7th COSSAN TRAINING COURSE - 08-10 April 2019

Introduction to Monte Carlo methodand reliability analysis

Edoardo Patelli, Matteo BroggiE: info@cossan.co.uk W: www.cossan.co.uk T: +44 01517944079

Introduction A brief overview

Outline

1 IntroductionA brief overviewEvaluation of DefiniteIntegralsEstimation of π

2 Reliability AnalysisOverviewPerformance function

3 Hands-on sessionCompute πCantilever Beam

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Introduction A brief overview

Monte Carlo method

Computation technique based onrandom numbersNumerical experiment by generating arandom sequence of number withprescribed probability distribution.Collecting quantity of interest

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Introduction A brief overview

Monte Carlo applications

Solution of integrals,differential equation,complex systemsetc...Simulation randomeventsCryptography,Decision-MakingGames

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Introduction A brief overview

Monte Carlo methodMain components

Probability distribution functions(describing the model)Random number generatorSampling rules (how to sample from PDFs)Error estimatorVariance reduction techniqueUse of High Performance Computing

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Introduction A brief overview

Monte Carlo methodOrigins

1777 Comte de Buffon - earliestdocumented use of randomsampling

P(needle intersects the grid) =2 ∗ Lπt

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Introduction A brief overview

Monte Carlo methodOrigins

1777 Comte de Buffon - earliestdocumented use of randomsampling

P(needle intersects the grid) =2 ∗ Lπt

1786 Laplace suggested toestimate π by random sampling

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Introduction A brief overview

Buffon’s experimentMonte Carlo simulation

1 Sample an u1 ∼ U[0,1) and u2 U[0,1)2 Calculate distance from a line:

d = u1 ∗ t3 Calculate angle between needle’s axis

and the normal to the linesφ = u2 ∗ π/2

4 if d ≤ Lcosφ the needle intercepts aline (update counter Ns = Ns + 1)

5 Repeat procedure N times6 Estimate probability intersection

Pi =2 ∗ Lπt

= limN→∞Ns

N

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Introduction A brief overview

Buffon’s experimentMonte Carlo simulation

1 Sample an u1 ∼ U[0,1) and u2 U[0,1)2 Calculate distance from a line:

d = u1 ∗ t3 Calculate angle between needle’s axis

and the normal to the linesφ = u2 ∗ π/2

4 if d ≤ Lcosφ the needle intercepts aline (update counter Ns = Ns + 1)

5 Repeat procedure N times6 Estimate probability intersection

Pi =2 ∗ Lπt

= limN→∞Ns

N

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 7 / 30

Introduction A brief overview

Buffon’s experimentMonte Carlo simulation

1 Sample an u1 ∼ U[0,1) and u2 U[0,1)2 Calculate distance from a line:

d = u1 ∗ t3 Calculate angle between needle’s axis

and the normal to the linesφ = u2 ∗ π/2

4 if d ≤ Lcosφ the needle intercepts aline (update counter Ns = Ns + 1)

5 Repeat procedure N times6 Estimate probability intersection

Pi =2 ∗ Lπt

= limN→∞Ns

N

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 7 / 30

Introduction A brief overview

Buffon’s experimentMonte Carlo simulation

1 Sample an u1 ∼ U[0,1) and u2 U[0,1)2 Calculate distance from a line:

d = u1 ∗ t3 Calculate angle between needle’s axis

and the normal to the linesφ = u2 ∗ π/2

4 if d ≤ Lcosφ the needle intercepts aline (update counter Ns = Ns + 1)

5 Repeat procedure N times6 Estimate probability intersection

Pi =2 ∗ Lπt

= limN→∞Ns

N

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 7 / 30

Introduction A brief overview

Buffon’s experimentMonte Carlo simulation

1 Sample an u1 ∼ U[0,1) and u2 U[0,1)2 Calculate distance from a line:

d = u1 ∗ t3 Calculate angle between needle’s axis

and the normal to the linesφ = u2 ∗ π/2

4 if d ≤ Lcosφ the needle intercepts aline (update counter Ns = Ns + 1)

5 Repeat procedure N times6 Estimate probability intersection

Pi =2 ∗ Lπt

= limN→∞Ns

N

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 7 / 30

Introduction A brief overview

Buffon’s experimentMonte Carlo simulation

1 Sample an u1 ∼ U[0,1) and u2 U[0,1)2 Calculate distance from a line:

d = u1 ∗ t3 Calculate angle between needle’s axis

and the normal to the linesφ = u2 ∗ π/2

4 if d ≤ Lcosφ the needle intercepts aline (update counter Ns = Ns + 1)

5 Repeat procedure N times6 Estimate probability intersection

Pi =2 ∗ Lπt

= limN→∞Ns

N

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 7 / 30

Introduction Evaluation of Definite Integrals

Outline

1 IntroductionA brief overviewEvaluation of DefiniteIntegralsEstimation of π

2 Reliability AnalysisOverviewPerformance function

3 Hands-on sessionCompute πCantilever Beam

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 8 / 30

Introduction Evaluation of Definite Integrals

Definite Integrals

G =

∫F

g(x)dx =

∫· · ·∫F

g(x1, . . . , xn)dx1 . . . dxn

Analytical solutionNumerical quadratureMonte Carlo estimation

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Introduction Evaluation of Definite Integrals

Evaluation of Definite Integrals

G =

∫g(x)f (x)dx

x can be seen as a random variablef (x) has characteristic of a probability density functiong(x) is also a random variable

E [g(x)] =∫

g(x)f (x)dx = GVar [g(x)] = E [g2(x)]−G2

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Introduction Evaluation of Definite Integrals

Monte Carlo darts method

G =

∫g(x)f (x)dx

1 Generate sample N points (xi)from f (x)

2 Evaluate function g(xi) (i.e. thescore of the i-th thrown)

3 Computed expected prise

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Introduction Evaluation of Definite Integrals

Monte Carlo darts method

G =

∫g(x)f (x)dx

Main componentsf (x)dx probability to hit a pointg(x) the scoreGN = 1

N

∑Ni=1 g(xi) Average score

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Introduction Evaluation of Definite Integrals

Estimation area of a circle

C =

∫∫F

f (x1, x2)dx1dx2 = π ∗ r2

f (x1, x2) = uniform distribution; F : x21 + x2

2 ≤ rThe integral can be rewritten as:

C =

∫∫H(x1, x2) · f (x1, x2)dx1dx2

H(x1, x2) =

{1 if x ∈ C; x2

1 + x22 ≤ r

0 otherwiseWe can use the dart game to estimate the area of the circle

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Introduction Estimation of π

Outline

1 IntroductionA brief overviewEvaluation of DefiniteIntegralsEstimation of π

2 Reliability AnalysisOverviewPerformance function

3 Hands-on sessionCompute πCantilever Beam

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Introduction Estimation of π

Estimation of π

Area Circle: π ∗ r2

Area of the square:2 ∗ r2

Ratio of the areas:π ∗ r2

4 ∗ r2 =π

4Tool: dart game

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Introduction Estimation of π

Estimation of πProcedure

1 Sample coordinate of a point

xi ∼ U [0, r ], yi ∼ U [0, r ]

2 Check if the sample is inside the circle ofradius r and update counterif x2

i + y2i < r then Nr = Nr + 1

3 Repeat steps 1-2 for N samples4 Compute π

π = 4 · Nr

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Reliability Analysis Overview

Outline

1 IntroductionA brief overviewEvaluation of DefiniteIntegralsEstimation of π

2 Reliability AnalysisOverviewPerformance function

3 Hands-on sessionCompute πCantilever Beam

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Reliability Analysis Overview

Reliability Analysis

The ability of a system or component to perform its requiredfunctions under stated conditions for a specified period of time.

Reliability is a probability

R(t) = Pr{T > t} =∫ ∞

tf (x)dx

where f (x) is the failure probability density function and t is thelength of the period of time

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Reliability Analysis Performance function

Outline

1 IntroductionA brief overviewEvaluation of DefiniteIntegralsEstimation of π

2 Reliability AnalysisOverviewPerformance function

3 Hands-on sessionCompute πCantilever Beam

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 19 / 30

Reliability Analysis Performance function

Performance function

Function of input quantities thatdescribe the status of thesystem: g(X1, · · · ,Xn)

Failure domain: g ≤ 0Safe domain: g ≥ 0Limit State Function:g(X1, · · · ,Xn) = 0(N − 1 dimension surface)

X1

X2

pdf countour

g(X1,X2) > 0

g(X1,X2) < 0

g(X1,X2) = 0

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Reliability Analysis Performance function

Performance function

Function of input quantities thatdescribe the status of thesystem: g(X1, · · · ,Xn)

Failure domain: g ≤ 0Safe domain: g ≥ 0Limit State Function:g(X1, · · · ,Xn) = 0(N − 1 dimension surface)

X1

X2

pdf countour

g(X1,X2) > 0

g(X1,X2) < 0

g(X1,X2) = 0

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 20 / 30

Reliability Analysis Performance function

Performance function

Function of input quantities thatdescribe the status of thesystem: g(X1, · · · ,Xn)

Failure domain: g ≤ 0Safe domain: g ≥ 0Limit State Function:g(X1, · · · ,Xn) = 0(N − 1 dimension surface)

X1

X2

pdf countour

g(X1,X2) > 0

g(X1,X2) < 0

g(X1,X2) = 0

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Reliability Analysis Performance function

Performance FunctionDemand - Capacity

Identify capacityIdentify demand

Can be random variables, orfunctions of random variables

Probability of failure: P(Demand > Capacity)

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Reliability Analysis Performance function

Performance FunctionExample

Failure: exceedance of the yieldstressg(θ) = σmax(θ)− σ(θ)

θ: structural and loaduncertainty vector

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Reliability Analysis Estimation of probability of failure

Failure quantificationFailure (F):Demand σ(θ) ≥ σmax(θ) CapacitySafe (S):Demand σ(θ) < σmax(θ) Capacity∫

FfX(x) dx =

∫IF(x) fX(x) dx

where:

IF(X) ={

0 ⇐⇒ X ∈ S1 ⇐⇒ X ∈ F

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Reliability Analysis Estimation of probability of failure

Failure quantification (”dart” game)f (x)dx probability to hit a point (generaterealisations of random variables)IF(x) the prize (evaluate the performance function)

Estimate: direct Monte Carlo simulation

Pf =

∫IF(x) fX(x) dx ≈ 1

N

N∑k=1

IF(X(k))

to meet specified accuracy: N ∝ 1Pf

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Hands-on session Gettting started

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Hands-on session Compute π

Outline

1 IntroductionA brief overviewEvaluation of DefiniteIntegralsEstimation of π

2 Reliability AnalysisOverviewPerformance function

3 Hands-on sessionCompute πCantilever Beam

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 26 / 30

Hands-on session Compute π

Estimation of πUsing COSSAN-X

1 Input: Define 1 parameter representing r2 Input: Define 2 Random variables representing random

coordinate of points3 Evaluator: MIO connector that computes d =

√x2

i + y2i

4 Performance Function: Defined as Capacity d and Demand r(Reliability analysis collects values of the performancefunction when Capacity > Demand)

5 Analysis: Perform the reliability analysis6 Result: pf =

π4

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Hands-on session Compute π

Buffon’s experimentUsing COSSAN-X

1 Define 1 random variable describing theangle φ

2 Define 1 random variable describingdistance d

3 Define 2 parameters (t and L)4 Define a function computing Lcosφ5 Define a performance function:

Demand (d) Capacity (t)6 Perform Monte Carlo simulation

(Reliability Analysis)E.Patelli M.Broggi COSSAN Training Course 8 April 2019 28 / 30

Hands-on session Cantilever Beam

Outline

1 IntroductionA brief overviewEvaluation of DefiniteIntegralsEstimation of π

2 Reliability AnalysisOverviewPerformance function

3 Hands-on sessionCompute πCantilever Beam

E.Patelli M.Broggi COSSAN Training Course 8 April 2019 29 / 30

Hands-on session Cantilever Beam

Reliability Analysis of a Cantilever BeamL,H, ρ,E random variablesDisplacement:

w =ρgBHL4

8EI+

FL3

3EI

I =BH3

12

F

H

BL

Failure: excidence maximum displacement wmax = 0.01

Define a probabilistic model and perform reliability analysis

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