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Adaptive strategies to improve the application of the generalized finite differences method in 2D and 3D
Author(s) -
Ureña Miguel,
Benito Juan José,
Ureña Francisco,
García Ángel,
Gavete Luis,
Benito Luis
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4675
Subject(s) - key (lock) , a priori and a posteriori , process (computing) , mathematics , mathematical optimization , cloud computing , finite element method , computer science , algorithm , philosophy , physics , computer security , epistemology , thermodynamics , operating system
The generalized finite differences method allows the use of irregular clouds of nodes. The optimal values of the key parameters of the method vary depending on how the nodes in the cloud are distributed, and this can be complicated especially in 3D. Therefore, we establish 2 criteria to allow the automation of the selection process of the key parameters. These criteria depend on 2 discrete functions, one of them penalizes distances and the other one penalizes imbalances. In addition, we show how to generate irregular clouds of nodes more efficient than finer regular clouds of nodes. We propose an improved and more versatile h‐adaptive method that allows adding, moving, and removing nodes. To decide which nodes to act on, we use an indicator of the error a posteriori. This h‐adaptive method gives results more accurate than those ones presented for the generalized finite differences method so far and, in addition, with fewer nodes. In addition, this method can be used in time‐dependant problems to increase the temporal step or to avoid instabilities. As an example, we apply it in problems of seismic waves propagation.