Illustrative example

To demonstrate the capabilities of the proposed method, a synthetic example is pre- sented. The ALS method is compared to a solution obtained via global optimisation. Both approaches are applied to calculate the interval failure probabilityp_F. With ALS the argument optima are detected using the information of the important direction as explained in Section 4.3.1.2. The sign of the important direction in the original space identifies the conjugate states where the extrema of the failure probability are located. With the global optimisation approach, the search is conducted as an iterative procedure. The examples are solved using both Genetic Algorithm (GA) [31] and BOBYQA (Bounded Optimization BY Quadratic Approximation) [56], as global and local searchers, respectively. With this approach, a thorough search in the sets Θh and

Table 4.1: Results from case (a), argument optima and associated failure probabilities obtained by means of ALS and Global Optimisation.

g“7`ξ´2x;

ALS Global Opt. (GA) (BOBYQA)

pµmin, µmaxq p2, 0q p2, 0.04q p2, 0.12q

pσmin, σmaxq p1.2, 2.3q p1.28, 2.3q p1.32, 2.3q

pxmin, xmaxq p1, 3q p1, 3q p1, 2.92q

pF r2.717 10´9, 0.332s r2.702 10´8, 0.322s r4.371 10´7, 0.134s

any iteration of GA/BOBYQA, a reliability analysis with ALS is performed. This ap- proach can be performed in reasonable time, because adaptive algorithm requires just a few evaluations of the performance function to complete an iteration. Replacing ALS with direct Monte Carlo would lead to thousands of evaluations of the performance function for each iteration, making the approach via global optimisation intractable.

Two cases are considered in this study, i.e. case (a) and case (b).

Case (a): The considered performance function gpξ, xq “ 7`ξ´2x, includes the parametric p-box,ξ PC, where

C“ tfΞpξ;µ, σq | µP r0.9,1.3s, σP r0.7,2.1su,

and the interval variablex“ r1, 3s. In this illustrative case the gradient ∇g“ p1,´2q,

provides an initial guess for the important direction as a “ p1,´2q{?5. ALS

leads to the bounds of the failure probability and the associated argument optima

pxmin, xmaxq “ px, xq, and pmin “ pµ, σq, pmax “ pµ, σq, as shown in Table 4.1. With

the global optimisation approach, using GA with a population size of50individuals, an

approximation of the of lower and upper bound was obtained after 52 iterations, while

BOBYQA provided a slightly less accurate estimate. In this case ALS coincides with the closed-form solution and it is preferable above the global optimisation approach.

Case (b): The multidimensional performance functiongpξ,xq “9`ξTb1´xTb2 is

considered, where b1 PR14, andb2 PR3. The parametric p-boxes ξ PR14 are defined

by the Credal set

C “ fΞpξ;µ,σq | µPµ,σPσ (

,

whereµ“ r0.1, 1s14, andσ “ r1.2, 2.3s3, while the interval variablesxPR14 are de-

fined by the bounded set x“ r1, 3s3. Because of the monotonicity, the ALS approach

provides numerically exact results for the failure probability (equal to the closed-form solution). As expected, the global optimisation approach provides only a rough ap- proximation of the solution, as shown in Table 4.2, as it becomes inefficient when the dimensionality of the search domain is too large.

Table 4.2: Results from case (b), interval failure probability obtained by means of ALS and Global Optimisation.

g“9`ξTb1´xTb2;

b1 “ p1,4,2,0.1,0.2,0.6,5,0.01,0.2,0.3,0.25,0.14,0.8,3q, b2 “ p´2,0.1,1q

ALS Global Opt. (GA) (BOBYQA)

pF r1.795 10´9, 0.1452s r7.302 10´6, 0.0053s r2.538 10´5, 0.0046s

L

P

v

Figure 4.2: Cantilever beam subject to point load and thermal gradient

Note on the application of the approach The presented numerical example is

meant to illustrate the performance of the methods without being bounded to any specific application. However, the functions presented in case (a) and (b) can be easily linked to an engineering problem. For example, in the context of structural engineering, the well-known problem of calculating the tip displacement of a cantilever beam (see Figure 4.2) with a tip point load and a thermal gradient resembles very much the function of case (a). The tip displacementv is obtained as a function of the point load P and the curvature of the beam χp∆Tq, function of the temperature difference ∆T. v can be obtained as the sum of the displacement due to the point load vP and the

displacement due to the thermal gradientvT, asv“vP ´vT, as

v“vP ´vT “ P L3 48EI ´ α∆T h{2 L 2 {2. (4.11)

In Eq. (4.11) the curvature of the beam,χpTq “ α_h∆_{2T is proportional to the temperature

difference T and the coefficient of thermal expansion α. By fixing a threshold on the tip displacement ˜v, the following performance function can be defined

gpP, Tq “v˜´v“v˜´ P L 3 48EI ` α∆T h{2 L 2 {2“v˜´aP `bT; (4.12)

where, the point load and the temperature are the state variables of the problem and can be modelled using p-boxes, as it has been done in the illustrative example.