eggn 598 – probabilistic biomechanics ch.7 – first order reliability methods anthony j petrella,...

EGGN 598 – Probabilistic Biomechanics

Ch.7 – First Order Reliability Methods

Anthony J Petrella, PhD

Review: Reliability Index

• To address limitations of risk-based reliability with greater efficiency than MC, we introduce the safety index or reliability index, b

• Consider the familiar limit state, Z = R – S, where R and S are independent normal variables

• Then we can write,

and POF = P(Z ≤ 0), which can be found as follows…

22; SRZSRZ

22

0

SR

SR

Z

Z

Z

Z

)(1)( POF0.00

0.15

0.30

0.45

-4 -3 -2 -1 0 1 2 3 4

f Z(z

)

Z Value (Limit State)

Review: MPP

FailureFailure

Safe

Safe

• Recognize that the point on any curve or n-dimensional surface that is closest to the origin is the point at which the function gradient passes through the origin

• Distance from the origin is the radius of a circle tangent to the curve/surface at that point (tangent and gradient are perpendicular)

Geometry of MPP

MPP → closest to the origin→ highest likelihood on joint PDF

in the reduced coord. space*

gradientdirection

gradient → perpendicularto tangent direction

Review: AMV Example

• For example, consider the non-linear limit state,

where,23

121

21 6)(

hip

fem

hip

ap

hiphip

hipfemaphip h

l

b

F

hb

hlF

I

Mc

)(65.836

)(08.0

)(012.006.0

)(05.066.0

NF

mb

mh

ml

ap

hip

hip

fem

Review: AMV Geometry

• Recall the plot depicts (1) joint PDF of l_fem and h_hip, and (2) limit state curves in the reduced variate space (l_fem’,h_hip’),

• To find g(X) at a certain prob level, we wish to find the g(X) curve that is tangent to a certain prob contour of the joint PDF – in other words, the curve that is tangent to a circleof certain radius b

• We start with the linearization ofg(X) and compute its gradient

• We look outward along the gradientuntil we reach the desired prob level

• This is the MPP because the linearg(X) is guaranteed to be tangent tothe prob contour at that point

-2

-1

0

1

2

-2 -1 0 1 2

h_hi

p' =

redu

ced

varia

te fo

r xse

ction

hei

ght

l_fem' = reduced variate for femur length

glinear_10%

glinear_90%

glinear_50%

1(0.9) 1.28

Review: AMV Geometry

• The red dot is (l_fem’*,h_hip’*), the tangency point for glinear_90%

• When we recalculate g90% at (l_fem’*,h_hip’*) we obtain an updated value of g(X) and the curve naturally passes through (l_fem’*,h_hip’*)

• Note however that the updatedcurve may not be exactly tangentto the 90% prob circle, so theremay still be a small bit of error(see figure below)

-2

-1

0

1

2

-2 -1 0 1 2

h_hi

p' =

redu

ced

vari

ate

for x

secti

on h

eigh

t


a

1(0.9) 1.28

MV10%

MV90%

AMV10%

AMV90%

Linear10%

Linear90%

AMV+ Method

• The purpose of AMV+ is to reduce the error exhibited by AMV• AMV+ simply translates to…“AMV plus iterations”• Recall Step 3 of AMV: assume an initial value for the MPP, usually at

the means of the inputs• Recall Step 5 of AMV: compute the new value of MPP

• AMV+ simply involves reapplying the AMV method again at the new MPP from Step 5

• AMV+ iterations may be continued until the change in g(X) falls below some convergence threshold

AMV+ Method (NESSUS)Assuming one is seeking values of the performance function (limit state) at various P-levels, the steps in the AMV+ method are:

1. Define the limit state equation

2. Complete the MV method to estimate g(X) at each P-level of interest, if the limit state is non-linear these estimates will be poor

3. Assume an initial value of the MPP, usually the means

4. Compute the partial derivatives and find alpha (unit vector in direction of the function gradient)

5. Now, if you are seeking to find the performance (value of limit state) at various P-levels, then there will be a different value of the reliability index bHL at each P-level. It will be some known value and you can estimate the MPP for each P-level as…

*

iXg

pHLi

p

iX a*

)(1 ppHL

AMV+ Method (NESSUS)

The steps in the AMV+ method (continued):

6. Convert the MPP from reduced coordinates back to original coordinates

7. Obtain an updated estimate of g(X) for each P-level using the relevant MPP’s computed in step 6

8. Check for convergence by comparing g(X) from Step 7 to g(X) from Step 2. If difference is greater than convergence criterion, return to Step 3 and use the new MPP found in Step 5.

• We will continue with the AMV example already started and extend it with the AMV+ method

Forward Difference TrialsTrial Variable Perturbation g dg/dX

ref = mean -- -- 1.1504E+07 --1 l_fem 0.05 1.2375E+07 1.7430E+072 h_hip 0.012 7.9889E+06 -2.9293E+08

Mean Value (MV) Method NESSUS 106 MCmean SD p g (hand) g g

1.1504E+07 3621533.28 0.05 5547090 5547074 63498870.1 6862800 6862856 71551180.5 11503982 11504017 114887430.9 16145164 16145012 20975184

0.95 17460874 17460908 25804032

AMV+ Example

Advanced Mean Value (MV) Method - Iteration 1 NESSUSdg/dX' alpha p g (hand) g

8.7151E+05 2.4065E-01 0.05 -- 6411184-3.5151E+06 -9.7061E-01 0.1 -- 7204743

0.5 -- 115039820.9 -- 20860680

0.95 -- 25572242

MPP Table: AMV Iteration 1, Level 4 (p = 0.90)0.05 0.10 0.50 0.90 0.95

-0.3958 -0.3084 0.0000 0.3084 0.3958 l_fem'1.5965 1.2439 0.0000 -1.2439 -1.5965 h_hip'0.6402 0.6446 0.6600 0.675420 0.6798 l_fem0.0792 0.0749 0.0600 0.045073 0.0408 h_hip

Mean Value Method AMV Method – Iteration 1

X = MPP-1

g(X)

• We will continue with the AMV example already started and extend it with the AMV+ method


8.7151E+05 2.4065E-01 0.05 -- 6411184-3.5151E+06 -9.7061E-01 0.1 -- 7204743

0.5 -- 115039820.9 -- 20860680

0.95 -- 25572242



Forward Difference Trials: AMV Iteration 1, Level 4 (p = 0.90)Trial Variable Perturbation g dg/dX

ref = MPP -- -- 2.0861E+07 --1 l_fem 0.05 2.2406E+07 3.0905E+072 h_hip 0.012 1.3011E+07 -6.5412E+08

AMV+ Example

AMV Method – Iteration 2AMV Method – Iteration 1


1.5453E+06 1.9316E-01 0.05 ---7.8494E+06 -9.8117E-01 0.1 --

0.5 --0.9 -- 20917711

0.95 --



X = MPP-2

g(X)

AMV+ Example

-2

-1

0

1

2

-2 -1 0 1 2

h_hi

p' =

redu

ced

varia

te fo

r xse

ction

hei

ght


a

1(0.9) 1.28

MV10%

MV90%

AMV10%

AMV90%

Linear10%

Linear90%


-2

-1

0

1

2

-2 -1 0 1 2

h_hi

p' =

redu

ced

vari

ate

for x

secti

on h

eigh

t


1(0.9) 1.28

MV10%

MV90%

AMV10%

AMV90%

Linear10%

Linear90%

AMV+ Example

-1.4

-1.3

-1.2

-1.1

-1.0

-0.9

0.0 0.1 0.2 0.3 0.4 0.5

h_hi

p' =

redu

ced

vari

ate

for x

secti

on h

eigh

t


90% Probability Contour

Linear-1 (about means)

Linear-2 (about MPP-1)

AMV-1 (g = 20860680)

AMV-2 (g = 20917711)

MPP-1MPP-2


-2

-1

0

1

2

-2 -1 0 1 2

h_hi

p' =

redu

ced

varia

te fo

r xse

ction

hei

ght


1(0.9) 1.28

MV10%

MV90%

AMV10%

AMV90%

Linear10%

Linear90%

AMV+ Example

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+0 2.5E+7 5.0E+7 7.5E+7

F G(g

)

Bending Stress in Hip (Pa)

10e6 Trials MC

3-Trial MV Method

8-Trial AMV Method

11-Trial AMV+ (90% only)

0.89

0.90

0.91

2.0E+7 2.1E+7 2.2E+7

F G(g

)

Bending Stress in Hip (Pa)

10e6 Trials MC

8-Trial AMV Method

11-Trial AMV+ (90% only)

eggn 598 – probabilistic biomechanics ch.7 – first order reliability methods anthony j petrella,...

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