Topology optimization for thermal-fluid applications using an unstructured finite volume scheme

Topology optimization is a method for developing optimized geometric designs that maximize a quantity of interest (QoI) subject to constraints. Unlike shape optimization, which optimizes the dimensions of a template shape, topology optimization does not start with a pre-conceived shape. Instead, the algorithm builds the geometry iteratively by placing material pixels in a specified background domain, aiming to maximize the QoI subject to a constraint on the volume of material or other constraints. The power of topology optimization lies in its ability to realize design solutions that are not initially apparent to the engineer. Topology optimization, though well established in structural applications, has not percolated to the thermal-fluids community to any great degree, and most published papers have not addressed sufficiently realistic engineering problems. However, the methodology has immense application potential in the area of fluid flow, heat and mass transfer and other transport phenomena at all length scales. In the literature, the solution methodology used for topology optimization is based mostly on finite element methods. However, unstructured finite volume methods are frequently the numerical method of choice in the industry for those addressing thermal-fluid or other transport problems. It is essential that methods for topology optimization work well in the finite volume framework if they are to find traction in industry. Regardless of the numerical method employed for forward solution, the most popular methodology employed for topology optimization is the solid isotropic material with penalization (SIMP) approach in conjunction with a gradient-based optimization algorithm. This optimization approach requires the calculation of sensitivity derivatives of the QoI with respect to design variables through a discrete adjoint method. The Method of Moving Asymptotes (MMA) is a widely-used algorithm for topology optimization. Thus the objective of this dissertation is to build a robust framework for topology optimization for thermal-fluid problems, employing SIMP and MMA, within the framework of industry-standard finite volume schemes.Towards realizing this goal, we first develop and demonstrate topology optimization for multidimensional steady heat conduction problems in a cell-centered unstructured finite volume framework. The fundamental methodologies for SIMP/RAMP interpolation of thermal conductivity and the basic optimization infrastructure using MMA are developed and tested in this chapter. The effect of including secondary gradients in sensitivity computations is evaluated for typical heat conduction problems. Topologies that maximize or minimize relevant quantities of interest in heat conduction applications with and without volumetric heat generation are presented. Industry standard finite volume codes for fluid flow are built on unstructured cell-centered formulations employing co-located pressure-velocity storage, and a sequential solution algorithm. This type of algorithm is very widely used, but poses a number of difficulties when used as the solution kernel for performing efficient gradient-based topology optimization. The complete Jacobian required for discrete adjoint sensitivity computation is never available in a sequential technique. Also, the complexities of co-located algorithms must be correctly reflected in the Jacobian and sensitivity computations if correct optimal structures are to evolve. We build an Automatic Differentiation library, christened 'Rapid', to compute accurate Jacobians and other necessary derivatives for the discrete adjoint method in the context of an unstructured co-located sequential pressure based algorithm. The library is designed to provide a problem-agnostic pathway to automatically computing all required derivatives to machine accuracy. With sensitivities obtained from the Rapid library, we next develop and demonstrate topology optimization for multidimensional laminar flow applications. We present a variety of test cases involving internal channel flows as well as external flows, for a range of Reynolds numbers. An essential feature of Rapid is that it is not necessary to write new code to find sensitivities when new physics, such as turbulence models, are added, or when new cost functions are considered. The next step is therefore to extend the topology optimization for flow problems to the turbulent regime. Based on the Spalart-Allmaras RANS turbulence model, the topology optimization methodology for steady state turbulent flow problems is developed and demonstrated for channel flow problems. Finally we develop topology optimization methodology for forced convection applications which requires the coupling of the Navier-Stokes and energy equations and which are typically solved sequentially in finite volume schemes. The coupled nature of the problem introduces the concept of multi-objective opposing cost functions from the two physical models, for example, minimizing pressure drop and simultaneously maximizing heat transfer. Techniques to obtain sensitivities for forced convection with laminar and turbulent flow with Rapid are presented. Challenges for topology optimization resulting from multi-objective cost functions are discussed. We believe this is the first time that a complete topology optimization framework using an unstructured finite volume method and the discrete adjoint method, fully generalizable to practical use in commercial solvers and for industrial applications, has been demonstrated in the open literature. The methodologies developed here provide a basis for performing topology optimization involving other transport phenomena, more complex cost functions and more realistic constraints.