Summary of the invention
1, purpose: the objective of the invention is to deficiency to existing physics of failure method; A kind of electronic product life model randomization method (PPoF) based on bayesian theory is provided; This method is a kind of high reliability long life electronic product Failure Probability Model based on the stress damage model; It is to cause the dispersiveness of the factor such as various material properties, size, stress of electronic product fault through analysis, and studies these dispersed describing methods; On existing physics of failure model based, consider to add these dispersed factors; Set up the relation between failure probability and time, stress, structure, the material; Realize the randomization of physics of failure model, for describing fault more accurately, the storage life of prediction product provides a kind of new way.
2, technical scheme: the present invention realizes through following technical scheme; At first according to residing environmental baseline of electronic product and condition of work; Confirm the main failure mechanism of each components and parts, parts, confirm stress condition and physics of failure model that various mechanism are corresponding; Analyze source and characterizing method dispersed in the failure mechanism then, obtain the prior distribution of dispersed parameter, and obtain the prior distribution of life-span obedience through monte carlo method; Utilize bayesian theory to upgrade dispersed parameter, combine, obtain the probability describing method in life-span with existing physics of failure model; Utilize the method for Monte Carlo simulation to find the solution the randomization physics of failure model in life-span at last, obtain the probability density distribution and the relevant reliability index of Single Point of Faliure.
A kind of electronic product life model randomization method of the present invention based on bayesian theory, its concrete steps are following:
Step 1: the confirming of main failure mechanism and physical model.According to residing environmental baseline of electronic product and condition of work, confirm to cause the dominant mechanism of product failure, and select appropriate physics of failure model.Failure mechanism is meant the physics that causes inefficacy, chemistry, biological or other process; Physics of failure model is meant in the reliability physics to a certain specific failure mechanism, the formula of basic physics, chemistry or other principles and (or) on the basis of test regression formula, the mathematical function model of (or time of origin) and relations such as material, structure, stress takes place in the reflection fault of setting up quantitatively; General type is following: and TTF=f (D, M, E); Wherein, TTF is the fault time of origin, and D is a parameters of structural dimension; M is a material parameter, and E is a stress parameters.Environmental stress mainly comprises temperature, vibration, humidity, electromagnetism etc.Temperature stress is divided into constant temperature, temperature cycles, temperature shock again, and vibration is divided into periodic vibration and random vibration again.Working stress mainly is meant electric stress.
Step 2: confirm the source and the characterizing method of various dispersivenesses in the main failure mechanism, mainly comprise:
A. because physics of failure model might be structure, stress, the isoparametric implicit function of material; The key parameter that causes losing efficacy does not directly embody in model; Therefore to carry out disaggregate approach to selected physics of failure model; Parameter is divided into each several parts such as material properties, physical dimension, defective workmanship and stress; Whether analyze these parameters and whether all have dispersiveness, whether dispersiveness directly embodies with Model parameter, also need further to decompose; Whether have correlativity (like the correlativity between the temperature and humidity, between electric stress and the temperature etc.) between each parameter, and confirm the dispersiveness source of key parameter in the physics of failure model through THE PRINCIPAL FACTOR ANALYSIS.
B. confirm the characterizing method that key parameter is dispersed.Although the uncertainty of various parameters makes it can not represent that its value is not rambling with a certain occurrence, but in certain scope, fluctuates, obey certain distribution.Therefore need analyze the characterizing method of dispersiveness on engineering that obtains these parameters through extensive investigation or to existing experimental data, i.e. each parameter distribution and characteristic parameter thereof of obeying, this distribution is the prior distribution of parameter.Finally set up main failure mechanism, mechanism model, model parameter, main dispersed factor, the dispersed relational matrix that characterizes (distribution pattern) and characteristic parameter thereof of each factor, its form is as shown in table 1.
The relational matrix that table 1 obtains
Step 3: confirm the Life Distribution that failure mechanism is obeyed.The distribution that parameters such as material, structure, stress, defective workmanship are obeyed in the physics of failure model that investigation is obtained is as the prior distribution of parameter; Be designated as π (Θ); Θ representes parameters such as material, structure; Utilize monte carlo method from the prior distribution of parameter, to obtain N group parameter combinations, the physics of failure model in conjunction with selected obtains N fail data.Resulting data are carried out fitting of distribution, obtain distribution, be designated as f (t| Θ) in the known obedience of following life-span of condition of parameter prior distribution π (Θ).
Step 4: distribute based on the bayesian theory undated parameter, and combine physics of failure model, utilize the numerical solution of the method acquisition randomization life model of Monte Carlo sampling.Mainly comprise:
The distribution f (t| Θ) and the physics of failure model of a. obeying according to the life-span that obtains in the
step 3; Obtain combining the bayesian theory undated parameter to distribute at parameter prior distribution π (Θ) likelihood function of known following out-of-service time of condition
; Obtain its posteriority distribution π (Θ | t), Bayesian formula is following:
Following formula generally is difficult to directly obtain its convergence solution with the method for resolving, and samples with Markov chain-Monte Carlo (MCMC) usually on the engineering and finds the solution Bayesian prior distribution.Existing at present a lot of ripe special software such as WinBUGS are used for Bayesian inference.
B. the posteriority according to the parameter that obtains distributes; The probability model that forms the life-span in conjunction with physics of failure model is described; The parameter that will soon have dispersiveness in the physical model distributes with its posteriority and explains, and utilizes the Monte Carlo simulation method that this model is carried out numerical solution, the data that obtain is carried out fitting of distribution analyze the posteriority distribution that just can obtain life model; Be the probability density function of Single Point of Faliure, its process and step 3 are roughly the same.And according to the relation between fiduciary level, failure probability, the probability density function, the reliability index that obtains being correlated with.
Through above four steps, promptly can directly obtain the probability density distribution and the relevant reliability index of Single Point of Faliure, thereby realize the randomization of physics of failure model by physics of failure model.
3. advantage and effect: a kind of electronic product life model randomization method based on bayesian theory of the present invention has the following advantages:
A. utilize " time ", " environment of setting up ", the relation of " probability ", indexs such as the fiduciary level of expectation product, failure probability, the relation that provides product reliability design improvement and index to change.Physics of failure model through randomization; Set up the relation between failure probability and time, stress, structure, the material; Can directly obtain the Life Distribution of product, and obtain its fiduciary level equiprobability index, not need extra test figure or historical data through further analysis; Thereby for design saves time and cost, and the relation that provides product reliability design improvement and index to change.
B. for complex product Modelling of Reliability Data Source is provided based on the fault behavior.The probabilistic information of each trouble spot is mainly derived from statistics in traditional fail-safe analysis, and this method is not reviewed the basic reason that causes product bug, can't satisfy the high reliability long life requirement of products.And the physics of failure of randomization is directly started with from the basic reason that causes fault; And consider the dispersed factor of each parameter; Set up the relation between failure probability and time, stress, structure, the material, thereby for based on the fail-safe analysis of fault behavior Data Source being provided.
C. for the design technology parameter information is provided.Because the physics of failure model of randomization has been considered information such as physical dimension, defective workmanship, so the data that get the physics of failure (PPoF) analytical calculation according to randomization can provide information for the design technology parameter.
Embodiment
To combine accompanying drawing and embodiment that the present invention is done further detailed description below.
Following examples are the randomization analyses to the Surface Mount solder joint thermal fatigue failure mechanism of slice component 0805; Be to implement according to flow process as shown in Figure 1, mainly comprise confirm failure mechanism and physics of failure model, confirm dispersed source and characterizing method thereof in the failure mechanism, confirm the life-span prior distribution, utilize the numerical solution that the bayesian theory undated parameter distributes, the utilization Monte Carlo simulation is realized the randomization physical model.
See Fig. 1, a kind of electronic product life model randomization method of the present invention based on bayesian theory, concrete steps are following:
Step 1: the confirming of failure mechanism and physical model.The temperature cycles section of this slice component 0805 is as shown in Figure 2, and high temperature is 125 ℃, and low temperature is-55 ℃, and high low temperature respectively stops 12min.The periodicity break-make of circuit and the cyclical variation meeting of environment temperature make solder joint stand the temperature cycles process; When temperature variation; Mismatch in coefficient of thermal expansion between electron device and the circuit board can cause that solder joint bears the cyclic stress strain; When plastic strain is accumulated to a certain degree, will fatigue damage at first take place and finally cause the solder joint thermal fatigue failure in the solder joint stress concentration zones.Solder joint thermal fatigue failure physical model is a lot, and like Coffin-Manson model, Engelmaier model, full strain model etc., table 2 has been listed the usable range of part solder joint lifetimes assessment models.
Table 2 part solder joint lifetimes assessment models and applicability thereof
Wherein, the Coffin-Manson model is the most widely used a kind of low-cycle fatigue life model, and it has provided the relation of the range of strain in fatigue lifetime and the circulation, and as far as elastic model, the range of shear strain during stress relaxation is easy to obtain.
Step 2: confirm the source and the characterizing method of various dispersivenesses in the main failure mechanism, mainly comprise:
A. the life-span physical model is about fault (or time of origin) and the mathematical function model that material, structure, stress etc. concern to take place, and possibly be the relation of implicit function, and the key parameter that material properties, physical dimension etc. causes losing efficacy is not embodied directly in the model.Therefore need decompose selected model, obtain the dispersiveness of each parameter.
The expression formula of Coffin-Manson model is:
Wherein, N
fThe times of thermal cycle that is experienced when take place destroying for solder joint, i.e. fatigue lifetime, Δ γ is a range of strain, and is relevant with packing forms, physical dimension, material properties, load history, ε
fFor fatigue extension coefficient, for the eutectic solder of extensive employing, ε
f=0.325, c is tired ductility index, is the parameter relevant with the temperature cycles section.In model, directly do not embody of the influence of parameters such as material properties, physical dimension, stress to losing efficacy.Therefore need decompose Δ γ and c.Through investigating domestic and international research about the Surface Mount solder joint, the expression formula that obtains range of strain Δ γ is following:
Wherein h is a solder joint height, L
DBe device length, Δ α=α
c-α
s, Δ T=T
Max-T
Min, wherein, α
c, α
sBe respectively the thermal expansivity of device and substrate, Δ T is the temperature variation in the thermal cycle process, and F is the experience correction factor, and value is between 0.5~1.5, and classical value is generally about 1.
The formula of tired ductility index is following:
Wherein, T
sBe the thermal cycle medial temperature, t
DBe the maximum temperature residence time in the cycle, unit is min.In real work, T
sAnd t
DDispersiveness be not very big to Fatigue Life, and can be controlled in the more accurate scope, so not consider T in this example
sAnd t
DDispersiveness, the value that can get tired ductility index according to the section of Fig. 2 is-0.44.
Through above-mentioned decomposition analysis, the parameter that obtains influencing solder joint failure mainly contains: physical dimension such as solder joint height, device length; The thermal expansivity of material properties such as substrate and device; Temperature variation in stress level such as the thermal cycle process etc.This moment, Coffin-Manson can be expressed as:
B. confirm the characterizing method that key parameter is dispersed, obtain the prior distribution of parameter.
After having confirmed to have in the model dispersed key parameter, need research how dispersiveness to be showed and be attached in the physical model, i.e. the dispersed characterizing method of key parameter obtains the prior distribution of parameter.Through investigating domestic and international material properties, physical dimension, stress, the isoparametric research of defective workmanship, obtain height h, the device length L of Surface Mount solder joint
DNormal Distribution, the ratio range of standard deviation and average are got μ at present between 0.1~0.3
h=0.3mm,
σ/μ=0.15.The temperature range of device work is at-55 ℃~125 ℃, temperature inversion amount Δ T Normal Distribution, μ
Δ T=180 ℃, σ
Δ T/ μ
Δ T=0.1, the dispersiveness of thermal expansivity is difficult to definite, and can be along with temperature variation; According to the viewpoint of bayesian theory, can regard it as and evenly distribute, the difference Δ α that therefore gets the thermal expansivity of device and substrate obeys evenly and distributes; Be Δ α~U (6,9).Through above-mentioned analysis, the relational matrix that obtains solder joint thermal fatigue failure mechanism is as shown in table 3:
The correlation matrix of table 3 solder joint thermal fatigue failure mechanism
Step 2: the prior distribution that obtains the life-span:
A. the isoparametric prior distribution of difference with the thermal expansivity of device length, solder joint height, temperature variation, device and substrate (is designated as I (μ
i, σ
i), i represents above-mentioned parameter) be updated in the Coffin-Manson model, obtain following expression formula:
B. utilize the Monte Carlo that (5) formula is sampled, setting frequency in sampling is 90000 times, obtain 90000 fatigue lifetime N
f, and these data are carried out Fitting Analysis, obtain the known following life-span of condition of parameter prior distribution obey be distributed as lognormal distribution, shown in Figure 4 is probability density function, i.e. f (N
f| Θ), Θ representes parameters such as material, structure, and the logarithm average that obtains is 9.69073, and the logarithm variance is 0.31357, and mean lifetime is 16981.Wherein the Monte Carlo simulation flow process is seen Fig. 3.The main part of program is following:
Step 4: distribute based on the bayesian theory undated parameter, and combine physics of failure model, utilize the numerical solution of the method acquisition randomization life model of Monte Carlo sampling.Mainly comprise:
A. according to the Life Distribution that obtains, obtain fatigue lifetime likelihood function L (Θ | N
i).Concrete steps are following:
Prior distribution according to parameter has obtained the thermal fatigue life obeys logarithm normal distribution, that is:
f(N
f)=Ln(μ,σ) (6)
Wherein, μ, σ are respectively logarithm average and logarithm standard deviation.Theoretical following formula (4) has been represented the average level of fatigue lifetime, so the logarithm average in the formula (6) can be expressed as:
Formula (7) is updated in the formula (6), can obtains the condition logarithm distribution function of thermal fatigue failure:
Obtain the likelihood function of fatigue lifetime in view of the above:
B. investigation obtains the thermal fatigue test data of similar (material, size, environment section etc. the same) Surface Mount solder joint, with this as the cycle index N that under the known condition of each parameter prior distribution, obtains
iThe experimental data of using is as shown in table 4:
Table 4 calculates the experimental data that likelihood function is used
2224 |
1627 |
1842 |
1209 |
501 |
745 |
1399 |
906 |
2141 |
1411 |
1231 |
2209 |
1424 |
1916 |
2071 |
562 |
1143 |
1123 |
2162 |
1574 |
707 |
983 |
1188 |
1672 |
1154 |
766 |
793 |
1110 |
900 |
2415 |
1526 |
2199 |
2265 |
2397 |
2346 |
990 |
1405 |
759 |
1011 |
1531 |
2113 |
1194 |
2054 |
1844 |
813 |
877 |
951 |
1810 |
1953 |
2151 |
2208 |
2376 |
1976 |
1173 |
1726 |
2171 |
2196 |
1635 |
729 |
1586 |
2312 |
1269 |
1740 |
2034 |
1171 |
1786 |
1394 |
2099 |
1239 |
1248 |
2216 |
680 |
2185 |
1039 |
1742 |
1976 |
2051 |
1981 |
1608 |
647 |
1161 |
1457 |
2311 |
1322 |
706 |
938 |
1455 |
1063 |
1778 |
1693 |
2090 |
1851 |
1974 |
2196 |
646 |
2211 |
783 |
937 |
1766 |
2352 |
2352 |
1154 |
2396 |
2210 |
1509 |
838 |
1816 |
1470 |
796 |
1319 |
1425 |
1792 |
1100 |
1263 |
823 |
2154 |
1995 |
1142 |
1962 |
1461 |
1144 |
1697 |
2091 |
2068 |
1213 |
1621 |
1531 |
1244 |
1921 |
1258 |
2085 |
681 |
2351 |
1448 |
977 |
1537 |
1944 |
904 |
1989 |
1721 |
710 |
1225 |
1821 |
1998 |
1023 |
1472 |
1225 |
1112 |
2309 |
1135 |
1080 |
1445 |
2191 |
647 |
608 |
1829 |
1231 |
1474 |
1041 |
1752 |
1319 |
1593 |
2113 |
1492 |
1099 |
1800 |
2085 |
2157 |
938 |
2001 |
2208 |
1034 |
2020 |
1068 |
1685 |
2053 |
2327 |
1898 |
619 |
1802 |
1013 |
1329 |
1815 |
707 |
2171 |
1319 |
1262 |
2171 |
1083 |
1405 |
1470 |
718 |
1511 |
1704 |
1580 |
1685 |
782 |
783 |
1740 |
1520 |
888 |
1690 |
262 |
1008 |
1752 |
1011 |
334 |
346 |
1057 |
662 |
976 |
1469 |
775 |
1009 |
976 |
472 |
322 |
461 |
1783 |
1341 |
553 |
404 |
697 |
371 |
1166 |
529 |
511 |
809 |
802 |
1835 |
616 |
822 |
444 |
745 |
800 |
797 |
681 |
706 |
445 |
507 |
1519 |
187 |
943 |
423 |
1579 |
2287 |
1098 |
1387 |
1209 |
735 |
682 |
858 |
1618 |
1334 |
1723 |
1723 |
1789 |
858 |
911 |
1158 |
749 |
1211 |
501 |
233 |
938 |
444 |
736 |
1237 |
740 |
529 |
1236 |
1403 |
1249 |
1698 |
542 |
913 |
1907 |
1378 |
2159 |
971 |
1940 |
1440 |
2131 |
1587 |
820 |
1613 |
1876 |
983 |
2032 |
1198 |
758 |
999 |
1013 |
505 |
397 |
521 |
661 |
361 |
1013 |
900 |
786 |
172 |
687 |
1345 |
421 |
604 |
244 |
528 |
421 |
1115 |
905 |
940 |
636 |
1951 |
1177 |
1432 |
824 |
615 |
1885 |
925 |
1407 |
650 |
1586 |
1448 |
1617 |
847 |
1679 |
728 |
737 |
820 |
1016 |
998 |
1013 |
1509 |
851 |
1384 |
1514 |
1194 |
597 |
1857 |
1139 |
788 |
950 |
1763 |
2116 |
1061 |
859 |
558 |
2351 |
900 |
821 |
712 |
1624 |
667 |
822 |
1663 |
1757 |
2104 |
2322 |
2229 |
1251 |
1509 |
1016 |
875 |
605 |
758 |
647 |
981 |
1632 |
2166 |
1993 |
1035 |
1959 |
1801 |
812 |
1714 |
1982 |
1204 |
1171 |
2147 |
2223 |
1151 |
1438 |
1017 |
1241 |
1554 |
2214 |
2048 |
1324 |
1518 |
1319 |
2183 |
2378 |
1144 |
1608 |
1522 |
2000 |
1873 |
1329 |
1424 |
1755 |
1976 |
887 |
2163 |
2323 |
951 |
1512 |
1348 |
1990 |
1509 |
2052 |
1530 |
1735 |
1509 |
1969 |
2353 |
851 |
780 |
2087 |
2169 |
975 |
1426 |
1238 |
2255 |
1655 |
1088 |
1249 |
2327 |
1186 |
1524 |
1000 |
1673 |
1138 |
857 |
1146 |
2173 |
1945 |
1779 |
1628 |
1046 |
652 |
849 |
855 |
2256 |
639 |
1336 |
1098 |
1584 |
1043 |
1324 |
671 |
614 |
1111 |
1339 |
1386 |
1207 |
983 |
899 |
275 |
1156 |
653 |
531 |
1263 |
504 |
929 |
353 |
1364 |
951 |
510 |
710 |
1087 |
911 |
650 |
1007 |
1400 |
1117 |
840 |
1394 |
680 |
500 |
C. the test figure of utilizing likelihood function and combining investigation to obtain is utilized more parameter distributions in the new model of bayesian theory, and the posteriority that obtains parameter distributes, be designated as π (Θ | N
i)=π (h, L
D, Δ T, Δ α, σ | N
f), wherein σ is the logarithm standard deviation:
Just can upgrade each parameter distributions through (10) formula, obtain its posteriority and distribute.Be difficult to directly try to achieve the analytic solution of (10) formula generally speaking with the method for integration; Usually sample with Markov chain-Monte Carlo (MCMC) and find the solution Bayesian prior distribution; Used WinBUGS software in this example; It is a special software that utilizes monte carlo method to carry out Bayesian inference, below is the main part of program:
The posteriority of each parameter that the continuous sampling iteration of process software can obtain distributes, and sees Fig. 5 (a) and (b), (c), (d), and each characteristic parameter that distributes is seen table 5.
The posteriority of each parameter of table 5 distributes
The parameter title |
Device size |
Solder joint height |
Temperature variation |
Thermal expansivity poor |
The distribution of obeying |
N(1.4577,0.4696) |
N(0.28,0.09324) |
N(180,3.3154) |
?N(7.387,0.327) |
D. turn back in the above-mentioned steps three, the posteriority that can obtain the life-span distributes, and is designated as f (N
f), see shown in the figure 6, obtain the obeys logarithm normal distribution fatigue lifetime this moment, and the logarithm average is 9.42599, and the logarithm variance is 0.223659, and average is 12721.This moment, the posteriority based on the parameter that obtains distributed, and corresponding Monte Carlo simulation program also can change to some extent, changes part as follows:
L=random(′norm′,mu1,sigma1,n1,1);
h=random(′norm′,mu2,sigma2,n3,1);
T=random(′norm′,mu3,sigma3,n1,1);
alpha=random(′norm′,mu4,sigma4,n2,1);
E. according to the relation between probability density function and fiduciary level, failure probability etc., can obtain corresponding reliable probability index, wherein fiduciary level is:
Failure probability is:
Fig. 7~8 are respectively failure probability curve and the fiduciary level curve of solder joint under this section.
The present invention has set up the electronic product life model method for calculating probability based on bayesian theory; Utilize this method, can have no under the situation of experimental data, utilize existing physics of failure model; Obtain the distribution that parameter is obeyed in the model according to investigation or to data analysis in the past; The utilization bayesian theory is parameter distributions in the new model more, the method for the Monte Carlo simulation that combines extensively to use on the engineering, the reliable probability index that just can directly obtained being correlated with by physics of failure model; Remedied the deficiency of traditional reliability engineering technology, for the expectation and the assessment of reliability provides new method.
Quoting the physical significance such as the following table of letter among the present invention explains:
N
f |
Fatigue lifetime |
Δγ |
Range of strain |
ε
f |
Tired ductility coefficient |
c |
Tired ductility index |
F |
The experience correction factor |
Δα |
The thermal expansivity of device and substrate poor |
ΔT |
Temperature variation in the thermal cycle process |
h |
Solder joint height |
L
D |
Device length |
T
s |
The thermal cycle medial temperature |
t
D |
The maximum temperature residence time in semiperiod |
f(N
f)
|
Fatigue lifetime probability density function |
F(N
f)
|
Failure probability |
R(N
f)
|
Reliability Function |
λ(N
f)
|
Crash rate |