Introduction to Parametric Optimization and Robustness Evaluation with Optislang

Introduction to Parametric Optimization and Robustness Evaluation with optiSLang

Dynardo GmbH

1 © Dynardo GmbH

1. Introduction 2. Process to optiSLang integration

6. Training 3. Sensitivity analysis

5. Robustness 4. Parametric analysis Optimization

Introduction to the parametric optimization and robustness evaluation with 2 optiSLang © Dynardo GmbH

1. Introduction 2. Process to optiSLang integration

6. Training 3. Sensitivity analysis

5. Robustness 4. Parametric analysis Optimization

Introduction to the parametric optimization and robustness evaluation with 3 optiSLang © Dynardo GmbH

Dynardo

• Founded: 2001 (Will, Bucher, CADFEM International) • More than 50 employees, offices at Weimar and Vienna • Leading technology companies Daimler, Bosch, E.ON, Nokia, Siemens, BMW are supported

Software Development

CAE-Consulting • Mechanical engineering • Civil engineering & Dynardo is engineering specialist for Geomechanics CAE-based sensitivity analysis, • Automotive industry optimization, robustness evaluation • Consumer goods industry and robust design optimization • Power generation

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Robust Design Optimization (RDO) in virtual product development optiSLang enables you to: • Identify optimization potentials • Improve product performance • Secure resource efficiency • Adjust safety margins without limitation of input parameters • Quantify risks • Save time to market

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Excellence of optiSLang

• optiSLang is an algorithmic toolbox for • sensitivity analysis, • optimization, • robustness evaluation, • reliability analysis • robust design optimization (RDO) • functionality of stochastic analysis to run real world industrial applications • advantages: • predefined workflows, • algorithmic wizards and • robust default settings

Introduction to the parametric optimization and robustness evaluation with 6 optiSLang © Dynardo GmbH Robust Design Optimization with optiSLang

2nd Multidisciplinary Optimization Adaptive Response Surface, Evolutionary Algorithm, Pareto Optimization

Introduction to the parametric optimization and robustness evaluation with 7 optiSLang © Dynardo GmbH

1. Introduction 2. Process to optiSLang integration

6. Training 3. Sensitivity analysis

5. Robustness 4. Parametric analysis Optimization

Introduction to the parametric optimization and robustness evaluation with 8 optiSLang © Dynardo GmbH

Process Integration

Parametric model as base for • User defined optimization (design) space • Naturally given robustness (random) space

Design variables Entities that define the design space

Response variables The CAE process Outputs from the Generates the system Scattering variables results according Entities that define the to the inputs robustness space

Introduction to the parametric optimization and robustness evaluation with 9 optiSLang © Dynardo GmbH optiSLang Integrations & Interfaces

Direct integrations

ò ANSYS Workbench

ò MATLAB

ò Excel

ò Python

ò SimulationX

Supported connections

ò ANSYS APDL

ò Abaqus

ò Adams

ò AMESim

ò …

Arbitary connection of ASCII file based solvers

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Full Integration of optiSLang in ANSYS Workbench

• optiSLang modules Sensitivity , Optimization and Robustness are directly available in ANSYS Workbench

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Example: Optimization of a Steel Hook

Deterministic Optimization • Minimize the mass • The maximum stress should not exceed 300MPa • Initially a safety factor of 1.5 is defined • 10 geometry parameters are used for the design variation

Robustness requirement • Proof for the optimal design that the failure stress limit is not exceeded with a 4.5 sigma safety margin • 16 scattering parameters are considered (geometry and material properties and the load components)

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Example: Simulation Model in ANSYS Mechanical

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Example: The Design Parameters

A Outer_Diameter 25-35 mm B Connection_Length 20-40 mm C Opening_Angle 10-30 ° D Upper_Blend_Radius 18-22 mm E Lower_Blend_Radius 18-22 mm F Connection_Angle 120-150 ° G Lower_Radius 45-55 mm H Fillet_Radius 2-4 mm I Thickness 15-25 mm Depth 15-25 mm

Introduction to the parametric optimization and robustness evaluation with 14 optiSLang © Dynardo GmbH

1. Introduction 2. Process to optiSLang integration

6. Training 3. Sensitivity analysis

5. Robustness 4. Parametric analysis Optimization

Introduction to the parametric optimization and robustness evaluation with 15 optiSLang © Dynardo GmbH

Flowchart of Sensitivity Analysis

Design of Regression Sensitivity Experiments Methods Evaluation • Deterministic • 1D regression • Correlations • (Quasi)Random • nD polynomials • Reduced regression • Sophisticated • Variance-based metamodels Solver 1. Design of Experiments generates a specific number of designs, which are all evaluated by the solver 2. Regression methods approximate the solver responses to understand and to assess its behavior 3. The variable influence is quantified using the regression functions

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Response Surface Method

• Approximation of response variables as explicit function of all input variables

• Approximation function can be used for sensitivity analysis and/or optimization

• Global methods ( Polynomial regression , Neural Networks, …)

• Local methods (Spline interpolation, Moving Least Squares , Radial Basis Functions, Kriging, …)

• Approximation quality decreases with increasing input dimension

• Successful application requires objective measures of the prognosis quality

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Metamodel of Optimal Prognosis (MOP)

• Approximation of solver output by fast surrogate model • Reduction of input space to get best compromise between available information (samples) and model representation (number of inputs) • Determination of optimal approximation model • Assessment of approximation quality • Evaluation of variable sensitivities

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Definition of the Design Parameter Bounds

• Specify the ranges of the design parameters • You may choose continuous and discrete/binary optimization variables

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Example: Results of the Sensitivity Analysis

• For the mass 6 important inputs are detected by the MOP • For the maximum stress only 3 inputs are important • Thickness, depth and lower radius are important for both responses • Prognosis quality of both response values is very good (99%)

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Example: Results of the Sensitivity Analysis

• Both responses show slightly nonlinear and monotonic behavior and can be explained with a prognosis quality of 99% ‹ Optimization should be straight forward

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1. Introduction 2. Process to optiSLang integration

6. Training 3. Sensitivity analysis

5. Robustness 4. Parametric analysis Optimization

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Optimization with MOP pre-search

Optimization

Optimizer Optimizer Sensitivity analysis • Gradient • Gradient • EA/GA • ARSM DOE MOP • EA/GA

Solver SolverMOP Solver

• Full optimization is performed on MOP by approximating the solver response • Optimal design on MOP can be used as – final design (verification with solver is required!) – as start value for second optimization step with direct solver

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optiSLang Optimization Algorithms

Gradient-based Adaptive Response Nature inspired Methods Surface Method Optimization • Most efficient method if • Attractive method for • GA/EA/PSO imitate gradients are accurate a small set of mechanisms of nature to enough continuous variables improve individuals (<20) • Consider its restrictions • Method of choice if like local optima, only • Adaptive RSM with gradient or ARSM fails continuous variables default settings is the • Very robust against and noise method of choice numerical noise, non- linearity, number of variables,… Start

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Definition of the Objective and Constraints

• All design parameters, responses and help variables can be used within mathematical formulations for objectives and constraints • Minimization and maximization tasks with constraints are possible

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Optimization Wizard

• Previous Sensitivity study may provide required information • By a few settings, optiSLang suggests the most promising algorithm • All algorithms come with robust default settings

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Example: Initial vs. Optimal Design Initial Design Optimal Design

Mass = 790g Mass = 588g Equivalent Stress = 439MPa Equivalent Stress = 200MPa

1. Introduction 2. Process to optiSLang integration

6. Training 3. Sensitivity analysis

5. Robustness 4. Parametric analysis Optimization

Robustness in terms Robustness in terms of constraints of the objective

• Safety margin (sigma level) of one • Performance (objective) of robust or more responses y: optimum is less sensitive to input uncertainties • Minimization of statistical • Reliability (failure probability) with evaluation of objective respect to given limit state: function f (e.g. minimize mean and/or standard deviation):

Robustness Analysis

1) Define the robustness space using 2) Scan the robustness space by scatter range, distribution and producing and evaluating n correlation designs

5) Identify the most 3) Check the variation important scattering 4) Check the variables explainability of the model

Definition of the Parameter Uncertainties

• The definition of different distribution types is possible (Normal, Uniform, Truncated-Normal, Log-normal, Gumbel, Weibull, …) • Mean value, Standard deviation or Coefficient of Variation have to be specified • Correlations between the random variables can be considered as well

Robustness Postprocessing

Traffic light plot Histogram & Statistical Data

MOP/CoP Sensitivities

Example: Results of the Robustness Evaluation

• Statistical Evaluation of the Maximum Stress : • Safety distance to failure stress of 300MPa is estimated with a sigma level of only 3.18 ‹ Attention: Requirement of a 4.5 sigma level is not fulfilled

Example: Results of the Robustness Evaluation

• Force in main direction is the most important input parameter for the maximum stress ‹ Attention: Scatter of this uncertainty is difficult to be reduced ‹ Therefore design has to be changed to reduce mean value of maximum stress and to fulfill the robustness requirement ‹ Safety factor is increased and deterministic optimization is performed again

Iterative Robustness Design Optimization

• Adapt the constraint condition to move the mean away from the limit • Robustness evaluation is performed again for new optimal design • Only 2 to 3 iterations steps are necessary to obtain robust design

Example: Robustness of Second Optimum

• Stress : Safety margin to failure limit of 300MPa is estimated with a sigma level of 4.82 (would fulfill the robustness requirement) • The sigma level of 4.82 corresponds to a failure probability of 7.1*10 -7 if the response is perfect normally distributed

Example Initial Design Deterministic Robust Summary: Optimum Optimum

Mass 790 g 588 g 666 g Stress 439 MPa 200 MPa 176 MPa Sigma Level - 3.3 4.8 FailureProbability >0.5 10^-3 10^-6

1. Introduction 2. Process to optiSLang integration

6. Training 3. Sensitivity analysis

5. Robustness 4. Parametric analysis Optimization

Further Training optiSLang 4 Basics 3 day introduction to process integration, sensitivity, optimization, calibration and robustness analysis optiSLang inside ANSYS Workbench 2 day introduction seminar to parameterization in ANSYS Workbench, sensitivity analysis and optimization optiSLang 4 and ANSYS Workbench 1 day introduction to the integration of ANSYS Workbench projects in a optiSLang 4 solver chain, parameterization of signals via APDL output Parameter Identification 1 day seminar on basics of model calibration, application of sensitivity analysis and optimization to calibration problems Robust Design and Reliability Analysis 1 day seminar on basics of probability, robustness and reliability analysis, robust design optimization

See our website: http://www.dynardo.de/en/trainings.html

12 th Weimar Optimization and Stochastic Days 2015

November 5-6 cc neue weimarhalle

Conference for CAE-based parametric optimization, stochastic analysis and Robust Design Optimization

Thanks for your attention!

Introduction to the parametric optimization and robustness evaluation with 41 optiSLang