Non Deterministic Approaches

In engineering design, the traditional deterministic design optimization model has been successfully applied to systematically improve the system design process, yielding a reduction of the costs and an improvement of the final quality of the products. However, uncertainties in either engineering simulations and/or manufacturing processes exist. This calls for different optimization models that can yield not only an improvement in the design, but also a higher level of confidence. Thus, a reliability-based design optimization (RBDO) model for robust and cost-effective designs can be defined using mean values of the random system variables as design parameters and optimizing the cost subject to prescribed probabilistic constraints (like probabilities of failure) by solving a mathematically nonlinear programming problem.

The general RBDO model can be defined as:

${(min_{d}{Cost[d(\mu_{X})]}),("subject to " P_{f,j}=P(G_{j}(x)<0)lt= bar P_{f,j} " with " j=1...m ):}$

(4)

where:

• the cost function can be any function of the mean values $mu_{X}$ of the input parameters and
  
• $bar P_{f,j}$ is the probabilistic constraint that can be defined for each failure mode j and needs to be satisfied.

For this optimization problem, the constraint definition is expressed in terms of probability distributions and thus needs to be evaluated, for each optimization step, within the probability framework. Thus, for each iteration of the optimization loop, an estimation of the probabilistic constraint in terms of its multidimensional integral (see Eq. (1) and (2) in this section) has to be computed. For this purpose, different methods exist. Most of them apply a transformation of the input parameter space X to the standard normal space Y. In this space, if the limit state functions are linear, each probability of failure can be represented in terms of the reliability index $beta_{j}$ as:

$P_{f,j}=P(G_{j}(x)<0) rArr P_{f,j}=P(G_{j}(y)<0)= Phi(-beta_{j}) rArr beta_{j}=-Phi^{-1}(P_{f,j})$

(5)

Where $Phi(*)$ is the standard normal CDF* (zero mean and standard deviation 1).

This equation can be generalized as:

$F_{G_{j}}(g_{j})=P(G_{j}(x)<g_{j})=Phi(-beta_{G_{j}})$

(6)

Where $F_{G_{j}}(*)$ is the CDF* of the $j^{th}$ system response.

The same approach can be used also to express the probabilistic constraint of Eq.(4) in a different notation. In this case, the probability of failure $P_{f,j}$ will be the target probability of failure $bar P_{f,j}$ and the reliability index $beta_{j}$ the target reliability index $beta_{t,j}$.

$bar P_{f,j}=Phi(-beta_{t,j}) rArr beta_{t,j}=-Phi^{-1}(bar P_{f,j})$

(7)

Using Eq.(6), the second condition of Eq.(4) can be rewritten as:

$P_{f,j}=F_{G_{j}}(0)=Phi(-beta_{j})lt=Phi(-beta_{t,j})=bar P_{f,j} rArr beta_{j}gt=beta_{t,j}$

(8)

The relation expressed in Eq.(8) in terms of the reliability index, can also be expressed in terms of the performance measure through inverse transformation. In fact, using Eq.(7) and Eq.(8), the target probability of failure can be expressed in terms of the target performance measure as

$bar g_{j} = F_{G_{j}}^{-1}(bar P_{f,j})=F_{G_{j}}^{-1}[Phi(-beta_{t,j})]$

(9)

where $bar g_{j}$ is named "target probabilistic performance measure". It represents the value of the performance function "equivalent" to the target reliability index $beta_{t,j}$.

The expression of the probabilistic constraint of Eq.(8) and (9) can be used in the optimization problem of Eq.(4) to equivalently replace the original definition.