Introduction to Monte Carlo method and reliability analysis · 2019. 4. 8. · Introduction A brief...
38
7 th COSSAN TRAINING COURSE - 08-10 April 2019 Introduction to Monte Carlo method and reliability analysis Edoardo Patelli, Matteo Broggi E: [email protected] W: www.cossan.co.uk T: +44 01517944079
Transcript of Introduction to Monte Carlo method and reliability analysis · 2019. 4. 8. · Introduction A brief...
7th COSSAN TRAINING COURSE - 08-10 April 2019
Introduction to Monte Carlo methodand reliability analysis
Edoardo Patelli, Matteo BroggiE: [email protected] 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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 2 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 3 / 30
Introduction A brief overview
Monte Carlo applications
Solution of integrals,differential equation,complex systemsetc...Simulation randomeventsCryptography,Decision-MakingGames
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 4 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 5 / 30
Introduction A brief overview
Monte Carlo methodOrigins
1777 Comte de Buffon - earliestdocumented use of randomsampling
P(needle intersects the grid) =2 ∗ Lπt
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 6 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 6 / 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 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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 9 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 10 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 11 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 12 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 13 / 30
Introduction Estimation of 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 14 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 15 / 30
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
NE.Patelli M.Broggi COSSAN Training Course 8 April 2019 16 / 30
Reliability Analysis Overview
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 17 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 18 / 30
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
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
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 20 / 30
Reliability Analysis Performance function
Performance FunctionDemand - Capacity
Identify capacityIdentify demand
Can be random variables, orfunctions of random variables
Probability of failure: P(Demand > Capacity)
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 21 / 30
Reliability Analysis Performance function
Performance FunctionExample
Failure: exceedance of the yieldstressg(θ) = σmax(θ)− σ(θ)
θ: structural and loaduncertainty vector
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 22 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 23 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 24 / 30
Hands-on session Gettting started
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 25 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 27 / 30
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
E.Patelli M.Broggi COSSAN Training Course 8 April 2019 30 / 30
