Spline-based meshfree method and its application to the analysis of bulk metal forming processes

Abstract

In the present study, the application of the SBMFM in nonlinear analyses is performed. The SBMFM is implemented in the nonlinear analysis of bulk metal forming processes. When the SBMFM is employed as a method of the nonlinear analysis, iterative analyses and consecutive curve movements arise. In some particular cases, the analysis boundary represented by trimming curves can produce extremely small trimmed elements. This comes from the properties of the SBMFM: the background mesh of the SBMFM (NURBS patch) is uniform rectangular grid and trimming curves are freely move in the background mesh. The extremely small trimmed elements can induce an excessively large condition number in global system matrix and numerical instabilities can occur. To resolve the instabilities, the technique of the weighted extended B-spline (WEB-spline) is implemented in the SBMFM. The basis functions with very small trimmed supports are extrapolated by neighboring basis functions with some special scheme so that those basis functions can be condensed in the solution process. In order to impose essential boundary conditions in the SBMFM with extended basis, Nitsche’s method is implemented to avoid an undesirable relation with other spaces or a large condition number of system. Also, the implementation strategy of the extension process in the SBMFM is presented. Through the numerical examples, the presented SBMFM with extended basis proved its validity and effectiveness. Moreover, the condition number of a system is well managed guaranteeing the stability of the numerical analysis. The application of the proposed method to the nonlinear problems with a large deformation is performed to show the effectiveness of the method: The rigid-plastic analysis for bulk metal forming processes is analyzed. The analysis is based on a full Eulerian framework and solving convective transport equation is avoided by employing trimming curves to represent the boundaries of analysis domain. Basic formul...