Adaptive smoothed FEM for forming simulations
A. H. van den Boogaard∗, W. Quak
University of Twente, Dept. Engineering Technology P.O. Box 217, 7500 AE Enschede, The Netherlands
a.h.vandenboogaard@utwente.nl
Keywords: ASFEM, forming simulation, large deformation, meshless. ABSTRACT
FEM simulation 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 Local Maximum Entropy approach [1] with nodal inte-gration, 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.
(a) Cloud of nodes (b) Delaunay triangles (c) Integration cells
Figure 1: Computational geometrical objects used for the analysis.
The extrusion process is characterized by large local deformations near the die exit. In ordinary FEM calculations very frequent remeshing or Eulerian formulations are required. With the ASFEM method, a Lagrangian formulation without remeshing can be used, simplifying the application of history de-pendent variables in material models. Results for a schematic extrusion simulation with the ASFEM method for 1891 nodes are given in Figure 2.
(a) t = 1.5 (b) t = 3.0 (c) t = 4.5 (d) t = 6.0
Figure 2: The deformed shapes of the billet at four different times.
Extrusion force, velocity distribution and equivalent strain distribution are in close agreement with a Eulerian FEM simulation with 6044 elements that was used as a reference model. Despite heavy deformations, that would lead to extreme mesh distortions in a Lagrangian FEM analysis, the results for ASFEM are very good [5], while calculation efficiency is maintained.
References
[1] M. Arroyo, M.A. Ortiz: Local maximum-entropy approximation schemes: a seamless bridge be-tween finite elements and meshfree methods. International Journal for Numerical Methods in
Engineering, 65 (2006), 2167–2202.
[2] W. Quak, A.H. van den Boogaard, D. Gonz´alez, E. Cueto: A Comparative study on the perfor-mance of meshless approximations and their integration. Computational Mechanics, 48 (2011), 121–137.
[3] J.S. Chen, C.T. Wu, S. Yoon, Y. You: A stabilized conforming nodal integration for Galerkin mesh-free methods. International Journal for Numerical Methods in Engineering, 50 (2001), 435–466. [4] M.A. Puso, J. Solberg: A stabilized nodally integrated tetrahedral. International Journal for
Nu-merical Methods in Engineering, 67 (2006), 841–867.
[5] W. Quak: On meshless and nodal-based numerical methods for forming processes. PhD thesis, University of Twente, 2011, http://dx.doi.org/10.3990/1.9789036532389
