Abstract
This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet located underneath the rectangular vessel. The rectangular magnet provides a spatially non-uniform magnetic field that is known analytically. The magnet oscillation creates a steady mean flow (steady streaming) similar to those observed from oscillating rigid bodies. In the optimization problem, randomly distributed mass-less particles are advected by the flow to achieve a homogeneous distribution (objective function 1) while keeping the work done to move the permanent magnet minimal (objective function 2). A single evaluation of these two objective functions may take more than two hours. For that reason, to save computational time, the proposed method uses interpolation models on trust-regions for finding descent directions. We show that, even for our significantly simplified model problem, the mixing patterns vary significantly with the control parameters, which justifies the use of improved optimization techniques and their further development.
Highlights
The use of electromagnetic induction to manipulate electrically conducting fluids is common in industrial applications, most notably in metallurgy, where timedependent magnetic fields are used to generate a stirring motion inside a molten metal that is supposed to mix additives
To show that the optimization problem is well-defined, we performed a test where two variables, and KC, are varied while the Hartmann number is kept constant at Ha = 30
We presented results from an optimization study of the mixing process in electrically conducting fluids, where Lorentz forces generate the flow due to an oscillating permanent magnet
Summary
The use of electromagnetic induction to manipulate electrically conducting fluids is common in industrial applications, most notably in metallurgy, where timedependent magnetic fields are used to generate a stirring motion inside a molten metal that is supposed to mix additives. A possible scenario of generating a stirring motion inside a liquid metal is either by moving one or more permanent magnets with respect to the liquid metal (Prinz et al 2016; Rivero et al 2016; Beltrán et al 2010) or by injecting an electric current that interacts with the magnetic field of a permanent magnet (Lara et al 2017). When the motion of the conducting fluid is driven by external forces, such as an applied pressure gradient, and passes a region of a non-uniform magnetic field, vorticity is generated, and the flow behaves to hydrodynamic flow past a solid obstacle (Cuevas et al 2006)
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