Optimization of Wall Electrodes for Electro-Hydrodynamic Control of Natural Convection during Solidification

Abstract

This article presents a numerical procedure to reduce and possibly control the natural convection effects in a cavity filled with a molten material by applying an external electric field whose intensity and spatial distributions are obtained by the use of a hybrid optimizer. This conceptually new approach to manufacturing could be used in creation of layered and functionally graded materials and objects. In the case of steady electro-hydrodynamics (EHD), the flow-field of electrically charged particles in a solidifying melt is influenced by an externally applied electric field while the existence of any magnetic field is neglected. Solidification front shape, distribution of the charged particles in the accrued solid, and the amount of accrued solid phase in such processes can be influenced by an appropriate distribution and orientation of the electric field. The intensities of the electrodes along the boundaries of the cavity were described using B-splines. The inverse problem was then formulated to find the electric boundary conditions (the coefficients of the B-splines) in such a way that the gradients of temperature along the horizontal direction are minimized. For this task we used a hybrid optimization algorithm that incorporates several of the most popular optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, the quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). The transient Navier-Stokes and Maxwell equations were discretized using the finite volume method in a generalized, curvilinear nonorthogonal coordinate system. For the phase change problems, we used the enthalpy method.

**This paper was presented at the International Mechanical Engineering Congress and Exhibition (IMECE'03), Washington, D.C., USA, November 16–21, 2003 as American Society of Mechanical Engineers (ASME) paper IMECE 2003–41703.

Keywords:

• Solidification
• Electrohydrodynamics
• Numerical methods
• Optimization
• Manufacturing processes

Acknowledgments

The first author is grateful for the postdoctoral fellowship received from University of Texas at Arlington and from CNPq, a Brazilian council for scientific and technological development.

The second author is grateful for the partial support provided for this research from the grant NSF DMS-0073698 administered through the Computational Mathematics program.

Notes

**This paper was presented at the International Mechanical Engineering Congress and Exhibition (IMECE'03), Washington, D.C., USA, November 16–21, 2003 as American Society of Mechanical Engineers (ASME) paper IMECE 2003–41703.