General Reliability problem

   In the reliability theory, variabilities of an engineering design can be characterized by the variations of a random system parameter set. These random variables are modeled with probability distribution functions and are representative of the uncertain model parameters. Thus, given a set of random variables $x=[x_{i}]^{T}$ , with i=1…n, the probability distribution of each random variable $x_{i}$ is described either by its cumulative distribution function (CDF*, $F_{x_{i}}(x_{i})$) or by its probability density function (PDF*, $f_{x_{i}}(x_{i})$) and is often bounded by tolerance limits on the system parameter values. Because of these limits, unbounded probability density functions, like the Gaussian distribution, are generally not suggested, as they allow events with non-zero probability density even at values that are far off the nominal value.

   For each realization of the set of input random variables, the performance of the system has to be evaluated. This performance is generally described by Performance Functions (PF, $G_{j}(x)$ with j=1..m) that, for a structural design, are usually the selected failure criteria. Thus, if $G_{j}(x)$ is one of the m system PFs, the system is considered to fail if $G_{j}(x)<0$ for at least one index j. The probability of failure of the system for every jth performance function is then the multi dimensional integral of the joint PDF* function of the set of random variables over the failure domain $\Omega_{j}$. This is expressed by the integral:

$F_{G_{j}}(0)=P[G_{j}(x)<0]=int_{\Omega_{j}}...int f_{X}(x)dx_{1}...dx_{n}$ where $\Omega_{j}:{x in RR^{n}:G_{j}(x)<0}$

(1)

   The event space $\Omega_{j}$ is the region of the stochastic space where only failure events occur. Thus, the integral of the joint probability density function $f_{X}(x)$ over $\Omega_{j}$ yields the probability of failure $P_{f,j}$ of the structure for the j^th failure criterion.

$P_{f,j}=F_{G_{j}}(0)=P(G_{j}(x)<0)$ with j=1...m

   In general, given a fixed performance index (or measure) $g_{j}$ of the system, Eq (1) can be generalized by computing the probability of exceedance of the selected $g_{j}$ threshold.

$F_{G_{j}}(g_{j})=P[G_{j}(x)<g_{j}]=int_{\Omega_{G_{j}}}... int f_{X}(x)dx_{1}...dx_{n}$ where $\Omega_{G_{j}}:{x in RR^{n}:G_{j}<g_{j}}$

(2)

   In this case, the event space $\Omega_{G_{j}}$ represents the region of the stochastic space where the performance parameter of the structure is below the prescribed quantity $g_{j}$.

   The main aim of the reliability analysis is to estimate the probability of failure of a structure, given a set of input random variables and a set of failure criteria defining the safe and fail regions $\Omega_{j}$ or $\Omega_{G_{j}}$. This estimation is usually carried on by approximating the multi dimensional integral with appropriate techniques. These include sampling methods and limit state approximations, usually computed by transforming the problem definition from the parameter space X into the standard normal space Y.

   The total probability of failure of the structure is given by the integral of the joint PDF* $f_{X}(x)$ over the union of all the failure domains. These failure domains are defined by the corresponding Performance Functions $G_{j}(x)$ and their threshold values $g_{j}$ (performance indexes or measures).

Limit State Approximations

 In most industrial cases, the integrals of Eq.(1) and of Eq.(2) (ref. this section) cannot be evaluated in closed form. Only for simple academic cases $f_{X}$ can be defined analytically. In such a case, the performance function $G_{j}(x)$ is available for each specific failure mode of the structure so that $\Omega_{j}$ can be defined. The boundary of the domain $\Omega_{j}$ is also called limit-state surface and the function $G_{j}(x)$ is called the limit-state function (LSF).

 In the most widely used reliability methods, approximations are made in the space of standard uncorrelated normal variates, Y, obtained from a transformation of the basic variables

$Y=T(X) $

(3)

 where the transformation T is expressed in terms of the distributions of X.

 In the space Y, denoted as the standard normal space, approximations of the probability of failure $P_{f,j}$ of Eq.(1) (ref. this section) are obtained by replacing the limit state surface with first or second order approximating surfaces. These surfaces are fitted to the limit state surface at points with minimal distance to the origin (design points).

The Reliability-Based Design Optimization (RBDO) Model

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.

Introduction to Monte Carlo methods

Numerical methods that are known as Monte Carlo methods can be loosely described as statistical simulation methods. A statistical simulation can be defined, in general terms, to be any method that utilizes sequences of random numbers to perform the simulation.

Monte Carlo methods have been used for centuries, but only in the past several decades this technique has gained the status of a full-fledged numerical method capable of addressing the most complex applications. The name "Monte Carlo" was coined by Metropolis (inspired by Ulam's interest in poker) during the Manhattan Project of World War II, because of the similarity of statistical simulation to games of chance, and because the capital of Monaco was a center for gambling and similar pursuits.

Monte Carlo is now used routinely in many diverse fields, from the simulation of complex physical phenomena such as radiation transport in the earth's atmosphere to more common cases, such as the simulation of a Bingo game. The analogy of Monte Carlo methods to games of chance is a good one but, in engineering applications, the "game" is a physical system, a Finite Element model or a black-box function, and the outcome of the game is a solution to some problem. The scientist judges the value of his results on their intrinsic worth, rather than the extrinsic worth of his holdings.

Statistical simulation methods may be contrasted to conventional numerical discretization methods, which typically are applied to ordinary or partial differential equations that describe some underlying physical or mathematical system.

In many applications of Monte Carlo, the physical process is simulated directly, and there is no need to even write down the differential equations that describe the behavior of the system. The only requirement is that the physical (or mathematical) system be described by probability density functions (PDFs).

From now on, we will assume that the behavior of a system can be described by PDFs. Once the PDFs are known, the Monte Carlo simulation can proceed by random sampling from the PDFs. Many simulations are then performed (multiple "trials" or "histories") and the desired result is taken as an average over the number of observations (which may be a single observation or perhaps millions of observations). In many practical applications, one can predict the statistical error (the "variance") in this average result, and hence an estimate of the number of Monte Carlo trials that are needed to achieve a given error.

Assuming that the evolution of the physical system can be described by probability density functions (PDFs), then the Monte Carlo simulation can proceed by sampling from these PDFs, which necessitates a fast and effective way to generate random numbers uniformly distributed on the interval [0,1]. The outcomes of these random samplings, or trials, must be accumulated or tallied in an appropriate manner to produce the desired result, but the essential characteristic of Monte Carlo is the use of random sampling techniques (and perhaps other algebra to manipulate the outcomes) to arrive at a solution of the physical problem. In contrast, a conventional numerical solution approach would start with the mathematical model of the physical system, discretizing the differential equations and then solving a set of algebraic equations for the unknown state of the system.

It should be kept in mind though that this general description of Monte Carlo methods may not directly apply to some applications. It is natural to think that Monte Carlo methods are used to simulate random, or stochastic, processes, since these can be described by PDFs. However, this coupling is actually too restrictive because many Monte Carlo applications have no apparent stochastic content, such as the evaluation of a definite integral or the inversion of a system of linear equations. However, in these cases and others, one can pose the desired solution in terms of PDFs, and while this transformation may seem artificial, this step allows the system to be treated as a stochastic process for the purpose of simulation and hence Monte Carlo methods can be applied to simulate the system.

Therefore, a broad view of the definition of Monte Carlo methods can be taken and can include in the Monte Carlo rubric all methods that involve statistical simulation of some underlying system, whether or not the system represents a real physical process.