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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 (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.

This paper presents a thin plate formulation with nodal integration for bending analysis using three-node triangular cells and linear interpolation functions. The formulation was based on the classic thin plate theory, in which only deflection field was required and dealt with as the field variables. They were assumed to be piecewisely linear and expressed using a set of three-node triangular cells. Based on each node, the integration domain has been further derived, where the curvature in the domain was computed using a gradient smoothing technique (GST). As a result, the curvature in each integration domain is a constant whereby the deflection is compatible in the whole problem domain. The generalized smoothed Galerkin weak form is then used to create the discretized system equations where the system stiffness is obtained using simple summation operation. The essential rotational boundary conditions are imposed in the process of constructing the curvature field in conjunction with imposing the translational boundary conditions in the same way as undertaken in the standard FEM. A number of numerical examples were studied using the present formulation, including both static and free vibration analyses. The numerical results were compared with the reference ones together with those shown in the state-of-art literatures published. Very good accuracy has been achieved using the present method.

This paper presents and compares continuous assumed gradient (CAG) methods when applied to structural elasticity. CAG elements are finite elements where the strain, i.e., the deformation gradient, is replaced by a C⁰-continuous interpolation. Similar approaches are found in nodal integration and SFEM. Recently, interpolation schemes for a continuous assumed deformation gradient were proposed for first order tetrahedral and hexahedral finite elements. These schemes try to balance accuracy and numerical efficiency. At the same time, the stability of the interpolation with respect to hourglassing and spurious low energy modes is ensured. This paper recalls the fundamentals of CAG elements, i.e., the formulation and linearization. Furthermore, it extends the approach to second order finite elements. Examples prove convergence and accuracy of the quadratic elements. Two interpolation schemes, one being supported by finite element nodes and interior points and the other being a higher-order tensor-product polynomial, are identified to be most accurate.

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). ${\alpha }_c$-value of 3.00 and $q$ value of 1.03 in radial basis functions give good results in all cases.

This paper presents a new particle formulation for extreme material flow analyses in the bulk forming applications. The new formulation is first established by an introduction of a smoothed displacement field to the standard Galerkin formulation to eliminate zero-energy modes in conventional particle methods. The discretized system of linear equations is consistently derived and integrated using a direct nodal integration scheme. The linear formulation is next extended to the large deformation quasi-static analysis of inelastic materials. As quasi-static Lagrangian simulation proceeds in the severe deformation range, the analysis method is switched to explicit dynamics formulation and an adaptive Lagrangian kernel approach is preformed to reset the reference configuration and maintain the injective deformation mapping at the particles. Both nonconvex and convex meshfree approximations are investigated in this study. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.