Optimal Material Design for Transport and Multi-Criteria Problems Using Topology Optimization

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Optimal multi-material layouts are critical to achieving the desired functionality of structures and devices. Topology optimization, a mathematically driven methodology, has developed into an indispensable tool to obtain optimal material layouts in diverse fields, from engineering to biomedical sciences at scales ranging from large-scale structural applications to nano-scale diagnostic devices. In this dissertation, we present developments in non-linear and multi-criteria topology optimization. The main contributions are: First, we propose a design framework with an alternative objective function, appropriate for all analysis models, to derive optimal designs for flow through porous media. We highlight the effect of pressure, inertia, and viscous shearing stresses on optimal layouts. Second, we develop a design framework to obtain optimal material layouts for structural applications to satisfy multiple objectives at macroscales. For applications at microscales, we investigate the effect of pressure-dependent viscosity, inertial forces, and viscous shear, often ignored by using the simple Darcy model to obtain optimal designs for flow through porous media. Many studies use the minimization of power as an objective function which is not universally applicable. We propose an alternative objective function -- the rate of mechanical dissipation -- valid even for non-linear models. Using canonical examples, we derive analytical solutions, which facilitate verification of computer implementations. We bring out the highly disparate optimal layouts under the non-linear models, which are more likely to reflect the true pressure and velocity fields within the domain. Employing these designs will positively impact the accuracy of microfluidic devices, which is critical to their functionality. At the macroscale, we apply topology optimization to obtain optimal layouts for a structural wall application. Structural elements (e.g., load-bearing walls) having high strength and low thermal conductivity are difficult to construct with a single material. Construction materials having high strength also exhibit high conductivity (e.g., steel), and materials having low conductivity exhibit low strength (e.g., concrete). Hence, it is imperative to utilize more than one material to achieve the requisite property. The design of a domain with materials having conflicting properties poses an enormous challenge for engineers. We use multi-criteria optimization techniques, pioneered by Francis Edgeworth and Vilfredo Pareto, combined with topology optimization using the homogenization technique to simultaneously realize such multi-criteria objectives.

Topology Optimization, Optimal material design, Multi-criteria optimization, Transport problems
Portions of this document appear in: T. Phatak and K. B. Nakshatrala. On optimal designs using topology optimization for flow through porous media applications. Transport in Porous Media, pages 1-41, 2021b. DOI: 10.1007/s11242-021-01616-z; and in: T. Phatak and K. B. Nakshatrala. Effect of viscous shearing stresses on optimal material designs for flow of fluids through porous media. Physics of Fluids, 33(6):063109, 2021a. DOI: 10.1063/5.0054527.