A novel meshless approach – Local Kriging (LoKriging) method with two-dimensional structural analysis

This paper develops a novel meshless approach, called Local Kriging (LoKriging) method, which is based on the local weak form of the partial differential governing equations and employs the Kriging interpolation to construct the meshless shape functions. Since the shape functions constructed by this interpolation have the delta function property based on the randomly distributed points, the essential boundary conditions can be implemented easily. The local weak form of the partial differential governing equations is obtained by the weighted residual method within the simple local quadrature domain. The spline function with high continuity is used as the weight function. The presently developed LoKriging method is a truly meshless method, as it does not require the mesh, either for the construction of the shape functions, or for the integration of the local weak form. Several numerical examples of two-dimensional static structural analysis are presented to illustrate the performance of the present LoKriging method. They show that the LoKriging method is highly efficient for the implementation and highly accurate for the computation.

[1]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[2]  François Trochu,et al.  Application of dual kriging to structural shape optimization based on the boundary contour method , 1998 .

[3]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

[4]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[5]  F. Trochu.  A contouring program based on dual kriging interpolation , 1993, Engineering with Computers.

[6]  Ricardo A. Olea,et al.  Geostatistics for Engineers and Earth Scientists , 1999, Technometrics.

[7]  Ted Belytschko,et al.  A coupled finite element-element-free Galerkin method , 1995 .

[8]  Mark A Fleming,et al.  Smoothing and accelerated computations in the element free Galerkin method , 1996 .

[9]  Hyun Gyu Kim,et al.  A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods , 1999 .

[10]  YuanTong Gu,et al.  A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids , 2001 .

[11]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[12]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

[13]  Guirong Liu,et al.  A LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FREE VIBRATION ANALYSES OF 2-D SOLIDS , 2001 .

[14]  Satya N. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics , 2000 .

[15]  S. Timoshenko,et al.  Theory of Elasticity (3rd ed.) , 1970 .

[16]  G. R. Liu,et al.  A local point interpolation method for stress analysis of two-dimensional solids , 2001 .

[17]  D. G. Krige,et al.  A Review of the Development of Geostatistics in South Africa , 1976 .

[18]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[19]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[20]  Joseph J Monaghan,et al.  An introduction to SPH , 1987 .

[21]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[22]  Guirong Liu,et al.  Point interpolation method based on local residual formulation using radial basis functions , 2002 .