The aim of a tolerance analysis of an assembly is to evaluate the cumulative effect due to the tolerances, that are assigned to the assembly components, on the functional requirements of the whole assembly. Each functional requirement is schematized through an equation, that is usually called stack-up function, whose variables are the model parameters that are function of the dimensions and the tolerances assigned to the assembly components. It looks like

FR = %, p2,…,pn) (1)

where FR is the considered functional requirement, p1,.,pn are the model parameters and f(p) is the stack-up function, that is usually not linear.

A functional requirement is usually a characteristic that relates two features. Its analytical expression is obtained by applying the equations of the Euclidean geometry to the features that define the functional requirement or to the points of the features that define the functional requirement.

A stack-up function has to model two possible assembly variations. The first variation is due to the tolerances assigned to the features of the assembly components. The obtained model (that is called "local model") has to be able to schematize all the tolerance kinds, i. e. dimensional, form, and so on, but in the same time it has to be able to represent the Envelope Principle (Rule # 1 of ASME standard) or the Independence Principle (according with ISO 8015 standard) applied to different dimensions of the same part. The local model has to define the range of variation of the model’s parameters from the assigned tolerances and it has to schematize the interaction among the assigned tolerance zones. The second variation is due to the contact among the assembly components. The variability of the coupled features, by which the link among the parts is made, gives a deviation in the location of the coupled parts. The resulting model (that is called "global model") has to be able to schematize the joints with contact and the joints with clearance between the coupled features.

Once modeled the stack-up functions, they may be solved by means of a worst case or a statistical approach (Creveling, 1997). To carry out a worst case approach, it is needed to define the worst configurations of the assembly (i. e. those configurations due to the cumulative effect of the smallest and the highest values of the tolerances assigned to the assembly components) that satisfy its assigned tolerances. This means to solve a problem of optimization (maximization and/or minimization) under constraints due to the assigned tolerances. Many are the methods developed by the literature to carry out a worst case approach (see Luenberger, 2003). To carry out a statistical approach, it is needed to translate each tolerance assigned to an assembly component into one or more parameters of the stack – up function. Therefore, a Probability Density Function (PDF), that is function of both the manufacturing and the assembly processes, is assigned to each parameter. Being the definition of the relationship among the production and assembly processes and the probability density function of the tolerance of the component strongly hard to estimate, the commonly used assumption is to adopt a Gaussian probability density function. Moreover, a further assumption is to consider independent the parameters used to represent the variability of the features delimiting each dimensional tolerance. The variation of the FR is obtained by means of a Monte Carlo simulation technique (Nigam & Turner, 1995; Nassef & ElMaraghy, 1996) it is usually calculated as ± three times the estimated standard deviation (three sigma paradigm of (Creveling, 1997)).