Sparse Narrow-Band Topology Optimization for Large-Scale Thermal-Fluid Applications

Abstract: We propose a fluid-based topology-optimization methodology for convective heat-transfer problems that can manage an extensive number of design variables, enabling the fine geometric features required for the next generation of heat-exchanger designs. Building on the classical Borrvall--Petersson formulation for Stokes flow, we develop a narrow-band optimization algorithm that concentrates computational effort on the fluid--solid interface, where it is most needed. To address the high cost of repeated forward and adjoint analyses, we utilize a flow solver specifically optimized for high-resolution voxel grids. The solver reduces memory usage and computational time by removing solid voxels from the analyses and directly imposing the no-slip boundary condition at the fluid--solid interface. It also employs an efficient preconditioner built on the Algebraic Multigrid method that ensures fast and reliable convergence for intricate flow configurations. The discretization uses a staggered-grid finite-difference scheme (marker-and-cell) for the Stokes--Brinkman model and an upwind finite-difference scheme for the heat convection--diffusion equation, ensuring stability at high Peclet numbers. We demonstrate the method on several examples, including the optimization of a two-fluid heat exchanger at $Pe = 10^{4}$ on a $370^{3}$ grid comprising $5 \times 10^{7}$ design variables using only a single desktop workstation. The framework shows considerable promise for advancing large-scale thermal-fluid applications and constitutes an important step toward a full conjugate-heat-transfer design methodology for high-Reynolds-number Navier--Stokes flows.