New boundary constraints for elliptic systems used in grid generation problems

Abstract

New boundary constraints for elliptic partial differential equations as used in grid generation problems in generalized curvilinear coordinate systems are proposed in this paper. These constraints, based on the principle of local conservation of thermal energy in the vicinity of the boundaries, are derived using the Green’s Theorem. They uniquely determine the so called decay parameters in the inhomogeneous terms of these elliptic systems. These constraints 1 Invention under US Patent application process. 1 are designed for boundary clustered grids where large gradients in physical quantities need to be resolved adequately. It is observed that the present formulation also works satisfactorily for mild clustering. Therefore, a closure for the decay parameter specification for elliptic grid generation problems has been provided resulting in a fully automated elliptic grid generation technique. Thus, there is no need for a parametric study of these decay parameters since the new constraints fix them uniquely. It is also shown that for Neumann type boundary conditions, these boundary constraints uniquely determine the solution to the internal elliptic problem thus eliminating the nonuniqueness of the solution of an internal boundary value grid generation problem with Neumann boundary conditions.