Statistical Analysis for Robust Design

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Copyright © ESI Group, 2017. All rights reserved.Copyright © ESI Group, 2017. All rights reserved.
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Scilab
Statistical Analysis for Robust Design
Hugues- Arthur Garioud & Yann Debray – Scilab – ESI Group
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All models are wrong,
But some are useful.
– George E.P. Box –

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Copyright © ESI Group, 2017. All rights reserved.
Agenda
• Introduction:
‣ Uncertainty Quantification
• Statistical Analysis for:
‣ Engineering
‣ Manufacturing
‣ In-Service
• Some theory behind:
‣ Design of Experiment – History of classic
‣ Computer experiment – Modern design

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Introduction
Uncertainty Quantification

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Uncertainty Quantification Methods
NAFEMS Benchmark January 2019

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Uncertainty propagation process

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Design rules and Uncertainty Quantification

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Cantilever beam with tip load

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Statistics 4
Engineering

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Crash, NVH & Material Calibration
Simulation Workflow
Design of Experiment
Finite Element
Simulation
Model Order
Reduction
Stats & Optim

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Material Calibration

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Material Calibration

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HVAC system
Battery
Interior Solution – Design Optimization
Control System
PHVAC
TCAB
PHVAC

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Interior Solution
Maximize Comfort (Tressenti)
Minimize Consumption (-> Battery)
Run Level
Temp_Am
biante
Temp_Air
_Injecte
1 0 -15 5
2 0 -15 20
3 0 40 5
4 0 40 20
5 1 -15 12,5
6 1 40 12,5
7 1 12,5 5
8 1 12,5 20
9 2 -15 7,1967
10 2 -15 17,8033
11 2 40 7,1967
12 2 40 17,8033
13 2 12,5 12,5
14 2 -6,94544 5
15 2 -6,94544 20
16 2 31,9454 5
17 2 31,9454 20

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Heat exchanger
Perform using FreeFEM driven by the Scilab toolbox:
https://atoms.scilab.org/toolboxes/SciFreeFEM

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Airfoil design
What are the factors (main effect) or combination
of factors (interactions) that affect the response
the most?
Statistical measures
•Analysis of variance (ANOVA)
•Signifiance testing
•Main effect
What sample of the population shall one study to
ensure catching all the effects ?

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Airfoil design
First 4 dominant POD modes for pressure DOE case pressure field projection on 4 modes POD
basis
4 modes
99.96% of the
global energy
Modesenergy

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Robust design process
OpenFOAM + Scilab = <3
Mesh Perturbations
- Hicks-Henne Sine Bumps -
Mesh Generation
- Mesh Morphing -
DOE Generation Surrogate Modeling
DOE Simulations
- simpleFoam -
Model reduction
- POD -
Optimization
Optimization
- Gradient/GA -
Validation
- simpleFoam -

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CEM One Work In Progress

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Statistics 4
Manufacturing

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Stamp – Robustness analysis
Pre-processing
Finite Element
Simulation
Post-processing
Stats & Optim
PAM-STAMP

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Thinning
Run

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Design parameters:
• Material
• 3D Geometry
Process parameters:
• Force
• Friction
(lubrication)
Objectives:
→ Thinning (<30%)
→ Friction (<rupture)
Beating Autoform SIGMA
Friction
Thinning
Trade-off Line
Theoretical optimum
Design space Objective space
f
Pareto Front
Dominant PointsDominant Points
Pareto Set

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Additive Manufacturing
Distortion Analysis

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Bracket_1 – DOE
Post-processing – Color map
Orientation 2 ( -39.6° ; 57.6°)
Orientation 1 ( 90° ; -162° )

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Work In Progress

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Statistics 4
In-Service

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Wind Speed Analysis
https://www.scilab.org/wind-speed-analysis

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Factor of failure in the gearbox
Helicopter Twin
The crash was caused by a crack in the gearbox
initiated by several factors:
• Very limited spalling not detected during inspection,
• Fatigue on rotating parts (defined by a maximum
operating time),
• Contact pressure on the bearings (directly linked to
the design and manufacturing by different suppliers),
• External damage occurring to the main gearbox,
• Contamination by impurity during the maintenance.

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93 Cars on Sale in the USA in 1993

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93 Cars on Sale in the USA in 1993

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The Wealth of Nations

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Hot dogs and buns Eating Contest

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Scilab Community
Some analytics
//Without Personal/Education/Research
Opened in 2018 (by country)

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Statistics
Some Theory

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https://archive.org/details/in.ernet.dli.2015.502684/page/n21

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I. Design of Experiment – History of classic
II. Computer experiment – Modern design
Some theory behind

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Screening variables
What are the factors (main effect) or combination
of factors (interactions) that affect the response
the most?
Statistical measures
•Analysis of variance (ANOVA)
•Signifiance testing
•Main effect
What sample of the population shall one study to
ensure catching all the effects ?

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Design space exploration, prediction & optimization
What sample of the population shall one study to ensure catching the system behavior
with low variance around zone of interest ?

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Design of Experiment purpose
Planning experiment in order to:
1/ Maximize learning
-Maximum experimental variance
-Control extraneous variance
-Minimum error variance
2/ Using minimum of resources

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Factorial design
RUNS X1 X2 X3
#1 1 1 1
#2 1 1 -1
#3 1 -1 1
#4 1 -1 -1
#5 -1 1 1
#6 -1 1 -1
#7 -1 -1 1
#8 -1 -1 -1
3 factors, at 2 levels  23 = 8 runs
Benefits:
Every effects (main and
interactions) are investigated
Drawback:
A lot of runs => Expensive

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Fractional factorial design
RUNS X1 X2 X3=X1.X2
#1 1 -1 -1
#2 -1 1 -1
#3 -1 -1 1
#4 1 1 1
3 factors, at 2 levels  23-1 = 4 runs
Benefits:
Less runs => Cheap
Drawback:
Effects are confounded (aliased)
with other main effects or
interactions effects

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Plackett-Burman
12 runs for 11 factors
Benefits:
Unbiased estimates of all main
effects in the smallest design
possible
Drawback:
Does not exists for every number of
factors
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
1 1 1 1 1 1 1 1 1 1 1
-1 1 -1 1 1 1 -1 -1 -1 1 -1
-1 -1 1 -1 1 1 1 -1 -1 -1 1
1 -1 -1 1 -1 1 1 1 -1 -1 -1
-1 1 -1 -1 1 -1 1 1 1 -1 -1
-1 -1 1 -1 -1 1 -1 1 1 1 -1
-1 -1 -1 1 -1 -1 1 -1 1 1 1
1 -1 -1 -1 1 -1 -1 1 -1 1 1
1 1 -1 -1 -1 1 -1 -1 1 -1 1
1 1 1 -1 -1 -1 1 -1 -1 1 -1
-1 1 1 1 -1 -1 -1 1 -1 -1 1
1 -1 1 1 1 -1 -1 -1 1 -1 -1

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Optimization objective
Response Surface Methodology (RSM)
1/ Screening
Keep only the important variable and update
range of the other
2/ Steepest ascent
First order Response Surface
Keep experimenting until no more improvement
New model
Loop until lack of fit (compared to pure error)
3/ Optimization
Second order Response Surface

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Computer Experiment

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Generalisation
Physic = Non-deterministic vs Simulation = Deterministic + numerical noise
i.e. Simulator produced the same answer if run twice for the same inputs
But output of computer simulator can exhibit bias because of omission in physics
or lack of accuracy in mathematical model implementation.
Randomization, replication, blocking have no more sense
Nature Extraneous variance Error variance
Physic Non Deterministic Randomization / replication Blocking
Simulation Deterministic Simulator bias

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Conclusion: Modern design
No random error & true response trend is unknown
+ Space filling to extract trend on unknown system

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Design of experiments techniques
•Static/System-Free: No
information on the system. Two
measure for uniformity
•Static/System-Aided: A little
information are taking into account
•Adaptive/Exploratory: Design
samples are set one after the
others for design space exploration
•Adaptive/Hybrid: System
information are taken into samples
positionning

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Workflow

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Regression based RSM - Polynomials
1st order
𝑦𝑖𝑗 = β0 + β1 𝑥1𝑖 + β2 𝑥2𝑖 + ⋯ + β 𝑞 𝑥 𝑞𝑖 + ε𝑖𝑗
2nd order
𝑦𝑖𝑗 = β0 + β 𝑘 𝑥 𝑘𝑖 + β 𝑢𝑣 𝑥 𝑢𝑖 𝑥 𝑣𝑖 + β 𝑤 𝑥 𝑤𝑖² + ε𝑖𝑗
Benefits:
•Ease of fitting and interpretability
•Can also be built up gradually, adding terms to reduce prediction error
Drawbacks:
•Polynomials can be non-robust when adding high order terms to fit given area
•Regression (vs. Interpolation) might be unreliable because of insufficient shape
capture

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Interpolation based RSM - RBF
𝑓 𝑥𝑖 = ෍
𝑘=1
𝑀
β 𝑘Φ(𝑥𝑖, 𝑥 𝑘) + p xi + ε𝑖
•Φ 𝑥, 𝑥 𝑘 = Φ(𝑟𝑘) kth basis function using kernel of center 𝑥 𝑘, 𝑟𝑘 = 𝑥 − 𝑥 𝑘
•Eventual relaxation polynomial p
Kernel Formula
Thin plate spline (TPS) Φ 𝑟 = 𝑟2
ln(𝑟)
Gaussian Φ 𝑟 = exp(−(𝑟/σ)²)
Inverse Multi-quadric
Φ 𝑟 = 1/ 1 + (𝑟/σ)²

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Response Surfaces
RBF interpolation
p1
p2
objective

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Linear estimation with lowest
variance (unbiased)
Benefits:
•More accurate for non linear
problem
•Statistical interpretation that
allows one to construct an
estimate of the potential error
in the interpolator
•Interpolant
Drawback:
•Hard to obtain
Interpolation based RSM - Kriging

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Modal based – POD
First 4 dominant POD modes for pressure DOE case pressure field projection on 4 modes
POD basis
4 modes
99.96% of the
global energy
Modesenergy

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Training data set 𝑥1, 𝑦1 , … , 𝑥𝑙, 𝑦𝑙
Goal in ε-SV regression is to:
•Find function f(x) that has at most ε deviation from the
targets for all the training data
•As flat as possible
Flatness means searching for small w
Machine learning – SVM/SVR

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For nonlinear function regression
Finding a hyperplan (linear separator) in a higher
dimensions space.
Projection kernels:
Machine learning – SVM/SVR
Kernels Formula
Homogeneous polynomial 𝑘 𝑥, 𝑥′
= 𝑥, 𝑥′ 𝑝
Hyperbolic tangent 𝑘 𝑥, 𝑥′
= tanh(ϑ + φ 𝑥, 𝑥′
)
Gaussian
𝑘 𝑥, 𝑥′
= exp(−
𝑥 − 𝑥′ ²
2σ²
)

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Machine learning – Neural network

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Layer of Neurons
Ouput layer is the last one. Others are hidden layers
Machine learning – Neural network

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Neural Networks function approximation

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Neural Networks function approximation

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Deep Learning
https://www.scilab.org/understanding-deep-learning-convolutional-neural-network

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Sources
•Data mining - DATA ANALYSIS AND STATISTICS – Openeering
https://www.scilab.org/tutorials/data-mining-%E2%80%93-tutorial
•Data Fitting – Interpolation & Approximation – Openeering
https://www.scilab.org/tutorials/data-fitting-%E2%80%93-tutorial
•Basic Statistics and Probability with SCILAB - Gilberto E. Urroz – 2001
https://www.scilab.org/basic-statistics-and-probability
•Classification – Logistic regression https://www.scilab.org/tutorials/machine-learning-
%E2%80%93-logistic-regression-tutorial
•Classification - Support Vector Machines
https://www.scilab.org/tutorials/machine-learning-%E2%80%93-classification-svm
•Neural Networks function approximation https://www.scilab.org/tutorials/machine-
learning-%E2%80%93-neural-network-function-approximation-tutorial

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Additional Resources
•ANN Toolbox Based on “Matrix techniques for ANN” Ryurick Hristev, 2000
•Neural Network Module http://atoms.scilab.org/toolboxes/neuralnetwork/2.0
•libSVM (Support Vector Machine) http://atoms.scilab.org/toolboxes/libsvm
This tool provides a simple interface to LIBSVM, a library for support
vector machines (http://www.csie.ntu.edu.tw/~cjlin/libsvm).
•NaN Toolbox https://atoms.scilab.org/toolboxes/nan
https://pub.ist.ac.at/~schloegl/matlab/NaN/
for data with and w/o MISSING VALUES encoded as NaN’s.
•Massive Open Online Course (MOOC) of Andrew NG, Stanford, on
Coursera https://www.coursera.org/learn/machine-learning
•For more content refer to the tutorials on scilab.io:
http://scilab.io/category/machine-learning/

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Data mining

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Data fitting

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Classification
Logistic regression

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Classification: Logistic Regression
Import your data
t=csvRead("data_classification.csv");
// Data generated randomly as follows
b0 = 10;
t = b0 * rand(100,2);
t = [t 0.5+0.5*sign(t(:,2)+t(:,1)-b0)];
b = 1;
flip = find(abs(t(:,2)+t(:,1)-b0)<b);
t(flip,$)=grand(length(t(flip,$)),1,"uin",0,1);

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Represent your data
Before representing your data, you need to split
them into two classes t0 and t1 as followed:
t0 = t(find(t(:,$)==0),:);
t1 = t(find(t(:,$)==1),:);
Then simply plot them:
clf(0);scf(0);
plot(t0(:,1),t0(:,2),'bo')
plot(t1(:,1),t1(:,2),'rx')

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Build a classification model
We want to build a classification model that estimates the probability that a new, incoming data belong to
the class 1.
First, we separate the data into features and results:
x = t(:, 1:$-1); y = t(:, $);
[m, n] = size(x);
Then, we add the intercept column to the feature matrix
// Add intercept term to x
x = [ones(m, 1) x];

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Build a classification model
The logistic regression hypothesis is defined as:
h(θ, x) = 1 / (1 + exp(−θTx) )
It’s value is the probability that the data with the features x belong to the class 1.
The Cost Function in logistic regression is
J = [−yT log(h) − (1−y)T log(1−h)]/m
where log is the “element-wise” logarithm, not a matrix logarithm.

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Gradient descent
If we use the gradient descent algorithm, then
the update rule for the θ is
θ → θ − α ∇J = θ − α xT (h − y) / m
// Initialize fitting parameters
theta = zeros(n + 1, 1);
// Learning rate and number of iterations
a = 0.01;
n_iter = 10000;
for iter = 1:n_iter do
z = x * theta;
h = ones(z) ./ (1+exp(-z));
theta = theta - a * x' *(h-y) / m;
J(iter) = (-y' * log(h) - (1-y)' * log(1-h))/m;
end

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Visualize the results
Now, the classification can be visualized:
// Display the result
disp(theta)
u = linspace(min(x(:,2)),max(x(:,2)));
clf(1);scf(1);
plot(t0(:,1),t0(:,2),'bo')
plot(t1(:,1),t1(:,2),'rx')
plot(u,-(theta(1)+theta(2)*u)/theta(3),'-g')

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Convergence of the model
The graph of the cost at each iteration is:
// Plot the convergence graph
clf(2);scf(2);
plot(1:n_iter, J');
xtitle('Convergence','Iterations','Cost')

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Classification
Support Vector Machines

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Classification: Support Vector Machines

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Model hypothesis

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Kernel
•0 : Linear
•1 : Polynomial
•2 : Radial Basis Function
•3 : Sigmoid
SVM type
• 0 : C-SVC (class separation)
• 1 : Nu-SVC (nu-classification)
• 2 : One class SVM
• 3 : Epsilon-SVR (regression)
• 4 : Nu-SVR (regression)
Model optimization In our example
with
•nSV : Number of support vector
•αi.li = sv_coef
•xi : Support vectors SVs
•ρ = – b

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