Robust Design with Random Field Noise via Computer Experimentation

Description

The research objective of this project is to develop robust design and tolerance designmethods for time consuming computer models with random field inputs.Robust design and tolerance design are important methods for building quality into aproduct. Due to the rapid increase in computing power, computer models of physicalsystems are increasingly being employed for product design. In such models, the outputsare physical quantities that are functions of space and time. Each physical quantity is asolution of a partial differential equation (PDE) with boundary and initial conditions,and the equations are solved numerically with the finite element or finite differencemethod. Inputs to the computer model include the geometry of the physical system,parameters of the PDE (e.g., material properties and source term), and boundary andinitial conditions. Most of these inputs are uncertain functions of space that can bemodeled as random fields. For example, the thermal conductivity of a solid body can bemodeled as a lognormal random field.The input uncertainty of main interest in this research represents uncertainty aboutthe actual properties of a unit of manufactured product or environmental noise. In theformer case, the mean function of the random field input is a control factor, and thevariance function of the input represents inherent variability in the manufacturingprocess, which can be reduced by improving the manufacturing process. The randomfield variation about its mean, whether it represents product property or environmentalvariation, is always noise. The problem considered in this research is that of robustdesign (choice of random field mean) and tolerance design (choice of variance) in thepresence of random field noise. We will develop methodologies for choosing the controlfactor settings (means and variances) to minimize the expected total cost (sum of qualityand manufacturing cost) due to variation in the response caused by noise. Our researchwill focus on time consuming computer models that require the use of metamodels/surrogates constructed with a computer experiment.The proposed project is an extension of current work on uncertainty quantification(UQ), which focuses on estimating the probability distribution of scalar summaries of thesolution of a PDE induced by input uncertainty. UQ has recently received widespreadinterest from applied mathematicians, engineers, and statisticians. Our project willprovide novel and computationally tractable statistical modeling and optimizationmethods for robust and tolerance design with random field inputs.