A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

2014

Int. J. Numer. Model.

This paper presents a meshfree‐enriched finite element formulation using nodal integration for electrostatic analysis. The meshfree‐enriched finite element method, originally proposed to solve the incompressible constraint in mechanical problem, is revisited in this paper and applied to the analysis of electrostatic problems to improve the solution accuracy of conventional finite element method. A novel nodal integration scheme based on the meshfree‐enriched finite element mesh is developed for the integration of discrete equation and is shown to pass the linear exactness in the Galerkin approximation. To demonstrate the accuracy of the proposed formulation, two numerical examples are studied and comparisons are made to several other finite element formulations. Copyright © 2014 John Wiley & Sons, Ltd.

Section: Variational Formulation and Discrete Governing Equationmentioning

confidence: 82%

“…Assume a convex hull C ( P ) of a node set

P = \{ξ_{i} \in {\overset{Q}{true}}_{e}, i = 1, \dots, 5\} \subset {frakturR}^{2}

defined by []

C (P) \equiv \{\sum_{i = 1}^{5} α_{i} ξ_{i} |ξ_{i} \in P, α_{i} \in {frakturR}^{+} \cup \{0\}, \sum_{i = 1}^{5} α_{i} = 1\},

the parametric convex meshfree method is to construct convex approximations of a given function u in the form

u^{h} (bold-italicξ) = \sum_{i = 1}^{5} {normalΨ}_{i} (bold-italicξ) u_{i} \forall ξ \in {\overset{Q}{true}}_{e}

with the generating function

{normalΨ}_{i} : C (P) \to R

satisfying the following polynomial reproduction property

\sum_{i = 1}^{5} {normalΨ}_{i} (ξ) ξ_{i} = ξ \forall ξ \in C (P) .

…”

Section: Review On Meshfree‐enriched Finite Element Methodsmentioning

confidence: 99%

“…The first‐order convex Generalized Meshfree (GMF) approximation in two‐dimensional extended parametric space is expressed as

{normalΨ}_{i} (ξ, λ_{r}) = \frac{κ_{i}}{κ} = \frac{φ_{a} (ξ; ξ - ξ_{i}) B_{i} (ξ - ξ_{i}, λ_{r})}{true \sum_{j = 1}^{5} φ_{a} ((), ξ; ξ - ξ_{j}) B_{j} ((), ξ - ξ_{j}, λ_{r})}

and is subjected to linear constraints

Z_{r} (ξ, λ_{r}) = \sum_{i = 1}^{5} {normalΨ}_{i} \cdot (ξ - ξ_{i}) = 0,

where

κ_{i} = φ_{a} (ξ; ξ - ξ_{i}) B_{i} (ξ - ξ_{i}, λ_{r})

κ = \sum_{i = 1}^{5} κ_{i} = \sum_{i = 1}^{5} φ_{a} (ξ; ξ - ξ_{i}) B_{i} (ξ - ξ_{i}, λ_{r}) .

…”

Section: Review On Meshfree‐enriched Finite Element Methodsmentioning

confidence: 99%

\sum_{i = 1}^{5} {normalΨ}_{i} (x) = 1, Ψ_{i} (x) \geq 0 \forall x \in Q_{e}

{normalΨ}_{5} (bold-italicx) = 0 \forall x \in \partial Q_{e} .

…”

Section: Review On Meshfree‐enriched Finite Element Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Meshfree-enriched electromagnetic finite element formulation using nodal integration

2014

Int. J. Numer. Model.

A parametric convex meshfree formulation for approximating the Helmholtz solution in circular coaxial waveguide

Int J Numerical Modelling

2014

The application of convex meshfree approximation to the time-harmonic electromagnetic wave propagation analysis of a waveguide with non-convex cross section such as the circular coaxial waveguide remains unsolved. This paper introduces a parametric convex meshfree formulation for the circular coaxial waveguide analysis. The present method reformulates the convex meshfree approximation on the basis of a special parametric space-an extended parametric domain. The new parametric domain ensures a one-to-one geometric mapping using the convex meshfree approximation and allows the convex meshfree method to be applied to the oscillatory type of Helmholtz equation for circular coaxial waveguide analysis. Both transverse electric and transverse magnetic mode studies are conducted using the present method, and results are compared with the standard bilinear finite element method.where ∇ 2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 denotes the two-dimensional Laplacian operator. When ϕ = E z , it represents the TE field. For this case, the field E z (x, y) is independent of z. Similarly, ϕ = H z accounts for the TM field. The symbol k c is the cutoff wavenumber. 552L.-F. WANG

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

Numerical Meth Engineering

2013

SUMMARYIn this work, an enhanced cell-based smoothed finite element method (FEM) is presented for the ReissnerMindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge-based smoothing technique through a simple area-weighted smoothing procedure on MITC4 assumed shear strain field. A three-field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement-based formulation via an assumed strain method defined by the edge-smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell-based smoothed FEM for Reissner-Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.