Error analysis of a meshless weak form method based on radial point interpolation technique for Sivashinsky equation arising in the alloy solidification problem

The transient thermo-elastic problems are solved by a cell-based smoothed radial point interpolation method (CS-RPIM). For this method, the problem domain is first discretized using triangular cells, and each cell is further divided into smoothing cells. The field functions are approximated using RPIM shape functions which have Kronecker delta function property. The system equations are derived using the generalized smoothed Galerkin (GS-Galerkin) weak form. At first, the temperature field is acquired by solving the transient heat transfer problem and it is then employed as an input for the mechanical problem to calculate the displacement and stress fields. Several numerical examples with different kinds of boundary conditions are investigated to verify the accuracy, convergence rate and stability of the present method.

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Using Meshfree Weak-Strong Form Method for a 2-D Heat Transfer Problem

Volume 9: Heat Transfer, Fluid Flows, and Thermal Systems, Parts A, B and C ◽

10.1115/imece2009-12525 ◽

2009 ◽

Cited By ~ 2

Author(s):

S. Zahiri ◽

F. Daneshmand ◽

M. H. Akbari

Keyword(s):

Heat Transfer ◽

Boundary Conditions ◽

Transfer Problem ◽

Interpolation Method ◽

Meshfree Method ◽

Weak Form ◽

Heat Transfer Problem ◽

Radial Point Interpolation ◽

Point Interpolation ◽

Local Weak Form

In this work, the numerical simulation of 2-D heat transfer problem is studied by using a meshfree method. The method is based on the local weak form collocation and the meshfree weak-strong (MWS) form. The goal of the paper is to find the temperature distribution in a rectangular plate. The results obtained are compared by those obtained by use of other numerical methods. Two types of boundary conditions are considered in this paper: Dirichlet and Neumann types. The Local Radial Point Interpolation Method (LRPIM) is used as the meshfree method. It is shown that the essential boundary conditions can be easily enforced as in the Finite Element Method (FEM), since the radial point interpolation shape functions posses the Kronecker delta property. It is also shown that the natural (derivative) boundary conditions can be satisfied by using the MWS method and no additional equation or treatment are needed. The MWS method as presented in this paper works well with local quadrature cells for nodes on the natural boundary and can be generated without any difficulty.

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A LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD FOR SOLID MECHANICS PROBLEMS

International Journal of Computational Methods ◽

10.1142/s0219876206001132 ◽

2006 ◽

Vol 03 (04) ◽

pp. 401-428 ◽

Cited By ~ 106

Author(s):

G. R. LIU ◽

Y. LI ◽

K. Y. DAI ◽

M. T. LUAN ◽

W. XUE

Keyword(s):

Solid Mechanics ◽

Interpolation Method ◽

Weak Form ◽

Delta Function ◽

Field Function ◽

Radial Point Interpolation Method ◽

Radial Point Interpolation ◽

Integration Techniques ◽

Point Interpolation Method ◽

Point Interpolation

A linearly conforming radial point interpolation method (LC-RPIM) is presented for stress analysis of two-dimensional solids. In the LC-RPIM method, each field node is enclosed by a Voronoi polygon, and the displacement field function is approximated using RPIM shape functions of Kronecker delta function property created by simple interpolation using local nodes and radial basis functions augmented with linear polynomials to guarantee linear consistency. The system equations are then derived using the Galerkin weak form and nodal integration techniques, and the essential boundary conditions are imposed directly as in the finite element method. The LC-RPIM method is verified via various numerical examples and an extensive comparison study is conducted with the conventional RPIM, analytical approach and FEM. It is found that the presented LC-RPIM is more stable, more accurate in stress and more efficient than the conventional RPIM.

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A Local Radial Point Interpolation Method for Two-Dimensional Schrödinger Equation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.268-270.1888 ◽

2012 ◽

Vol 268-270 ◽

pp. 1888-1893

Author(s):

Wei Gang Zhai ◽

Xing Hui Cai ◽

Jiang Ren Lu ◽

Xin Li Sun

Keyword(s):

Schrödinger Equation ◽

Analytical Solutions ◽

Schrodinger Equation ◽

Interpolation Method ◽

Weak Form ◽

Time Dependent ◽

Radial Point Interpolation Method ◽

Radial Point Interpolation ◽

Point Interpolation Method ◽

Point Interpolation

A local radial point interpolation method is employed to the simulation of the time dependent Schrödinger equation with arbitrary potential function. Local weak form of the time dependent Schrödinger equation is obtained and radial point interpolation shape functions are applied in the space discretization. Computations are carried out for an example of time dependent Schrödinger equation having analytical solutions. Numerical results agreed with analytical solutions very well.

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Radial point interpolation collocation method based approximation for 2D fractional equation models

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2021.05.007 ◽

2021 ◽

Vol 97 ◽

pp. 153-161

Author(s):

Qingxia Liu ◽

Pinghui Zhuang ◽

Fawang Liu ◽

Minling Zheng ◽

Shanzhen Chen

Keyword(s):

Collocation Method ◽

Radial Point Interpolation ◽

Point Interpolation ◽

Fractional Equation

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Simulation of the viscoplastic extrusion process using the radial point interpolation meshless method

Proceedings of the Institution of Mechanical Engineers Part L Journal of Materials Design and Applications ◽

10.1177/1464420720988583 ◽

2021 ◽

pp. 146442072098858

Author(s):

ROSS Costa ◽

J Belinha ◽

RM Natal Jorge ◽

DES Rodrigues

Keyword(s):

Meshless Method ◽

Fused Deposition Modeling ◽

Interpolation Method ◽

Extrusion Process ◽

Fused Deposition ◽

Radial Point Interpolation ◽

Large Growth ◽

Point Interpolation Method ◽

Point Interpolation ◽

Deposition Modeling

Additive manufacturing is an emergent technology, which witnessed a large growth demanded by the consumer market. Despite this growth, the technology needs scientific regulation and guidelines to be reliable and consistent to the point that is feasible to be used as a source of manufactured end-products. One of the processes that has seen the most significant development is the fused deposition modeling, more commonly known as 3D printing. The motivation to better understand this process makes the study of extrusion of materials important. In this work, the radial point interpolation method, a meshless method, is applied to the study of extrusion of viscoplastic materials, using the formulation originally intended for the finite element method, the flow formulation. This formulation is based on the reasoning that solid materials under those conditions behave like non-Newtonian fluids. The time stepped analysis follows the Lagrangian approach taking advantage of the easy remeshing inherent to meshless methods. To validate the newly developed numerical tool, tests are conducted with numerical examples obtained from the literature for the extrusion of aluminum, which is a more common problem. Thus, after the performed validation, the algorithm can easily be adapted to simulate the extrusion of polymers in fused deposition modeling processes.

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