basic concepts of robust design optimization (rdo) · basics concepts of robust design optimization...

1

Basic Concepts of Robust Design

Optimization (RDO)

WOST15 2018

Veit Bayer

Christian Bucher

Dynardo Weimar Vienna

2Basics Concepts of Robust Design Optimization

WOST 15 Weimar June 21 2018

Overview

Introduction

Definitions of Probability

Random Variables and Vectors

Simulation

Estimation

Robustness and Reliability Analysis

Variance-based sigma levels reliability-based

Methods of Reliability Analysis

Examples



Introduction



Model CalibrationsIdentify important model parameter

for the best fit between simulation

and measurement



and measurement

Design ImprovementOptimize design performance

Design QualityEnsure design robustness

and reliability


and reliability

Design UnderstandingInvestigate parameter sensitivities

reduce complexity and

generate best possible meta models




CAE-Data

Measurement

Data

Robust Design


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bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties

bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary

Uncertainties in Optimization

1 Search optima with

flat surrounding

(stability)






2 Keep safety distance

from infeasible domain

(safety quality)



Probability



bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)

bull Event that one realization of parameters falls into subdomain

Kolmogorov axioms

Events and Probabilities



Complimentary events

bull An event can happen ( ) or not happen ( )

bull Complimentary events cannot happen at the same time




Conditional Probability

Independence




Total Probability

Bayesrsquo Theorem

Purpose Testing model updating conclude from a measurement eg to product safety or quality

Decomposition of Event Space



Random Variables



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Statistical Characterization of Random Variables

bull Expectation operator

bull Mean value

bull Variance

bull Coefficient of variation



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Probability Distribution and Density Function



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Distribution Types

Uniform Normal Log-normal

Exponential Weibull Rayleigh



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Statistical Characterization of Random Vectors

bull Arbitrary number k of random variables can be arranged in a vector

bull Mean value vector

bull Coefficient of correlation between two random variables

bull Covariance matrix of a random vector



bull Random generators produce numbers uniformly distributed in [01]

bull Mapping to prescribed marginal distribution

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Simulation of Random Variables

fU

u FX

fX

x



bull For each random variable the original marginal distribution is

transformed to an uncorrelated standard normal variable by the CDF

bull Assume a correlated joint Normal distribution for the random vector

bull Iterate the correlation coefficients of Zi Zj to match the original ones

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Simulation of Random Vectors



bull Example

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Standard normal space Original space



Estimation



bull Estimate an unknown parameter from independent observations

bull Example mean value

bull Consistency

bull (Asymptotic) Unbiasedness

Remarks

bull The true parameter is usu not known the available information is the

sample

bull Any estimate from a finite sample contains statistical uncertainty

which can be reduced by an increased sample size

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Estimation



bull An estimator from a random sample is a random variable by itself

bull Variance of the estimator

bull Estimator for variance

bull Estimate the variance of the estimator

serves to assess the confidence of the estimate

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Estimator Variance



bull Statistical error (or standard error) of the estimator

bull If the distribution of the error is known (eg assume Normal)

then the confidence interval can be established

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Confidence Interval



Robustness Analysis



bull Intuitively The performance of a robust design is largely unaffected by random perturbations

bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables

bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)

bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value

How to Define the Robustness of a Design



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Robustness in terms of limits

bull Safety margin (sigma level) of one or more responses y

bull Reliability (failure probability) with respect to given limit state

Robustness in terms of stability

bull Performance (objective) of robust optimum is less sensitive to input uncertainties

bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)



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Taguchi loss functions

bull Target value m is optimal (k scaling factor for costs)

bull Minimum is optimal (requires positive objective)

bull Maximum is optimal (requires strictly positive objective)



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Variance based Robustness Analysis

1) Define the robustness space using scatter range distribution and correlation

2) Scan the robustness space by producing and evaluating ndesigns

3) Check the variation 4) Check the

explainability of the model

5) Identify the most important scattering variables



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Exceedance Probability

bull Probability of reaching values above a limit for Gaussian distribution

m x

fX(x)

x



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Sigma Level vs Failure Probability

bull The sigma level can be used to estimate the probability of exceeding

a certain response limit

bull Since the distribution type of the response is generally unknown

this estimate may be very inaccurate for small probabilities

(sigma levels larger than 3)

bull The sigma level deals with single limit values whereas the failure

probability quantifies the event that any of several limits is exceeded

Reliability analysis should be applied to proof the required safety level

Distribution Required sigma level (CV=20)

pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475

Log-normal 277 404 757

Rayleigh 272 376 611

Weibull 203 254 349



Example Optimized Damped Oscillator

bull Robustness evaluation at

the deterministic optimum

bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin

taken as normally distributed random variables



bull Robustness analysis with respect to damped eigen-frequency and

maximum amplitude

ndash Check CV of objective and constraints

ndash Check if safety constraint safety = 85 rads

is outside of 45 level

ndash Check importance of input variables

ndash Check explainability by MOPCoP

Example Damped OscillatorVariance based Robustness Analysis



Constraint equation (omega)

bull CoD and CoP is 100

bull k is most important m is minor

bull Mean is close to deterministic

value

bull CV is 27

bull Safety limit is 238 which is

smaller as the required 45

Optimum is not robust in terms of

the constraint condition

Example Damped Oscillator

238



Objective function (xmax)


bull D is most important Ekin

and k are minor important

bull Mean is not close to

deterministic value

bull CV is 110

Optimum is not robust in terms

of the objective function




Reliability Analysis




Concept of Safety

bull Failure occurs if loading S exceeds the resistance R

bull Probability of failure




Partial Safety Factors

bull Definition of characteristical values for loading Sk and resistance Rk

bull Design values are obtained by

using partial safety factors

bull Final safety proof



bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)

bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system

(eg exceedance of allowable stress)

bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0

bull Multiple failure criteria (limit state functions) are possible

bull Series system

fails if one single component fails

g(x) = mini (gi (x))

bull Parallel system

fails if all components fail

g(x) = maxi (gi (x))

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FF

G



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Failure Probability

bull The probability of failure is the integral of the joint probability density

function over the failure domain

bull By introducing an indicator function

I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else

this can be computed as the expected value of I



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Monte Carlo Simulation

bull Robust for arbitrary limit state functions

bull Confidence of the estimate is very low for small failure probabilities

Sigma level le 2

Independent of number of random variables

X1

X2

g=0

Sigma

level

PF N for cov(PF) = 10

2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



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First Order Reliability Method (FORM)

bull Operates in the space of

standardized Gaussian variables

bull Search for failure point with

maximum probability density

(design point)

bull Equals the point in U on the limit state surface with minimal

distance to origin

bull Limit state function is linearized

around design point

bull Then failure probability can be

calculated analytically

bull Distance to origin (in U) is called

reliability index b

bull Can be interpreted as

generalization of sigma level



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Importance Sampling

bull Guide the sampling by making use of information about the failure

domain in order to increase the amount of failure events

bull To warrant correct statistics each sample is weighted by the ratio of

original to sampling density

bull Different strategies exist to estimate an ldquooptimalrdquo sampling density



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Importance Sampling Using Desing Point (ISPUD)

bull Based on FORM

bull Sampling density is centered at the design point

Requires continuously differentiable limit state function

Multiple design points (local minima) are not supported

May be able to mitigate error due to linearization in FORM

(oscillating limit state surface)

Moderate number of random variables

g(X) = 0

design point



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Adaptive Importance Sampling

bull Sampling density is defined by mean value vector and covariance

matrix of samples in the failure domain

bull Search for dominant failure region by 2-3 sampling iterations

Applicable for non-smooth and even discontinuous limit state functions

Limited to small to medium number of random variables



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Directional Sampling

bull Radial search for multiple ldquostar-shapedrdquo failure regions


Limited to small number of random variables

Few unsuccessful solver calls possible (as long as search is successful)



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Adaptive Response Surface Method

bull The limit state function is approximated by an Adaptive Response

Surface Method using a Moving Least Squares model

bull Directional Sampling is performed on the Response Surface

bull Additional supports are added near the limit state surface in regions of

high probability density

Applicable to a wide range of limit state functions

Efficient for a moderately high number of random variables



Overview of Methods

Recommended area of application

Approach Non-linearity Failure domains No parameters No solver runs

Monte Carlo

Simulation

arbitrary arbitrary many gt10^4 (3 sigma)

gt10^7 (5 sigma)

Directional

Sampling

arbitrary arbitrary lt= 10 1000-5000

Adaptive Importance

Sampling

arbitrary one dominant lt= 10 500-1000

FORM SORM

ISPUD

monotonic one dominant lt= 20 200-500

Adaptive Response

Surface Method

continuous few dominant lt= 20 200-500



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Example Damped OscillatorVerification of Robust Design by Reliability Analysis

bull Safety margin of 45 is equivalent to a failure probability of 3410-6

if responses were normally distributed

Reliability

Method Samples Failure probability Error Beta

FORM 65 1310-6 - 47

Adaptive Sampling 1500 1310-6 8410-8 47

Directional Sampling 600 1310-6 4910-7 47



Recommendations for Best Practice

bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified

bull Results with high sigma levels must be verified by reliability analysis

bull Choose proper reliability method due to dimension reliability level solver behavior

bull Reliability results shall be confirmed by a second method

bull When a reduced parameter set is used a confirmation with full parameter set is required

bull Use MOP (based on robustness samples) in order to

bull Monitor sampling

bull Monitor solver behavior

bull Analyze cause for non-robustness



Recommendations for best practice

bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on

bull Knowledge about uncertainty

bull Number of failed designs

bull Solver behavior

bull Sigma level



Robust Design Optimization





and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

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Iterative Robust Design Optimization

bull Decoupled optimization and

robustnessreliability analysis

bull For each optimization run the

safety margins are adjusted for

the critical model responses

bull Applicable to variance- and

reliability-based RDO

In our implementation variance-

based robustness analysis is

used inside the iteration and a

final reliability proof is performed

for the final design

Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



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Example Damped OscillatorVariance-based Iterative RDO

bull Safety margin of 45s to safety limit safety=85 rads

Sigma level requires (safety-mean) ge 45

Deterministic constraint is modified iteratively

Step 1 Step 2 Step 3 hellip

Optimization (global ARSM) Robustness (100 ALHS)

Constraint m k xmax Mean Sigma Sigma level

le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



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Coupled Robust Design Optimization

bull Fully coupled optimization and robustnessreliability analysis

bull For each design during the optimization procedure (nominal design)

the robustnessreliability analysis is performed

bull Applicable to variance- reliability- and Taguchi-based RDO

Our efficient implementation uses small sample variance-based

robustness measures during the optimization and a final

(more accurate) reliability proof

But still the procedure is often not applicable to complex CAE models

Definition of

design and

stochastic

variables

Sensitivity

analysisOptimization

Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



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Coupled RDO in optiSLang

bull Nested loop enables the coupled RDO

bull Optimizer has to handle statistical

errors of inner robustness analysis

bull Sigma level as constraint

See tutorial HelpTutorialsOscillatorOscillator_Robustness



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Example Damped OscillatorVariance-based Fully Coupled RDO

bull ARSM + robustness analysis

bull For each design 1+20 solver runs

bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0

Robust Design Optimization (ARSM+LHS)

m k xmax Mean Sigma Sigma level

ARSM+20LHS 087 500 028 76 020 45



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bull Approximation of model responses in

mixed optimizationstochastic space

bull Simultaneous RDO is performed on

a global response surface

bull Applicable to variance- reliability-

and Taguchi-based RDO

bull Approximation quality significantly

influences RDO results

Final robustnessreliability proof

is required

bull Pure stochastic variables have small

influence compared to design variables

Important local effects in the stochastic

space may be not represented

RDO on Global Response Surface



C Bucher Computational Analysis of Randomness in Structural

Mechanics Taylor amp Francis 2009

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Further Reading



Overview

Introduction

Definitions of Probability

Random Variables and Vectors

Simulation

Estimation

Robustness and Reliability Analysis

Variance-based sigma levels reliability-based

Methods of Reliability Analysis

Examples



Introduction





and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design


copy Dynardo GmbH







flat surrounding

(stability)








(safety quality)



Probability





Kolmogorov axioms











Independence




Total Probability

Bayesrsquo Theorem





Random Variables



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bull Mean value

bull Variance




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copy Dynardo GmbH

Distribution Types















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fU

u FX

fX

x







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bull Example

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Estimation





bull Consistency


Remarks


sample



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Estimation








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Estimator Variance






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Confidence Interval



Robustness Analysis










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m x

fX(x)

x



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pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






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FF

G



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Failure Probability








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Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



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(design point)


distance to origin


around design point




reliability index b





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Importance Sampling








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bull Based on FORM







g(X) = 0

design point



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copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



Introduction





and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design


copy Dynardo GmbH







flat surrounding

(stability)








(safety quality)



Probability





Kolmogorov axioms











Independence




Total Probability

Bayesrsquo Theorem





Random Variables



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bull Mean value

bull Variance




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copy Dynardo GmbH

Distribution Types















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fU

u FX

fX

x







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bull Example

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Estimation





bull Consistency


Remarks


sample



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Estimation








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Estimator Variance






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Confidence Interval



Robustness Analysis










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copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



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pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






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FF

G



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Failure Probability








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Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



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(design point)


distance to origin


around design point




reliability index b





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Importance Sampling








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bull Based on FORM







g(X) = 0

design point



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copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading





and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design


copy Dynardo GmbH







flat surrounding

(stability)








(safety quality)



Probability





Kolmogorov axioms











Independence




Total Probability

Bayesrsquo Theorem





Random Variables



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bull Mean value

bull Variance




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copy Dynardo GmbH

Distribution Types















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fU

u FX

fX

x







copy Dynardo GmbH




bull Example

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Estimation





bull Consistency


Remarks


sample



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Estimation








copy Dynardo GmbH

Estimator Variance






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Confidence Interval



Robustness Analysis










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copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



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pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






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FF

G



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Failure Probability








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Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



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(design point)


distance to origin


around design point




reliability index b





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Importance Sampling








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bull Based on FORM







g(X) = 0

design point



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copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading







flat surrounding

(stability)








(safety quality)



Probability





Kolmogorov axioms











Independence




Total Probability

Bayesrsquo Theorem





Random Variables



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bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

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Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading








(safety quality)



Probability





Kolmogorov axioms











Independence




Total Probability

Bayesrsquo Theorem





Random Variables



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bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

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Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



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pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








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bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



Probability





Kolmogorov axioms











Independence




Total Probability

Bayesrsquo Theorem





Random Variables



copy Dynardo GmbH



bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading





Kolmogorov axioms











Independence




Total Probability

Bayesrsquo Theorem





Random Variables



copy Dynardo GmbH



bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading










Independence




Total Probability

Bayesrsquo Theorem





Random Variables



copy Dynardo GmbH



bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



Total Probability

Bayesrsquo Theorem





Random Variables



copy Dynardo GmbH



bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



Random Variables



copy Dynardo GmbH



bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH



bull Mean value

bull Variance




copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH




copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH

Distribution Types















copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading













copy Dynardo GmbH


fU

u FX

fX

x







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading







copy Dynardo GmbH




bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



bull Example

copy Dynardo GmbH





Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



Estimation





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading





bull Consistency


Remarks


sample



copy Dynardo GmbH

Estimation








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading








copy Dynardo GmbH

Estimator Variance






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading






copy Dynardo GmbH

Confidence Interval



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



Robustness Analysis










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading










copy Dynardo GmbH









copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH







copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH









copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH



m x

fX(x)

x



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH











pF = 10-2 pF = 10-3 pF = 10-6

Normal 232 309 475



Weibull 203 254 349











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading











maximum amplitude













value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading







value

bull CV is 27






238








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading








deterministic value

bull CV is 110










Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading







Concept of Safety


















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading
















bull Series system






copy Dynardo GmbH


FF

G



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH

Failure Probability








copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH




Sigma level le 2


X1

X2

g=0

Sigma

level


2 23E-2 4 400

3 13E-3 74 000

45 34E-6 29 500 000



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH






(design point)


distance to origin


around design point




reliability index b





copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH

Importance Sampling








copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH


bull Based on FORM







g(X) = 0

design point



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH









copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH








copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH











Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



Overview of Methods



Monte Carlo

Simulation


gt10^7 (5 sigma)

Directional

Sampling


Adaptive Importance

Sampling


FORM SORM

ISPUD


Adaptive Response

Surface Method




copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH




Reliability


FORM 65 1310-6 - 47






















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading




















bull Sigma level








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading








and measurement



and measurement



and reliability


and reliability







CAE-Data

Measurement

Data

Robust Design

copy Dynardo GmbH




copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH














Definition of

design and

stochastic

variables

Sensitivity

analysis

Design

failure

Update

constraints

Deterministic

optimization

Variance-

based

robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH








le 8 078 500 799 025 799 022 233

le 7 104 500 694 029 694 019 82

le 763 086 491 756 028 755 020 466



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH










Definition of

design and

stochastic

variables

Sensitivity


Robustness

evaluation

Final

reliability

proof

Optimal and

robust

design



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH









copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH







ARSM+20LHS 087 500 028 76 020 45



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading



copy Dynardo GmbH










is required










copy Dynardo GmbH

Further Reading





copy Dynardo GmbH

Further Reading

basic concepts of robust design optimization (rdo) · basics concepts of robust design optimization...

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