Uncertainty modelling using software FReET
Transcript of Uncertainty modelling using software FReET
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D. Novak, M. Vorechovsky, R. Rusina
Uncertainty modelling using software FReET
Brno University of Technology Brno, Czech Republic
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Outline
• Introduction• Methods and main features• Software FReET• Selected applications
– Probabilistic analyses of concrete structures– Statistical size effect studies– Identification of computational model
parameters
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Methods and main features
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• Small-sample simulation of Monte Carlo type
• Imposing statistical correlation
• Sensitivity analysis
• Reliability analysis
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Latin hypercube sampling
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• The range (0; 1) of PDF Φ(Yi) of
each random variable Yi is divided
into N non-overlapping intervals of
equal probability 1/N (McKay et al.
1979. Iman & Conover 1980, Iman
& Shortencarier 1984).
• The centroids are selected
randomly based on random
permutations of integers.
• Every interval of each variable is
used only once during the
simulation process.
variable
sim
ula
tion
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LHS: Step 1 - simulation
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,i kx ( )b
Sim
a
N x f x dx= ⋅ ⋅∫
Huntington & Lyrintzis(1998)
• Mean value: accurately
• Stand. deviation: significant improvement
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LHS: Step 2 – imposing statistical correlation
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x1 y 1 … z1
x2 y 2 … z2
x3 y 3 … z3
x4 y 4 … z4
x5 y 5 … z5
x6 y 6 … z6
x7 y 7 … z7
x8 y 8 … z8
… … … …
xNSim yNSim … zNSim
variable
sim
ula
tion
• Simulated annealing: Probability to escape from local minima
• Cooling - decreasing of system excitation
• Boltzmann PDF, energetic analogy
( ) b
E
k TrP E e
− ∆ ⋅ ≈
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LHS: Step 2 – imposing statistical correlation
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x1 y 1 … z1
x2 y 2 … z2
x3 y 3 … z3
x4 y 4 … z4
x5 y 5 … z5
x6 y 6 … z6
x7 y 7 … z7
x8 y 8 … z8
… … … …
xNSim yNSim … zNSim
variable
sim
ula
tion
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Sensitivity analysis
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Nonparametric rank-order correlation between
input variables ane output response variable
• Kendall tau
• Spearman
• Robust - uses only orders
• Additional result of LHS simulation, no extra effort
• Bigger correlation coefficient = high sensitivity
• Relative measure of sensitivity (-1, 1)
R1x1,1
……
……
……
R, Nx1,N
OUTPUTINPUT
p1q1,1
……
……
……
p Nq1,N
OUTPUTINPUT
( ) Nj,pqττ jjii ,,2,1, K==
( )( )11
61 1
2
+−−=
∑=
nnn
dr
n
ii
s
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Reliability analysis
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• Simplified – rough estimates,
as constrained by extremally small
number of simulations (10-100)!
• Cornell safety index
• Curve fitting
• FORM, importance sampling
response surface…
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FREET
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Response/Limit state function• Closed form (direct) using implemented
Equation Editor (simple problems)• Numerical (indirect) using user-defined DLL function prepared practically in..any programming language (C++, Fortran, Delphi, etc.)• General interface to third-parties software using user-defined *.BAT or *.EXE
Probabilistic techniques• Crude Monte Carlo simulation• Latin Hypercube Sampling (3 types)• First Order Reliability Method (FORM)• Curve fitting• Simulated Annealing• Bayesian updating
http://www.freet.cz
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FREET
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Stochastic model (inputs)• Friendly Graphical User Environment (GUE)• 30 probability distribution functions (PDF),mostly 2-parametric, some 3-parametric,two 4-parametric (Beta PDF and normalPDF with Weibullian left tail)• Unified description of random variablesoptionally by statistical moments orparameters or a combination of momentsand parameters• PDF calculator• Statistical correlation (also weightingoption)• Categories and comparative values forPDFs• Basic random variables visualization,including statistical correlation in bothCartesian and parallel coordinates
http://www.freet.cz
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FREET – Stochastic model
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FREET – Correlation, simulation
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FREET – Assessment
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Software tools: SARA Studio
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+
Probabilistic software FReET
http://www.freet.cz
Software for nonlinear fracture
mechanics analysis ATENA
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ATENA
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• Well-balanced approach for practical applications of advanced FEM in civil engineering
• Numerical core – state-of-art background• User friendly Graphical user environment
visualization + interaction
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Material models for concrete: ATENA software
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Numerical core – advanced nonlinear material modelsconcrete in tension• tensile cracks• post-peak behavior• smeared crack approach• crack band method• fracture energy• fixed or rotated cracks• crack localization• size-effect is captured
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Software tools: SARA Studio
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Probabilistis analyses of concrete structures – Coll de Isarco viaduct
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0
5
10
15
20
25
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0 25 50 75 100 125 150 175 200 225 250
Line Load [kN/m]
Safe
ty in
dex
ββ ββ
CoV = 0.15
CoV = 0.12
CoV = 0.10
CoV = 0.075
CoV = 0.05
Eurocode
Design Load
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Probabilistis analyses of concrete structures – Coll de Isarco viaduct
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Random variable description Symbol Units Mean value COVDistribution type Reference Concrete grade B500Modulus of elasticity Ec GPa 36.95 0.15 Lognormal 6
Poisson's ratio µ - 0.2 0.05 Lognormal EstimationTensile strength ft MPa 3.257 0.18 Weibull 6
Compressive strength fc MPa 42.5 0.10 Lognormal 6,7
Specific fracture energy Gf N/m 81.43 0.20 Weibull 8
Uniaxial compressive strain εc - 0.0023 0.15 Lognormal 6
Reduction of strength cRed - 0.8 0.06 Rectangular EstimationCritical comp displacement wd m 0.0005 0.10 Lognormal EstimationSpecific material weight ρ MN/m3 0.023 0.10 Normal 9
Prestressing strandsModulus of elasticity Es GPa 200.0 0.03 Lognormal 10
Yield stress fy MPa 1600.0 0.07 Lognormal 10
Prestressing force F MN 21.85 0.04 Normal 9
Area of strands As m2 0.0237 0.001 Normal 9
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Probabilistis analyses of concrete structures – soil structure interaction
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• Stability of concrete tunnel tube in
complicated geological conditions
• Influence of spatial variability of
Young modulus and material
constants of Drucker-Prager
criterion (based on cohesion and
angle of internal friction)
• Analyzed part 50 x 60m, diametr of
tunnel 11m, wall thickness 0.5m
• Plain strain state, 5000 finite
elements
5.657E-026.300E-027.350E-028.400E-029.450E-021.050E-011.155E-011.260E-011.365E-011.470E-011.575E-011.680E-011.718E-01
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Statistical size effect studies
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Malpasset dam (failed 1959)
Calculation for different sizes, microplane M4 model
Bažant, Vořechovský, Novák - Icossar 2005
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Statistical size effect studies
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Statistical size effect studies
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5045
050
0
300
300
300
300
1000 92
575
250
225
25 125
110
15BN100 BN50 BN25 BN12
400
600
800
1000
1200
1400
1600
0.0 0.2 0.4 0.6 0.8 1.0
Size (depth of beam) [m]
Nom
inal
stre
ngth
[kN
/m2]
ExperimentsStatistical simulationACIEurocode 2CollinsBažant
Shear failure, verification of design formulas
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Identification of computational model parameters
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Stochastic calculation (LHS) – training set for calibration of synaptic weights and biases
Materialmodelparameters
Structural response
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Identification of computational model parameters – shear wall test
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VariableSymb
olUnit
Mean value
COV
Modulus of elasticity E GPa 30 0.10
Tensile strength ft MPa 2.5 0.10
Compressive strength fc MPa 30 0.10
Fracture energy GF N/m 75 0.20
Compressive strain εc - 0.0025 0.20
Max. comp. displacement
wd m 0.003 0.30
Bilinear diagram of steel for smeared reinforcement
x1 m 0.0027 0.10
fx1 kN 574 0.10
x2 m 0.015 0.10
fx2 kN 764 0.10
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Identification of computational model parameters
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Identification of computational model parameters
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0
200
400
600
800
1000
0 2 4 6 8 10 12 14Horizontal displacement [mm]
Hor
izon
tal f
orce
[kN
] .
S2 Experiment
ATENA 6 parameters identified
ATENA 10 parameters identified
DLNNET 6 par. 10 par.
E [MPa] 29,9 33,0
ft [MPa] 2,47 2,47
fc [MPa] 34,51 35,3
Gf [MN/m] 75,0 77,85
ec [-] 2,51E-03 2,57E-03
wd [m] 3,00E-03 3,10E-03
x1 2,72E-03 2,74E-03
fx1 566,9 570,7
x2 1,50E-02 1,47E-02
fx2 764 768,8
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Conclusions
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• FREET - software for statistical, sensitivity andreliability analysis
• Suitable for analysis of computationally intensiveproblems (eg. continuum mechanics, FEM)
• Software development:• 1) Stand alone module - definition of reliability problem
(user-defined response/limit state function)
• 2) Integration with software ATENA - nonlinear fracturemechanics of reinforced concrete structures (ČervenkaConsulting)
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Thank You for Your attention!
http://www.freet.cz
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