Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (March 1, 2007) Vita. Includes bibliographical references.

Nonsmoothness appears in various applications in chemical engineering, including multi-stream heat exchangers, nonsmooth flash calculation, process integration. In terms of numerical approaches, convex/concave relaxations of static and dynamic systems may also exhibit nonsmoothness. These relaxations are used in deterministic methods for global optimization. This thesis presents several new theoretical results for nonsmooth sensitivity analysis, with an emphasis on convex relaxations. Firstly, the "compass difference" and established ODE results by Pang and Stewart are used to describe a correct subgradient for a nonsmooth dynamic system with two parameters. This sensitivity information can be computed using standard ODE solvers. Next, this thesis also uses the compass difference to obtain a subgradient for the Tsoukalas-Mitsos convex relaxations of composite functions of two variables. Lastly, this thesis develops a new general subgradient result for Tsoukalas-Mitsos convex relaxations of composite functions. This result does not limit on the dimensions of input variables. It gives the whole subdifferential of Tsoukalas-Mitsos convex relaxations. Compare to Tsoukalas-Mitsos’ previous subdifferential results, it does not require additionally solving a dual optimization problem as well. The new subgradient results are extended to obtain directional derivatives for Tsoukalas-Mitsos convex relaxations. The new subgradient results and directional derivative results are computationally approachable: subgradients in this article can be calculated both by the vector forward AD mode and reverse AD mode. A proof-of-concept implementation in Matlab is discussed. / Thesis / Master of Applied Science (MASc)

The geometrically nonlinear constant moment triangle based on the von Karman theory of thin plates is first described. This finite element, which is believed to be the simplest possible element to pass the totality of the von Karman patch test, is employed throughout the present work. It possesses the special characteristic of providing a tangent stiffness matrix which is accurate and without approximation. The stability of equilibrium of discrete conservative systems is discussed. The criteria which identify the critical points (limit and bifurcation), and the method of determination of the stability coefficients are presented in a simple matrix formulation which is suitable for computation. An alternative formulation which makes direct use of higher order directional derivatives of the total potential energy is also presented. Continuation along the stable equilibrium solution path is achieved by using a recently developed Newton method specially modified so that stable points are points of attraction. In conjunction with this solution technique, a branch switching method is introduced which directly computes any intersecting branches. Bifurcational buckling often exhibits huge structural changes and it is believed that the computation of the required switch procedure is performed here, and for the first time, in a satisfactory manner. Hence, both limit and bifurcation points can be treated without difficulty and with continuation into the post buckling regime. In this way, the ability to compute the stable equilibrium path throughout the load-deformation history is accomplished. Two numerical examples which exhibit bifurcational buckling are treated in detail and provide numerical evidence as to the ability of the employed techniques to handle even the most complex problems. Although only relatively coarse finite element meshes are used it is evident that the technique provides a powerful tool for any kind of thin elastic plate and shell problem. The thesis concludes with a proposal for an algorithm to automate the computation of the unknown parameter in the branch switching method.