Date of Award

Summer 8-23-2018

Document Type


Degree Name

Doctor of Philosophy (PhD)


Applied Mathematics

First Advisor

John A. Evans

Second Advisor

Stephen Becker

Third Advisor

Bengt Fornberg

Fourth Advisor

Alireza Doostan

Fifth Advisor

Gregory Beylkin

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.


In this dissertation, we present a methodology for understanding the propagation and control of geometric variation in engineering design and analysis. This work is comprised of two major components: (i) novel discretizations and associated solution strategies for rapid numerical solution over geometric parametrizations of the linear and nonlinear thin-shell equations, and (ii) efficient surrogate modeling techniques and algorithms towards the control of geometric variation. While the methodologies presented are in the setting of structural mechanics, particularly Nitsche's method in the context of linearized membranes, Kirchhoff-Love plates, and Kirchhoff-Love shells, they are applicable to any system of parametric partial differential equations. We present a design space exploration framework that elucidates design parameter sensitivities used to inform initial and early-stage design and a novel tolerance allocation algorithm for the assessment and control of geometric variation on system performance. Both of these methodologies rely on surrogate modeling techniques where various designs throughout the design space considered are sampled and used in the construction of approximations to the system response. The design space exploration paradigm enables the visualization of a full system response through the surrogate model approximation. The tolerance allocation algorithm poses a set of optimization problems over this surrogate model restricted to nested hyperrectangles represents the effect of prescribing design tolerances, where the maximizer of this restricted function depicts the worst-case member, i.e. design. The loci of these tolerance hyperrectangles with maximizers attaining the performance constraint represents the boundary to the feasible region of allocatable tolerances. The boundary of the feasible set is elucidated as an immersed manifold of codimension one, over which optimization routines exist and are employed to efficiently determine an optimal feasible tolerance with respect to a user-specified measure. Examples of these methodologies for problems of various complexities are presented.