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Model order reduction for meshfree solution of Poisson singularity problems
Author(s) -
Chen JiunShyan,
Marodon Camille,
Hu HsinYun
Publication year - 2014
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4743
Subject(s) - singularity , mathematics , projection (relational algebra) , gravitational singularity , basis function , galerkin method , reduction (mathematics) , projection method , kernel (algebra) , mathematical analysis , mathematical optimization , dykstra's projection algorithm , finite element method , algorithm , geometry , physics , combinatorics , thermodynamics
Summary Model order reduction (MOR) techniques for enriched reproducing kernel meshfree methods are proposed for analysis of Poisson problems with mild and strong singularities. The employment of an integrated singular basis function method (ISBFM), in conjunction with the selection of harmonic near‐tip asymptotic basis functions, leads to a Galerkin formulation in which the non‐smooth near‐tip basis functions appear only on the boundaries away from the singularity point. This approach avoids the need of integrating the derivatives of non‐smooth functions near the singularity point and yields a discrete system that allows effective MOR procedures. Under this framework, a decomposed reduction method equipped with two distinct projections for smooth and non‐smooth parts of the finite‐dimensional space is proposed. Compared with the uniform reduction approach using a single projection operator, the decomposed projection on ISBFM discrete system preserves the singularity behavior of the fine‐scale solution in its lower‐dimensional approximation. Analytical error estimation and stability analysis show that ISBFM with the decomposed projection can achieve better accuracy with only a slight increase of condition number compared with the uniform reduction approach in the reduced‐order solution of singularity problems. The numerical tests validate the effectiveness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.