Z. H. Zhong

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.

MID-FREQUENCY ACOUSTIC ANALYSIS USING EDGE-BASED SMOOTHED TETRAHEDRON RADIALPOINT INTERPOLATION METHODS

An edge-based smoothed tetrahedron radial point interpolation method (ES-T-RPIM) is formulated for the 3D acoustic problems, using the simplest tetrahedron mesh which is adaptive for any complicated geometry. In present ES-T-RPIM, the gradient smoothing operation is performed with respect to each edge-based smoothing domain, which is also serving as building blocks in the assembly of the stiffness matrix. The smoothed Galerkin weak form is then used to create the discretized system equations. The acoustic pressure is constructed using radial point interpolation method, and two typical schemes of selecting nodes for interpolation using RPIM have been introduced in detail. It turns out that the ES-T-RPIM provides an ideal amount of softening effect, and significantly reduces the numerical dispersion error in low- to mid-frequency range. Numerical examples demonstrate the superiority of the ES-T-RPIM for 3D acoustic analysis, especially at mid-frequency.

An inversion procedure for determination of variable binder force in U-shaped forming

In this article, a neural network procedure is suggested to inversely determine the variable binder force in U-shaped forming from the deforming shape of the part. The finite element method (FEM) is used to simulate the forming process. The inputs of the neural network (NN) model come from the deformation shape of the part. The outputs of the NN are the stepped variable binder force parameters. To reach the desired deformation, the NN model will go through a progressive retraining process. A constant binder force simulation method is also recommended to obtain the desired deformation shape. The numerical result for model NUMISHEET’93 demonstrates the efficiency of the present procedure.

Identification of geometric parameters of drawbead in metal forming processes

A computational inverse technique is presented for identification of geometric parameters of drawbead in sheet forming processes. The explicit dynamic finite element method (FEM) is employed as the forward solver to calculate the maximal effective stress, maximal effective strain and maximal thinning ratio of sheet thickness for known drawbead geometric parameters. A neural network (NN) is adopted as the inverse operator to determine the geometric parameters of circular drawbead. A sample design method with the strategy of updating training sample set is developed for the fast convergence in the training process of NN model. Once the training sample set is updated, the NN structure will be optimized using the genetic algorithm (GA). The numerical examples are presented to demonstrate the efficiency of the technique.

IDENTIFICATION OF GEOMETRIC PARAMETERS OF DRAWBEAD USING NEURAL NETWORKS

In this paper, a neural network (NN) model was designated to identify the geo- metric parameters of drawbead in sheet forming processes. The genetic algorithm (GA) was used to determine the neuron numbers of the hidden layers of the neural network, and a sample design method with the strategy of updating training sam- ples was also used for the convergence. The NN model goes through a progressive retraining process and the numerical study shows that this technique can give a good result of the parameter identification of drawbead.