A Method for the Validation of Predictive Computations Using a Stochastic Approach

The task of model validation deals with quantifying the extent to which predictions from a particular model can be relied upon as representatives of the true behavior of the system being modeled. This issue is of great importance in assessing the reliability and safety of structures since in most cases their quantification relies on predictions from sophisticated probabilistic models. The paper describes a formalism that will extend the realm of the model to include all aspects of data collection and parameter calibration. Error estimators are developed that permit the quantification of the value of computational efforts (mesh refinement) versus analytical efforts (model refinement) and experimental effort (data acquisition and analysis). Starting with the hypothesis that the material properties of a given medium can be modeled within the framework of probability theory, a very rich mathematical setting is available to completely characterize the probabilistic behavior and evolution of the associated random medium under an external disturbance. The probability measure on the material properties is uniquely transformed into a probability measure on the state of the medium. A computational implementation of related concepts has been developed in the framework of stochastic finite elements and applied to a number of problems. Clearly, great value can be attributed to the ability of performing the forward analysis whereby the probability measure on the state of the system is completely characterized by the measure on the material properties. The assumed probability measure on the material properties, however, is greatly dependent on the amount and quality of data used to synthesize this measure. As this measure is updated, estimates of the performance of the underlying natural or physical system change. Significant interest exists therefore in developing the capability of controlling the error in the probabilistic estimates through designed data collection. The mathematical setting adopted in this paper for describing random variables is ideally suited for treating this problem as one of data refinement. A close parallel will be delineated between this concept and that of adaptive mesh refinement, well established in deterministic finite elements. Underlying this latter problem, are issues related to error estimation that are relied upon to guide the adaptation of the refinement. The paper develops a similar “data refinement” concept and highlights the basic underlying principles.