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

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 pro...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering Vol. 190; no. 29; pp. 3797 - 3818
Main Authors: Patra, Abani, Gupta, Ashish
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 13.04.2001
Elsevier
Subjects:
Applied sciences
Computational techniques
Exact sciences and technology
Finite-element and galerkin methods
Mathematical methods in physics
Mathematical programming
Operational research and scientific management
Operational research. Management science
Physics
Gradient
Refinement method
Irregularity
Automatic mesh generation
Adaptive method
Optimization
Finite element method
Problem solving
Convergence rate
Nonlinear programming
Program design
Mesh generation
Mathematical programming
ISSN:0045-7825, 1879-2138
Online Access:Get full text
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Summary: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 User's Guide, Technical Report SOL 83-20R, Stanford University, Stanford, 1987, Revised February 1995], was used as the NLP optimizer.
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ISSN:0045-7825
1879-2138
DOI:10.1016/S0045-7825(00)00298-X