Multi-step Inverse Method Of Sheet Metal Forming Based On Isogeometric Analysis

In the early stage of design,the rapid estimation of part formability is essential.The incremental method needs a lot of parameters in the modeling of this process,such as blank shape and size,which will significantly affect the quality of formed parts.Since the detailed design parameters cannot be determined,the inverse method,which can quickly verify the formability of a given part,is widely used.Traditional one-step inverse method and multi-step inverse method are based on finite element method(FEM).Although this method has many advantages,much of the time needed by FEM is spent in the mesh generation stage.Tedious mesh generation has become a bottleneck restricting the development of FEM.In recent years,it has become a hot research topic to quickly and efficiently simulate sheet metal forming process by bridging the gap between design,analysis and optimization.For this reason,an isogeometric analysis(IGA)method is used to conduct geometric modeling and forming calculation by using NURBS basis function in this study.In other words,this method only requires one modeling and analytical representation compared with FEM.The main research contents are as follows:(1)IGA is used to replace the traditional finite element method in this paper.A multi-step inverse IGA method is proposed by combining the isogeometric analysis method with the multi-step inverse forming method.This method can remedy the defect caused by ignoring the deformation loading history in one-step inverse IGA method.The sliding constraint surface is constructed according to the position and contour of die and punch in forming process.The mesh mapping method is adopted to construct the initial solution of intermediate configuration.The stress-strain relationship is derived through incremental theory.The stress is updated by classical return mapping algorithm.(2)The isogeometric membrane element is adopted here.The local coordinate system and coordinate transformation matrix in multi-step inverse IGA method is deduced.The NURBS element stiffness matrix and internal force solution formula in multi-step inverse IGA method have been derived.The Newton-Raphson algorithm is used for nonlinear plastic iteration.Finally,the new proposed algorithm is applied to calculate the thickness,strain and stress distribution of square box and S-rail models.The results of the new method is compared with the incremental method and the multi-step inverse FEM.The results of these two examples illustrate that the multi-step inverse IGA algorithm can predict the distribution of these physical quantities well.The precision of the new method is as good as that of multi-step inverse FEM method and incremental method.The maximum error is within 5%.However,compared with these two methods,the new method based on IGA saves a lot of computational cost.It does not require a meshing process,nor does it require a large number of design parameters.