Clay Sanders, Duke University

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Vibration effects have crucial implications for the stability and longevity of structures. Emerging classes of metamaterials, with applications in acoustic sensing or vibration control, likewise are designed with consideration for dynamic behavior. PDE-constrained optimization techniques, such as topology optimization, are useful strategies to prototype and refine structural designs in order to control dynamic responses while also abiding by material or fabrication constraints. However, design problems for dynamics possess significant difficulties, as objectives functions are highly non-convex and gradient-based optimization schemes can converge to poor designs. We present a strategy to improve the robustness of steady-state response topology optimization problems with respect to initial guesses and considered loading conditions. We extend the modified error-in-constitutive-equations (MECE) method, used previously in material parameter inverse problems, for the reformulated structural design problem. The main idea of the framework is to relax the PDE constraints in order to smooth the objective function and facilitate convergence to superior local minima. We permit violation of the constitutive laws relating stresses and strains and relating inertial forces and structural displacements, laws normally enforced exactly in PDE constraint evaluation. Instead, error functionals measuring violation of these laws are added to the optimization objective. Minimization of the error terms is weighted against minimization of the structural design objective, e.g., minimized vibration response, allowing for design objectives to be achieved while regularizing the objective function to suppress singularities.

We show this approach yields an objective with improved convexity, helping avoid convergence toward poor local minima and obtaining superior designs to optimization approaches enforcing exact satisfaction of the PDE. The MECE strategy integrates into a density-based topology optimization framework for void-solid or two-phase material structural design. We highlight the merits of our approach in a variety of scenarios for steady-state response design, considering multiple frequency cases and a variety of structural objectives.

Abstract Author(s): Clay Sanders, Timothy Walsh, Julian Norato, Wilkins Aquino