Construction of entropy-based prior and posterior probability distributions with partial information for fatigue damage prognostics
View Patent ↗A method for predicting fatigue crack growth in materials includes providing a prior distribution obtained using response measures from one or more target components using a fatigue crack growth model as a constraint function, receiving new crack length measurements, providing a posterior distribution obtained using the new crack length measurements, and sampling the posterior distribution to obtain crack length measurement predictions.
1. A computer-implemented method for predicting fatigue crack growth in materials, comprising:
providing, via a processor, a prior distribution obtained using response measures from one or more target components using a fatigue crack growth model as a constraint function;
receiving, via the processor, new crack length measurements;
generating, via the processor, a posterior distribution based on the new crack length measurements;
sampling, via the processor, the posterior distribution for generating crack length measurement predictions,
wherein the prior distribution is expressed as p 0 (θ)∝exp{λM(θ)}, wherein M is the fatigue crack growth model, θ is a fatigue crack growth model parameter, M(θ) is the output of the fatigue crack growth model, and λ is a Lagrange multiplier, and the constraint function is expressed as E p 0 (θ) [M(θ)]= α , wherein α is a mean of the response measures from one or more target components; and
the posterior distribution is expressed as
p
(
θ
)
∝
exp
[
λ
M
(
θ
)
]
exp
{
-
1
2
∑
i
=
1
n
[
a
i
-
M
i
(
θ
)
σ
ɛ
]
2
}
,
where a i represents new crack length measurements associated with the one or more target components, σ ε is a standard deviation of Gaussian likelihood, and n is a total number of new crack length measurements; and
wherein the Lagrange multiplier λ is obtained by solving, via the processor,
∂
ln
∫
λ
M
(
θ
)
ⅆ
θ
∂
λ
=
a
_
;
and
predicting, via the processor, fatigue crack growth in the material based on the posterior distribution.
2. The computer-implemented method of claim 1 , wherein the posterior distribution is sampled using a Markov-chain Monte-Carlo simulation.
3. The computer-implemented method of claim 1 , wherein σ ε =√{square root over (σ ε 1 2 +σ ε 2 2 )}, wherein σ ε 1 is a standard deviation associated a statistical uncertainty of the fatigue crack growth model M, and σ ε 2 is a standard deviation associated with a measurement uncertainty.
4. The computer-implemented method of claim 1 , further comprising updating the posterior distribution as new crack length measurements are received.
5. The computer-implemented method of claim 1 , wherein the fatigue crack growth model is Paris' model, expressed as
ⅆ
a
ⅆ
N
=
c
(
Δ
K
)
m
,
wherein a is a crack size, N is a number of applied cyclic loads,
Δ
K
=
π
a
Δ
σ
g
(
a
w
)
is a range of a stress intensity factor during one cycle, Δσ is a range of the applied stress during one cycle, g(a/w) is a geometric correction term, w is a width of the specimen, and c and m are model parameters.
6. The computer-implemented method of claim 1 , wherein the fatigue crack growth model is Forman's model, expressed as
ⅆ
a
ⅆ
N
=
c
(
Δ
K
)
m
(
1
-
R
)
K
cr
-
Δ
K
,
wherein a is a crack size, N is a number of applied cyclic loads,
Δ
K
=
π
a
Δ
σ
g
(
a
w
)
is a range of a stress intensity factor during one cycle, Δσ is a range of the applied stress during one cycle, g(a/w) is a geometric correction term, w is a width of the specimen, K cr is a fracture toughness of the material, R is a load ratio, and c and m are model parameters.
7. The computer-implemented method of claim 1 , wherein the fatigue crack growth model is McEvily's model, expressed as
ⅆ
a
ⅆ
N
=
c
(
Δ
K
-
Δ
K
th
)
2
[
1
-
Δ
K
K
cr
-
K
max
]
,
wherein a is a crack size, N is a number of applied cyclic loads,
Δ
K
=
π
a
Δ
σ
g
(
a
w
)
is a range of a stress intensity factor during one cycle, Δσ is a range of the applied stress during one cycle, g(a/w) is a geometric correction term, w is a width of the specimen, K cr is a fracture toughness of the material, ΔK th is a threshold stress intensity range below which cracks either propagate at an extremely low rate or do not propagate at all, K max is a maximum stress intensity in one cyclic load, and c is a model parameter.
8. The computer-implemented method of claim 1 , wherein the posterior distribution is sampled using a Markov-chain Monte-Carlo simulation.
9. A computer-implemented method for predicting fatigue crack growth in materials, comprising:
providing, via a processor, a prior distribution subject to a constraint function obtained using response measures from one or more target components, wherein the prior distribution is expressed as p 0 (θ)∝exp{λM(θ)}, wherein M is the fatigue crack growth model, θ is a fatigue crack growth model parameter, M(θ) is the output of the fatigue crack growth model, and λ is a Lagrange multiplier is obtained by solving
∂
ln
∫
λ
M
(
θ
)
ⅆ
θ
∂
λ
=
a
_
,
wherein α is a mean of the response measures from one or more target components, and a constraint function is expressed as E p 0 (θ) [M(θ)]= α ;
generating, via the processor, a posterior distribution based on the prior distribution function, and
predicting, via the processor, fatigue crack growth based on the posterior distribution.
10. The computer-implemented method of claim 9 , further comprising:
receiving new crack length measurements;
providing a posterior distribution obtained using the new crack length measurements, wherein the posterior distribution is expressed as
p
(
θ
)
∝
exp
[
λ
M
(
θ
)
]
exp
{
-
1
2
∑
i
=
1
n
[
a
i
-
M
i
(
θ
)
σ
ɛ
]
2
}
,
wherein a i represents new crack length measurements associated with the one or more target components, σ ε =√{square root over (σ ε 1 2 +σ ε 2 2 )} is a standard deviation of Gaussian likelihood wherein σ ε 1 is a standard deviation associated a statistical uncertainty of the fatigue crack growth model M and σ ε 2 is a standard deviation associated with a measurement uncertainty, and n is a total number of new crack length measurements; and
updating the posterior distribution as new crack length measurements are received.
11. The computer-implemented method of claim 10 , further comprising sampling the posterior distribution to obtain crack length measurement predictions, wherein the posterior distribution is sampled using a Markov-chain Monte-Carlo simulation.
12. A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executed by the computer to perform the method steps for predicting fatigue crack growth in materials, the method comprising the steps of:
providing a prior distribution obtained using response measures from one or more target components using a fatigue crack growth model as a constraint function;
receiving new crack length measurements;
generating a posterior distribution based on the new crack length measurements;
sampling the posterior distribution for generating crack length measurement predictions,
wherein the prior distribution is expressed as p 0 (θ)∝exp{λM(θ)}, wherein M is the fatigue crack growth model, θ is a fatigue crack growth model parameter, M(θ) is the output of the fatigue crack growth model, and λ is a Lagrange multiplier, and the constraint function is expressed as E p 0 (θ) [M(θ)]= α , wherein α is a mean of the response measures from one or more target components; and
the posterior distribution is expressed as
p
(
θ
)
∝
exp
[
λ
M
(
θ
)
]
exp
{
-
1
2
∑
i
=
1
n
[
a
i
-
M
i
(
θ
)
σ
ɛ
]
2
}
,
where a i represents new crack length measurements associated with the one or more target components, σ ε is a standard deviation of Gaussian likelihood, and n is a total number of new crack length measurements; and
wherein the Lagrange multiplier λ is obtained by solving
∂
ln
∫
λ
M
(
θ
)
ⅆ
θ
∂
λ
=
a
_
;
and
predicting fatigue crack growth based on the posterior distribution.
13. The computer readable program storage device of claim 12 , wherein the posterior distribution is sampled using a Markov-chain Monte-Carlo simulation.
14. The computer readable program storage device of claim 12 , wherein σ ε =√{square root over (σ ε 1 2 +σ ε 2 2 )}, wherein σ ε 1 is a standard deviation associated a statistical uncertainty of the fatigue crack growth model M, and σ ε 2 is a standard deviation associated with a measurement uncertainty.
15. The computer readable program storage device of claim 12 , the method further comprising updating the posterior distribution as new crack length measurements are received.
16. The computer readable program storage device of claim 12 , wherein the fatigue crack growth model is Paris' model, expressed as
ⅆ
a
ⅆ
N
=
c
(
Δ
K
)
m
,
wherein a is a crack size, N is a number of applied cyclic loads,
Δ
K
=
π
a
Δ
σ
g
(
a
w
)
is a range of a stress intensity factor during one cycle, Δσ is a range of the applied stress during one cycle, g(a/w) is a geometric correction term, w is a width of the specimen, and c and m are model parameters.
17. The computer readable program storage device of claim 12 , wherein the fatigue crack growth model is Forman's model, expressed as
ⅆ
a
ⅆ
N
=
c
(
Δ
K
)
m
(
1
-
R
)
K
cr
-
Δ
K
,
wherein a is a crack size, N is a number of applied cyclic loads,
Δ
K
=
π
a
Δ
σ
g
(
a
w
)
is a range of a stress intensity factor during one cycle, Δσ is a range of the applied stress during one cycle, g(a/w) is a geometric correction term, w is a width of the specimen, K cr is a fracture toughness of the material, R is a load ratio, and c and m are model parameters.
18. The computer readable program storage device of claim 12 , wherein the fatigue crack growth model is McEvily's model, expressed as
ⅆ
a
ⅆ
N
=
c
(
Δ
K
-
Δ
K
th
)
2
[
1
-
Δ
K
K
cr
-
K
max
]
,
wherein a is a crack size, N is a number of applied cyclic loads,
Δ
K
=
π
a
Δ
σ
g
(
a
w
)
is a range of a stress intensity factor during one cycle, Δσ is a range of the applied stress during one cycle, g(a/w) is a geometric correction term, w is a width of the specimen, K cr is a fracture toughness of the material, ΔK th is a threshold stress intensity range below which cracks either propagate at an extremely low rate or do not propagate at all, K max is a maximum stress intensity in one cyclic load, and c is a model parameter.