Does the Partition of Unity Influence the Convergence of Schwarz Methods?

2020 ◽  
pp. 3-15 ◽  
Author(s):  
Martin J. Gander
Keyword(s):  
Partition Of Unity ◽  
Schwarz Methods

scholarly journals Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2288
Author(s):  
Hongming Luo ◽  
Guanhua Sun
Keyword(s):  
Structural Dynamics ◽  
Interpolation Method ◽  
Mass Matrix ◽  
Time Integration ◽  
Partition Of Unity ◽  
Implicit Time ◽  
Lumped Mass ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Consistent Mass Matrix

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

VARIATIONAL MULTISCALE ANALYSIS OF SOLIDS WITH STRAIN GRADIENT PLASTIC EFFECTS USING MESHFREE APPROXIMATION

2005 ◽  
Vol 02 (04) ◽  
pp. 601-626 ◽  
Author(s):  
JEOUNG-HEUM YEON ◽  
SUNG-KIE YOUN
Keyword(s):  
Strain Gradient ◽  
Partition Of Unity ◽  
Meshfree Method ◽  
Internal Variable ◽  
Plasticity Theory ◽  
Fine Scale ◽  
Meshfree Approximation ◽  
Gradient Effect ◽  
Classical Plasticity ◽  
Variational Form

A meshfree multiscale method is presented for efficient analysis of solids with strain gradient plastic effects. In the analysis of strain gradient plastic solids, localization due to increased hardening of strain gradient effect appears. Chen-Wang theory is adopted, as a strain gradient plasticity theory. It represents strain gradient effects as an internal variable and retains the essential structure of classical plasticity theory. In this work, the scale decomposition is carried out based on variational form of the problem. Coarse scale is designed to represent global behavior and fine scale to represent local behavior and gradient effect by using the intrinsic length scale. From the detection of high strain gradient region, fine scale region is adopted. Each scale variable is approximated using meshfree method. Meshfree approximation is well suited for adaptivity. As a method of increasing resolution, partition of unity based extrinsic enrichment is used. Each scale problem is solved iteratively. The proposed method is applied to bending of a thin beam and bimaterial shear layer and micro-indentation problems. Size effects can be effectively captured in the results of the analysis.

Application of Extended Finite Element Method (XFEM) to Stress Intensity Factor Calculations

2015 ◽  
Author(s):  
Do-Jun Shim ◽  
Mohammed Uddin ◽  
Sureshkumar Kalyanam ◽  
Frederick Brust ◽  
Bruce Young
Keyword(s):  
Finite Element Method ◽  
Stress Intensity Factor ◽  
Finite Element ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Degrees Of Freedom ◽  
Extended Finite Element Method ◽  
Partition Of Unity ◽  
Extended Finite Element ◽  
Element Method

The extended finite element method (XFEM) is an extension of the conventional finite element method based on the concept of partition of unity. In this method, the presence of a crack is ensured by the special enriched functions in conjunction with additional degrees of freedom. This approach also removes the requirement for explicitly defining the crack front or specifying the virtual crack extension direction when evaluating the contour integral. In this paper, stress intensity factors (SIF) for various crack types in plates and pipes were calculated using the XFEM embedded in ABAQUS. These results were compared against handbook solutions, results from conventional finite element method, and results obtained from finite element alternating method (FEAM). Based on these results, applicability of the ABAQUS XFEM to stress intensity factor calculations was investigated. Discussions are provided on the advantages and limitations of the XFEM.

scholarly journals *—Inductive limits and partition of unity

1989 ◽  
Vol 12 (3) ◽  
pp. 429-434
Author(s):  
V. Murali
Keyword(s):  
Inductive Limit ◽  
Partition Of Unity ◽  
Vector Spaces ◽  
Inductive Limits ◽  
Topological Vector Spaces ◽  
Limit Space ◽  
Direct Sum

In this note we define and discuss some properties of partition of unity on *-inductive limits of topological vector spaces. We prove that if a partition of unity exists on a *-inductive limit space of a collection of topological vector spaces, then it is isomorphic and homeomorphic to a subspace of a *-direct sum of topological vector spaces.

The Partition of Unity Finite Element Method for the simulation of waves in air and poroelastic media

2014 ◽  
Vol 135 (2) ◽  
pp. 724-733 ◽  
Author(s):  
Jean-Daniel Chazot ◽  
Emmanuel Perrey-Debain ◽  
Benoit Nennig
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partition Of Unity ◽  
Poroelastic Media ◽  
Element Method

Asynchronous optimized Schwarz methods with and without overlap

2017 ◽  
Vol 137 (1) ◽  
pp. 199-227 ◽  
Author(s):  
Frédéric Magoulès ◽  
Daniel B. Szyld ◽  
Cédric Venet
Keyword(s):  
Schwarz Methods ◽  
Optimized Schwarz Methods

Optimized Schwarz Methods with Elliptical Domain Decompositions

2021 ◽  
Vol 86 (2) ◽  
Author(s):  
Xin Chen ◽  
Martin J. Gander ◽  
Yingxiang Xu
Keyword(s):  
Schwarz Methods ◽  
Optimized Schwarz Methods
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