An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics

2015 ◽  
Vol 107 (7) ◽  
pp. 603-630 ◽  
Author(s):  
Michael Hillman ◽  
Jiun-Shyan Chen
Keyword(s):  
Meshfree Methods ◽  
Nonlinear Mechanics ◽  
Nodal Integration ◽  
Linear And Nonlinear ◽  
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.

A GALERKIN MESHFREE METHOD WITH STABILIZED CONFORMING NODAL INTEGRATION FOR GEOMETRICALLY NONLINEAR ANALYSIS OF SHEAR DEFORMABLE PLATES

2011 ◽  
Vol 08 (04) ◽  
pp. 685-703 ◽  
Author(s):  
DONGDONG WANG ◽  
YUE SUN
Keyword(s):  
Nonlinear Analysis ◽  
Large Deflection ◽  
Lagrangian Description ◽  
Geometrically Nonlinear ◽  
Geometrically Nonlinear Analysis ◽  
Nodal Integration ◽  
Galerkin Meshfree ◽  
Shear Deformable ◽  
Shear Deformable Plates ◽  
Stabilized Conforming Nodal Integration

A Galerkin meshfree approach formulated within the framework of stabilized conforming nodal integration (SCNI) is presented for geometrically nonlinear analysis of large deflection shear deformable plates. This method is based upon a Lagrangian curvature smoothing in which the smoothed curvature is constructed within a nodal representative domain on the initial configuration. It is shown that the Lagrangian smoothed nodal gradients of the meshfree shape function is capable of exactly representing arbitrary constant curvature fields in the discrete sense of nodal integration. The consistent linearization is performed on the weak form of large deflection plate in the context of the total Lagrangian description. Subsequently, the discrete incremental equations are obtained by the method of SCNI in which to relieve the locking as well as ensure the stability of the present scheme, the bending contribution is evaluated using the smoothed nodal gradients, while the membrane and shear contributions are computed with the direct nodal gradients. The effectiveness of the present method is thoroughly demonstrated through several numerical examples.

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

Efficient Meshfree Formulation for Metal Forming Simulations

2000 ◽  
Vol 123 (4) ◽  
pp. 462-467 ◽  
Author(s):  
Sangpil Yoon ◽  
Cheng-Tang Wu ◽  
Hui-Ping Wang ◽  
Jiun-Shyan Chen
Keyword(s):  
Computational Efficiency ◽  
Metal Forming ◽  
Integration Method ◽  
Galerkin Approximation ◽  
Meshfree Methods ◽  
Nodal Integration ◽  
Strain Smoothing ◽  
Forming Analysis ◽  
Forming Simulations ◽  
Integration Constraint

A stabilized conforming (SC) nodal integration method is developed for elastoplastic contact analysis of metal forming processes. In this approach, strain smoothing stabilization is introduced to eliminate spatial instability in collocation meshfree methods. The gradient matrix associated with strain smoothing satisfies the integration constraint (IC) of linear exactness in the Galerkin approximation. Strain smoothing formulation and numerical procedures for history-dependent problems are introduced. Applications to metal forming analysis are presented, with the results demonstrating a significant improvement in computational efficiency without loss of accuracy.

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