An arbitrary order variationally consistent integration for Galerkin meshfree methods

2013 ◽  
Vol 95 (5) ◽  
pp. 387-418 ◽  
Author(s):  
Jiun-Shyan Chen ◽  
Michael Hillman ◽  
Marcus Rüter
Keyword(s):  
Arbitrary Order ◽  
Meshfree Methods ◽  
Galerkin Meshfree ◽  
Variationally Consistent Integration

Galerkin Meshfree Methods: A Review and Mathematical Implementation Aspects

2019 ◽  
Vol 5 (4) ◽  
Author(s):  
Bhaumik Nagevadiya ◽  
Bhavik Vaghasia ◽  
Nikunj Rachchh ◽  
Ramesh Bhoraniya
Keyword(s):  
Meshfree Methods ◽  
Implementation Aspects ◽  
Galerkin Meshfree

Application of Galerkin meshfree methods to nonlinear thermo-mechanical simulation of solids under extremely high pulsed loading

2013 ◽  
Vol 88 (9-10) ◽  
pp. 2744-2747 ◽  
Author(s):  
Daniel Iglesias Ibáñez ◽  
Juan C. García Orden ◽  
B. Brañas ◽  
J.M. Carmona ◽  
J. Molla
Keyword(s):  
Meshfree Methods ◽  
Mechanical Simulation ◽  
Pulsed Loading ◽  
Galerkin Meshfree

A Boundary Enhancement for the Stabilized Conforming Nodal Integration of Galerkin Meshfree Methods

2015 ◽  
Vol 12 (02) ◽  
pp. 1550009 ◽  
Author(s):  
Dongdong Wang ◽  
Ming Sun ◽  
Pinkang Xie
Keyword(s):  
Transformation Method ◽  
Meshfree Methods ◽  
Shape Functions ◽  
Problem Domain ◽  
Nodal Integration ◽  
Support Size ◽  
Galerkin Meshfree ◽  
Integration Constraint ◽  
Linear Field ◽  
Stabilized Conforming Nodal Integration

The stabilized conforming nodal integration (SCNI) has been successfully developed for Galerkin meshfree methods based upon the linear exactness requirement. In this study, it is shown that for a given problem domain, when the support of the meshfree shape functions associated with the interior nodes do not cover the essential boundary, the linear exactness can be perfectly achieved by the standard SCNI formulation. On the other hand, when the essential boundary lies in the support of the meshfree shape functions of the interior nodes, a linear field may not be exactly obtained with the original SCNI formulation where the essential boundary conditions are enforced via the nodally exact transformation method, and the error even becomes more pronounced with the increase of support size. To resolve this issue, a flux term associated with the essential boundary is recovered in the variational formulation and it turns out to be proper to keep this term since the meshfree shape functions of interior nodes usually do not vanish on the boundary. Consequently the original SCNI integration constraint is revised and the stiffness matrix is enhanced by an additional stiffness contribution from the flux integration along the essential boundary. It is demonstrated that the proposed enhanced formulation is capable of exactly reproducing linear fields regardless of the support sizes. Moreover, several benchmark examples reveal that the present SCNI formulation with boundary enhancement yields better accuracy compared with the original SCNI approach, particularly for meshfree discretizations with larger support sizes.

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