Computer-implemented method for the probabilistic estimation of a probability of failure of a component, a data processing system, a computer program product and a computer-readable storage medium
A computer-implemented method for probabilistic quantification of probability of failure of a component, especially a gas turbine component, which during operation is subjected to cyclic stress, wherein the component is divided virtually in one or more domains. The method includes: providing or determining for at least one domain, a domain probability density function for crack initiation and providing or determining for the considered domains a domain probability density function for subsequent crack propagation induced failure. Determining for each considered domain a combined domain cumulative distribution function for failure or its probability density function is done by convoluting either both the considered domain probability density functions for crack initiation induced failure and the respective domain probability density function for subsequent crack propagation induced failure, or their integral function. Alternatively, numerical methods for said component failure probabilities include domain-based Monte-Carlo schemes.
1. A computer-implemented method for probabilistic estimation of probability of failure PoF(n) of a component, especially a gas turbine component, which during operation is subjected to cyclic stress, wherein the component is divided virtually in multiple domains, the method comprising:
a. providing or determining for at least one domain, preferably for each domain, a domain probability density function for crack initiation PDF CI i (n) and providing or determining for the considered domains a domain probability density function for subsequent crack propagation induced failure PDF CPF i (n),
b.
b1) determining for each considered domain a combined domain probability density function for failure PDF Fail i (n) according to
PDF i Fail ( n )=PDF i CI ( n )×PDF i CPF ( n ),
wherein X designates a convolution operator between the two PDFs, and b2) determining for each considered domain a combined domain cumulative distribution function for failure CDF Fail i (n) based on the respective combined domain probability density function for failure PDF Fail i (n), wherein the CDF is a cumulative distribution function of the PDF,
OR
b3) determining for each considered domain a domain cumulative distribution function for crack initiation CDF CI i (n) and cumulative distribution function for subsequent crack propagation induced failure CDF CPF i (n) based on the respective domain probability density function for crack initiation PDF CI i (n) and subsequent crack propagation induced failure PDF CPF i (n), and b4) determining for each considered domain a combined domain probability density function for CDF Fail i (n) according to
CDF
i
Fail
(
n
)
=
d
dn
(
CDF
i
CI
(
n
)
XCDF
i
CPF
(
n
)
)
,
wherein—CDF CI i (n) designates the domain cumulative distribution function for crack initiation, —CDF CPF i (n) designates the domain cumulative distribution function for subsequent crack propagation induced failure, —X designates the convolution operator between the two CDFs,
c. determining ( 110 ) the total probability of failure PoF(n) of the component according to
PoF( n )=CDF Fail =1−Π i=1 N [1−CDF Fail i ( n )],
wherein a step of determining crack propagation data either as each considered domain a domain probability density function for subsequent crack propagation induced failure PDF CPF i (n) or as crack propagation cycle N ij CPF considers at least one of crack growth and failure relevant material properties comprising fatigue growth rate FCGR, creep crack growth rate CCGR, crack corrosion pitting, erosion rate-fracture toughness K1 c , ΔK threshold , or tensile properties or any combination thereof,
with a failure criterion which is based on at least one of stress intensity factor K exceeding the fracture toughness K1 c , ΔK exceeding, stress intensity factor range ΔK exceeding a fatigue crack growth stress intensity range threshold ΔK threshold , a crack length exceeding a critical crack length or exceeding a safe region of a two parameter failure assessment diagram (FAD) based on properties listed above, especially based on British R6 criteria which are based on two parameters load ratio Lr and a fracture ratio Kr, and
wherein a step of defining of domains of the component comprises a definition of a number of domains of equally sized voxels or the step of defining of domains of the component comprises the definition of a number of domains, wherein each domain represents a zone of different functions of the component.
2. The method according to claim 1 ,
wherein the step of determining crack initiation data either as each considered domain a domain probability density function for crack initiation PDF CI i (n) or as crack initiation cycle N ij CPF is based on at least of one of Low-Cycle fatigue (LCF), High-Cycle fatigue (HCF), Thermo-Mechanical fatigue (TMF), creep crack propagation or oxidation or the like or any combination thereof.
3. The method according to claim 2 ,
wherein the step of determining for each considered domain a domain probability density function for crack initiation PDF CI i (n) is based on a stochastic distribution, especially a Weibull distribution, or on a result of a numeric simulation, especially a Monte-Carlo-Simulation.
4. The method according to claim 1 ,
wherein the crack formation in surface regions is mainly considered by
η
=
(
∫
A
1
N
det
m
dA
)
-
1
m
.
5. The method according to claim 1 ,
wherein the component is embodied as one of the group of blades, vanes, vane carrier, rotor disk, especially its hub region or attachment region for attaching rotor blade, casing components of either a gas turbine, of a steam turbine or of a generator or as a combustor transitions of a gas turbine.
6. The method according to claim 1 ,
wherein the crack initiation process considers surface related defects of the component and/or nucleating flaws located below the components surface.
7. A method for operating a component under cyclic stress, comprising:
scheduling a downtime or maintenance of said component considering a probability of failure PoF(n) of said component as estimated by the method according to claim 1 .
8. A non-transitory computer-readable storage medium, comprising:
instructions stored thereon which, when executed by a computer, cause the computer to carry out the method of claim 1 .
9. A computer-implemented method for probabilistic estimation of probability of failure PoF(n) of a component, especially a gas turbine component, which during operation is subjected to cyclic stress, wherein the component is divided virtually in multiple domains i, wherein N is a number of domains, the step of defining of domains of the component comprises a definition of a number of domains of equally sized voxels on a step of defining of domains of the component comprises the definition of a number of domains, wherein each domain represents a zone of different functions of the component, the method comprising:
providing data regarding material of the component, its structure and regarding a loading of the component,
defining a number S of Monte-Carlo-Samples j for a Monte-Carlo-Simulation,
providing nested loops, in particular an outer loop and an inner loop, to traverse the domains N and the Monte-Carlo-Samples S, wherein in particular the outer loop traverses through the one of both samples S and domains N and the inner loop traverses through the other of the both the samples S and the domains N,
determining within both the inner loop and the outer loop a crack initiation cycle to failure N ij CI ,
determining within both the inner loop and the outer loop a subsequent crack propagation cycle to failure N ij CPF for domain i and for sample j, especially based on fracture mechanical properties drawn from respective distributions and considering stress/temperature and geometry of fracture location for domain j,
calculating within both the inner loop and the outer loop the cycles to failure for domain i and sample j: N ij Fail =N ij CI +N ij CPF ,
determining minimum failure cycle of all domains for sample j, especially according to: if N ij Fail <N j Fail set N j Fail =N ij Fail ,
and
calculating a total probability of failure PoF(n) as a function of cycles n based on S f (n)/S, wherein S f (n)=Number of samples failed until cycle n,
wherein a step of determining crack propagation data either as each considered domain a domain probability density function for subsequent crack propagation induced failure PDF CPF i (n) or as crack propagation cycle N ij CPF considers at least one of crack growth and failure relevant material properties such as fatigue crack growth rate FCGR, creep crack growth rate CCGR, crack corrosion pitting, erosion rates, fracture toughness K1 c , ΔK threshold , or tensile properties or any combination thereof,
with a failure criterion which can be based on at least one of stress intensity factor K exceeding the fracture toughness K1 c , ΔK exceeding, stress intensity factor range ΔK exceeding a fatigue crack growth stress intensity range threshold ΔK threshold , a crack length exceeding a critical crack length or exceeding a safe region of a two parameter failure assessment diagram (FAD) based on properties listed above, especially based on British R6 criteria which are based on two parameters load ratio Lr and the fracture ratio Kr.
10. The method according to claim 9 ,
wherein the step of determining crack initiation data either as each considered domain a domain probability density function for crack initiation PDF CI i (n) or as crack initiation cycle N ij CPF is based on at least of one of Low-Cycle fatigue (LCF), High-Cycle fatigue (HCF), Thermo-Mechanical fatigue (TMF), creep crack propagation or oxidation or any combination thereof.
11. The method according to claim 10 ,
wherein the step of determining for each considered domain a domain probability density function for crack initiation PDF CI i (n) is based on a stochastic distribution, especially a Weibull distribution, or on a result of a numeric simulation, especially a Monte-Carlo-Simulation.
12. The method according to claim 9 ,
wherein the crack formation in surface regions is mainly considered by
η
=
(
∫
A
1
N
det
m
dA
)
-
1
m
.
13. The method according to claim 9 ,
wherein the component is embodied as one of the group of blades, vanes, vane carrier, rotor disk, especially its hub region or attachment region for attaching rotor blade, casing components of either a gas turbine, of a steam turbine or of a generator or as a combustor transitions of a gas turbine.
14. The method according to claim 9 ,
wherein the crack initiation process considers surface related defects of the component and/or nucleating flaws located below the components surface.
15. A method for operating a component under cyclic stress, comprising:
scheduling a downtime or maintenance of said component considering a probability of failure PoF(n) of said component as estimated by the method according to claim 9 .
16. A non-transitory computer-readable storage medium, comprising:
instructions stored thereon which, when executed by a computer, cause the computer to carry out the method of claim 9 .