Newton Linearization of the Curvature Operator in Structured Grid Generation with Sample Solutions

Abstract

Elliptic grid generation equations based on the Laplacian operator have the well-known property of clustering the mesh near convex boundaries and declustering it near concave boundaries. In prior work, a new differential operator was derived and presented to address this issue. This new operator retains the strong smoothing properties of the Laplacian without the latter’s adverse curvature effects. However, the new operator exhibits slower convergence properties than the Laplacian, which can lead to increased turnaround times and in some cases preclude the achievement of convergence to machine accuracy. In the work presented here, a Newton linearization of the new operator is presented, with the objective of achieving more robust convergence properties. Sample solutions are presented by evaluating a number of solvers and preconditioners and assessing the convergence properties of the solution process. The efficiency of each solution method is demonstrated with applications to two-dimensional airfoil meshes.