Topology Optimization for Eigenvalue Problems with Applications to Phononic Crystals and Stochastic Dynamics

Abstract

Topology optimization is the process of exploring the optimal layout of material within a design domain. It is a free-form technique as material can be added or removed from any location, making it more general than sizing and shape optimization. Although the rst topology optimization paper was written in late 1980s, it has experienced extremely rapid expansion over the last decade. It has been applied to nd optimal solutions for various engineering problems governed by diverse mechanics. However, only a relatively limited number of works have focused on problems governed by eigenvalues, and most of them have assumed deterministic eigenvalues and symmetric matrices. Therefore, this dissertation proposes topology optimization algorithms for general eigenvalue problems with and without considering uncertainties and applies them to the design of materials and structures.

The topology optimization formulation for eigenvalue problems is rstly presented and the numerical challenges are subsequently discussed. Next, the sensitivity of complex eigenvalues and eigenvectors are derived using perturbation method. Then the proposed algorithm combined with a fast mixed variational eigenvalue solver and distributed Graphic Processing Unit computations developed by collaborators is used to reveal 3-D phononic structures which exhibit the largest normalized all-angle allmode band gaps reported to date.

Uncertainties are considered in this dissertation for mitigating dynamic response under stochastic dynamic excitations. Stochastic equations are formulated in the standard manner by using second order di erential equations and state space in which they are described by rst order di erential equations. Later they are solved both in frequency domain and using state space analysis. It has been found that using state space formulation and further solving in frequency domain requires the least computational e ort. In addition, the by-product of this formulation is that it is capable of incorporating non-classical damping. Numerical results are presented to illustrate the comparisons between topologies optimized for stochastic ground motion loading and topologies optimized under free vibration.

Lastly, this dissertation addresses the design of reinforced concrete structure by developing a stress-dependent truss-continuum topology optimization algorithm. Sti - ness is formulated such that truss elements carry only tensile forces and thus represent straight steel rebar, while the continuum elements carry only compression forces and thus represent concrete compression load paths. Constructability of reinforcement is also discussed by replacing the volume constraint with a total cost constraint.