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

Adaptive refinement usually involves refining or enriching a fraction of mesh elements by one level based on a cut-off criterion, requiring several costly intermediate solutions before a mesh that yields an acceptable solution is obtained. We avoid this by formulating and solving the mesh design problem as a mathematical program. Our approach simultaneously modifies both mesh size (h) and local polynomial order (p) to yield an “optimal” mesh for a target error or given computational cost with gradients from local convergence rates. Constraints such as the one irregularity rule during mesh refinement 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 analytical gradients derived from initial mesh calculations. Finally, we apply our method to three model problems showing that complex hp-adaptive grids can be obtained directly from a uniform coarse grid. A commercial optimization software, MINOS [B.A. Murtagh, M.A. Saunders, MINOS 5.4 Userʹs Guide, Technical Report SOL 83-20R, Stanford University, Stanford, 1987, Revised February 1995], was used as the NLP optimizer.