K. Y. Dai

A LINEARLY CONFORMING POINT INTERPOLATION METHOD (LC-PIM) FOR 2D SOLID MECHANICS PROBLEMS

A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

A LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD FOR SOLID MECHANICS PROBLEMS

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.

A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates

A mesh-free method is presented to analyze the static deflection and natural frequencies of thin and thick laminated composite plates using high order shear deformation theory. In the present method, the problem domain is represented by a set of properly scattered nodes and no element conformability is required. Moving least-squares method is applied to construct the shape functions. Variational principle is used to derive the discrete system equations based on the third order shear deformation theory (TSDT) of Reddy. Essential boundary conditions are efficiently implemented by a penalty technique for both the static deflection and natural frequency analysis. Several examples are solved to demonstrate the convergence, accuracy and validity of the proposed method. The present solutions are verified with those available values by analytical as well as finite element method. The results from classical plate theory and first order shear deformation theory are also computed and compared with those of TSDT. The effects of the material coefficients, side-to-thickness ratio, nodal distribution and shear correction factor are discussed.

Selective smoothed finite element method

Abstract The paper examines three selective schemes for the smoothed finite element method (SFEM) which was formulated by incorporating a cell-wise strain smoothing operation into the standard compatible finite element method (FEM). These selective SFEM schemes were formulated based on three selective integration FEM schemes with similar properties found between the number of smoothing cells in the SFEM and the number of Gaussian integration points in the FEM. Both scheme 1 and scheme 2 are free of nearly incompressible locking, but scheme 2 is more general and gives better results than scheme 1. In addition, scheme 2 can be applied to anisotropic and nonlinear situations, while scheme 1 can only be applied to isotropic and linear situations. Scheme 3 is free of shear locking. This scheme can be applied to plate and shell problems. Results of the numerical study show that the selective SFEM schemes give more accurate results than the FEM schemes.

A radial point interpolation method for simulation of two-dimensional piezoelectric structures

A meshfree, radial point interpolation method (RPIM) is presented for the analysis of piezoelectric structures, in which the fundamental electrostatic equations governing piezoelectric media are solved numerically without mesh generation. In the present method, the problem domain is represented by a set of scattered nodes and the field variable is interpolated using the values of nodes in its support domain based on the radial basis functions with polynomial reproduction. The shape functions so constructed possess a delta function property, and hence the essential boundary conditions can be implemented with ease as in the conventional finite element method (FEM). The method is successfully applied to determine deflections or electric potentials of a bimorph beam and mode shapes and natural frequencies of transducers. The present results agree well with those of experiments as well as the FEM by ABAQUS. Some shape parameters are also investigated thoroughly for the future convenience of applying the RPIM for smart materials and structures without the use of elements.