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

Abstract

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.

Claims

1.-14. (canceled)

15. 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 more domains, the method comprising: a. providing or determining (104) for at least one domain, preferably for each domain, a domain probability density function for crack initiation PDF.sup.CI.sub.i(n) and providing or determining (104) for the considered domains a domain probability density function for subsequent crack propagation induced failure PDF.sup.CPF.sub.i(n), b. b1) determining (106) for each considered domain a combined domain probability density function for failure PDF.sup.Fail.sub.i(n) according to
PDF.sub.i.sup.Fail(n)=PDF.sub.i.sup.CI(n)×PDF.sub.i.sup.CPF(n), wherein X designates the convolution operator between the two PDFs, and b2) determining (108) for each considered domain a combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n) based on the respective combined domain probability density function for failure PDF.sup.Fail.sub.i(n), wherein the CDF is a cumulative distribution function of the PDF, OR b3) determining (206) for each considered domain a domain cumulative distribution function for crack initiation CDF.sup.CI.sub.i(n) and cumulative distribution function for subsequent crack propagation induced failure CDF.sup.CPF.sub.i(n) based on the respective domain probability density function for crack initiation PDF.sup.CI.sub.i(n) and subsequent crack propagation induced failure PDF.sup.CPF.sub.i(n), and b4) determining (208) for each considered domain a combined domain probability density function for CDF.sup.Fail.sub.i(n) according to ${\mathrm{CDF}}_i^{\mathrm{Fail}}\left(n\right)=\frac{d}{\mathrm{dn}}\left({\mathrm{CDF}}_i^{\mathrm{CI}}\left(n\right){\mathrm{XCDF}}_i^{\mathrm{CPF}}\left(n\right)\right),$ wherein—CDF.sup.CI.sub.i(n) designates the domain cumulative distribution function for crack initiation, —CDF.sup.CPF.sub.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.sup.Fail=1−Π.sub.i=1.sup.N[1−CDF.sup.Fail.sub.i(n)] wherein the step of determining crack propagation data either as each considered domain a domain probability density function for subsequent crack propagation induced failure PDF.sup.CPF.sub.i(n) or as crack propagation cycle N.sub.ij.sup.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 rate-fracture toughness K1.sub.c, ΔK.sub.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.sub.c, ΔK exceeding, stress intensity factor range ΔK exceeding a fatigue crack growth stress intensity range threshold ΔK.sub.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 the British R6 criteria which are based the two parameters load ratio Lr and the fracture ratio Kr, and wherein the step of defining of domains of the component comprises the 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.

16. 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 more domains i, wherein N is the number of domains, the step of defining of domains of the component comprises the 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, the method comprising: providing data regarding material of the component, its structure and regarding the 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, determining within both the inner loop and the outer loop a crack initiation cycle to failure N.sub.ij.sup.CI, determining within both the inner loop and the outer loop a subsequent crack propagation cycle to failure N.sub.ij.sup.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.sub.ij.sup.Fail=N.sub.ij.sup.CI+N.sub.ij.sup.CPF, determining minimum failure cycle of all domains for sample j, especially according to: if N.sub.ij.sup.Fail≤N.sub.j.sup.Fail set N.sub.j.sup.Fail=N.sub.ij.sup.Fail, and calculating the total probability of failure PoF(n) as a function of cycles n based on S.sub.f(n) S, wherein S.sub.f(n)=Number of samples failed until cycle n, wherein the step of determining crack propagation data either as each considered domain a domain probability density function for subsequent crack propagation induced failure PDF.sup.CPF.sub.i(n) or as crack propagation cycle N.sub.ij.sup.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.sub.c, ΔK.sub.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.sub.c, ΔK exceeding, stress intensity factor range ΔK exceeding a fatigue crack growth stress intensity range threshold ΔK.sub.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 the British R6 criteria which are based the two parameters load ratio Lr and the fracture ratio Kr.

17. The method according to claim 15, wherein the step of determining crack initiation data either as each considered domain a domain probability density function for crack initiation PDF.sup.CI.sub.i(n) or as crack initiation cycle N.sub.ij.sup.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.

18. The method according to claim 17, wherein the step of determining for each considered domain a domain probability density function for crack initiation PDF.sup.CI.sub.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.

19. The method according to claim 15, wherein the crack formation in surface regions is mainly considered by $\eta ={\left({\int }_A\frac{1}{N_{\det }^m}\mathrm{dA}\right)}^{-\frac{1}{m}}.$

20. The method according to claim 15, 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.

21. The method according to claim 15, wherein the crack initiation process considers surface related defects of the component and/or nucleating flaws located below the components surface.

22. 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 15.

23. A data processing system, comprising: means for carrying out the method of claim 15.

24. 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 15.

25. The method according to claim 16, wherein the step of determining crack initiation data either as each considered domain a domain probability density function for crack initiation PDF.sup.CI.sub.i(n) or as crack initiation cycle N.sub.ij.sup.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.

26. The method according to claim 25, wherein the step of determining for each considered domain a domain probability density function for crack initiation PDF.sup.CI.sub.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.

27. The method according to claim 16, wherein the crack formation in surface regions is mainly considered by $\eta ={\left({\int }_A\frac{1}{N_{\det }^m}\mathrm{dA}\right)}^{-\frac{1}{m}}.$

28. The method according to claim 16, 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.

29. The method according to claim 16, wherein the crack initiation process considers surface related defects of the component and/or nucleating flaws located below the components surface.

30. 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 16.

31. A data processing system, comprising: means for carrying out the method of claim 16.

32. 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 16.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0077] The present invention is further described hereinafter with reference to illustrated embodiments shown in the accompanying drawings, in which:

[0078] FIG. 1 is flowchart of a first exemplary embodiment of the invention,

[0079] FIG. 2 is flowchart of a second exemplary embodiment of the invention,

[0080] FIG. 3 is a schematic illustration of a turbine blade as an exemplary embodiment of a component, virtually divided into four domains,

[0081] FIG. 4 is a third exemplary embodiment of the invention based on a Monte-Carlo scheme and voxel representation of component,

[0082] FIG. 5 is an example of a 2D-voxel representation of a gas turbine rotor disk,

[0083] FIG. 6 shows three charts illustrating the convolution of domain cumulative distribution function for crack initiation CDF.sup.CI.sub.i(n) and subsequent crack propagation induced failure CDF.sup.CPF.sub.i(n) to a combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n),

[0084] FIG. 7 shows schematically the graphs of FIG. 6 combined into a single chart, for a first domain,

[0085] FIG. 8 shows schematically the graphs of FIG. 6 combined into a single chart, for a second domain,

[0086] FIG. 9 shows schematically a chart displaying all determined domain probability density functions for failure CDF.sup.Fail.sub.i(n) of all considered domains and the resulting probability density function for component failure CDF.sup.Fail(n),

[0087] FIG. 10 shows schematically a chart illustrating decreasing CDF.sup.Fail(n) for an increasing number of domains,

[0088] FIG. 11 shows schematically the exemplary embodiment of the invention with the aid of pictograms and

[0089] FIG. 12 shows the two phases of the failure mechanism the crack nucleation and subsequent crack growth.

[0090] FIG. 13 shows two maps of local risks of a gas turbine rotor disc for different failure phases, that are convoluted to a resulting failure map,

[0091] FIG. 14 shows a schematic of a heavy-duty rotor disk with a section from axis-symmetrical considerations,

[0092] FIG. 15 illustrates a heavy-duty rotor disk modelled in 2D with a typical stress-field plot,

[0093] FIG. 16 shows schematically a pore within a component's material and its stress distribution.

[0094] FIGS. 17, 18 shows two possible realizations of a direct Monte-Carlo approach,

[0095] FIG. 19 shows a direct Monte-Carlo loop, in which all the statistical parameters are randomly sampled,

[0096] FIG. 20 shows a risk contour plot of the results of the MC loop on a case-study for a rotor disk of a gas turbine,

[0097] FIG. 21 shows the dependency of probabilities of failure from the number of load cycles and

[0098] FIG. 22 illustrates a system 80 for fatigue crack life estimation of a component.

DETAILED DESCRIPTION OF INVENTION

[0099] FIG. 1 is a flowchart of a first exemplary embodiment of a computer-implemented method 100 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. The computer-implemented method can be executed by a data processing system comprising appropriate means for carrying out and/or supporting one or more of steps of the invention. The first exemplary embodiment utilizes steps a), b1), b2) and c) of the invention.

[0100] According to the first exemplary embodiment in a first step 102 the user of the method defines in the computer one or more different areas of the component that are subject to different loadings, different crack flaws, different operation conditions, etc. The number of areas, in the following called domains, being selected and defined depend on the individual requirements of the user, as explained in detail with the aid of FIG. 3.

[0101] FIG. 3 shows a turbine blade of a gas turbine as an exemplary component for which a probability of failure PoF(n) shall be determined. In a first approach the turbine blade could be considered as a single domain, so that a global approach 102a is made. This means that the same data, like crack flaw data, expected propagation rate, the stochastic distributions, thermal and mechanical load, etc. are applied for every part of the whole component. This is beneficial as fast computing results could be expected, however, with lower accuracy of the total failure probability estimation.

[0102] Instead of a global approach and according to a zone-based approach 102b the component could be divided virtually into a smaller number of domains. Crack flaw data, crack initiation and/or crack propagation, operational loads, etc. differs from zone to zone and it is assumed that within each zone these parameters are identical. Hence, for each zone the same calculation steps can be performed, however with different data. In the exemplary embodiment as shown in FIG. 3 four zones or domains are defined: the first two zones or domains comprise the tip and the remaining part of the airfoil of the turbine blade, the third zone or domain comprises only the platform and the fourth zone or domain comprises only the root of the turbine blade. In this example the zones are defined based on local regions of the component with identical functions.

[0103] Instead of this, the zones could also be defined based on regions that have similar features and/or based on specific symmetries, or the like. This zone-based approach balances the need of a more accurate assessment of the failure probability estimation and the effort for implementing and executing the methods described herein.

[0104] Most accurate but accompanied with most effort in designing and computing is a third approach, in which a larger number of voxels are defined as domains each representing a local volume of the component. The third approach is also called voxel-based approach 102c. A voxel, also known as volume pixel element, represents in a grid a two- or three-dimensional space, usually a square or a cube with a predetermined length of edges. Then, for each voxel the data i.e. crack flaw data, expected propagation rate, etc. and stochastic distribution, fracture toughness K1.sub.c, etc. are to provide. It is self-explaining that the voxel-based approach is that approach in which one or more voxels could be omitted in the estimation process when these specific voxels can be identified in advance with lowest or no likelihood of failure.

[0105] Further, it is also possible that for each domain different FEA-meshes, different loading and/or operating conditions are applied, if suitable and indicated. Further, it is also possible to exclude single of multiple domains from consideration when expected that these regions are definitely not stressed enough that critical defects occur for all cycles n.

[0106] After the domains have been defined according to one of the approaches as mentioned above and turning back to FIG. 1, in step 104 the computer-implemented method determines for each considered domain, a domain probability density function PDF.sup.CI.sub.i(n) for crack initiation. In one example this could be done in a conventional manner. The crack initiation probability model at a given number of cycles for a whole component or a domain of said component can be described by a Weibull distribution

[00003]$\begin{array}{cc}{F}_N\left(n\right)=1-\exp \left[-{\left(\frac{n}{\eta }\right)}^m\right]& \mathrm{eq}.\left(10\right)\end{array}$

[0107] With the Weibull shape m being a material parameter, i.e. the scatter, and the Ndet being the local deterministically calculated life cycles to crack initiation the Weibull scale parameter η can be described e.g. by an integral over the relevant surface area A of the component, when crack formation in surface regions is mainly considered:

[00004]$\begin{array}{cc}\eta ={\left({\int }_A\frac{1}{N_{\det }^m}\mathrm{dA}\right)}^{-\frac{1}{m}}& \mathrm{eq}.\left(11\right)\end{array}$

[0108] The determined initiated cracks can either have a fixed assumed crack size (as measure from experiments) or a specific distribution of crack sizes and shapes. Note that eq. (11) is just one example, the type of distribution can be different including non-analytical distribution. Also, the integral might not be limited to a surface integration as described in eq. (11).

[0109] Further, the method determines for at least one domain, preferably for each domain, a domain probability density function for subsequent crack propagation induced failure PDF.sup.CPF.sub.i(n) as a second model. These determinations could be done in a conventional way, usually by the consideration of fracture mechanics, as exemplarily explained in the U.S. Pat. No. 9,280,620 B2 of Amann, Gravett, and Kadau, which complete content is herewith incorporated by reference. They described a probabilistic fracture mechanics approach which is using the Monte-Carlo methodology.

[0110] It is noted that the crack propagation relevant material properties such as fatigue crack grow rate (FCGR), fracture toughness K1c, tensile properties, etc. can also vary as described in the referenced patent. When the calculation of the PDF.sup.CPF.sub.i(n) is based on fracture mechanics considerations, this calculation requires a failure criterion which can the stress intensity factor K exceeding the fracture toughness K1c, or another threshold value such as fatigue crack growth stress intensity threshold K.sub.th. Other failure criteria such as reaching a critical crack length can be applied as well.

[0111] In a next step 106 (FIG. 1) for each considered domain, preferably for all domains N, a combined domain probability density function for failure PDF.sup.Fail.sub.i(n) is determined (108) by convoluting both the domain probability density function for crack initiation PDF.sup.CI.sub.i(n) and domain probability density function for subsequent crack propagation induced failure PDF.sup.CPF.sub.i(n). Then, the combined domain probability of failure density PDF.sup.Fail.sub.i(n) is transformed in step 108 into a combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n) by integration of the first.

[0112] If only one domain is defined or considered, then the calculated combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n), with i=1 represents already the total probability of failure PoF(n). This means that step 110 can be omitted.

[0113] Only if more than one domain is defined and considered, the component has to be considered fail when any of the considered domains N have failed. Therefore, subsequently all considered combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n) are utilized in the last step 110 for determining for each considered domain the total probability of failure PoF(n) according to the formula:

PoF(n)=1−Π.sub.i=1.sup.N[1−CDF.sup.Fail.sub.i(n)]  eq. (12)

[0114] A second exemplary embodiment of the invention is depicted in FIG. 2 and utilizes steps a), b3), b4) and c) of the invention as defined above. According to FIG. 2, the sequence of convolution and of transforming the PDF(n) into the CDF(n) could swapped. Hence, the same result could be achieved when first for at least one domain a domain cumulative distribution function for failure and for crack initiation CDF.sup.CI.sub.i(n) and subsequent crack propagation induced failure CDF.sup.CPF.sub.i(n) are determined based on the respective domain probability density functions PDF.sup.CI.sub.i(n) and PDF.sup.CPF.sub.i(n), and then the convolution of the CDFs with subsequent differentiation with respect to cycles n is performed.

[0115] FIG. 4 is a schematic de-composition of a generic component Ω into twelve voxels, whereas FIG. 5 is a more detailed example of a 2D-voxel representation in cartesian coordinate systems (X/Y) of a gas turbine rotor disk as an example of component for which the total probability of failure shall be determined. In FIG. 5 the cross section of the gas turbine rotor disc 130 is hatched. According to the example each voxel edge has a length A. The size of the gas turbine disk along y-axis is Y and along the x-axis is X, the number of 2D-voxels can be calculated to 240.

[0116] An example of a convolution of a domain cumulative distribution function for crack initiation CDF.sup.CI.sub.i(n) and for subsequent crack propagation induced failure CDF.sup.CPF.sub.i(n) for a single domain is shown in FIG. 6. In this example the variable n represents the cycles, e.g. the number of starts of a gas turbine in which the turbine blade is embedded. For the considered domain it is assumed that the total probability of failure for crack initiation is based on a Weibull-distribution due to LCF. The graph of the left chart of FIG. 6 shows the cumulative distribution function of the domain cumulative distribution function crack initiation CDF.sup.CI.sub.i(n) for. The chart for the same domain shown in the middle is the numerical fatigue crack growth as cumulative distribution function for subsequent crack propagation induced failure CDF.sup.CPF.sub.i(n), which could be determined by a Monte-Carlo-Simulation. Both CDF.sup.CI.sub.i(n) and CDF.sup.CPF.sub.i(n) are combined by convolution and subsequent derivation wrt n according to eq. (3) to determine the combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n), which is displayed in the chart located at the right-hand side of FIG. 6.

[0117] What is shown in FIG. 6 for a first domain in three charts is combinedly shown in FIG. 7 in a single chart, however with graphs having a different shape than in FIG. 6. Both, the cumulative distribution function of the domain for crack initiation CDF.sup.CI.sub.i(n) and the cumulative distribution function of the domain for subsequent crack propagation induced failure CDF.sup.CPF.sub.i(n) are drawn in full line whereas the resulting combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n) is shown in dashed style.

[0118] FIG. 8 shows in principle the same chart as FIG. 7, but because of the different data that are applied for the determination of the combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i(n), the graphs for the second domain are different. The respective data are selected according to the load, stresses, material properties, crack flaw data, etc. as being relevant for the second domain.

[0119] Then FIG. 9 shows in one chart all combined domain cumulative distribution functions for failure CDF.sup.Fail.sub.i(n) of all considered domains i, for i=1 to N, with N>1 which are all combined according to eq. (5) to compute the total probability of failure of a component PoF(n) represented in the dashed line.

[0120] FIG. 10 shows the dependency of the total probability of failure PoF(n) of a component from the number of considered domains. With increasing number of domains, as indicated by the arrow, a more accurate estimation for the total probability of failure PoF(n) of a component is achieved on costs of calculation time that the computer needs for performing the steps proposed. The dashed line represents the converged probability of failure PoF(n) of the component for a large number of domains.

[0121] The whole procedure of which most is done by a computer, is shown again in FIG. 11 as exemplary embodiment of the invention with the aid of pictograms. According to this third exemplary embodiment of the method the turbine rotor blade is virtually separated into three domains (N=3), a first domain comprises the airfoil of the turbine rotor blade, the second domain comprises the platform and the third domain comprises the root of the turbine rotor blade. As the zone-approach is shown in FIG. 11, these domains could also be mentioned as zones. The domain representing the platform of the turbine blade is ignored in this example. Hence, only for two of the three zones, namely the airfoil and the root, the crack initiation probabilities and the crack propagation probabilities are determined, convoluted and further processed for the determination of the total probability of failure PoF(n) of the turbine blade. If needed, these data can also be used to determine a probability of failure of a turbine stage PoF.sup.stage(n) comprising of a number of identical not correlated components. Furthermore, the probability of failure for the whole engine PoF.sup.Engine(n) or a subsystem thereof PoF.sup.System(n), e.g. turbine can be quantified as well by calculating the individual component and stage probabilities.

[0122] The before-mentioned explanation of the invention was mainly directed to cracks newly formed in surfaces. In the following the invention will be explained in detail again for flaws in the material of the component, which are embedded below its surface. Only for the sake of easy in the following the first phase of failure mechanism, crack initiation, will be called crack nucleation in the following, without generating any differences.

[0123] FIG. 12 shows again the two phases of the failure mechanism, now explained with aid of a material imperfection, e.g. a forging flow induced by a non-metallic inclusion. Under cyclic and or static loading the area of a non-metallic inclusions first nucleates into a crack before subsequent crack growth will lead to component failure. The crack growth part is typically described by engineering fracture mechanics. As the local lifetime of the component regarding crack nucleation N.sub.Nuc depends significantly on the type of flaw (inclusion, separation, etc.), which is typically unknown, the invention proposes to use a probabilistic model to describe the nucleation of the flaw into a crack.

[0124] The total local failure probability can then be described by the local convolution of the probability for nucleation and for fracture mechanics life as exemplary described in eq. (2) resp. eq. (3), accompanied by required calculations mentioned above, as indicated in step c).

[0125] The nucleation modeling process is a function of the number of cycles N, the applied min./max. stress σ.sub.min/σ.sub.max, the temperature T, the flaw size and geometry A and the flaw type. The total flaw nucleation probability over all flaw types can be calculated with a mean type occurrence rate ρ.sub.i with the following formula:

[00005]$\begin{array}{cc}{\mathrm{PoF}}_{\mathrm{Nuc}}\left(N,\sigma ,T,A\right)={.Math.}_{i=1}^n\frac{F_{\mathrm{Nuc}}\left(N,\sigma ,T,A\right)}{{\rho }_i}& \mathrm{eq}.\left(13\right)\end{array}$

[0126] Eq. (3), (4) and (13) are applicable in the general case where no correlation between a) N.sub.CI and b) N.sub.CPF respectively N.sub.FM are expected. In this case a parameter m is defining the different flaw types which can occur within the material of the component, e.g. in a gas turbine rotor disc 130 (FIG. 4) and parameter i represent one of the domain N that is at present considered.

[0127] FIG. 13 shows local risk maps of a gas turbine rotor disc 130 after a high number of cycles for local crack nucleation probability (which is a domain cumulative distribution function for crack initiation CDF.sup.CI.sub.i(n)), local fracture mechanics failure probability (which is a domain cumulative distribution function for subsequent crack propagation induced failure CDF.sup.CPF.sub.i(n)), and the resulting local total failure probability (which is the combined domain probability density function for CDF.sup.Fail.sub.i(n)) according to the invention. For the crack nucleation probability, a nucleation model F.sub.Nuc as expressed in eq. (13) has been determined and applied.

[0128] In FIG. 13 this approach is refined with a local crack nucleation model which accounts for the local stress and temperatures within the component. As can be seen the local combination of the crack nucleation probability and the fracture mechanics failure probability strongly depends on the local conditions and can significantly reduce the calculated overall risk in a non-homogeneous fashion.

[0129] FIG. 14 shows a schematic of a heavy-duty rotor disk with a section from axis-symmetrical considerations, like FIG. 5 does also. In FIG. 15 the disk modelled in 2D and with a typical stress-field plot for such a component is illustrated. It is obtained from the thermo-mechanical transient Finite Element Analyses (FEA) for a specific operational profile.

[0130] The present invention can utilize very efficiently parallel computing platforms with thousands of CPUs to solve direct simulation Monte-Carlo schemes involving up to billions of individual fatigue crack growth simulations. Depending on problem size and involved number of processors the numerical solution can be obtained in minutes. It is focused on a probabilistic description of the crack formation phase and its integration into the probabilistic fatigue crack propagation to failure. The following describes a first nucleation model based on both the aforementioned experimental characterization as well as micromechanical modeling aspects.

[0131] A first forging flaw nucleation model based on numerical FEA of a simplified elliptical shaped flaw geometry is embedded in the steel matrix of the considered heavy duty gas turbine disk 130. The flaw is modelled as a pore in the center of a representative volume. The FEA cell is uni-axially loaded with a uniform stress equivalent to the load applied in the laboratory tests or resulting from the mechanical models of the component design.

[0132] The stress field around the pore causes micro-cracks to initiate. FIG. 16 shows an example of such a FEA performed with ANSYS. In this example the equator zone 152 of the pore 150 experiences the highest stresses due to its shape and the applied loading direction. Regions with higher stresses contribute more to crack formation and the model inherently describes the statistical size effect. Larger flaws experience a shorter nucleation life compared to smaller flaws subjected to the same cyclic load. The modelled flaw size can vary from 10 μm to 5 mm in diameter. Also, the loading ranges and the temperatures can vary, according to values relevant to rotor disks in heavy-duty gas turbines, typically from 0-900 MPa and 0-600° C. respectively.

[0133] In the nucleation model, the nucleation process is a function of the number of cycles until crack nucleation, N, the applied min/max stress, the temperature, the flaw size and geometry and the flaw type. Especially the flaw typology and geometry are subject to future studies, as they are expected to influence the nucleation life considerably. The resulting probability of crack nucleation around the flaw can, e.g., be described by a two-parameter Weibull distribution:

[00006]$\begin{array}{cc}{\mathrm{PoF}}_{\mathrm{Nuc}}^i\left(N,\sigma ,T,A\right)=1-\exp \left(-{\left(\frac{N}{{\eta }_{\sigma ,T,A}}\right)}^m\right)& \mathrm{eq}.\left(14\right)\end{array}$

[0134] where the shape parameter m is an inherent material property describing the scatter in LCF life and the scale parameter, η, is a geometry (A), load (σ) and temperature (T) dependent variable. This example model is valid for a specific flaw type, i.

[0135] The total flaw nucleation probability over all flaw types can be calculated with a mean type occurrence rate p, with the following formula:

[00007]$\begin{array}{cc}{\mathrm{PoF}}_{\mathrm{Nuc}}\left(N,\sigma ,T,A\right)={.Math.}_{i=1}^n\frac{{\mathrm{PoF}}_{\mathrm{Nuc}}^i\left(N,\sigma ,T,A\right)}{{\rho }_i}& \mathrm{eq}.\left(15\right)\end{array}$

[0136] Depending on the type of material imperfection, the component loading, and other material properties, also other models can be selected. The proposed framework needs a probabilistic description of the failure mechanisms, be it numerical or analytical. In this example, a limitation is made to one single flaw type and reduce herewith the modeling complexity.

[0137] Both the nucleation life and the crack propagation are, on one hand, dependent of material specific parameters such as SN-curves,

[00008]$\frac{\mathrm{da}}{\mathrm{dN}}$

curves,

[00009]$\frac{\sigma }{s}$

relationships, flaw sizes and geometries and their statistical distribution along the component. On the other hand, they depend of the loading conditions, such as temperature transient and stress field. The flowchart in FIG. 19 shows the brute force Monte-Carlo loop, by which all the statistical parameters are randomly sampled. Each simulation consists of a realization of one flaw with its own nucleation life (N.sub.Nuc) and subsequent fracture mechanical life (N.sub.FM). The two are both locally computed and, as such, dependent of the local temperatures and stresses. By summation of the two one obtains the total lifetime of the flaw (N.sub.Flaw). The variations in the material properties come from the scattering of material test data. The parameters on the loading side are generated with mechanical and thermal FEA. At the bottom of the flowchart the probabilistic life of the component with respect to the probability of the presence of a flaw, its nucleation and subsequent growth can be computed with eq. (16).

[0138] FIGS. 17 and 18 shows two possible realizations of a direct Monte-Carlo approach to realize an equivalent embodiment to the direct convolution of CDFs or PDFs. In this example the crack propagation induced failure is based on fracture mechanics (FM) considerations. The calculation of the amount of cycles for crack nucleation and crack growth can be either calculated independent in a parallel structure (FIG. 17) or dependent on the nucleation calculation in a serial structure (FIG. 18). The calculation in FIG. 17 is only applicable if the fracture mechanics part of the evaluation does not depend on the nucleation phase. This means there is no correlation between them, which then corresponds to the convolution approaches as described above. However, the calculation in FIG. 18 allows for correlations between the nucleation processes and the subsequent crack growth phase. Specifically, after a specific instance the nucleation cycle is evaluated in step 18a, the subsequent crack growth phase can then be dependent on the simulated crack nucleation process 18b. The calculation in FIG. 18 is therefore more generic. In both examples the two components the nucleation life N.sub.Nuc,j and the fracture mechanics life N.sub.FM,j can be calculated and added to a life N.sub.tot,j for each Monte-Carlo loop j.

[0139] The proposed process can be applied for the integration over the whole component or only over a zone or over multiple voxels of the component (not shown in FIGS. 17 and 18). Also, probabilistic input can be spatially varying, incorporate analytical and non-analytical distributions, as the proposed scheme is generically designed.

[0140] Referring back to FIG. 5 each location of the component, i.e. each domain, zone or voxel, can have different local nucleation and fracture mechanics failure probabilities. The proposed method allows for the integration over the whole component and calculate the total probability of failure for the complete component. This procedure can be performed by applying eq. (5), or directly in a Monte-Carlo loop and checking all location for failure and applying the following eq. (16) and (17).

[0141] Eq. (16) and (17) define important quantities as they are obtained from the proposed direct Monte-Carlo scheme.

[00010]$\begin{array}{cc}\mathrm{PoF}\left(N\right)=\frac{S_f\left(N\right)}{S}& \mathrm{eq}.\left(16\right)\end{array}$

[0142] Where S is the total number of simulated Monte-Carlo-Samples, and S.sub.f(N) is the number of samples that failed after N cycles. A Hazard rate H is defined by

[00011]$\begin{array}{cc}H\left(N\right)=\frac{\mathrm{PoF}\left(N+1\right)-\mathrm{PoF}\left(N\right)}{1-\mathrm{PoF}\left(N\right)}& \mathrm{eq}.\left(17\right)\end{array}$

[0143] where H is used as a measure for the risk of failure within the next cycle under the condition that no failure has occurred before.

[0144] FIG. 19 shows again the direct Monte-Carlo approach to realize the suggested summation of nucleation life and fracture mechanical life.

[0145] In general, the two processes of crack nucleation and crack propagation do not have the same definitions of critical temperatures and stresses along the load transient. The nucleation process e.g. is accelerated by higher temperatures and larger stress ranges. The crack growth might be limited to critical transient time points exposing a low fracture toughness for low temperatures.

[0146] Also, crack growth might be more dependent on the crack plane and stress orientation. For each process two or more different contour plots of the significant temperatures and stress ranges are obtained. FIG. 19 illustrates the overall direct MC algorithm. Each MC sample has a location i that experiences different relevant temperatures and loads for N.sub.Nuc and N.sub.FM, respectively. Each MC instance has randomly assigned material properties and loading properties, whereas the convolution is established by adding the local values N.sub.Nuc,i and N.sub.FM,i to N.sub.flaw,i. Then millions to billions of MC loops j need to be performed in order to quantify the low probabilities of failure according to eq. (7).

[0147] The total probability of failure PoF(n) of the entire component after N cycles can be computed by the weakest link theory, by which the entire component fails if one crack grows beyond critical. The following expressions define important quantities used for component lifing. The total probability of failure of the entire component could be calculated again according eq. (16).

[0148] In the first step the influence of nucleation can be studied without considering its dependence on flaw size, temperature, stress and location. Instead, effective shape and scale parameters (m and η) are selected and applied to the entire component in a global fashion. The system computes the probabilistic fracture mechanical life and, by convolution, the global nucleation life can be added to it.

[0149] The following pseudo code shows an example of a numerical Monte-Carlo method according to the beforementioned examples. The outer loop is used to traverse the number of Monte-Carlo-Sample S. The inner loop is used to traverse the number of domains N. With slight modification the inner and outer loop can be exchanged without any impacts on result and performance.

[0150] Pseudo Code: [0151] For j=1 . . . S (outer loop; S total number of Monte-Carlo-Samples) [0152] Set N.sub.j.sup.Fail=LARGE [0153] For i=1 . . . N (inner loop; N number of domains of the component Ω): [0154] compute crack initiation cycle N.sub.ij.sup.CI for domain i and sample j (for example by drawing from a Weibull distribution with Weibull shape and scale parameter describing domain i, see eq. (10). For some crack initiating failure mechanism, the initiation might be dominant to the surface of the component, for others it might be the volume, or both. In another example including pre-existing manufacturing related flaws such as forging flaws an occurrence probably can be utilized to probe the existence of the flaw in this domain for this instance. Once a flaw, its size and shape are established in this instance, the nucleation cycle can be established by eq. (13)-(15)) [0155] compute fracture mechanics calculation for sample j based on fracture mechanical properties drawn from respective distributions (crack size, fatigue crack growth rate, fracture toughness, etc.). Calculate fracture life for domain i and sample j N.sub.ij.sup.CPF. Consider stress/temperature and/or geometry of fracture location for domain i. This can dependent on crack initiation process as described in the previous step in order to include correlations between the two processes [0156] Calculate cycles to failure for domain i and [0157] sample j: N.sub.ij.sup.Fail=N.sub.ij.sup.CI+N.sub.ij.sup.CPF [0158] if N.sub.ij.sup.Fail=N.sub.j.sup.Fail set N.sub.j.sup.Fail=N.sub.ij.sup.Fail [0159] (this ensures that we capture the minimum failure cycle of all domains for sample j) [0160] Calculate total probability of failure PoF(n) as a function of number of cycles n based on S.sub.f(n)/S:

PoF(n)=S.sub.f(n)/S [0161] wherein: [0162] S.sub.f(n)=Number of samples failed until cycle n (utilize the individual failure cycles to failure N.sub.j.sup.FAIL from the individual samples j. The values that have been calculated in the above nested loop.) [0163] From the PoF(n) each relevant probabilistic/stochastic value (such as the hazard function [0164] H(n) [PoF(n+1)−PoF(n)]/[1−Pof(n)]]) can be calculated. In such numerical approaches enough samples S should be utilized in order to obtain converging results.

[0165] The addition of local crack initiation life N.sub.ij.sup.CI and local crack propagation life N.sub.ij.sup.CPF to local failure life N.sub.ij.sup.Fail looks at a first glance deterministically. However, as this addition is embedded in inner loops of both, the local representation and within the Monte-Carlo-Simulation, this approach integrates all individual instances representing possible scenarios in operation. Hence, this could be understood as a convolution of probabilities as the uncorrelated case of both crack initiation and subsequent crack propagation induced failure. One advantage of the presented numerical method is the straightforward implementation of correlations between the two processes as described above and shown in FIG. 18.

[0166] FIG. 20 shows a risk contour plot of the results of the MC loop on a case-study for a rotor disk of a gas turbine. For this demonstration, theoretical operational data and FEA are used as input to the simulations. The material used is a high-quality rotor steel. Each flaw that led to failure after n.sub.0 load cycles is represented in these plots. Since failure from a flaw in a well-designed component is an extremely rare event, the amount of load cycles was set unrealistically high to visualize the local spread of probabilities. From these figures it results that some areas that first looked riskier, are in reality—after implementing the nucleation life—less risky.

[0167] From these local risks the global probability of failure on component level can be obtained by integration over the component and is equivalent to eq. (16).

[0168] FIG. 21 shows the results of several computations where first no nucleation was considered (full line 160). The line 170 illustrated as bold full line 170 shows the result of computation with nucleation considered. Then different effective scale parameters η with constant shape parameter m were used. Remarkable for the total probability of failure PoF(n) for smaller number of flaws η (dashed line 180) does not lead to a significant different result for than a larger number of flaws η (bolt dashed line 190).

[0169] Infant mortality causes high rates of failure at the very begin of the lifetime. This phenomenon represents all those flaws which are sampled with disadvantageous material properties and fail after only few load cycles when no nucleation is accounted for. The effect of nucleation is that this initial peak is flattened and shifted to the right. With higher scale parameters this effect accentuates. These studies illustrate the influence of a nucleation model and the shown trends can easily be understood.

[0170] FIG. 21 shows how these probabilities develop with an increasing number of load cycles. The PoF at n.sub.0 cycles of the central plot reflects the situation of FIG. 20. Also interesting are the results for lower numbers of cycles in the upper plot. For comparison also two simulations with the global effective nucleation model are plotted. The model with η=n.sub.0/12, e.g., compares well with the local nucleation scheme at n.sub.0/6 cycles, but they diverge for higher amounts of cycles. These results show the importance of modeling the nucleation life locally. Considering the nucleation life globally would over-estimate the local nucleation lives in areas where, due to low stresses or temperatures, flaws eventually never nucleate, thus also never propagate.

[0171] Hence, as explained with the aid of the material flaws embedded in component's material the invention proposes a method for probabilistic estimation of the component life limited by cyclic and time dependent damage mechanisms wherein [0172] a probabilistic nucleation model is defined that combines an amount of local forging flaw crack nucleation processes, and [0173] a fracture mechanic model is defined that combines an amount of local crack growth failure probabilities, and [0174] convolution of the probabilistic nucleation model and the fracture mechanic model locally, and [0175] calculation of the total probability of failure for the hole component, and [0176] probabilistically predict the total life of a component under a specific load spectrum

[0177] The method has been explained according to the second embodiment for the example of forging flaws. However, the presented method is generic and can be applied to many other manufacturing imperfections where a nucleation process is followed by a crack propagation phase. These examples include AM parts including a variety of imperfections including voids (or imperfectly closed voids and separations after HIP), as well as cast porosity in turbine blades. In both of those instances the flaw to be instantaneously to be a crack neglecting the nucleation (or initiation) phase. Other examples are manufacturing defects in the semiconductor industry including chipsets and boards. Here, thermos mechanical loading can initiate a crack at manufacturing defects and subsequent crack growth can eventually fail the component.

[0178] Core of the invention is the combination of a crack nucleation model for forging flaws with fracture mechanics methods in a probabilistic manner. The invention can be realized via a direct Monte-Carlo scheme or local convolution. The competitive advantage is an increase in the design lifetime of components. This method can be applied during new apparatus design as well as for lifetime extension of service frames.

[0179] FIG. 22 schematically illustrates a system 80 for fatigue crack life estimation of a component based on direct simulation probabilistic fracture mechanics, according to one embodiment of the present invention. The illustrated system 80 is a computer system comprising a memory 81, processing means 82 and an input-output device 83. The memory 81 stores all material property scatter data, flaw-size scatter data and all other relevant data as mentioned above. Additionally, the memory 81 may also store stress-temperature fields. The processing means 82 includes a plurality of functional blocks or modules which may be implemented in hardware and/or software, typically a combination of both. These modules include a fracture mechanics module 84, an FEA module 88, a computation module 85, an I/O module 86, and a plurality of libraries 87, including libraries that handle the memory management of the stress/temperature field, libraries that handle material properties, libraries to handle NDE (such as ultrasound) and flaw information, and high performance look-up table libraries. The libraries may be linked together into an executable.

In detail the computation module 85 is able to support and/or execute the method steps described herein.

[0180] In brief the invention relates to a computer-implemented method 100 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 one or more domains i, the method comprising the steps of: providing or determining 104 for at least one domain, preferably for each domain, a domain probability density function for crack initiation PDF.sup.CI.sub.i and providing or determining 106 for the considered domains a domain probability density function for subsequent crack propagation induced fracture PDF.sup.CPF.sub.I. For providing an improved method for probabilistic estimation of a probability of failure of a component, especially a gas turbine component, designated for being subjected to cyclic stress, it is proposed to determine for each considered domain a combined domain cumulative distribution function for failure CDF.sup.Fail.sub.i or its probability density function PDF.sup.Fail.sub.i by convoluting either the considered domain probability density function for crack initiation PDF.sup.CI.sub.i and the considered domain probability density function for subsequent crack propagation induced failure PDF.sup.CPF.sub.i, or their integral functions CDF.sup.CI.sub.i and CDF.sup.CPF.sub.i. Also, correlations between the crack initiation and subsequent crack propagation can be incorporated if needed.

[0181] The method described herein is explained on the basis of a turbine blade and rotor disk. However, the method can be applied to all components where failure due to an initiation event and subsequent (crack) progression is relevant. Components include but are not limited to gas turbines, steam turbine, generators and their components like blades, vanes, transitions, vane carrier, rotor disks, casing components, electrical conductors or electrical connections or the like.

[0182] While specific embodiments have been described in detail, those with ordinary skill in the art will appreciate that various modifications and alternatives to those details could be developed in light of the overall teachings of the disclosure. For example, elements described in association with different embodiments may be combined. Accordingly, the particular arrangements disclosed are meant to be illustrative only and should not be construed as limiting the scope of the claims or disclosure, which are to be given the full breadth of the appended claims, and any and all equivalents thereof. It should be noted that the term “comprising” does not exclude other elements or steps and the use of articles “a” or “an” does not exclude a plurality.