Efficient ALE mesh management for 3D quasi‐Eulerian problems

Abstract

SUMMARY In computational solid mechanics, the ALE formalism can be very useful to reduce the size of finite element models of continuous forming operations such as roll forming. The mesh of these ALE models is said to be quasi‐Eulerian because the nodes remain almost fixed—or almost Eulerian—in the main process direction, although they are required to move in the orthogonal plane in order to follow the lateral displacements of the solid. This paper extensively presents a complete node relocation procedure dedicated to such ALE models. The discussion focusses on quadrangular and hexahedral meshes with local refinements. The main concern of this work is the preservation of the geometrical features and the shape of the free boundaries of the mesh. With this aim in view, each type of nodes (corner, edge, surface and volume) is treated sequentially with dedicated algorithms. A special care is given to highly curved 3D surfaces for which a CPU‐efficient smoothing technique is proposed. This new method relies on a spline surface reconstruction, on a very fast weighted Laplacian smoother with original weights and on a robust reprojection algorithm. The overall consistency of this mesh management procedure is finally demonstrated in two numerical applications. The first one is a 2D ALE simulation of a drawbead, which provides similar results to an equivalent Lagrangian model yet is much faster. The second application is a 3D industrial ALE model of a 16‐stand roll forming line. In this case, all attempts to perform the same simulation by using the Lagrangian formalism have been unsuccessful. Copyright © 2012 John Wiley & Sons, Ltd.

3D numerical models of FSW processes with non-cylindrical pin

“…Indeed, the Eulerian formulation (used in the region 3) is a particular case of the ALE formulation. In the ALE formalism, unlike in the Lagrangian case which is commonly used in Solid Mechanics, the mesh no longer follows the material motion [2,3,9,10,11]. Consequently, a new grid coordinate system is defined and the conservation laws able to describe the FSW process are rewritten in terms of the new coordinates : Mass:…”

“…The ALE formulation used in the solid approach is described in more details in [9,10,11]. To simplify the solution procedure and remain competitive against Lagrangian models, the system of ALE equations (Eq.…”

“…Consequently, classical numerical simulation techniques have to be extended in order to track the correct material deformations. One of the possiblilities is to use the Arbitrary Lagrangian Eulerian (ALE) formulation [3,9,10,11]. This formulation is used to keep the mesh motion under control regardless of the real material displacements.…”

Comparison of a Fluid and a Solid Approach for the Numerical Simulation of Friction Stir Welding with a Non‐Cylindrical Pin

“…Indeed, the Eulerian formulation (used in the region 3) is a particular case of the ALE formulation. In the ALE formalism, unlike in the Lagrangian case, which is commonly used in solid mechanics, the mesh no longer follows the material motion . Consequently, a new grid coordinate system is defined and the conservation laws able to describe the FSW process are rewritten in terms of the new coordinates :…”

3D numerical models using a fluid or a solid formulation of FSW processes with a non-cylindrical pin

“…Consequently, classical numerical simulation techniques have to be extended in order to track the correct material deformations. One of the possibilities is to use the Arbitrary Lagrangian Eulerian (ALE) formulation [15][16][17][18]. This formulation is used to keep the mesh motion under control regardless of the real material displacements.…”