A systematic strategy for simultaneous adaptive hp finite element mesh modification using nonlinear programming

Adaptive refinement usually involves refining or enriching a fraction of me sh elements by one level based on a cut-off criterion, requiring several co stly intermediate solutions before a mesh that yields an acceptable solutio n is obtained. We avoid this by formulating and solving the mesh design pro blem as a mathematical program. Our approach simultaneously modifies both m esh size (h) and local polynomial order (p) to yield an "optimal" mesh for a target error or given computational cost with gradients from local conver gence rates. Constraints such as the one irregularity rule during mesh refi nement are systematically incorporated in this formulation. The design task leads to a mixed integer nonlinear program (MINLP), that is relaxed to an NLP. To reduce the computations for the NLP, we employ simplified analytica l gradients derived from initial mesh calculations. Finally, we apply our m ethod to three model problems showing that complex hp-adaptive grids can be obtained directly from a uniform coarse grid. A commercial optimization so ftware, MINOS [B.A. Murtagh, M.A. Saunders, MINOS 5.4 User's Guide, Technic al Report SOL 83-20R, Stanford University, Stanford, 1987, Revised February 1995], was used as the NLP optimizer. (C) 2001 Elsevier Science B.V. All r ights reserved.