The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions

Jaśkowiec, Jan; Milewski, Sławomir

doi:10.1016/j.camwa.2015.06.020

Cited by 15 publications

(3 citation statements)

References 28 publications

(63 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In such a manner, resulting approximation coefficients have simple interpretation as local derivatives, up to the assumed pth order. Fundamentals and general remarks concerning the MWLS technique may be found in [18,25,31], whereas details of its application for the second-order differential equations are given in [45,46,48]. Comparison of MWLS (with singular weight functions) and MLS (with non-singular weight functions) techniques with other meshless approximation methods may be found in [29,31,33,35].…”

Section: Determination Of Direction Selection Probabilitiesmentioning

confidence: 99%

A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method

Milewski

2019

Numer Algor

View full text Add to dashboard Cite

This paper is devoted to the development of an innovative Matlab software, dedicated to the numerical analysis of two-dimensional elliptic problems, by means of the probabilistic approach. This approach combines features of the Monte Carlo random walk method with discretization and approximation techniques, typical for meshless methods. It allows for determination of an approximate solution of elliptic equations at the specified point (or group of points), without a necessity to generate large system of equations for the entire problem domain. While the procedure is simple and fast, the final solution may suffer from both stochastic and discretization errors. The attached Matlab software is based on several original author's concepts. It permits the use of arbitrarily irregular clouds of nodes, non-homogeneous right-hand side functions, mixed type of boundary conditions as well as variable material coefficients (of anisotropic materials). The paper is illustrated with results of analysis of selected elliptic problems, obtained by means of this software. Keywords Monte Carlo method • Random walk technique • Meshless methods • Elliptic problems • Finite difference method • Implementation in Matlab

show abstract

Section: Determination Of Direction Selection Probabilitiesmentioning

confidence: 99%

A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method

Milewski

2019

Numer Algor

View full text Add to dashboard Cite

show abstract

“…Currently, a number of studies have been performed to develop coupled methods of FEM and meshless methods. A majority of existing studies focused on coupling of FEM and weak-form meshless methods such as elementfree Galerkin (EFG) method [16][17][18], meshless local Petrov-Galerkin (MLPG) method [19,20], and meshless finite difference method [21,22]. Most of these studies [16,17,19,20] constructed interface elements and established special approximation functions in these elements to achieve a continuous transition between FEM and meshless regions.…”

Section: Introductionmentioning

confidence: 99%

An Explicit Coupled Method of FEM and Meshless Particle Method for Simulating Transient Heat Transfer Process of Friction Stir Welding

Xiao

2020

Mathematical Problems in Engineering

View full text Add to dashboard Cite

Friction stir welding (FSW) is a favorable welding technology for aluminum alloys. The FSW process involves complex heat and mass transfer. Explicit meshless particle methods are currently popular methods for simulating the process, but they require expensive computational cost. Coupling explicit finite element method (FEM) and meshless particle methods can ease the problem by making use of high efficiency of FEM and advantages of meshless particle methods. Though many efforts have been made to couple FEM and meshless particle methods for transient dynamics problems, coupling them for transient heat transfer problems is seldom addressed. In this work, we focus on treating this problem. We developed an explicit coupled method of FEM and the meshless particle method presented in a previous work and used it to simulate the thermal process during FSW. In the method, FEM using lumped heat capacity matrix and low-order numerical integration is constructed to obtain high efficiency. A new coupling algorithm is proposed to link thermal calculations of the weak-form FEM and the strong-form meshless particle method. Forward Euler method is used for time integration to achieve an explicit algorithm. The coupled method is used to calculate a numerical example having analytical solution. Calculated results show that it can achieve a good accuracy. The method is employed to simulate FSW of Al 6061-T6 plates. It predicts thermal cycles in good agreement with experimental results. It shows an accuracy comparable to that of the meshless particle method while having a higher efficiency than the latter.

show abstract

“…In addition to the previous three main groups of methods, some other procedures have been proposed. For example, Jagkowiec and Milewski laid off a thin layer of material along the essential boundary or the material interface and utilized finite difference to approximate partial derivatives of the field variables in the thin layer, leading to an asymmetric coefficient matrix. By direct splitting the meshfree function space, Schweitzer implemented a conforming local treatment of essential boundary data, where the PU functions must be flat‐top.…”

Section: Introductionmentioning

confidence: 99%

Exact imposition of essential boundary condition and material interface continuity in Galerkin‐based meshless methods

Zheng

2016

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary Represented by the element free Galerkin method, the meshless methods based on the Galerkin variational procedure have made great progress in both research and application. Nevertheless, their shape functions free of the Kronecker delta property present great troubles in enforcing the essential boundary condition and the material continuity condition. The procedures based on the relaxed variational formulations, such as the Lagrange multiplier‐based methods and the penalty method, strongly depend on the problem in study, the interpolation scheme, or the artificial parameters. Some techniques for this issue developed for a particular method are hard to extend to other meshless methods. Under the framework of partition of unity and strict Galerkin variational formulation, this study, taking Poisson's boundary value problem for instance, proposes a unified way to treat exactly both the material interface and the nonhomogeneous essential boundary as in the finite element analysis, which is fit for any partition of unity‐based meshless methods. The solution of several typical examples suggests that compared with the Lagrange multiplier method and the penalty method, the proposed method can be always used safely to yield satisfactory results. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions

Cited by 15 publications

References 28 publications

A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method

A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method

An Explicit Coupled Method of FEM and Meshless Particle Method for Simulating Transient Heat Transfer Process of Friction Stir Welding

Exact imposition of essential boundary condition and material interface continuity in Galerkin‐based meshless methods

Contact Info

Product

Resources

About