Speaker: Dr. John M. Finn Lawrence Livermore National laboratory
Abstract: I will present a new cell-area equidistribution method for grid adaptation, based on Monge-Kantorovich optimization (or Monge-Kantorovich optimal transport). The method is based on a rigorous variational principle, in which the L_2 norm of the grid displacement is minimized, constrained locally to produce a prescribed positive-definite cell volume distribution. The procedure involves solving the Monge-Ampere equation,a single, nonlinear, elliptic scalar equation with no free parameters, and with proved solution existence and uniqueness theorems. We show that, for sufficiently small grid displacement, this method also minimizes the mean grid-cell distortion, measured by the trace of the covariant metric tensor. We solve the Monge-Ampere equation numerically with a Jacobian-Free Newton-Krylov method. The ellipticity property of the Monge-Ampere equation allows multigrid preconditioning techniques to be used effectively, delivering a scalable algorithm under grid refinement. Several challenging test cases demonstrate that this method produces optimal grids in which the constraint is satisfied numerically to truncation error. We also compare this method to the well known deformation method [G. Liao and D. Anderson, Appl. Anal., v.44, p.285 (1992)]. We show that the new method achieves the desired equidistributed grid using comparable computational time, but with considerably better grid quality than the deformation method. I will present recent work with more general boundaries in 2D and in a cube in 3D.