A meshless integral method based on regularized boundary integral equation

Bodin, Anthony; Ma, Jianfeng; Xin, X. J.; Krishnaswami, Prakash

doi:10.1016/j.cma.2005.12.005

Cited by 12 publications

(14 citation statements)

References 61 publications

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“…Representative applications of the LBIE method are found in two-dimensional linear elasticity [17,[22][23][24][25], in thermoelasticity [26], in micropolar elasticity [27], in two-dimensional elastodynamic problems [28][29][30][31], in two-dimensional Navier-Stokes flows [32] and in three-dimensional heat conduction and elasticity with geometry axisymmetry [33,34].…”

Section: Introductionmentioning

confidence: 99%

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

Vavourakis

2009

Comput Mech

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A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points' cloud a "background" mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring "background" triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements' shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements' interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.

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Section: Introductionmentioning

confidence: 99%

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

Vavourakis

2009

Comput Mech

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“…The system equation is obtained based on the Total Lagrangian (TL) approach. Hu et al [13] used meshless local Petrov-Galerkin method for large deformation contact analysis of elastomeric components. A nonlinear formulation of meshless local Petrov-Galerkin finite-volume mixed method was developed by Han et al to analyze static and dynamic large deformation problems [14].…”

Section: Introductionmentioning

confidence: 99%

“…A Galerkin SPH method was utilized by Wang to solve large deformation fracture problems [23]. Reproducing kernel particle methods were used to analyze large deformation problems in [24][25][26]. In [27], Li et al used an element-free Galerkin method to solve die forging problems.…”

Section: Introductionmentioning

confidence: 99%

“…

In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi's theory over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation.

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confidence: 99%

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Elastoplastic Large Deformation Using Meshless Integral Method

Xin

2012

WJM

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<p style="text-align:justify;"> In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi’s theory over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. Green-Naghdi’s theory starts with the additive decomposition of the Green-Lagrange strain into elastic and plastic parts and considers aJ2elastoplastic constitutive law that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. A simple, generalized collocation method is proposed to enforce essential boundary conditions straightforwardly and accurately, while natural boundary conditions are incorporated in the system governing equations and require no special handling. The solution algorithm for large deformation analysis is discussed in detail. Numerical examples show that meshless integral method with large deformation is accurate and robust. </p>

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“…However, the meaning of "sufficiently" is not made precise, and their solution for avoiding the problem is acknowledged to be computationally expensive and numerically poor if the number of data points included is not "considerably" greater than the theoretical minimum. A useful clue to the cause of singularities is provided by the work of Bodin et al [3], who note that singularities occur when the data points are arranged in a "degenerate pattern", citing an arrangement along a straight line as an example. However, no further discussion is provided of the general conditions under which singularities occur.…”

Section: Introductionmentioning

confidence: 99%

A singularity-avoiding moving least squares scheme for two-dimensional unstructured meshes

Chenoweth

Soria

Ooi

2009

Journal of Computational Physics

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Moving least squares interpolation schemes are in widespread use as a tool for numerical analysis on scattered data. In particular, they are often employed when solving partial differential equations on unstructured meshes, which are typically needed when the geometry defining the domain is complex. It is known that such schemes can be singular if the data points in the stencil happen to be in certain special geometric arrangements, however little research has addressed this issue specifically. In this paper, a moving least squares scheme is presented which is an appropriate tool for use when solving partial differential equations in two dimensions, and the precise conditions under which singularities occur are identified. The theory is then applied in the form of a stencil building algorithm which automatically detects singular stencils and corrects them in an efficient manner, while attempting to maintain stencil symmetry as closely as possible. Finally, the scheme is used in a convection-diffusion equation solver, and the results of a number of simulations are presented.

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A meshless integral method based on regularized boundary integral equation

Cited by 12 publications

References 61 publications

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

Elastoplastic Large Deformation Using Meshless Integral Method

A singularity-avoiding moving least squares scheme for two-dimensional unstructured meshes

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