Generalized viscoelastic material design with integro-differential equations and direct optimal control

Rheological material properties are examples of functionvalued quantities that depend on frequency (linear viscoelasticity), input amplitude (nonlinear material behavior), or both. This dependence complicates the process of utilizing these systems in engineering design. In this article, we present a methodology to model and optimize design targets for such rheological material functions. We show that for linear viscoelastic systems simple engineering design assumptions can be relaxed from a conventional spring-dashpot model to a more general linear viscoelastic relaxation kernel, K(t). While this approach expands the design space and connects system-level performance with optimal material design functions, it entails significant numerical difficulties. Namely, the associated governing equations involve a convolution integral, thus forming a system of integro-differential equations. This complication has two important consequences: 1) the equations representing the dynamic system cannot be written in a standard state space form as the time derivative function depends on the entire past state history, and 2) the dependence on prior time-history increases time derivative function computational expense. Previous studies simplified this process by incorporating parameterizations of K(t) using viscoelastic models such as Maxwell or critical gel models. While these simplifications support efficient solution, they limit the type of viscoelastic materials that can be designed. This article introduces a more general approach that can explore arbitrary K(t) designs using direct optimal control methods. In this study, we analyze a nested direct optimal control approach to optimize linear viscoelastic systems with no restrictions on K(t). The study provides new insights into efficient optimization of systems modeled using integro-differential equations. The case study is based on a passive vibration isolator design problem. The resulting optimal K(t) functions can be viewed as early-stage design targets that are material agnostic and allow for creative material design solutions. These targets may be used for either material-specific selection or as targets for later-stage design of novel materials.