Adaptive smoothed FEM for forming simulations

FEMsimulation of large deformations as occur in metal forming processes is usually accompanied with highly distorted meshes. This leads first to a reduction of accuracy and later to loss of convergence when implicit solvers are used. Remeshing can be used to reduce element distortion, but repeated remeshing will result in smoothing of data like equivalent plastic strain, due to averaging and interpolation. A meshless method circumvents the problem of mesh distortion, but depending on the integration of the weak formulation of equilibrium mapping of data and hence smoothing of data still remains unless a nodal integration scheme is used. Starting with a LocalMaximum Entropy approach [1] with nodal integration, we end-up with a smoothed Finite Element formulation in the limit of local approximations [2]. It is straightforward to adapt the triangulation in every increment, yielding an Adaptive Smoothed Finite Element Method, in which large deformations can be modelled with a Lagrangian description without the necessity to map data from one step to the other. A cell based stabilized conforming nodal integration method (SCNI) [3] is used. Depending on the configuration of nodes, nodal integration can yield singular stiffness matrices, resulting in spurious displacement modes [4]. A stabilization is used, based on minimizing the difference between a ‘linear assumed’ and the consistent strain field. The cells are based on the Delaunay triangulation, connecting mid-sides and centres of gravity of the triangles (Figure 1). Especially at the outer boundary, this yields a simpler formulation than using the dual Voronoi tesselation