ART for safety assessment

In its current implementation, ART can perform the following:

• First Order Reliability Method (FORM)

• Second Order Reliability Method (SORM)

• Monte Carlo Simulation (MCS)

• Importance Sampling (IS)

• Subset Simulation (SubsetSim)

• Line Sampling (LS)

• ARBIS

• Simulation of correlated vector random variables

• MCMC using Metropolis-Hastings and component-wise Metropolis-Hastings algorithms.

Downloads:    art 1.07 Mac  ,  art 1.07 Linux ,  art 1.07 Windows          

If using MinGW, please comment #include "getline.h" in main.c and art.c files.

Click here for a short video that guides Windows 10 users to install and run ART using MinGW

Practice examples   

ART Theory 

For example, consider three random variables (mean,coefficient of variation)

X1~Lognormal(40,0.125)                  X2~Lognormal(50,0.05)         X3~Gumbel(1000,0.2)

The input file should be as follows:

3

2      2      5

40     50     1000

0.125  0.05   0.2

1      0.4    0

0.4    1      0

0      0      1

Identity matrix is assumed by default if the correlation matrix is not provided. When the random variables are standard normal and there exists no correlation between the random variables, the input file may contain only two lines. This feature is especially useful while solving time variant reliability problems when only the external excitation is random - for example, the white noise or filtered white noise can be represented in terms of standard normal random variables.

Line 1: dimension of the random variable vector

Line 2: 0

NOTE: The input file should contain only numeric values.

Please make sure that the first random variable is correlated with any of the other random variables when dealing with correlated random variables.

Go through these examples to get a hang of defining complicated performance functions.

Example 1 - a simple performance function

Highly nonlinear performance function from V. S. Sundar and M . D. Shields, 2016, Surrogate-enhanced stochastic search algorithms to identify implicitly defined functions for reliability analysis, Structural Safety, 62, 1-11.

G=4-x[0]/4+sin(5*x[0])-x[1];

Example 2 - performance function using loops

S. Engelund, R. Rackwitz, 1993, A benchmark study on importance sampling techniques in structural reliability, Structural safety 12 (4), 255–276.

int i;

double summ=0;

for (i=0;i<n;i++)

    summ=summ+x[i];

G=3-summ/sqrt(n);

Example 3 - complicated performance function

2dof primary secondary system from Der Kiureghain and Stephano, 1991, Efficient algorithm for second-order reliability analysis, J. Eng. Mech., 117(12): 2904-23.

double mp=x[0];

double ms=x[1];

double kp=x[2];

double ks=x[3];

double etap=x[4];

double etas=x[5];

double Fs=x[6];

double S0=x[7];

double omgp=sqrt(kp/mp);

double omgs=sqrt(ks/ms);

double gamma_val=ms/mp;

double omga=(omgp+omgs)/2;

double etaa=(etap+etas)/2;

double theta=(omgp-omgs)/omga;

double E=(pi*S0/(4*etas*pow(omgs,3)))*(etaa*etas)*(etap*pow(omgp,3)+etas*pow(omgs,3))*omgp/((etap*etas*(4*pow(etaa,2)+pow(theta,2))+gamma_val*pow(etaa,2))*4*etaa*pow(omga,4));

G=Fs-3*ks*sqrt(E);

Running ART

After downloading the package, please compile it once.

In the command line type:

Mac:     gcc art.c -o art

Linux:   gcc art.c -o art -lm

Windows: gcc art.c -o art

This will create an ART executable file. Now you can run the procedures using the below commands:

Mac & Linux:  ./art FLAG VALUE

Windows:      art FLAG VALUE

You can combine the above two into a single step as well

Mac & Linux:  gcc art.c -o art && ./art FLAG VALUE

Windows:      gcc art.c -o art && art FLAG VALUE

Now use the following flags to obtain estimates using various methods. 

MONTE CARLO SIMULATION

./art -input input.txt -pfn pfn.txt -method mcs -N 10000

FIRST ORDER RELIABILITY METHOD

./art -input input.txt -pfn pfn.txt -method form -algo hlrf -ndp 4

SECOND ORDER RELIABILITY METHOD

./art -input input.txt -pfn pfn.txt -method sorm -ndp 2

SUBSET SIMULATION

./art -input input.txt -pfn pfn.txt -method subsetsim -ntrial 10 -N 500 -pp 0.1 -proposal uniform -width 1 -factor 0.9

./art -input input.txt -pfn pfn.txt -method subsetsim -ntrial 10 -N 500 -pp 0.1 -proposal normal   -width 1 -factor 0.9

IMPORTANCE SAMPLING

End results from FORM, Subset Simulation, and ARBIS are used to define the importance sampling pdfs.

ART 1.04 onward has an option of using Markov chains to explore the space and obtaining the important regions.

When using subset simulation, use a big number (>5) for the number of design points. This is because, the method yields correlated samples, and it is likely that the best samples are not unique!

./art -input input.txt -pfn pfn.txt -method impsamp -ntrial 10 -N 1000 -dpmethod form     -ndp 3

./art -input input.txt -pfn pfn.txt -method impsamp -ntrial 10 -N 1000 -dpmethod subsetsim -ndp 10

./art -input input.txt -pfn pfn.txt -method impsamp -ntrial 10 -N 1000 -dpmethod arbis     -ndp 7

./art -input input.txt -pfn pfn.txt -method impsamp -ntrial 10 -N 1000 -dpmethod mc       -ndp 5

LINE SAMPLING

End results from FORM, Subset Simulation, and ARBIS are used to define the important directions.

ART 1.04 onward has an option of using Markov chains to explore the space and obtaining the important regions.

When using subset simulation, use a big number (>5) for the number of design points. This is because, the method yields correlated samples, and it is likely that the best samples are not unique!

./art -input input.txt -pfn pfn.txt -method linesamp -ntrial 10 -N 1000 -dpmethod form     -ndp 3

./art -input input.txt -pfn pfn.txt -method linesamp -ntrial 10 -N 1000 -dpmethod subsetsim -ndp 10

./art -input input.txt -pfn pfn.txt -method linesamp -ntrial 10 -N 1000 -dpmethod arbis     -ndp 7

./art -input input.txt -pfn pfn.txt -method linesamp -ntrial 10 -N 1000 -dpmethod mc         -ndp 5

ARBIS

./art -input input.txt -pfn pfn.txt -method arbis -N 10000

SIMULATING CORRELATED NON-GAUSSIAN RANDOM VARIABLES

./art -input input.txt -method simulate -N 10000 -output /Users/Sundar/xsim.txt

MCMC USING METROPOLIS-HASTINGS

The input file should contain two lines -

Line 1: number of variables used to define the target density function

Line 2: 0

For the target density function p=w1*N(x;0,1)+w2*N(x;4,1), the input.txt looks like this:

1

0

./art -input input.txt -method mcmc -target target.txt -sampler mh   -proposal normal   width 1 -N 10000 -nlag 1000 -output /Users/Sundar/xsim_mh.txt

./art -input input.txt -method mcmc -target target.txt -sampler mmh -proposal uniform  width 1 -N 10000 -nlag 100    -output /Users/Sundar/xsim_mmh.txt

References

Au SK and Beck JL, 2001, Estimation of small failure probabilities in high dimensions by subset simulation, Probabilistic Engineering Mechanics, 16, 263-277.

Breitung K, 1984, Asymptotic approximation for multinormal integrals, Journal of Engineering Mechanics, 110(3), 357-366.

Engelund S and Rackwitz R, 1993, A benchmark study on importance sampling techniques in structural reliability, Structural Safety, 12, 255-276.

Grooteman F, 2008, Adaptive radial-based importance sampling method for structural reliability, Structural Safety, 30, 533-542.

Liu JS, 2001, Monte Carlo Strategies in Scientific Computing, Springer, New York.

Madsen HO, Krenk S and Lind NC, 1986, Methods of Structural Safety, Prentice-Hall, New Jersey.

Melchers RE, 1999, Structural Reliability Analysis and Prediction, 2nd edition, John Wiley & Sons, Chichester.

Schuëller GI, Pradlwarter, HJ, and Koutsourelakis PS, 2004. A critical appraisal of reliability estimation procedures for high dimensions. Probabilistic engineering mechanics, 19(4), 463-474.

Sundar VS, 2013, Monte Carlo simulations with variance reduction for structural reliability modeling, updating and testing, PhD Thesis, Department of Civil Engineering, Indian Institute of Science, Bangalore.

Acknowledgements

Professor C S Manohar, IISc Bangalore, my PhD advisor.

Dr. Kayalvizhi Madhivanan, UCSD (past TSRI), my wife and fellow researcher.

Updates

April 2, 2018

• ART  version 1.02 onward uses flags to specify the parameters for the methods.

• FORM,SORM and Importance Sampling can now handle multiple design points.

• Line Sampling and ARBIS needs fine tuning.

• Practice examples.

April 9, 2018

• Markov chain based method to explore the space and determine the important regions for importance and line sampling.

  • Option "mc" for -dpmethod

• MCMC simulation using Metropolis-Hastings algorithm.

  • Option "mh" for -sampler

• Improvement in speed due to better memory allocation.

April 11, 2018

• MCMC simulation using component-wise Metropolis-Hastings algorithm.

  • Option "mmh" for -sampler

July 12, 2018

• SORM code updated.

• Included numerical illustration from Breitung (1984) in Practice examples.

July 26, 2018

• Included numerical illustration on reliability of structures subjected to fire.

June 7, 2020

• Included source code compatible for Windows gcc compiler (say MinGW)

Jan 8, 2021 New!

• Updated QR algorithm for SORM.

Citation:

If you use ART in any published work, please cite the software as an electronic resource/URL.

V. S. Sundar, 2018, ART for safety assessment, available from https://sites.google.com/site/sundarvelkur/art-for-safety-assessment.