AN EFFICIENT SMOOTHED POINT INTERPOLATION METHOD FOR DYNAMIC ANALYSES

2013 ◽  
Vol 10 (01) ◽  
pp. 1340007 ◽  
Author(s):  
DELFIM SOARES ◽  
ANNE SCHÖNEWALD ◽  
OTTO VON ESTORFF
Keyword(s):  
Stiffness Matrix ◽  
Shape Function ◽  
Interpolation Method ◽  
Mass Matrix ◽  
Dynamic Analyses ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Support Domain ◽  
Edge Based

In this work, a new procedure to compute the mass matrix in the smoothed point interpolation method is discussed. Therefore, the smoothed subdomains are employed to evaluate the mass matrix, which have already been computed for the construction of the stiffness matrix, rendering a more efficient methodology. The procedure is discussed, taking into account the edge-based, cell-based, and node-based smoothed point interpolation methods, as well as different T-schemes for the construction of the support domain of the approximating shape function, which is here formulated based on the radial point interpolation method. Numerical results of different dynamic analyses are presented, illustrating the potentialities of the proposed methodology.

DYNAMIC ELASTOPLASTIC ANALYSES BY SMOOTHED POINT INTERPOLATION METHODS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350030 ◽  
Author(s):  
DELFIM SOARES
Keyword(s):  
Interpolation Method ◽  
Mass Matrix ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Internal Forces ◽  
Elastoplastic Materials ◽  
Edge Based ◽  
The Time Domain ◽  
Raphson Method

In this work, meshfree techniques based on weakened weak formulations are presented for the solution of dynamic problems considering elastoplastic materials. Nonlinear internal forces are computed taking into account edge-based, cell-based, and node-based smoothed domains. T-schemes are applied for the construction of the support domains of the approximating shape functions, which are here formulated based on the radial point interpolation method. The mass matrix is also computed considering smoothed domains and their quadrature points. For the time-domain solution of the nonlinear system of equations, the Newmark/Newton–Raphson method is adopted. Numerical results illustrate the accuracy and efficiency of the discussed methodologies.

MESHFREE ELASTODYNAMIC ANALYSIS OF THREE-DIMENSIONAL SOLIDS USING RADIAL POINT INTERPOLATION METHOD

2011 ◽  
Vol 02 (01) ◽  
pp. 83-95 ◽  
Author(s):  
KYOKO HASEGAWA ◽  
SUSUMU NAKATA ◽  
SATOSHI TANAKA
Keyword(s):  
Stiffness Matrix ◽  
Interpolation Method ◽  
Three Dimensional ◽  
Time Dependent ◽  
Partial Differential ◽  
Radial Point Interpolation Method ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Matrix Construction

Meshfree methods are effective tools for solving partial differential equations. The radial point interpolation method, a partial differential equation solver based on a meshfree approach, enables accurate imposition of displacement boundary conditions and has been successfully applied to elastostatic analysis of various kinds of three-dimensional solids. In this method, stiffness matrix construction accounts for the majority of CPU time required for the entire process, resulting in high computational costs, especially when higher-order numerical integration is applied for accurate matrix construction. An alternative method, modified radial point interpolation, was proposed to overcome this shortcoming and has accomplished fast computation of elastostatic solid analysis. The purpose of this study is to develop an algorithm for time-dependent simulation of three-dimensional elastic solids. We show that the modified radial point interpolation method also accelerates the construction of the mass matrix required for time-dependent analysis in addition to that of the stiffness matrix. In our approach, the problem domain is assumed to have an implicit function representation that can be constructed from a set of surface points measured using a three-dimensional scanning system. Several numerical tests for elastodynamic analysis of complex shape models are presented.

On the Usage of Tetrahedral Background Cells in Nodal Integration of RPIM for 3D Elasto-Static Problems

2015 ◽  
Vol 12 (06) ◽  
pp. 1550036
Author(s):  
M. M. Yavuz ◽  
B. Kanber
Keyword(s):  
Taylor Series ◽  
Interpolation Method ◽  
Shape Parameters ◽  
Radial Point Interpolation Method ◽  
Nodal Integration ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Support Domain ◽  
Fifth Order

In this paper, tetrahedral background cells are used in nodal integration of radial point interpolation method (RPIM). The nodal integration is based on Taylor series terms and it is originally applied for the solutions of 2D problems in literature. Therefore, in this study, it is attempted that the tetrahedral integration cells are used in the solution of 3D elasto-static problems. The accuracy is seriously affected by order of Taylor series terms and it is investigated up to fifth order. A methodology is developed for prevention of negative volumes and calculation problems in subdivision of integration cells for each node. Three different case studies are solved with different support domain sizes and shape parameters. The best accuracy is achieved with fourth-order Taylor terms in nodal integration radial point interpolation method (NI-RPIM). [Formula: see text]-value of 3.00 and [Formula: see text] value of 1.03 in radial basis functions give good results in all cases.

scholarly journals An improved radial point interpolation method applied for elastic problem with functionally graded material

2015 ◽  
Vol 18 (2) ◽  
pp. 131-138
Author(s):  
Viet Quoc Phung ◽  
Nha Thanh Nguyen ◽  
Thien Tich Truong
Keyword(s):  
Interpolation Method ◽  
Functionally Graded ◽  
Thin Plates ◽  
Graded Materials ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Graded Material ◽  
Point Interpolation ◽  
Support Domain ◽  
The One

A meshless method based on radial point interpolation was developed as an effective tool for solving partial differential equations, and has been widely applied to a number of different problems. Besides its advantages, in this paper we introduce a new way to improve the speed and time calculations, by construction and evaluation the support domain. From the analysis of two-dimensional thin plates with different profiles, structured conventional isotropic materials and functional graded materials (FGM), the results are compared with the results had done before that indicates: on the one hand shows the accuracy when using the new way, on the other hand shows the time count as more economical

MID-FREQUENCY ACOUSTIC ANALYSIS USING EDGE-BASED SMOOTHED TETRAHEDRON RADIALPOINT INTERPOLATION METHODS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350103 ◽  
Author(s):  
Z. C. HE ◽  
G. Y. LI ◽  
ERIC LI ◽  
Z. H. ZHONG ◽  
G. R. LIU
Keyword(s):  
Acoustic Analysis ◽  
Interpolation Method ◽  
Building Blocks ◽  
Numerical Dispersion ◽  
Softening Effect ◽  
Radial Point Interpolation Method ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Edge Based

An edge-based smoothed tetrahedron radial point interpolation method (ES-T-RPIM) is formulated for the 3D acoustic problems, using the simplest tetrahedron mesh which is adaptive for any complicated geometry. In present ES-T-RPIM, the gradient smoothing operation is performed with respect to each edge-based smoothing domain, which is also serving as building blocks in the assembly of the stiffness matrix. The smoothed Galerkin weak form is then used to create the discretized system equations. The acoustic pressure is constructed using radial point interpolation method, and two typical schemes of selecting nodes for interpolation using RPIM have been introduced in detail. It turns out that the ES-T-RPIM provides an ideal amount of softening effect, and significantly reduces the numerical dispersion error in low- to mid-frequency range. Numerical examples demonstrate the superiority of the ES-T-RPIM for 3D acoustic analysis, especially at mid-frequency.

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.

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

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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