This work is concerned with synthesis methods for nonlinear structures and the investigation of their dynamic response. Nonlinear resonant structures are capable of displaying a wide range of phenomena including but not limited to internal resonances. However, there are not many systematic synthesis methods for their design which often leads to a very constrained design space. This shortage of options may often lead to a sub-optimal design being selected for a particular application. Topology optimization is a blanket term used for a variety of computational synthesis methods which have been used for the design of various types of structures such as compliant mechanisms. This thesis describes new topology optimization methods developed for the design of nonlinear structures, specifically for structures satisfying given internal resonances. The objective of the research proposed here is to unify the two disparate fields of topology optimization and nonlinear dynamics, thus providing nonlinear dynamics with a sorely missing design tool and at the same time extending the application space of topology optimization into an entirely new domain. This convergence would pave the way for the design of optimal, sensitive and robust Micro Electro-Mechanical Systems (MEMS) as well as meso-scale resonators and meta-materials which could see wide applications in diverse fields such as frequency dividers and filters in electronic circuits and as mass and chemical sensors for defense and industrial applications. This research could also lead to the development of insight and tools which can be used to produce a new generation of designs and lead to a wider appreciation of the myriad ways in which nonlinearities can be "designed into" devices for increased effectiveness and efficiency.
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