Adaptive remeshing based on a posteriori error estimation for forging simulation

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Abstract

This paper presents a fully automatic 3D adaptive remeshing procedure and its application to non-steady metal forming simulation. Remeshing, here, is considered as the improvement of an existing mesh rather than a complete rebuilding process. The mesh optimization technique is described. It is based on the combination of local improvement of the neighbourhood of nodes and edges. The surface and the volume remeshing are coupled by using a layer of virtual boundary elements. The mesh adaptation is performed by the optimization of the shape factor. The mesh size map enforcement is accounted for working in a locally transformed space. The size map is provided by a Zienkiewicz–Zhu type error estimator. Its accuracy is evaluated in the frame of a velocity/pressure formulation, viscoplastic constitutive equation and 3D linear tetrahedral elements, by numerical experiments. The adaptive remeshing procedure is applied to non-steady forging. Several complex 3D examples show the reliability of the proposed approach to automatically produce optimal meshes at a prescribed computational cost.

Section snippets

Introduction: Large deformation and adaptive remeshing

Unstructured mesh generation is a general solution for the construction of a mesh in complex geometries. In a certain number of domains as in forming process simulation, the industrial geometries are really complex. The meshing technique used in this work was developed to solve the remeshing stage in large deformations [1], [2]. In such cases, the mesh deforms with the material domain (Lagrangian approach) and thus it degenerates rapidly. In fact, this meshing method is a complete solution in

Mesh topology

The mesh optimization technique used in this paper is based on a simple local mechanism applying on mesh topologies. For that purpose let us introduce notations allowing to introduce precisely the mesh topology. A mesh is determined by a set of coordinates (the mesh node coordinates) and by a set of elements, each element being completely defined by the node numbers of its vertices. The mesh connectivity by means of the element node relations will be called the mesh topology. It can be

A posteriori error estimation

For a flow formulation, the error due to the finite element approximation can be expressed in any suitable norm as the difference between the exact velocity solution field v and the finite element one vh:eh=v-vh.The energy norm is generally preferred for engineering applications. For incompressible materials, the error related to the spherical part of the stress tensor, the pressure p, is neglected [12], [17], so the selected norm is written asehE=s-shE=Ω(s-sh):(ε˙-ε˙h)dω1/2=Ως(s)-1(s-

Superconvergent patch recovery (SPR)

The SPR method is based on a continuous polynomial expansion of the stress tensor on a finite element patch (see Fig. 5). This expansion is obtained by a local least square fit of the stresses at supposingly superconvergent Gauss points of the patch. For some problem, it has been shown [20], [21] that the convergence rate of the derivatives is higher than the global finite element rate at some points. However, for triangles and tetrahedra, the location of superconvergent points is not fully

Application to 3D forging

In this section error estimation and determination of optimized meshes are carried out. The size map for the mesh generator is constructed by nodal averaging of the element sizes computed from (34). The remeshing procedure is activated by different remeshing criteria:

  • Remeshing due to an increase of the total strain since last remeshing, which allows adapting the mesh when important deformation occurs.

  • Remeshing due to volume element quality when an element of the mesh is distorted.

  • Periodic

Conclusion

In 3D, for a velocity/pressure formulation and a viscoplastic constitutive law, the SPR procedure suggested by Zienkiewicz and Zhu produces enhanced stress fields that allow building a satisfactory error estimator. However, due to the inelastic behaviour of the material, superconvergence properties are not observed as in the elastic frame, and therefore the efficiency index of the estimation is less than 1. A correcting factor of 1.25 has to be introduced, as earlier noticed for 2D applications

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