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EFFECT OF SLIGHT PERTURBATION IN FORM/SORM RESULTS

Table of Contents

  1. Description

  2. Limit State Function

  3. Random Variables

  4. Analysis

  5. Results

  6. Reference

 

Description:

Consider the limit-state functions defined by Case 1 and Case 2. This problem demonstrates the effects of slight perturbation on the efficiency, stability and accuracy of FORM/SORM when the limit-state function is highly nonlinear. To examine the geometric effect of limit-state function, the problem is defined directly in the standard transformed space and the random variables are assumed Standard Normal.

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Limit-State Function:

Without Perturbation:                                         

g7(X) = 3 - (X1)2 + 2(X1)4 - X2

( 1)

 With Perturbation: 

g8(X) = 3 - (X1 + 0.01)2 + 2(X1 + 0.01)4-X2

( 2)

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Random Variables:

X1 ~ Normal(m=0, s=1)

X2 ~ Normal(m=0, s=1)

X1 and X2 are independent

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 Analysis:

Anal1: Perform Probability Analysis

Anal1a: Compute the failure probability for the limit-state function defined in Eq. 1 using different MPP Identification methods in FORM.

            Case 1: U Based U Linearized MPP 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)

Anal1b: Compute the failure probability for the limit-state function defined in  Eq. 2 using different MPP Identification methods.

             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: Perform Inverse Probability Analysis

Anal2a: Perform the Inverse Probability Analysis to find the level of limit-state function defined in Eq. 1 by FORM for Pf = .0002.

            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 12: Non-Gradient Based Method (Simulation Method)

Anal2b: Perform the Inverse Probability Analysis to find the level of limit-state function defined in Eq. 2 by FORM for Pf = .0002.

            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 12: Non-Gradient Based Method (Simulation Method)

Anal3: Perform Probability Analysis to verify the results of Anal2.

Anal3a: Perform Probability Analysis for g(x)£ g(x*) where x* is the MPP found in Anal2a.

             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 12: Non-Gradient Based Method (Simulation Method)

Anal3b: Perform Probability Analysis for g(x)£ g(x*) where x* is the MPP found in Anal2b.

                 Case 1: U Based U-Linearized MPPL Method

                 Case 2: Modified U Based U-Linearized MPPL Method

             Case 4: Modified U Based X-Linearized MPPL Method 

             Case 12: Non-Gradient Based Method (Simulation Method)

Anal4: Probability Analysis by SORM to investigate the effect of perturbation on the second-order failure probability.

Anal4a: With limit-state function defined in Eq. 1.

   Case1: Repeat Anal1a using the SORM Curvature-Fitting method.

Case2: Repeat Anal1a using the SORM Point-Fitting method.

Anal4b: With limit-state function defined in Eq. 2.

Case1: Repeat Anal1b using the SORM Curvature-Fitting method.

Case2: Repeat Anal1b using the SORM Point-Fitting method.

 Anal5: Probability Analysis by Simulation Methods to investigate the effect of perturbation on the second-order failure probability.

Anal5a: Repeat Anal1a, i.e., Eq. 1, using the Simulation method (exact solution).

Anal5b: Repeat Anal1b, i.e., Eq. 2,  using the Simulation method (exact solution).

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Results:

The following figure shows the plot of the two Limit State function in X-space (same as Normal Space). Note that the functions are almost identical. The Limit State function in Case 1 is an even function and symmetric with respect to X2 axis. But Limit State function in Case 2 is not symmetric with respect to origin.

 

The green curve is G7 and the red curve is G8. Note also that because of the curvilinear nature of the Limit State function there is a possibility that more than one MPP points exist on the Limit State function with similar distances from origin (that is same reliability index beta).

Figure 1-1Limit State Function G7 and G8

 Analysis 1a: FORM probability analysis on the Limit State function G7 and their comparison 

CASE

MPP Method

Gradient Convergent

MPP of G7

Pf of G7

No of g

No ofÑg

No of Iter.

Case 1

U/U

Yes

(0.0003,3.0000)

1.3498981E-03

6

2

2

Case 2

Mod. U/U

Yes

(0.0003,3.0000)

1.3498981E-03

6

2

2

Case 3

U/X

Yes

(0.0003,3.0000)

1.3498981E-03

4

1

2

Case 4

Mod. U/X

Yes

(0.0003,3.0000)

1.3498981E-03

4

1

2

Case 6

HL-RF

Yes

(0.0003,3.0000)

1.3498981E-03

6

2

2

Case 7

Mod. HL-RF

Yes

(0.0003,3.0000)

1.3498981E-03

8

3

2

Case 8

Imp. HL-RF

Yes

(0.0012,3.0000)

1.3498989E-03

12

3

2

Case 9

SQM

Yes

(0.45535,2.87854)

1.7822549E-03

98

23

12

Case 10

GPM

Yes

(0.0003,3.0000)

1.3498981E-03

12

3

2

Case 11

Mod. GPM

Yes

(0.0003,3.0000)

1.3498981E-03

4

1

2

Case 12

Sim. Method

NA

(-0.41083,2.88819)

1.7655804E-03

41

0

4

 

Analysis 1b: FORM probability analysis on the Limit State function G and their comparison

CASE

MPP Method

Gradient Convergent

MPP of G8

Pf of G8

No of g

No ofÑg

No. of Iter

Case 1

U/U

Yes

Not-Converging

Not-Converging

 

 

20

No

(0.06025,2.99869)

1.3530271E-03

4

1

2

Case 2

Mod. U/U

Yes

(0.445529,2.8786)

 
1.7904780E-03

29

5

5

No

(0.06025,2.99869)

1.3530271E-03

4

1

2

Case 3

U/X

Yes

(0.06025,2.99869)

1.3530271E-03

4

1

2

No

(0.06025,2.99869)

1.3530271E-03

4

1

2

Case 4

Mod. U/X

Yes

(0.06025,2.99869)

1.3530271E-03

4

1

2

No

(0.06025,2.99869)

1.3530271E-03

4

1

2

Case 6

HL-RF

Yes

Not-Converging

Not-Converging

 

 

20

Case 7

Mod. HL-RF

Yes

(0.445534,2.88837)

1.7359893E-03

199

71

15

Case 8

Imp. HL-RF

Yes

(0.44987,2.87847)

1.791802E-03

92

15

15

Case 9

SQM

Yes

(0.44861,2.87853)

1.7915713E-03

83

19

10

Case 10

GPM

 Yes

Not-Converging

Not-Converging

 

 

20

Case 11

Mod. GPM

 Yes

(0.06025,2.99869)

1.3530271E-03

4

1

2

Case 12

Sim. Method

 NA

(-0.411226,2.89085)

1.7504344E-03

41

0

4

 Notes: 

  • Most of the MPP identification methods are affected by slight perturbation in the Limit State function because of symmetric and multi MPP properties. Some even do not converge as opposed to the convergence exhibited by the same method on the perturbed case. Also note that the simulation method gives similar values of MPP for both the cases of slight perturbed and without perturbation. Also the Pf is more from Simulation Method compared to that of the other methods.

  • Another important point to be noted is that the MPP point calculated by different methods varies a lot (See Figure 1). This is of not much consequence when using the FORM analysis because the reliability index in all these cases is quite close.

  •  Some observations in the Figure 1

    Limit State Function – G7: 

          MPP1: U/U, Mod. U/U, U/X, Mod. U/X, HL-RF, GPM, Mod. GP

MPP2: Mod. HL-RF           

     MPP3: SQM

    MPP4: Simulation Method

                  Limit State Function – G8: 

        MPP1: U/U, Mod. U/U, U/X, Mod. U/X 

   MPP2: Mod. HL-RF, Imp. HL-RF, SQM    

   MPP4: Simulation Method

       Other methods not listed along with G8 do not converge.

 Analysis 2a: Inverse probability analysis  of G7 by FORM Pf = 0.0002

CASE

MPP Method

Gradient Convergent

G7 Level

No. of g

No of Ñ

No of Iter.

 ase 1

U/U

Yes

-0.54008513

6

2

2

Case 2

Mod. U/U

Yes

-0.54008513

6

2

2

Case 3

U/X

Yes

-0.54008513

6

2

2

Case 4

Mod. U/X

Yes

-0.54008513

6

2

2

Case 12

Sim. Method

NA

-0.62823018

46

0

5

Analysis 2b: Inverse probability analysis of G8 by FORM Pf = 0.0002

CASE

MPP Method

Gradient Convergent

G8  Level

No. of g

No of Ñg

No. of Iter

Case 1

U/U

No

-0.54586339

7

2

    2

Case 2

Mod. U/U

Yes

-0.63344389

28

5

    5

Case 3

U/X

No

-0.54586339

7

2

    2

Case 4

Mod. U/X

Yes

-0.63344389

28

5

     5

Case 12

Sim. Method

NA

-0.63285789

46

0

5

Notes: Note from the results provided in the table above, the perturbation has affected stability of the inverse probability analysis for U/U and U/X MPPL Method. Also note that the results of U/U and U/X are identical because the random variables are standard normal distributed.

 Analysis 3a: Probability Verification of previous Inverse probability analysis (Analysis 2a)

CASE

MPP Method

Gradient Convergent

Pf of G7

No. of g

No of Ñg

No. of Iter

Case 1

U/U

Yes

1.9999900E-04

6

2

2

Case 2

Mod. U/U

Yes

1.9999900E-04

6

2

2

Case 3

U/X

Yes

1.9999900E-04

6

2

2

Case 4

Mod. U/X

Yes

1.9999900E-04

6

2

2

Case 12

Sim. Method

NA

2.0121351E-04

44

0

4

Analysis 3b:  Probability Verification of previous Inverse probability analysis (Analysis 2b)

CASE

MPP Method

Gradient Convergent

Pf of G8

No. of g

 No of Ñg

No. of Iter

Case 1

U/U

Yes

1.9627084E-04

4

1

2

Case 2

Mod. U/U

Yes

1.9999434E-04

30

5

5

Case 3

U/X

Yes

1.9627084E-04

4

1

2

Case 4

Mod. U/X

Yes

1.4028403E-04

4

1

2

Case 12

Sim. Method

NA

1.9750281E-04

44

0

  4

 Notes: 

  •   Anal3a – cases 1, 2, 3, 4, and 12 give the correct Pf value by using the corresponding G-value from Analysis 2a

  •  Anal3b – cases 1, 2, 3, 4, 12 give correct probability values for the corresponding G-levels from Analysis 2b.

Analysis 4a and 4b:  SORM Probability analysis to study effect of perturbation

CASE

MPP Method

SORM Method

Pf of G7

Pf of G8

Case 1

Mod. U/X

Curvature Fit

2.49227690E-0

2.15585630E-02

Case 2

Mod. U/X

Point Fit

 6.67892970E-04

6.69183610E-04

Notes: 

  • Curvature fit gives a much more sensitive to the perturbation of limit state function than Point Fitting Method.

  • From the Figure 1 it can be seen that around MPP1 the g-function has concave down curvature (negative curvature) and the fourth-order term of x1 will not affect the curvature for the Curvature Fitting method. Hence when curvature approximation of the g-function is done around MPP1 the approximate curve will be concave down and thus leading to much conservative estimate of Pf. 

Analysis 5: Probability analysis using Simulation Method 

ANALYSIS 

Sim. Method

Pf

No. of G-func

Analysis 5a

Monte Carlo

9.950000E-04

200000

Analysis 5b

Monte Carlo

1.000000E-03

200000

Notes:

  •    The simulation method gives the most accurate value of the probability of failure value.

  • From all the above analysis it is seen that the SORM-point fit gives Pf quite close to that of the value given by Simulation method.

  • There is not much effect of perturbation on Pf value using Simulation method.

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Reference:

D.H Ebbeler et al. “Alternative Computational Approaches for Probabilistic Fatigue Analysis.” 36th AIAA/ASME/ASCE Conference on Structures, Structural Dynamics and Materials, Section on Probabilistic Applications. New Orleans, April 1995.

 

 

Last Updated 02/08/10

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