Rheologically complex materials are described by function-valued properties with dependence on a timescale (linear viscoelasticity), input amplitude (nonlinear material behavior), or more generally both (nonlinear viscoelasticity). This complexity presents a difficulty when trying to utilize these material systems in engineering designs. Here, we focus on linear viscoelasticity and a methodology to identify the desired viscoelastic behavior. This is an early-stage design step to optimize target (function-valued) properties before choosing or synthesizing a real material. In linear viscoelasticity, it is not obvious which properties can be treated as independent design variables. Thus, it is nontrivial to select the most design-appropriate constitutive model, to be as general as possible, but not violate fundamental restrictions. We use the Kramers–Kronig constraint to show that frequency-dependent moduli (e.g., shear moduli G′(ω) and G″(ω)) cannot be treated as two independent design variables. Rather, a single function such as the relaxation modulus (e.g., K(t) for force-relaxation or G(t) for stress relaxation) is an appropriate function-valued design variable. A simple case study is used to demonstrate the framework in which we identify target properties for a vibration isolation system. Viscoelasticity improves performance. Different parameterizations of the kernel function are optimized and compared for performance. While parameterization may limit the generality of the kernel function, we do include a nonobvious representation (power law) that is found in real viscoelastic material systems and in the spring-dashpot paradigm would require an infinite number of components. Our methodology provides a means to answer the question, “What viscoelastic properties are desirable?” This ability to identify targeted behavior will be useful for subsequent stages of the design process including the selection or synthesis of real materials.

In the last several decades fluid power has been used extensively in diverse industries such as agriculture, construction, marine, offshore resource extraction, and even entertainment. With a vast and ever-increasing spectrum of potential applications, the design of efficient and leak-free components in fluid power systems has become essential. Previous experiments and studies have shown that the use of microtextured surfaces in hydraulic components achieves performance enhancement by reducing friction and leakage. This article aims to build on this recent work through a systematic optimization-based study of performance improvement through microtexture surface design. These studies evaluate the potential of Newtonian fluid properties, coupled with varying surface features, to achieve design objectives for efficiency. This early-stage design strategy aims to find optimal surface features that minimize apparent fluid viscosity (low friction) and the area of the microtexture. The resulting multi-objective optimization (MOO) problem involves a computationally intensive simulation of the system based on computational fluid dynamics (CFD). As a strategy to reduce overall computational expense, this paper describes the development of a new adaptive surrogate modeling strategy for multi-objective optimization. Two case studies are presented: a simple analytical case study illustrating the details of the method and a more sophisticated case study involving the two-dimensional CFD simulation of Newtonian fluids on symmetric surface textures. This design approach embraces the potential of using rheologically complex fluids in engineering system design and optimization. This study can be further extended to a more generalized problem by coupling both fluid and geometrical design decisions.

Rheological material properties are examples of function-valued 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.