The Partition of Unity Meshfree Method for Solving Transport-Reaction Equations on Complex Domains: Implementation and Applications in the Life Sciences

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.

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Enriched Meshfree Method for an Accurate Numerical Solution of the Motz Problem

Advances in Mathematical Physics ◽

10.1155/2016/6324754 ◽

2016 ◽

Vol 2016 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Won-Tak Hong

Keyword(s):

Numerical Solution ◽

Series Expansion ◽

Abrupt Change ◽

Partition Of Unity ◽

Meshfree Method ◽

Local Approximation ◽

Approximation Space ◽

Enrichment Technique ◽

Accurate Numerical Solution ◽

Function Enrichment

We present an enriched meshfree solution of the Motz problem. The Motz problem has been known as a benchmark problem to verify the efficiency of numerical methods in the presence of a jump boundary data singularity at a point, where an abrupt change occurs for the boundary condition. We propose a singular basis function enrichment technique in the context of partition of unity based meshfree method. We take the leading terms of the local series expansion at the point singularity and use them as enrichment functions for the local approximation space. As a result, we obtain highly accurate leading coefficients of the Motz problem that are comparable to the most accurate numerical solution. The proposed singular enrichment technique is highly effective in the case of the local series expansion of the solution being known. The enrichment technique that is used in this study can be applied to monotone singularities (of typerαwithα<1) as well as oscillating singularities (of typerαsin⁡(ϵlog⁡r)). It is the first attempt to apply singular meshfree enrichment technique to the Motz problem.

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Computation of stress intensity factors in functionally graded materials using partition-of-unity meshfree method

The Aeronautical Journal ◽

10.1017/s0001924000007636 ◽

2012 ◽

Vol 116 (1186) ◽

pp. 1263-1287 ◽

Cited By ~ 4

Author(s):

N. Muthu ◽

S. K. Maiti ◽

B. G. Falzon ◽

I. Guiamatsia

Keyword(s):

Stress Intensity ◽

Functionally Graded Materials ◽

Stress Intensity Factors ◽

Partition Of Unity ◽

Meshfree Method ◽

Absolute Error ◽

Functionally Graded ◽

Local Smoothing ◽

Intensity Factors ◽

Graded Materials

Abstract This paper describes the computation of stress intensity factors (SIFs) for cracks in functionally graded materials (FGMs) using an extended element-free Galerkin (XEFG) method. The SIFs are extracted through the crack closure integral (CCI) with a local smoothing technique, non-equilibrium and incompatibility formulations of the interaction integral and the displacement method. The results for mode I and mixed mode case studies are presented and compared with those available in the literature. They are found to be in good agreement where the average absolute error for the CCI with local smoothing, despite its simplicity, yielded a high level of accuracy.

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