Ph.D. (Mathematics) / Please refer to full text to view abstract

For any product development process, limited time and resources are always a focus for the engineer. However, will the overall program goals be achieved with the provided time and resources? Uncertainty analysis is a tool that is capable of providing the answer to that question. Product development process uncertainty analysis employs previous knowledge in modeling, experimentation, and manufacturing in an innovative approach for analyzing the entire process. This research was initiated with a pilot project, a four-bar-slider mechanism, and an uncertainty analysis was completed for each individual product development step. The uncertainty of the final product was then determined by combining uncertainties from the individual steps. The uncertainty percentage contributions of each term to the uncertainty of the final product were also calculated. The combination of uncertainties in the individual steps and calculation of the percentage contributions of the terms have not been done in the past. New techniques were developed to evaluate the entire product development process in an uncertainty sense. The techniques developed in this work will be extended to other processes in future work.

A methodology has been developed to quantify the simulation uncertainty of a computational model calibrated against test data. All test data used in the study undergoes an experimental uncertainty analysis. The modeling software ROCETS is used and its structure is explained. The way the model was calibrated is presented. Next, a general simulation uncertainty analysis methodology is shown that is valid for calibrated models. Finally the ROCETS calibrated model and its simulation uncertainty are calculated using the general methodology and compared to a second set of comparison test data. The simulation uncertainty analysis methodology developed and implemented can be used for any modeling with a calibrated model. The methodology works well for a process of incremental testing and recalibration of the model whenever new test data is available.

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This study has two parts. In the first part we develop a computational approach to the solution of an inverse modelling problem concerning the material properties of electrolytes used in Lithium-ion batteries. The dependence of the diffusion coefficient and the transference number on the concentration of Lithium ions is reconstructed based on the concentration data obtained from an in-situ NMR imaging experiment. This experiment is modelled by a system of 1D time-dependent Partial Differential Equations (PDE) describing the evolution of the concentration of Lithium ions with prescribed initial concentration and fluxes at the boundary. The material properties that appear in this model are reconstructed by solving a variational optimization problem in which the least-square error between the experimental and simulated concentration values is minimized. The uncertainty of the reconstruction is characterized by assuming that the material properties are random variables and their probability distribution estimated using a novel combination of Monte-Carlo approach and Bayesian statistics. In the second part of this study, we carefully analyze a number of secondary effects such as ion pairing and dendrite growth that may influence the estimation of the material properties and develop mathematical models to include these effects. We then use reconstructions of material properties based on inverse modelling along with their uncertainty estimates as a framework to validate or invalidate the models. The significance of certain secondary effects is assessed based on the influence they have on the reconstructed material properties. / Thesis / Doctor of Philosophy (PhD)

Inverse problems aim at estimating the unknown excitations or properties of a physical system based on available measurements of the system response. For example, wave tomography is used in geophysics for seismic waveform inversion; in biomedical engineering, optical tomography is used to detect breast cancer tissue; in structural engineering, inversion techniques are used for health monitoring and damage detection in structural safety evaluation. Inverse solvers depend on the type of measurement data the unknown parameters to be estimated. The work in this thesis focuses on structural parameter identification based on static and dynamic measurements. As an integral part of the formulated inverse solver, the associated forward problem is studded and deeply investigated. In reality, the data are associated with uncertainties caused by measurement devices or unfriendly environmental conditions during data acquisition. Traditional approaches use probability theory and model uncertainties as random variables. This approach has its own limitation due to a prior assumption on the probability structure of uncertainty. This is usually too optimistic or not realistic. However, in practice, it is usually difficult to reliably assess the statistical nature of uncertainties. Instead, only bounds on the uncertain variables and some partial information about their probabilities are known. The main source of uncertainty is due to the accuracy of measuring devices; these are designed to operate within specific allowable tolerances, as defined by National Institute of Standards and Technology (NIST). Tolerances are performance requirements that fix the limit of allowable error or departure from true performance or value. Thus closed intervals are the most realistic way to model uncertainty in measurements. In this work, uncertainties in measurement data are modeled as interval variables bounded by their endpoints. It is proven that interval analysis provides guaranteed enclosure of the exact solution set regardless of the underlying nature of the associated uncertainties. This work presents a solution of inverse problems under measurements uncertainty within the framework of Interval Finite Element Methods (IFEM) and adjoint-based optimization techniques. The solution consists of a two-step algorithm: first, an estimate of the parameters is obtained by means of a deterministic iterative solver. Then, the algorithm switches to a full interval solution, using the previous deterministic estimate as an initial guess. In general, the solution of an inverse problem requires iterative solutions of the forward problem. Efficient and accurate interval forward solutions in static and dynamic domains have been developed. In particular, overestimation due to interval dependency has been drastically reduced using a new decomposition of the load, stiffness, and mass matrices. Further improvements in the available interval iterative solvers have been achieved. Conjugate gradient and Newton-Raphson methods to gether with an inexact line search are used in the newly formulated optimization procedure. Moreover Tikhonov regularization is used to improve the conditioning of the ill-posed inverse problem. The developed interval solution for the inverse problem under uncertainty has been tested in a wide range of applications in static and dynamic domains. By comparing current solutions with other available methods in the literature, it is proven that the developed method provides guaranteed sharp bounds on the exact solution sets at a low computational cost. In addition, it contains those solutions provided by probabilistic approaches regardless of the used probability distributions. In conclusion, the developed method provides a powerful tool for the analysis of structural inverse problem under uncertainty.

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No description available.