it as success string

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1. Open objective section (double click)2. Create new objective function 3. Define mass as objective

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1. Open constraint section (double click)2. Create inequality constraint equation for each displacement value

1-|disp_node/20| (scaled to one)3. Create inequality constraint equation for each stress value

1-|stress_element/25000| (scaled to one)

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1. Close parametrization editor2. Check overview for inputs3. Check overview for outputs



1. Check overview for objectives2. Check overview for constraints


Design Of Experiments (DOE)

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1. Start a new DOE workflow2. Define workflow name and workflow identifier3. Enter problem file name



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1. Enter solver call (slang –b 10bar_truss.s)

2. Start DOE evaluation3. Generate 100 Latin

Hypercube samples

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1. Check input samples2. Check input distributions3. Check input correlations4. Continue to evaluate

response values


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• Reload file in optimization mode to find best design from DOE samples as possible start value for optimization task

• Best design with valid constraints: mass = 3285 (39% of initial mass)



Meta-Model of Optimal Prognosis (MOP)

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1. Start a new MOP workflow2. Define workflow name 3. Define workflow identifier4. Choose DOE result file5. Choose optional problem file

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1. CoP settings (sample splitting or cross validation)2. Investigated approximation models3. CoP - accepted reduction in prediction quality to simplify model4. Filter settings5. Selection of inputs and outputs

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• MOP indicates only a1, a3, a8 as important variables for maximum stress and displacements,but all inputs are important for objective function



• For single stress values used in constraint equations, each input variable occurs at least twice as important parameter

Reduction of number of inputs seems not possible

max_stressmax_dispstress10stress9stress8stress8stress6stress5stress4stress3stress2stress1

disp4disp2mass

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

MOP filter


Gradient-based optimization

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1. Start a new Gradient-based workflow2. Define workflow name and workflow identifier3. Enter problem file name4. Choose optimization method5. Enter solver call (slang –b 10bar_truss.s)6. Start gradient workflow

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1. Decrease size of diffentiation interval

2. Choose single sided differences to reduce number of solver calls

3. Choose best valid sample from DOE workflow as start value


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• Elements 2,5,6 and 10 are set to minimum

• Stresses in remaining elements reach maximum value

• 153 solver calls (+100 from DOE)


Adaptive response surface

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1. Start a new ARSM workflow2. Define workflow name and workflow identifier3. Enter problem file name4. Enter solver call (slang –b 10bar_truss.s)5. Start ARSM workflow

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3.1. GA & NLPQL as optimization

method2. Approximation settings:

keep polynomial regression3. Advanced settings: no recycle

of previous support points

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• Elements 2,6 and are set to minimum, 5 and 10 are close to minimum

• 360 solver calls


Evolutionary algorithm (global search)

1. Start a new NOA workflow2. Define workflow name and workflow identifier 3. Enter problem file name4. Choose optimization algorithm (EA with global search as default)5. Enter solver call (slang –b 10bar_truss.s) and start workflow

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1. Choose start population size2. Keep defaults for Selection,

Crossover and Mutation




• 392 solver calls


Evolutionary algorithm (local search)

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1. Keep defaults for Start population, Selection, Crossover and Mutation

2. Choose start design from global EA optimization

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Evolutionary algorithm (local search)


• 216 solver calls (+392 from global search)


Overview optimization resultsMethod Settings Mass Solver calls Constraints

violatedInitial - 8393 - -DOE LHS 3285 100 75%

NLPQL diff. interval 0.01%, single sided 1595 153(+100) 42%

ARSM defaults (local) 1613 360 80%EA global defaults 2087 392 56%EA local defaults 2049 216(+392) 79%

PSO global defaults 2411 400 36%GA global defaults 2538 381 25%SDI local defaults 1899 400 70%

• NLPQL with small differentiation interval with best DOE as start design is most efficient

• Local ARSM gives similar parameter set• EA/GA/PSO with default settings come close to global optimum• GA with adaptive mutation has minimum constraint violation


Parameterization of second objective

1. Start a new parametrize workflow2. Define workflow name3. Create a copy and modify it4. Open truss.pro and enter new problem file name

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Parameterization of second objective

1. Create a new objective2. Enter name, activate and enter function obj2 = max_stress3. Delete stress constraints4. Close editor and check objectives

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Pareto optimization with EA

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1. Start a new Pareto workflow2. Define workflow name and workflow identifier 3. Enter problem file name4. Choose EA as optimization algorithm 5. Enter solver call (slang –b 10bar_truss.s) and start workflow

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Find optimal designs for start population: • Open ARSM optimization results in statistic mode• Select anthill plot mass vs. max_stress• Select some designs around the optimum which seem

to be Pareto optimal • Remember the design numbers

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1. Select “specify start population” 2. Increase minimum and maximum number of generations3. Keep defaults for Selection, Crossover and Mutation

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1. Import design from ARSM optimization2. Choose 20 designs (preferred designs from anthill plot

and the remaining from the designs with minimal mass)

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Pareto front

Date post:	19-Mar-2016
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Tutorial 3, Part 1: Optimization of a linear truss structure

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