The main objective of this research is to develop a computational design tool for multilevel optimization of product-material systems under uncertainty. To accomplish this goal, an exponential penalty function (EPF) formulation based on method of multipliers is developed for solving multilevel optimization problems within the framework of Analytical Target Cascading (ATC). The original all-at-once constrained optimization problem is decomposed into a hierarchical system with consistency constraints enforcing the target-response coupling in the connected elements. The objective function is combined with the consistency constraints in each element to formulate an augmented Lagrangian with EPF. The EPF formulation is implemented using double-loop (EPF I) and single-loop (EPF II) coordination strategies and two penalty-parameter-updating schemes. The computational characteristics of the proposed approaches are investigated using different nonlinear convex and non-convex optimization problems. An efficient reliability-based design optimization method, Single Loop Single Vector (SLSV), is integrated with Augmented Lagrangian (AL) formulation of ATC for solution of hierarchical multilevel optimization problems under uncertainty. In the proposed SLSV+AL approach, the uncertainties are propagated by matching the required moments of connecting responses/targets and linking variables present in the decomposed system. The accuracy and computational efficiency of SLSV+AL are demonstrated through the solution of different benchmark problems and comparison of results with those from other optimization methods. Finally, the developed computational design optimization tool is used for design optimization of hybrid multiscale composite sandwich plates with/without uncertainty. Both carbon nanofiber (CNF) waviness and CNF-matrix interphase properties are included in the model. By decomposing the sandwich plate, structural and material designs are combined and treated as a multilevel optimization problem. The application problem considers the minimum-weight design of an in-plane loaded sandwich plate with a honeycomb core and laminated composite face sheets that are reinforced by both conventional continuous fibers and CNF-enhanced polymer matrix. Besides global buckling, shear crimping, intracell buckling, and face sheet wrinkling are also treated as design constraints.

Traditional methods in engineering design involve producing solutions at a single level. However, in complex engineering design problems, such as concurrent product and materials design, various levels of model complexity are considered. A design process in which design problems are defined and analyzed at various levels of design complexity is referred to as multilevel design. One example of multilevel design is the design of a material, product, assembly, and system. Dividing a design problem into multiple levels increases the possibility for introducing and propagating uncertainty. Design solutions that perform predictably in the presence of uncertainty are robust designs. Robust design concepts that were originally developed for designs at a single level can be applied to a multilevel design process. The Inductive Design Exploration Method (IDEM) is an existing design method used to produce robust multilevel design solutions. In this thesis, the strategy presented in IDEM is incorporated into design templates in order to extend its overall usefulness. Design templates are generic, reusable, modules that provide the theoretical and computational framework for solving design problems. Information collected, stored, and analyzed from design templates is leveraged for a variety of design problems. In this thesis, the possibilities of a template-based approach to multilevel design are explored. Two example problems, which employ the developed multilevel robust design template, are considered. Multilevel design templates are created for the design of a cantilever beam and its associated material and the design of a blast resistant panel. The design templates developed for example problems can be extended to facilitate a generic, modular, template-based approach to multilevel robust design.