Tutorial 1b – Limit-state function formulated directly in Fesslix

In this tutorial we formulate the limit-state function of the problem introduced in Tutorial 1 directly in Fesslix. Subset Simulation is employed to estimate the probability of failure.

1   Step by Step Instruction

1.1   Constants

As in Tutorial 1a, we start by defining scalar variables.

Fesslix parameter file
const N = 100;     # number of random variables used in the problem
const la = 1;      # parameter of the exponential distribution
const Nsmp = 1e3;  # number of samples used per level in Subset Simulation
const pthr = 10%;  # probability of the intermediate Subset levels

1.2   Random variables

Before we can actually define the random variables, we have to create a set that can hold an arbitrary number of random variables. We assign the name RVS to this set:

Fesslix parameter file
rbrv_set_new RVS;

Next, we define the 100 exponentially distributed random variables.

Fesslix parameter file
sfor (i;N) {              # this is a loop that runs from 1 to N.
  rbrv_set_addRV RVS +=   # add a random variable to the set RVS
   exponential(           # the type of the random variable is 'exponential'
    rv_ & {i},            # the name of the random variable is, e.g., rv_1, rv_2, ...
    lambda = la           # we assign the value of 'la' to the parameter lambda.

Finally, we need to finalize the set with the command rbrv_set_create:

Fesslix parameter file
rbrv_set_create RVS;

After the above command is executed, no more random variables can be added to the set.

1.3   Limit-state function

In this tutorial, we define the limit-state function directly within Fesslix.

In the following we use the keyword var to define a scalar function that does not take any explicit parameters. The difference to a variable defined with the const-keyword is: var stores a function and evaluates the function anew each time the variable is needed.

Fesslix parameter file
var lsf = mtxsum(
    rbrv_vec_get x: tmpx = "RVS";
) - 60;

In the code above, the command rbrv_vec_get retrievs a vector containing the realizations of all random variables in the set RVS and assigns it to vector tmpx. The function mtxsum returns the sum of all coefficients of vector tmpx.

1.4   Subset Simulation

We tackle the reliability problem with Subset Simulation. Subset Simulation is contained in the module RND which needs to be loaded explicitly:

Fesslix parameter file
loadlib "RND";       # Subset Simulation is contained in the module 'RND'.

Finally, we can start Subset Simulation:

Fesslix parameter file
sus (
  Nsmp*pthr;         # The number of chains used at each level in Subset Simulation.
  1/pthr;            # The length of each chain.
  lsf                # The limit-state function.
) {
  rbrvsets = "RVS";  # The set of random variables to use.

2   The complete parameter file

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