Two-dimensional surface-coupled modeling of electromagnetic shaping and levitation of liquid metals

The engineering of electromagnetic levitation, shaping, and confinement systems involves an understanding of the interaction between liquid metal and electromagnetic field structure. A basic problem is calculating the equilibrium free boundary geometry of liquid metal in an alternating magnetic field produced by a system of current sources. This is a coupled problem which requires a self-consistent solution. The magnetic field and induced eddy currents interact to exert magnetic pressures on the liquid metal which can change shape, and in turn the changing shape of the liquid metal results in a new magnetic field structure and magnetic pressure distribution. Another basic problem, which is more interesting from an engineering standpoint but has not been addressed in the engineering literature, is the inverse of the free boundary problem where a desired free boundary shape is specified and one solves for a system of source currents that produce the proper magnetic field structure to achieve this specified liquid metal shape. This is a practical problem to be solved since one typically specifies the product and then engineers the system to cast the desired product.; Specifically, this thesis describes general purpose two-dimensional surface-coupled methodologies for solving both of these problems. The methods are general purpose in the sense that any two-dimensional system may be analyzed and the methods may be used in computer-aided engineering systems. The surface-coupled modeling is applicable to high frequency systems where the skin depth is negligible with respect to the characteristic dimensions of the liquid metal and ignores the stirring effects of the Lorentz forces. A two-step iterative procedure involving finite element solution of the magnetic field structure and a robust general-purpose method for updating a discretized free boundary based on the stresses (magnetic, surface tension, gravitational pressure) acting on that free boundary is used for calculation of two-dimensional equilibrium shapes. The method for updating the free boundary is also shown to predict instability mechanisms. An integral method is developed for solving the inverse shaping problem and tested using the free boundary calculation procedure.