Arbitrary Lagrangian Eulerian (ALE) method in LS Dyna

The finite element method which we generally use is termed as a lagrangian formulation. Although the normal methods tends to be feasible for usage in short duration impact applications such as crash analysis, applications concerned with very short transient loadings, such as blast analysis require better simulation algorithms. The use of plain Lagrangian Formulation in large deformation problems leads to severe distortion of mesh rendering it incapable of yielding accurate results.

Also the time step size per iteration becomes very small and consequently increases the computational time. To some extent high deformation problems can be modeled by Lagrangian techniques by using 1) Adaptive re-meshing algorithm or 2) Element deletion method. But the adaptive re-mesh algorithm prohibitively increases the computational time and is also not fully developed for three dimensional problems. The deletion of elements results in significant loss of mass. Hence there is a need for an element formulation which would avoid the mesh distortion and also be computationally efficient. The Arbitrary Lagrangian Eulerian formulation also referred to as ALE formulation is used in large strain deformation problems, mainly because it does not face problems regarding mesh distortion. This method is a combination of Lagrangian as well as Eulerian methods.

In ALE formulation, an arbitrary referential domain is defined for the description of motion which is different from the material (lagrangian) and the spatial (eulerian) domain. In the lagrangian method the mesh moves along with the material where as in the Eulerian method the mesh remains fixed and the material moves through it. Also more than one material can be included in an Eulerian formulation. In cases where there are multiple Eulerian materials, Multi Material Groups (known as ALE-MMG) are required to be used. Each of the Eulerian material is assigned to a group using the LS Dyna keyword *ALE_MULTI_MATERIAL_GROUP. It is to be noted that each part within a specific multi material group has identical material properties.

The ALE formulation can be defined as an algorithm for performing an automatic rezoning. Manual rezone consists of the following three steps:

1. Stop Calculations after Mesh Distortion
2. Mesh Smoothing
3. Advection - Remap the solution from Distorted Mesh to Smoothened Mesh.

1. Stop Calculations after Mesh Distortion: The calculations are stopped after the distortion of mesh so as to perform the mesh smoothing.

2. Mesh Smoothing: There are many algorithms available for mesh smoothing. Some of them used in LS Dyna are listed below:

• Simple Averaging: In this algorithm the coordinates of a node is the simple average of the its surrounding node coordinates.
• Kikuchi’s Algorithm (Volume Weighted Averaging): This algorithm uses a volume-weighted average of the coordinates of the centroids of the elements surrounding a node.
• Equipotential Smoothing: Equipotential zoning is a method of making a structured mesh for finite difference or finite element calculations by using the solutions of Laplace equations as the mesh lines. This same method can also be used to smooth selected points in an unstructured three-dimensional mesh if it is at least locally structured.
• Surface Smoothing: The surfaces are smoothed by extending the two-dimensional equipotential stencils to three dimensions.
• Combining Smoothing Algorithms: In LS Dyna, the user can use a weighted average of the three algorithms (Equipotential, Simple averaging and Kikuchi‟s smoothing algorithm) to generate a composite algorithm.

3. Advection: The remap or the advection step maps the solution from a distorted Lagrangian mesh on to the new mesh. During the remap, the topology of the mesh is assumed to be fixed. Also the mesh motion during a step is assumed to be less than the characteristic lengths of the surrounding elements. In other words the Courant number should be less than or equal to one.

Where, C is the Courant‟s Number, u is Velocity, t is Timestep and x is the Length Interval. Generally during the advection the following steps are performed:

      

        a) Decide which nodes to move.

        b) Move the boundary nodes.

        c) Move the interior nodes.

        d) Calculate the transport of the element-centered variables.

        e) Calculate the momentum transport and update the velocity.

Each element solution variable needs to be transported. The total number of solution variables is dependent on the material models. A minimum of six variables need to be transported. The number of variables may go up to fifteen when the elements have strength, where components of the stress tensor and the plastic strain etc. must also be advected. The following schemes can be used for the purpose of advection:

• Van Leer advection scheme
• Donor Cell advection scheme

The Van Leer advection scheme is a Second order monotonic scheme whereas the Donor Cell is First order accurate scheme. During the transport calculations the errors may bring about the smoothing of the solution, reducing the peak solution variable values. The monotonic second order advection algorithms prevent the transport calculations from creating new spurious oscillations or the peak values (either minimum or maximum) for the variables. Hence Second order accuracy is required for transport of variables. The monotonic algorithms generally provide stability to the code. Also there are many constitutive material models which have a specific allowable range for their history variables. The constitutive models are defined only for the specific range. The second order advection scheme is generally used for supersonic Eulerian flows, where the first order advection schemes may be inaccurate. 

Note: This article is still in progress. Sorry for the complicated wordings. 

Reference: LSTC Theory Manual, J. Hallquist