scholarly journals A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition

2020 ◽  
Vol 121 (10) ◽  
pp. 2174-2205 ◽  
Author(s):  
R. Silva‐Valenzuela ◽  
A. Ortiz‐Bernardin ◽  
N. Sukumar ◽  
E. Artioli ◽  
N. Hitschfeld‐Kahler
Keyword(s):  
Integration Scheme ◽  
Galerkin Methods ◽  
Nodal Integration ◽  
Virtual Element

A Moving Kriging Interpolation Meshfree Method Based on Naturally Stabilized Nodal Integration Scheme for Plate Analysis

2019 ◽  
Vol 16 (04) ◽  
pp. 1850100 ◽  
Author(s):  
Chien H. Thai ◽  
H. Nguyen-Xuan
Keyword(s):  
Computational Cost ◽  
Meshfree Method ◽  
Weak Form ◽  
Delta Function ◽  
Current Approach ◽  
Kriging Interpolation ◽  
Integration Scheme ◽  
Nodal Integration ◽  
Moving Kriging Interpolation ◽  
Quadrature Scheme

A moving Kriging interpolation (MKI) meshfree method based on naturally stabilized nodal integration (NSNI) scheme is presented to study static, free vibration and buckling behaviors of isotropic Reissner–Mindlin plates. Gradient strains are directly computed at nodes similar to the direct nodal integration (DNI). Outstanding features of the current approach are to alleviate instability solutions in the DNI and to decrease computational cost significantly when compared with the traditional high-order Gauss quadrature scheme. The NSNI is a naturally implicit gradient expansion and does not employ a divergence theorem for strain fields as addressed in the stabilized conforming nodal integration method. The present formulation is derived from the Galerkin weak form and avoids a naturally shear-locking phenomenon without using any other techniques. Thanks to satisfied Kronecker delta function property of MKI shape function, essential boundary conditions (BCs) are easily and directly enforced similar to the finite element method. A variety of numerical examples with various geometries, stiffness ratios and BCs are studied to verify the effectiveness of the present approach.

scholarly journals Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part II

2019 ◽  
Vol 53 (2) ◽  
pp. 503-522 ◽  
Author(s):  
Philip L. Lederer ◽  
Christoph Lehrenfeld ◽  
Joachim Schöberl
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Integration Scheme ◽  
Galerkin Methods ◽  
Robust Analysis ◽  
Reconstruction Operator ◽  
The Impact

The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size h and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness (pressure independent velocity error estimates) using a modified force discretization. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed H(div)-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements, Lederer and Schöberl (IMA J. Numer. Anal. (2017)) and is derived by a direct approach instead of using a best approximation Céa like result. We further treat the impact of the reconstruction operator on the hp analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier–Stokes equations which is based on the methods recently presented in Lehrenfeld and Schöberl (Comp. Methods Appl. Mech. Eng. 307 (2016) 339–361) and includes the ideas of the reconstruction operator.

An efficient nodal integration with quadratic exactness for three-dimensional meshfree Galerkin methods

2016 ◽  
Vol 70 ◽  
pp. 99-113 ◽  
Author(s):  
Bingbing Wang ◽  
Qinglin Duan ◽  
Yulong Shao ◽  
Xikui Li ◽  
Dixiong Yang ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Galerkin Methods ◽  
Nodal Integration

Quadratically consistent nodal integration for second order meshfree Galerkin methods

2014 ◽  
Vol 54 (2) ◽  
pp. 353-368 ◽  
Author(s):  
Qinglin Duan ◽  
Bingbing Wang ◽  
Xin Gao ◽  
Xikui Li
Keyword(s):  
Second Order ◽  
Galerkin Methods ◽  
Nodal Integration

Meshfree Analysis of Higher-Order Gradient Crystal Plasticity Using Nodal Integration

2019 ◽  
Vol 794 ◽  
pp. 214-219
Author(s):  
Tota Niiro ◽  
Yuichi Tadano
Keyword(s):  
Size Effect ◽  
Crystal Plasticity ◽  
Meshfree Method ◽  
Higher Order ◽  
Integration Scheme ◽  
Plasticity Model ◽  
The Finite Element Method ◽  
Nodal Integration ◽  
Crystal Plasticity Model ◽  
Meshfree Analysis

The size effect of metallic materials is one of the important factors for understanding characteristics of material. The higher-order gradient crystal plasticity is a powerful model for describing the size effect. However, it is known that the finite element method sometimes provides an improper solution. In this study, we analyze the higher-order gradient crystal plasticity model using a meshfree method, and a nodal integration scheme is introduced to improve the analysis accuracy. The effectiveness and stability of the meshfree method for the higher-order gradient crystal plasticity model are quantitatively discussed.

Recent techniques for PDE discretizations on polyhedral meshes

2014 ◽  
Vol 24 (08) ◽  
pp. 1453-1455 ◽  
Author(s):  
N. Bellomo ◽  
F. Brezzi ◽  
G. Manzini
Keyword(s):  
Discontinuous Galerkin ◽  
Finite Differences ◽  
Discontinuous Galerkin Methods ◽  
Galerkin Methods ◽  
Special Issue ◽  
Polyhedral Meshes ◽  
Recent Developments ◽  
Virtual Element Methods ◽  
Mimetic Finite Differences ◽  
Virtual Element

This brief paper is an introduction to the papers published in a special issue devoted to survey on recent techniques for discretizing Partial Differential Equations on general polygonal and polyhedral meshes. The number of different techniques to deal with discretizations on polygonal and polyhedral meshes is quite huge, and their history is quite long. Here we concentrate on the most recent techniques, including Mimetic Finite Differences, Virtual Element Methods, and the recent developments, in this direction, of Finite Volumes and Discontinuous Galerkin Methods.

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