Tutorial 2, Part 1: Optimization of a damped oscillator

2 Tutorial 2, Part 1: Optimization

Damped oscillator

• Mass m, damping c, stiffness k and initial kinetic energy

• Equation of motion:

• Undamped eigen-frequency:

• Lehr's damping ratio D

• Damped eigen-frequency

3 Tutorial 2, Part 1: Optimization

Damped oscillator

• Time-dependent displacement function

• Optimization goal: Minimize maximum amplitude after 5s free vibration

• Optimization constraint:

• Optimization parameter bounds & constant parameters:

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Task description

• Parameterization of the problem

• Definition and evaluation of DOE scheme

• Definition and evaluation of MOP

• Single objective, constraint optimization

• Gradient based optimization

• Global response surface method

• Adaptive response surface method

• Evolutionary algorithm

• Multi objective optimization

• Pareto optimization with evolutionary algorithm

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Project manager

1. Open the project manager2. Define project name3. Create a new project directory4. Copy optiSLang examples/Oscillator into project director

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Parameterization of the problem

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

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Parameterization of the problem

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

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Parameterization of the problem

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

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Parameterization of the problem

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

(0.1 … 5.0)

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Parameterization of the inputs

1. Repeat procedure for k

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1. Click “open file” icon in parametrize editor2. Browse for the SLang output file oscillator_solution.txt 3. Choose file type as OUTPUT

Parameterization of the output signal

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Parameterization of the problem

1. Mark output value in editor2. Define omega as output parameter3. Repeat for x_max and x_max_env

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Parameterization of the problem

1. Create new objective function2. Define objective as x_max

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Parameterization of the problem

1. Create new constraint equation2. Define inequality constraint 0≤8-omega

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Parameterization of the problem

1. Check overview for inputs2. Check overview for outputs

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Parameterization of the problem

1. Check overview for objectives2. Check overview for constraints

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Design Of Experiments (DOE)

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1. Start a new DOE workflow2. Define workflow name and workflow identifier3. Enter problem file name4. Enter solver call (slang –b oscillator.s)5. Start DOE evaluation with 100 LHS samples

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Meta-Model of Optimal Prognosis (MOP)

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1. Start a new MOP workflow2. Define workflow name and workflow identifier3. Choose DOE result file4. Choose optional problem file

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Meta-Model of Optimal Prognosis (MOP)

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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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Meta-Model of Optimal Prognosis (MOP)

• Check approximation quality to identify solver problems and for a possible use of MOP for the optimization task

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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 oscillator.s)6. Start gradient workflow

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Gradient-based optimization

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1. Keep default settings and start optimization task Differentiation interval is too small for noisy objective Optimizer runs into local optimum

2. Change differentiation interval to 5% Slow convergence but global optimum is found

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Gradient-based optimization

1. Objective history 2. Best design input parameters3. Best design response data4. Objective and constraints data5. Parameter history

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Gradient-based optimization using MOP

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1. Start a new Gradient-based workflow2. Define workflow name, workflow identifier and problem file name3. Use MOP as solver and choose MOP data file4. Enter solver call (slang –b oscillator.s) to verify best design5. Start gradient workflow

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Gradient-based optimization using MOP

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1. Keep default settings and start optimization task

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Gradient-based optimization using MOP

1. Objective history 2. Best design input parameters (MOP and calculated)3. Best design response data (MOP and calculated)4. Objective and constraints data (MOP and calculated)

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Adaptive response surface (local)

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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 oscillator.s)5. Start ARSM workflow

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Adaptive response surface (local)

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1. NLPQL as optimization method2. Approximation settings:

keep polynomial regression3. Advanced settings: no recycle

of previous support points

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Adaptive response surface (local)

1. Objective history for each iteration step 2. Best design input parameters (ARSM and original)3. Best design response data (ARSM and original)4. Objective and constraints data (ARSM and original)5. Parameter history for each iteration step

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Adaptive response surface (global)

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1. GA & NLPQL as optimization method2. Approximation settings: choose Moving Least Squares

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Adaptive response surface (global)

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1. Increase stopping criteria to 0.1%2. Advanced settings: choose recycle previous support points

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Adaptive response surface (global)

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Evolutionary algorithm (EA)

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

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Evolutionary algorithm (EA)

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

Crossover and Mutation

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Evolutionary algorithm (EA)

1. Objective history for each design2. Best design input parameters 3. Best design response data 4. Penalized objective and constraints data5. Parameter history for each design,

new generations are marked

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Parameterization of second objective

1. Start a new parametrize workflow2. Define workflow name3. Create a copy and modify it4. Open Tutorial_Oscillator.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 = omega3. Delete omega constraint4. 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 oscillator.s) and start workflow

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Pareto optimization with EA

1. Choose start population size2. Keep defaults for Selection, Crossover and Mutation

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Pareto optimization with EA

1. Plot of objective values of designs including Pareto front 2. Best design input parameters 3. Best design response data 4. Objectives data5. Parameter history for each design, new generations are marked

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