Guangyao Li

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.

THE UPPER BOUND PROPERTY FOR SOLID MECHANICS OF THE LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD (LC-RPIM)

It has been proven by the authors that both the upper and lower bounds in energy norm of the exact solution to elasticity problems can now be obtained by using the fully compatible finite element method (FEM) and linearly conforming point interpolation method (LC-PIM). This paper examines the upper bound property of the linearly conforming radial point interpolation method (LC-RPIM), where the Radial Basis Functions (RBFs) are used to construct shape functions and node-based smoothed strains are used to formulate the discrete system equations. It is found that the LC-RPIM also provides the upper bound of the exact solution in energy norm to elasticity problems, and it is much sharper than that of LC-PIM due to the decrease of stiffening effect. An effective procedure is also proposed to determine both upper and lower bounds for the exact solution without knowing it in advance: using the LC-RPIM to compute the upper bound, using the standard fully compatible FEM to compute the lower bound based on the same mesh for the problem domain. Numerical examples of 1D, 2D and 3D problems are presented to demonstrate these important properties of LC-RPIM.

A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element

A general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations (BIEs). The method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located. The distance functions are redefined in two local coordinate systems. A new system denoted as (@a,@b) is introduced here firstly. Its implementation is simpler than that of the polar system and it also performs efficiently. Then a new distance transformation is developed to remove or weaken the near singularities. To perform integration on irregular elements, an adaptive integration scheme is applied. Numerical examples are presented for both planar and curved surface elements. The results demonstrate that our method can provide accurate results even when the source point is very close to the integration element, and can keep reasonable accuracy on very irregular elements. Furthermore, the accuracy of our method is much less sensitive to the position of the projection point than the conventional method.

ANALYSIS OF MINDLIN–REISSNER PLATES USING CELL-BASED SMOOTHED RADIAL POINT INTERPOLATION METHOD

In this paper, a formulation for the static and free vibration analysis of Mindlin–Reissner plates is proposed using the cell-based smoothed radial point interpolation method (CS-RPIM) with sub-domain smoothing operations. The radial basis functions augmented with polynomial basis are employed to construct the shape functions that have the Delta function property. The generalized smoothed Galerkin (GS-Galerkin) weakform is adopted to discretize the governing differential equations, and the curvature smoothing is performed to relax the continuity requirement and to improve the accuracy and the rate of convergence of the solution. The present CS-RPIM formulation is based on the first-order shear deformation plate theory, with effective treatment for shear-locking and hence is applicable to both thin and relatively thick plates. To verify the accuracy and stability of the present formulation, intensive comparison studies are conducted with existing results available in the literature and good agreements are obtained. The numerical examples confirm that the present method is shear-locking free and very stable and accurate even using extremely distributed nodes.

AN EXPLICIT SMOOTHED FINITE ELEMENT METHOD (SFEM) FOR ELASTIC DYNAMIC PROBLEMS

This paper presents an explicit smoothed finite element method (SFEM) for elastic dynamic problems. The central difference method for time integration will be used in presented formulations. A simple but general contact searching algorithm is used to treat the contact interface and an algorithm for the contact force is presented. In present method, the problem domain is first divided into elements as in the finite element method (FEM), and the elements are further subdivided into several smoothing cells. Cell-wise strain smoothing operations are used to obtain the stresses, which are constants in each smoothing cells. Area integration over the smoothing cell becomes line integration along its edges, and no gradient of shape functions is involved in computing the field gradients nor in forming the internal force. No mapping or coordinate transformation is necessary so that the element can be used effectively for large deformation problems. Through several examples, the simplicity, efficiency and reliability of the smoothed finite element method are demonstrated.

INTEGRATION OF SUBDIVISION METHOD INTO BOUNDARY ELEMENT ANALYSIS

Subdivision surface modeling, which is based on polygon mesh modeling, can generate a whole smooth geometry without limiting to the topology and connectivity. Meanwhile, the boundary element analysis (BEA), which is based on the boundary integral equation, requires only boundary discretization of the body in question. Thus, performing BEA directly on the subdivision surface models may be a promising way to realize the seamless integration. This work presents a unified framework for the BEA and CAD modeling based on the subdivision surface. Numerical examples for 3D potential problems have demonstrated that the implementation is successful.

A General Algorithm for Evaluating Domain Integrals in 2D Boundary Element Method for Transient Heat Conduction

A time-dependent boundary integral equation method named as pseudo-initial condition method is widely used to solve the transient heat conduction problems. Accurate evaluation of the domain integrals in the pseudo-initial condition formulation is of crucial importance for its successful implementation. As the time-dependent kernel in the domain integral is close to singular when small time step is used, a straightforward computation using Gaussian quadrature may produce large errors, and thus lead to instability of the analysis. To improve the computational accuracy of the domain integral, a coordinate transformation coupled with a domain cell subdivision technique is presented in this paper for 2D boundary element method. The coordinate transformation is denoted as (α, β) transformation, while the cell subdivision technique considers the position of the source point, the shape of the integration cell and the relations between the size of the cell and the time step. With the cell subdivision technique, more Gaussian points are shifted towards the source point, thus more accurate results can be obtained. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

A CURVATURE-CONSTRUCTED METHOD FOR BENDING ANALYSIS OF THIN PLATES USING THREE-NODE TRIANGULAR CELLS

This paper presents a curvature-constructed method (CCM) for bending analysis of thin plates using three-node triangular cells and assumed piecewisely linear deflection field. In the present CCM, the formulation is based on the classic thin plate theory, and only deflection field is treated as the field variable that is assumed piecewisely linear using a set of three-node triangular background cells. The slopes at nodes and/or the mid-edge points of the triangular cells are first obtained using the gradient smoothing techniques (GST) over different smoothing domains. Three schemes are devised to construct the curvature in each cell using these slopes at nodes and/or the mid-edge points. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. The essential rotational boundary conditions are imposed in the process of constructing the curvature field, and the translational boundary conditions are imposed in the same way as in the standard FEM. A number of numerical examples, including both static and free vibration analyses, are studied using the present CCM and the numerical results are compared with the analytical ones and those in the published literatures. The results show that outstanding schemes can obtain very accurate solutions.