Topology optimisation-based design of duct cross-sections for fully developed magnetohydrodynamic flows

• A novel topology optimization method for magnetohydrodynamics is proposed. • The method allows the design of cross-sections for fully developed flows. • Two different MHD models are used, potential-based and fully conservative. • Different field orientations and Hartmann numbers are studied. • The influence of the solid-to-fluid conductivity ratio is analysed. In this communication, duct cross-sections for fully developed inductionless magnetohydrodynamics (MHD) flows are designed via topology optimisation (TO). The objective function to be maximised is the volumetric flow rate that can be achieved along the duct under a prescribed pressure gradient, subject to an area constraint imposed on the cross-sectional geometry. The solid-liquid transition within the topology is modelled through a Darcy term, where the permeability is defined using the Carman-Kozeny equation as a function of an apparent liquid fraction, effectively enforcing zero velocity in solid regions. The well-known solid isotropic material with penalisation (SIMP) technique is used to define the apparent liquid fraction and to interpolate between the electrical conductivities of solid and fluid phases. A finite element framework is used for the multiphysics analysis, with a single design variable (artificial density) assigned to each finite element. Details of the adjoint-based sensitivity analysis within this multiphysics context are provided, and the influence of both the orientation and dominance of the magnetic field on the resulting topologies is also examined. The results demonstrate that the proposed optimisation strategy effectively yields solutions that maximise the flow rate, and that such topologies differ significantly from those expected in standard fluid dynamics problems, with strong dependence on magnetic field characteristics. Furthermore, the combined use of SIMP interpolation and the Carman-Kozeny-based Darcy term ensures stable convergence of the optimisation process within the inductionless MHD framework, even when accounting for the interpolation of electrical conductivity.