Model Order Reduction Methods for Meshfree Approximation of Problems with Singularities and Discontinuities
The modeling of problems with singularities and discontinuities, such as cracks in elastic media, is one of the most challenging problems in computational mechanics. The numerical solution of such problems necessitates sufficient resolution, which is very costly. In this work, model order reduction (MOR) techniques are developed for problems with singularities based on meshfree methods. Theoretical investigation on the performance of the proposed methods via error estimation, stability analysis, and complexity analysis is also carried out in this research.
First, the full discrete model is constructed using enriched meshfree approximation to properly approximate singularities and discontinuities. This requires high order quadrature rules for accurate integration of enrichment functions and their derivative. An integrated singular basis function method (ISBFM) is introduced to circumvent this difficulty and the resulting ISBFM Galerkin formulation only integrates the enrichment functions on the boundaries away from the singularity point. This ISBFM Galerkin formulation also allows effective MOR procedures proposed in this work.
Two MOR methods via projection of a discrete system obtained from the ISBFM Galerkin formulation have been proposed, the ISBFM-UR method based on a uniform projection and the ISBFM-DR method based on a decomposed projection of smooth and non-smooth DOF in order to retain the singular behavior of the full scale solution.
Error estimation and stability analysis show that the reduced solution from ISBFM-DR can achieve a much-enhanced accuracy with slight increase of condition number compared to ISBFM-UR. While the stability analysis shows that the two MOR methods provide better conditioning than the full scale system, it also suggests that the conditioning of the ISBFM-DR method is compromised by its better accuracy compared to that of the ISBFM-UR method. Nevertheless, the complexity analysis shows that the ISBFM-DR method provides a better efficiency in addition to the enhanced accuracy compared to the ISBFM-UR method.
The proposed MOR methods are applied Poisson problems with singularities and LEFM problems. The numerical tests validate the effectiveness of the MOR methods, and the numerical results on accuracy and stability of the proposed methods are shown to be in good agreement with the analytical predictions carried in this research.