The Research On Adaptive Reproducing Kernel Particle Method In The Metal Forming Process

Adopting Lagrangian FEM to simulate bulk forming problems, the mesh distortions maybe occur. If the computation is based on this distortion mesh, it will be divergent. Thus, it is difficult to simulate the whole metal forming process with a fixed mesh configuration for the complicated large deformation problems. The means to overcome these difficulties is to remesh in ertain step. This can, of course, introduce numerous difficulties such as the need to project between meshes in successive stages of problem, which leads to degradation of accuracy and complexity in the computer program, not to mention the burden associated with a large number of meshes. Currently, the rapid and credibility remesh method is the hard mission in computational mechanics fields. Meshless methods do not require costly mesh generation remeshing. Furthermore, meshless methods implement a functional basis and allow arbitrary placement of points, therefore the solution and its derivatives may be obtained directly where they are needed and with better accuracy than with finite element method where interpolation is required. Limitation of meshless method is the low computational efficiency due to high order interpolation. But as nodes need not to be structured, they may be moved, inserted, and deleted freely—this is particularly suitable for adaptive analysis. For this reason, the adaptive meshless method is systemically studied and apllied into the simulations for metal forming large deformation problem. It can speed up the meshless computation speed without lower the precision of the results.We summary innovative spots as follows:(1) According to the character of RKPM, A MCEE error estimation model is constructed to capture the high gradients of stresses behavior in large deformation. A reliable error estimation model is the first crucial issue must be considered in adaptive analysis. In MCEE error estimate, the interpolation error and the integration error which commonly exist in numerical approximation are both measured and it performs exactly the job that is required for adaptive analysis. The model is programmed and passed the benchmark test.(2) A domain enrichment algrithm based on the interplotion of the vertex of the back ground integral cell is proposed. In adaptive analysis, a problem domain may be refined according the error distribution if the desired accuracy is not achieved. The most suitable nodes placement is to essure the error in each background cell is balanced. The refinement method used in this paper is also h-type adaptivity but this method is more freely and flexibly.(3) In order to maintain the accuracy of numerical integration, a cell subdivion method based on hierarchy-territory algrorithm is proposed. RKPM need background (integration) cells for integration, and the test function is rational function not polynomial, the accuracy can not be guaranteed only by inserting some nodes, the background cells should be subdivided accordingly. Furthermore, the integration error can be minimized if the integration order matches the interpolation order.(4) The accuracy of interpolation and the efficiency of computation depend on the nodes in the support domain of the point of interest. Therefore, a suitable support domain dimension should be chosen to ensure a proper area of coverage for interpolation. In the mesh-free adaptive analysis for dynamic large deformation problem the nodal density can very dramatically in each time step. The use of the same support domain size based on the current point of interest can lead to an unbalanced selection of nodes for construction of shape functions. So how to determine the support domain size of each interest points in each time step is a very important problem in adaptive analysis. In this paper, a method based on the domain decomposition technique has been proposed to determine the dimension of the nodes support domain. The numerical examples show it performs well.(5) An adaptive meshfree RKPM3D program for metal forming simulation is developed and successfully applied into metal bulk forming and blank forming. The compare of the springback accuracy in sheet metal forming is also done by FEM and adaptive RKPM3D.