Structural topology optimization has been extensively studied in aeronautical, civil, and mechanical engineering applications in order to improve performance of systems. This thesis focuses on an optimal design of damping treatment using topology optimization, and the reduction of computational expense of the topology optimization procedure. This thesis presents mainly two works on topology optimization. In the first work, topology optimization is implemented to optimally design damping treatments in unconstrained-layer damping material. Since the damping effect relies on the placement of damping treatment, and the weight of damping material may be an important factor, the placement of damping material is optimally determined using topology optimization with an allowable maximum. Unconstrained-layer plate and shell structures are modeled. The damping layer on the unconstrained-layer structures is considered as the design domain. Using topology optimization, the damping layer is designed numerically, and then experimentally validated by comparing the damping effects. In the numerical example, the topological damping treatment usually provides much higher damping effects compared to other approaches such as strain energy distribution (SED) and an evolutionary structural optimization (ESO). In the second work, a numerical algorithm, named as reducible design variable method (RDVM) topology optimization, is proposed in order to efficiently reduce the computational expense. Since it usually requires thousands to millions of design variables and up to hundreds of iterations in topology optimization, the major difficulty is its computational expense. The RDVM topology optimization is implemented into static (minimization of compliance) and dynamic (maximization of the fundamental resonance frequency) problems. The RDVM significantly reduces computing time, as confirmed by numerical examples. / Thesis (Ph.D, Mechanical and Materials Engineering) -- Queen's University, 2011-07-08 10:10:20.606