|
NONLINEAR G-FUNCTION
-
Description
-
Limit
State Function
-
Random
Variables
-
Analysis
-
Results
This problem demonstrates the effect of higher order
terms of limit-state function in the MPP approach.
Limit-State Function:
g6(X)
= 4 - (X1+0.25)2 + (X1+0.25)3 + (X1+0.25)4
- X2
( 1)
X1 ~ Normal(m=0, s=1)
X2 ~ Normal(m=0, s=1)
X1 and X2 are independent
Anal1:
Calculate the failure probability using FORM.
Case 1: U Based U-Linearized MPPL Method
Case 2: Modified U Based U-Linearized MPPL Method
Case 3: U Based X-Linearized MPPL Method
Case 4: Modified U Based X-Linearized MPPL Method
Case 6: HL-RF Method
Case 7: Modified HL-RF Method
Case 8: Improved HL-RF Method
Case 9: Sequential Quadratic Method
Case 10: Gradient Projection Method
Case 11: Modified Gradient Projection Method
Case 12: Non-Gradient Based Method (Simulation Method)
Anal2:
Calculate the failure probability using the SORM Curvature-Fitting
Method.
Anal3:
Calculate the failure probability using the SORM Point-Fitting method.
Anal4:
Calculate the failure probability using Monte Carlo Simulation (exact
solution method).
Anal5:
CDF/PDF analysis with First-order Pf=(0.0001,0.9999) and minimum 10 points
in the curve.
Case 1: By Modified U Based/X Linearized MPPL Method.
Case 2: By Non-Gradient Based MPP Searching (Simulation) method.
The G-function in this case is
highly nonlinear and has more than one MPP. This nonlinearity and multi MPPs
might cause some the optimization techniques long time to converge or some might
even not converge. Note also that for simplicity the random variables are
assumed to standard normal variables. The X-space and U-space are identical. The
graph of G=0 is plotted in the next figure. This plot can be used to interpret
various results obtained from analysis using UNIPASS.
Figure 1
G-function Plot in X-space
The graph shows the highly nonlinear nature of the
G-function. The function quickly grows to infinity on either sides of the
origin. This might cause some trouble if the MPP search algorithms are in this
region.
Analysis 1: Failure Probability
using FORM
|
CASE
|
MPP
Method
|
Gradient
Convergent
|
MPP
|
first-order
Pf
|
No. of g
|
No. of Ñg
|
No. of
Iter.
|
|
Case
1
|
U/U
|
No
|
Not
Converging
|
Not
Converging
|
|
|
20
|
|
Case
2
|
Mod.
U/U
|
Yes
|
(-1.35507,2.92316)
|
6.3655965E-04
|
46
|
7
|
7
|
|
Case
3
|
U/X
|
No
|
Not
Converging
|
Not
Converging
|
|
|
20
|
|
Case
4
|
Mod.
U/X
|
Yes
|
(-1.35512,2.92306)
|
6.3671145E-04
|
46
|
7
|
7
|
|
Case
6
|
HL-RF
|
Yes
|
Not
Converging
|
Not
Converging
|
|
|
20
|
|
Case
7
|
Mod.
HL-RF
|
Yes
|
(0.160391,3.97726)
|
3.4385824E-05
|
41
|
15
|
5
|
|
Case
8
|
Imp.
HL-RF
|
Yes
|
Not
Converging
|
Not
Converging
|
|
|
20
|
|
Case
9
|
SQM
|
Yes
|
Not
Converging
|
Not
Converging
|
|
|
20
|
|
Case
10
|
GPM
|
Yes
|
(-0.28877,3.9979)
|
3.05798E-05
|
72
|
17
|
9
|
|
Case
11
|
Mod.
GPM
|
Yes
|
Not
Converging
|
Not
Converging
|
|
|
20
|
|
Case
12
|
Sim.
Method
|
NA
|
(-1.32596,2.93678)
|
6.359579E-04
|
47
|
0
|
4 |
Notes: Most of the methods fail to identify a MPP point and do
not converge. This is attributed to the fact that the function is highly
nonlinear and thus effecting convergence various algorithms. Again the
Simulation method (non-gradient based MPP searching method) provides an accurate
alternative to gradient based approaches for calculating failure probability and
MPP.
Analysis 2: Probability Analysis Using
SORM Curvature fitting.
Using SORM Curvature Fitting probability analysis the
second order failure probability is obtained as 2.0991475E-04. The MPP
identification is done using Mod. U-based/X-Linearization method. Note that the
failure probability is different compared to that of probability of failure
using FORM, although they are of the same order. FORM based approach is more
conservative but SORM is more accurate.
Analysis 3: Probability
Analysis Using SORM Point fitting.
Using SORM Point Fitting probability analysis the second
order failure probability is obtained as 2.9972265E-04. The MPP
identification is done using Mod. U-based/X-Linearization method. Note that the
failure probability is different compared to that of probability of failure
using FORM, although they are of the same order. FORM based approach is more
conservative but SORM is more accurate. In this particular problem probably the
point-fitting approximation of Limit State function is better approximation
compared to Curvature fitting of the Limit State Function.
Analysis 4: Probability Analysis Using
Simulation Method
Using the Monte Carlo Simulation
method the accurate value of failure probability can be calculated. The analysis
results from 2237690 sample points and G-function calls gives a failure
probability value of 1.7875577E-04 with COV equal to 4.9995542E-02.
This value can be considered the exact estimate of failure probability
for this highly nonlinear probability problem. The CDF/PDF of g is a by product
of Monte Carlo Simulation which is given as
SUMMARY OF PDF/CDF RESULTS BY SIMULATION *****
Based on 2237690 Monte Carlo simulations,
Limit-State Function: 6
Number of points in each curve: 75
|
g(x)-g_init
|
PDF
of g(x)-g_init
|
CDF
of g(x)-g_init
|
|
-1.3164467E+00
|
2.9293073E-07
|
1.8206894E-06
|
|
-7.2810635E-01
|
2.7462256E-06
|
2.0710342E-05
|
|
-1.3976604E-01
|
2.0541767E-05
|
1.6545515E-04
|
|
4.4857426E-01
|
1.2266474E-04
|
1.0555447E-03
|
|
1.0369146E+00
|
5.3441550E-04
|
5.1395786E-03
|
|
1.6252549E+00
|
1.7495654E-03
|
1.9335494E-02
|
|
2.2135952E+00
|
4.3883220E-03
|
5.7485082E-02
|
|
2.8019355E+00
|
8.3846294E-03
|
1.3687442E-01
|
|
3.3902758E+00
|
1.2143809E-02
|
2.6446743E-01
|
|
3.9786161E+00
|
1.3430618E-02
|
4.2342340E-01
|
|
4.5669564E+00
|
1.1396580E-02
|
5.7773502E-01
|
|
5.1552967E+00
|
7.6086193E-03
|
6.9586044E-01
|
|
5.7436370E+00
|
4.3113911E-03
|
7.6994838E-01
|
|
6.3319773E+00
|
2.4352796E-03
|
8.1188182E-01
|
|
6.9203176E+00
|
1.5669963E-03
|
8.3675767E-01
|
|
7.5086579E+00
|
1.1754578E-03
|
8.5380319E-01
|
|
8.0969982E+00
|
9.6527998E-04
|
8.6710879E-01
|
|
8.6853385E+00
|
8.1914416E-04
|
8.7819975E-01
|
|
9.2736788E+00
|
7.1204136E-04
|
8.8771672E-01
|
|
9.8620191E+00
|
6.2368613E-04
|
8.9601884E-01
|
|
1.0450359E+01
|
5.6041310E-04
|
9.0337852E-01
|
|
1.1038700E+01
|
5.0255928E-04
|
9.0998534E-01
|
|
1.1627040E+01
|
4.5499465E-04
|
9.1593695E-01
|
|
1.2215380E+01
|
4.0823558E-04
|
9.2130229E-01
|
|
1.2803721E+01
|
3.7231495E-04
|
9.2615375E-01
|
|
1.3392061E+01
|
3.4745246E-04
|
9.3062741E-01
|
|
1.3980401E+01
|
3.2248011E-04
|
9.3479133E-01
|
|
1.4568742E+01
|
2.9175900E-04
|
9.3860908E-01
|
|
1.5157082E+01
|
2.7268189E-04
|
9.4211733E-01
|
|
1.5745422E+01
|
2.5550883E-04
|
9.4540026E-01
|
|
1.6333762E+01
|
2.3427135E-04
|
9.4844445E-01
|
|
1.6922103E+01
|
2.2541019E-04
|
9.5130157E-01
|
|
1.7510443E+01
|
2.0640631E-04
|
9.5398549E-01
|
|
1.8098783E+01
|
2.0131664E-04
|
9.5651966E-01
|
|
1.8687124E+01
|
1.8388726E-04
|
9.5891387E-01
|
|
1.9275464E+01
|
1.7572182E-04
|
9.6114899E-01
|
|
1.9863804E+01
|
1.6272302E-04
|
9.6325257E-01
|
|
2.0452145E+01
|
1.5777981E-04
|
9.6524463E-01
|
|
2.1040485E+01
|
1.4503733E-04
|
9.6712677E-01
|
|
2.1628825E+01
|
1.4302343E-04
|
9.6891719E-01
|
|
2.2217166E+01
|
1.3412566E-04
|
9.7063979E-01
|
|
2.2805506E+01
|
1.2390970E-04
|
9.7224359E-01
|
|
2.3393846E+01
|
1.2109024E-04
|
9.7376637E-01
|
|
2.3982186E+01
|
1.1120383E-04
|
9.7521018E-01
|
|
2.4570527E+01
|
1.1285156E-04
|
9.7660278E-01
|
|
2.5158867E+01
|
1.0333131E-04
|
9.7794644E-01
|
|
2.5747207E+01
|
9.8388108E-05
|
9.7920022E-01
|
|
2.6335548E+01
|
9.3664600E-05
|
9.8039391E-01
|
|
2.6923888E+01
|
8.9160790E-05
|
9.8153024E-01
|
|
2.7512228E+01
|
8.5169609E-05
|
9.8261378E-01
|
|
2.8100569E+01
|
8.1691057E-05
|
9.8365089E-01
|
|
2.8688909E+01
|
7.7260479E-05
|
9.8463884E-01
|
|
2.9277249E+01
|
7.8285737E-05
|
9.8560563E-01
|
|
2.9865590E+01
|
7.2903135E-05
|
9.8654533E-01
|
|
3.0453930E+01
|
6.8802105E-05
|
9.8742609E-01
|
|
3.1042270E+01
|
6.9754129E-05
|
9.8828728E-01
|
|
3.1630610E+01
|
6.4774307E-05
|
9.8912343E-01
|
|
3.2218951E+01
|
6.3602584E-05
|
9.8992135E-01
|
|
3.2807291E+01
|
5.9977566E-05
|
9.9068945E-01
|
|
3.3395631E+01
|
5.6828561E-05
|
9.9141545E-01
|
|
3.3983972E+01
|
5.5327291E-05
|
9.9211255E-01
|
|
3.4572312E+01
|
5.3972486E-05
|
9.9279189E-01
|
|
3.5160652E+01
|
5.2434600E-05
|
9.9345326E-01
|
|
3.5748993E+01
|
4.9065897E-05
|
9.9408412E-01
|
|
3.6337333E+01
|
4.5770426E-05
|
9.9467357E-01
|
|
3.6925673E+01
|
4.7235080E-05
|
9.9525164E-01
|
|
3.7514013E+01
|
4.4745169E-05
|
9.9582334E-01
|
|
3.8102354E+01
|
4.3866376E-05
|
9.9637410E-01
|
|
3.8690694E+01
|
4.3060817E-05
|
9.9691439E-01
|
|
3.9279034E+01
|
4.1046918E-05
|
9.9743715E-01
|
|
3.9867375E+01
|
3.9472415E-05
|
9.9793761E-01
|
|
4.0455715E+01
|
3.9142868E-05
|
9.9842624E-01
|
|
4.1044055E+01
|
3.7934529E-05
|
9.9890531E-01
|
|
4.1632396E+01
|
3.4675675E-05
|
9.9935661E-01
|
|
4.2220736E+01
|
3.4419360E-05
|
9.9957054E-01
|
PDF of g by Monte Carlo Simulation: 
CDF of g by Monte Carlo simulation:
Analysis 5: CDF/PDF analysis with First-order
Pf=(0.0001,0.9999) and minimum 10 points in the curve
Case 1: By Modified U Based/Xlinearized MPPL Method.
*****
SUMMARY OF CDF ANALYSIS *****
First-Order CDF of Limit-State
Function 6
Number of points in each CDF curve: 17
|
1st-order
b
|
1st-order
Pf
|
G-Value
= g(x)-g_init
|
|
3.7190178E+00
|
9.9999482E-05
|
-5.4330453E-01
|
|
3.3262541E-01
|
3.6970853E-01
|
3.6161201E+00
|
|
-1.0805964E+00
|
8.6006167E-01
|
7.7641783E+00
|
|
-1.3408015E+00
|
9.1000755E-01
|
1.1931950E+01
|
|
-1.5142131E+00
|
9.3501410E-01
|
1.6093166E+01
|
|
-1.6484930E+00
|
9.5037423E-01
|
2.0253193E+01
|
|
-1.7597110E+00
|
9.6077159E-01
|
2.4412875E+01
|
|
-1.8554778E+00
|
9.6823599E-01
|
2.8572427E+01
|
|
-1.9400658E+00
|
9.7381415E-01
|
3.2732004E+01
|
|
-2.4116777E+00
|
9.9206034E-01
|
6.5975912E+01
|
|
-2.7205910E+00
|
9.9674173E-01
|
9.9270379E+01
|
|
-2.9574208E+00
|
9.9844888E-01
|
1.3255204E+02
|
|
-3.1523962E+00
|
9.9919032E-01
|
1.6583008E+02
|
|
-3.3195874E+00
|
9.9954925E-01
|
1.9910676E+02
|
|
-3.4668036E+00
|
9.9973666E-01
|
2.3238286E+02
|
|
-3.5988723E+00
|
9.9984020E-01
|
2.6565867E+02
|
|
-3.7190178E+00
|
9.9990000E-01
|
2.9893504E+02
|
CDF plot of g by Modified U Based X Linearized MPPL Method:
***** SUMMARY OF PDF ANALYSIS *****
First-Order PDF of Limit-State Function
6
|
1st-order
b
|
1st-order
Pf
|
G-Value
= g(x)-g_init
|
|
-9.2976187E-01
|
3.6804255E-04
|
-5.4330453E-01
|
|
-9.8283997E-01
|
3.7099476E-01
|
3.6161201E+00
|
|
-8.3075332E-02
|
1.8485099E-02
|
7.7641783E+00
|
|
-4.8818306E-02
|
7.9271413E-03
|
1.1931950E+01
|
|
-3.6035926E-02
|
4.5683733E-03
|
1.6093166E+01
|
|
-2.9096825E-02
|
2.9829894E-03
|
2.0253193E+01
|
|
-2.4660746E-02
|
2.0917120E-03
|
2.4412875E+01
|
|
-2.1547260E-02
|
1.5371210E-03
|
2.8572427E+01
|
|
-1.9225255E-02
|
1.1680768E-03
|
3.2732004E+01
|
|
-1.0946362E-02
|
2.3834742E-04
|
6.5975912E+01
|
|
-7.9753124E-03
|
7.8599067E-05
|
9.9270379E+01
|
|
-6.3924009E-03
|
3.2161107E-05
|
1.3255204E+02
|
|
-5.3915265E-03
|
1.4951990E-05
|
1.6583008E+02
|
|
-4.6942733E-03
|
7.5786089E-06
|
1.9910676E+02
|
|
-4.1770892E-03
|
4.0921382E-06
|
2.3238286E+02
|
|
-3.7762005E-03
|
2.3200629E-06
|
2.6565867E+02
|
|
-3.4552112E-03
|
1.3677317E-06
|
2.9893504E+02
|
PDF Plot of g by Modified U Based X Linearized MPPL Method:
No. of g-functions called = 202 No. of derivatives of g called = 39
Case 2: By Non-Gradient Based MPP Searching Method
*****
SUMMARY OF CDF ANALYSIS *****
First-Order CDF of Limit-State
Function 6
Number of points in each CDF curve: 17
|
1st-order
b
|
1st-order
Pf
|
G-Value
= g(x)-g_init
|
|
3.7190178E+00
|
9.9999482E-05
|
-5.4294696E-01
|
|
7.5830520E-01
|
2.2413415E-01
|
3.6164732E+00
|
|
-1.0812919E+00
|
8.6021636E-01
|
7.7842368E+00
|
|
-1.3416101E+00
|
9.1013877E-01
|
1.1925864E+01
|
|
-1.5144486E+00
|
9.3504395E-01
|
1.6089105E+01
|
|
-1.6485816E+00
|
9.5038331E-01
|
2.0250278E+01
|
|
-1.7596167E+00
|
9.6076359E-01
|
2.4416894E+01
|
|
-1.8556403E+00
|
9.6824757E-01
|
2.8570446E+01
|
|
-1.9402913E+00
|
9.7382785E-01
|
3.2723644E+01
|
|
-2.4119425E+00
|
9.9206611E-01
|
6.6010126E+01
|
|
-2.7205971E+00
|
9.9674179E-01
|
9.9286590E+01
|
|
-2.9574624E+00
|
9.9844909E-01
|
1.3254828E+02
|
|
-3.1524138E+00
|
9.9919037E-01
|
1.6582341E+02
|
|
-3.3195981E+00
|
9.9954926E-01
|
1.9909815E+02
|
|
-3.4668139E+00
|
9.9973667E-01
|
2.3237264E+02
|
|
-3.5988850E+00
|
9.9984021E-01
|
2.6564696E+02
|
|
-3.7190178E+00
|
9.9990000E-01
|
2.9892231E+02
|
CDF Plot of g by Non-Gradient Base MPP Searching Method:
The PDF Plot of g by Non-Gradient Based MPP Searching Method is
obtained by derivative of CDF curve as shown below.
No. of g-functions called = 501 No. of derivatives of g called = 0
| 
