Inverse-Problem Inspired Approaches in the Design of Solids for Frequency-Domain Dynamics
Dynamic conditions and harmful vibration are prevalent in many engineered systems. Computational optimization methods can usefully guide the design of structures and solid systems to obtain layouts with desired dynamic behaviors, such as minimized or tailored vibration response, while accounting for additional constraints. Due to resonance phenomena in steady-state dynamics, however, objectives used to control dynamic response possess many local minima that may represent poor solutions to a design problem, causing challenges for gradient-based optimization methods. Additionally, designs are typically characterized by high-dimensional parameter spaces, which require costly regularization operations and can lead to slow convergence. This dissertation presents strategies to address these obstacles in dynamic response design problems, seeking inspiration from solution strategies for the related class of medium inversion problems.
We propose an extension of the modified error in constitutive equations (MECE) method, used previously in material identification problems, as a new approach for frequency-domain dynamics topology optimization to mitigate solution dependence upon initial guess and considered loading conditions. The main idea of the MECE strategy is to relax the constitutive relations between stresses and strains and between inertial forces and displacements in the elastodynamic PDE-constraint. An additional penalty-like term measuring the discrepancy in these relations is added to the objective, so that both model error and the original response objective are minimized. The resulting optimization problem features an objective more amenable to minimization, as resonant responses are suppressed in the relaxed elastodynamic model. We show that this approach yields superior designs to conventional gradient-based optimization approaches that solely use a functional of displacements as the objective, while strictly enforcing the constitutive equations. We highlight the merits of our approach in a variety of scenarios for direct frequency response design, considering multiple frequency load cases and different frequency-response control objectives.
We next present an adaptive eigenspace basis (AEB) strategy for effective design dimensionality reduction in topology optimization problems. We construct the density field as a linear combination of eigenfunctions, computed for an elliptic operator defined over the design domain, and solve for the associated eigenfunction coefficients. Restriction to this finite-dimensional eigenspace drastically reduces the design dimension and imposes implicit regularization upon the solution. We furthermore develop a basis adaptation scheme, in which we iteratively recompute the eigenfunction basis to conform to the evolving solution field, enabling further dimension reduction and acceleration of the optimization process. We demonstrate the AEB method on topology optimization problems for static linear elasticity, benchmarking its performance against conventional TO implementations. We further present the formulation of an MECE frequency-response control problem using an AEB design parameterization. We demonstrate the combined methods provide the benefits of both dimensionality reduction and robust performance in multifrequency design scenarios.
We finally present a case study in dynamic design: a PDE-constrained optimization formulation for the design of continuously graded elastic cylinders to achieve passive cloaking from harmonic acoustic excitation, both at single frequencies and over extended bandwidths. We show that elastic parameters in a multilayered, constant-density cylinder may be selected such that the residual between the pressure field from an unobstructed spreading wave in a fluid and the pressure field produced by the cylindrical inclusion is minimized.
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