CAE-Companion-2018-2019

Theory WISSEN CAE

Arbitrary Lagrangian-EulerianMethod (ALE)

Introduction Before performing a numerical simulation of a multidi- mensional problem the choice for a suitable kinematical description of the continuum has to be made. When it comes to solid mechanics this choice normally leads to a Lagrangian formulation. Here the mesh nodes follow each motion of the material (material configuration) which makes it easy to track free surfaces or edges to treat with contact algorithms. However, when it comes to large deformations appearing for instance in forming simulations, the mesh distortion can lead to a strongly decreasing accuracy of the results. The nu- merical simulation of fluid dynamics usually uses an Eulerian formulation where the continuummoves through a fixed mesh (spatial configuration). However, this means that a boundary does not necessarily have to remain with the initial defining nodes which makes the imposition of boundary conditions more complicated. ALE, as the name insinuates, is a simulation approach for the coupled numerical simulation of interacting Lagrangian and Eulerian continua combining their strengths and ruling out their disadvantages as far as possible. In the following passages we first recall the princi- ples of Lagrangian as well as Eulerian continua. After this the necessary adaptations for an ALE approach are presented. In the end some examples for the practical use of the ALE method shall be given. Lagrange vs. Euler To describe the motion of particles in continuummechanics usually two domains are used. One is the material domain consisting of material particles X and the other one is the spatial domain consisting of spatial points x . In the Lagrangian description the reference configuration is identified via the material coordinates X . The relation be- tween the material coordinates X and the spatial coordinates x is realized through the motion of the material points. By means of a mapping it is possible to link X and x in time by the law of motion (see Fig. 1). The material ve- locity v for this formulation is . By the inversion it is possible to identify the reference position of any given material particle occupying a coordinate x at a given time t .

The disadvantage of the Lagrangian approach and in particu- lar of the coincidence of grid points and material points arises when it comes to problems with excessive mesh distortions, for instance explosions, fluid dynamics, etc. For such cases often the Eulerian description is used which is formulated based on the spatial coordinates x and time t using a contin- uumwhich moves through a fixed mesh. Due to this the Eu- lerian description only involves variables with significance for the current point of time. The current configuration serves as reference configuration which means that a deduction of a former point in time as it is possible with the Lagrangian approach is inhibited. In the Eulerian approach the material velocity of a node corresponds to the velocity of the material point which is coincident with the node in question at the considered time. It is expressed as with a reference to the fixed mesh but without a reference to the initial configuration of the continuum and thus without a reference to the initial material coordinates. The relative motion of the material opposite the fixed grid leads to advective effects. Arbitrary Lagrangian-Eulerian (ALE) Approach In the ALE approach an external reference system is intro- duced since neither the Lagrangian configuration nor the Eu- lerian configuration can be used as a reference. Fig. 2 shows the relations between the three configurations. The motion that was introduced as before can now be expressed as . The mapping from the reference configuration to the spatial domain which is equivalent to the motion of the mesh points in the spatial configuration yields the mesh velocity For practical purposes we can directly regard the mapping of which yields the velocity of . The latter representing the particle velocity in the reference configuration.

Figure 1: Lagrangian approach

Figure 2: Arbitrary Lagrangian-Eulerian (ALE) approach

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