Local Point Interpolation Method Research Articles

Spherical Indentation of a Micropolar Solid: A Numerical Investigation Using the Local Point Interpolation–Boundary Element Method

The load-penetration curve in elastic nanoindentation of an elastic micropolar flat by a diamond spherical punch is analyzed. The presented results are obtained by a specifically developed numerical tool based on a judicious combination of the conventional boundary element method and strong form local point interpolation method. The results show that the usual linear relationship between the material depression and the square of the radius of the contact area is also valid in this case of micropolar elastic material. It is also shown that the relation between the indentation stress (applied load over the contact surface) and the indentation strain (ratio of contact radius by the punch radius) is linear. The proportionality coefficient which is none other than the indentation stiffness varies with the coupling factor of the micropolar elastic medium. A relation between the indentation stiffness of a micropolar solid and that of a conventional solid with the same Young modulus and Poisson ratio is derived.

Coupling BEM and the Local Point Interpolation for the Solution of Anisotropic Elastic Nonlinear, Multi-Physics and Multi-Fields Problems

The pure boundary element method (BEM) is effective for the solution of a large class of problems. The main appeal of this BEM (reduction of the problem dimension by one) is tarnished to some extent when a fundamental solution to the governing equations does not exist as in the case of nonlinear problems. The easy to implement local point interpolation method applied to the strong form of differential equations is an attractive numerical approach. Its accuracy deteriorates in the presence of Neumann-type boundary conditions which are practically inevitable in solid mechanics. The main appeal of the BEM can be maintained by a judicious coupling of the pure BEM with the local point interpolation method. The resulting approach, named the LPI-BEM, seems versatile and effective. This is demonstrated by considering some linear and nonlinear elasticity problems including multi-physics and multi-field problems.

Application of Local Point Interpolation Method to Electromagnetic Problems With Material Discontinuities Using a New Visibility Criterion

In this paper, the local point interpolation method (LPIM) is used with a modified visibility criterion to handle material discontinuities. In general, visibility criterion is applied only to shape function generation over support nodes selection. We present a modified version where it is also applied to the integration process. The method is simpler and more robust than other techniques often employed on multimaterials problems, with a straight-forward implementation.

Isotropic-BEM coupled with a local point interpolation method for the solution of 3D-anisotropic elasticity problems

This paper presents a solution procedure for the three-dimensional linear elastic problem with anisotropic properties. The approach uses the partition of the displacement field into complementary and particular parts. The former is the solution of a differential equation similar to that of an isotropic elastostatic and is obtained by the isotropic boundary element method. The particular integral is obtained by solving the corresponding strong form differential equations, using the local radial point interpolation method. This promising approach is simple to implement and leads to highly accurate solutions in some simple tested situations.

A variation of local point interpolation method ( vLPIM) for analysis of microelectromechanical systems (MEMS) device

In this paper, a variation of local point interpolation method ( vLPIM) is presented for meshless analysis of microelectromechanical systems (MEMS) devices, in which the weight function is assigned by high-order spline functions, instead of the Galerkin formulation of the local point interpolation method (LPIM). The present vLPIM with Timoshenko beam theory are then used to analyze MEMS. In the conventional MEMS design and simulation, time-consuming mesh generation is required. Recently, a number of meshless techniques have been applied to MEMS simulations since they do not require mesh generation. As one of them, the LPIM is based on the local weak form of the partial differential equations and easily implements the boundary conditions due to the shape functions constructed by the PIM with the delta function properties. Several MEMS devices are studied by the present vLPIM. The simulated results are compared with those by other simulation approaches, and it is shown that the present meshless method is very efficient and accurate for the analysis of MEMS devices.

Meshless analysis of Timoshenko beams based on a locking-free formulation and variational approaches

A meshless method is developed, based on subdomain variational formulations and a local Petrov–Galerkin approximation, along with the locking-free formulation, for the analysis of bending of a thick beam. The local point interpolation method is employed to construct both trial and test functions. The present method is a truly meshless method based only on a number of randomly located nodes. No global background integration mesh is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. Problems of thick and thin beams under various loading, and boundary conditions are analysed by the proposed method and the numerical results are compared with analytical solutions. The contact problem for beam by means of subdomain variational inequalities and the meshless method has also been investigated.

Meshless analysis of the obstacle problem for beams by the MLPG method and subdomain variational formulations

In this paper, subdomain variational formulations are presented to describe the bending problem of thin beams, which is then analysed by using the MLPG meshless method. Particular attention is paid to the solution of the obstacle problem for beams by means of subdomain variational inequalities and the MLPG method. The local point interpolation method is employed to construct both trial and test functions. The present method is a truly meshless method based only on a number of randomly located nodes. No global background integration mesh is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Problems of thin beams under various loading, and boundary conditions, as well as the obstacle problem for beams are analysed by the proposed method and the numerical results are compared with analytical solutions. Some parameters which affect the performance of the present method are also investigated.

A matrix triangularization algorithm for the polynomial point interpolation method

A novel matrix triangularization algorithm (MTA) is proposed to overcome the singularity problem in the point interpolation method (PIM) using the polynomial basis, and to ensure stable and reliable construction of PIM shape functions. The present algorithm is validated using several examples, and implemented in the local point interpolation method (LPIM) that is a truly meshfree method based on a local weak form. Numerical examples demonstrate that LPIM using the present MTA are very easy to implement, and very robust for solving problems of computational mechanics. It is shown that PIM with the present MTA is very effective in constructing shape functions. Most importantly, PIM shape functions possess Kronecker delta function properties. Parameters that influence the performance of them are studied in detail. The convergence and efficiency of them are thoroughly investigated.

Comparisons of two meshfree local point interpolation methods for structural analyses

As truly meshless methods, the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM), are based on the point interpolations and local weak forms integrated in a local domain of very simple shape. LPIM and LR-PIM are examined and compared with each other. They are also compared with the established FEM and the meshless local Petrov-Galerkin (MLPG) method. The numerical implementations of these two methods are discussed in detail. Parameters that influence the performance of them are detailedly studied. The convergence and efficiency of them are thoroughly investigated. LPIM and LR-PIM formulations are developed for structural analyses of 2-D elasto-dynamic problems and 1-D Timoshenko beam problems in the first time. It is found that LPIM and LR-PIM are very easy to implement, and very efficient obtaining numerical solutions to problems of computational mechanics.

A local point interpolation method for static and dynamic analysis of thin beams

The local point interpolation method (LPIM) is a newly developed truly meshless method, based on the idea of meshless local Petrov–Galerkin (MLPG) approach. In this paper, a new LPIM formulation is proposed to deal with fourth-order boundary-value and initial-value problems for static and dynamic analysis (stability, free vibration and forced vibration) of beams. Local weak forms are developed using weighted residual method locally. In order to introduce the derivatives of the field variable into the interpolation scheme, a technique is proposed to construct polynomial interpolation with Kronecker delta function property, based only on a group of arbitrarily distributed points. Because the shape functions so-obtained possess delta function property, the essential boundary conditions can be implemented with ease as in the conventional finite element method (FEM). The validity and efficiency of the present LPIM formulation are demonstrated through numerical examples of beams under various loads and boundary conditions.

A local point interpolation method for stress analysis of two-dimensional solids

A local point interpolation method (LPIM) is presented for the stress analysis of two-dimensional solids. A local weak form is developed using the weighted residual method locally in two-dimensional solids. The polynomial interpolation, which is based only on a group of arbitrarily distributed nodes, is used to obtain shape functions. The LPIM equations are derived, based on the local weak form and point interpolation. Since the shape functions possess the Kronecker delta function property, the essential boundary condition can be implemented with ease as in the conventional finite element method (FEM). The presented LPIM method is a truly meshless method, as it does not need any element or mesh for both field interpolation and background integration. The implementation procedure is as simple as strong form formulation methods. The LPIM has been coded in FORTRAN. The validity and efficiency of the present LPIM formulation are demonstrated through example problems. It is found that the present LPIM is very easy to implement, and very robust for obtaining displacements and stresses of desired accuracy in solids.