Error in Constitutive Equations Approach to Structural and Material Optimization
Clay Sanders, Duke University
Design optimization is an increasingly important step in the development of novel engineered structures and materials. Optimization algorithms are used in tailoring metamaterials for acoustic and mechanical vibration control applications while improved fabrication methods, such as additive manufacturing, have expanded the complexity and precision with which these engineered structures can be created. Topology optimization, which seeks the optimal shape or distribution of material within a structural domain, is a well-explored field for statically loaded structures. Work remains, however, to develop robust methods to design dynamically loaded structures and to capture potential errors in underlying material modeling. We explore these shortcomings by proposing a new topology optimization framework for dynamically loaded structures using an "error-in-constitutive-equations" (ECE) approach. In this method, borrowed from material identification inverse problems, we optimize a structure by minimizing an objective function represented as the sum of two functionals: the least-squares error between the design's displacement and a target displacement pattern; and an ECE term allowing relaxation of the constitutive laws relating stresses and strains. We solve a partial differential equation-constrained optimization problem to obtain the optimal design, characterized by the distribution of a density parameter interpolating between distinct material phases. The performance of the algorithm is demonstrated for different structural design scenarios for mechanical vibration reduction. We show that the use of the ECE term provides robustness for the method with respect to both initial design and structural loading. We also explore the proper selection of a weighting parameter for the ECE term to capture modeling error and to improve convergence properties. Finally, we demonstrate a Quasi-Newton method for solving the optimization problem using new derivations for the second-order operators of the ECE objective.
Abstract Author(s): Clay Sanders, Wilkins Aquino, Timothy Walsh