Tracking Initial Cell Coordinates and Material Rotation

The first component of the anisotropic strength model is the ability to track the orientation of the material relative to fixed coordinate directions. For a transversely isotropic material, the important orientation is that which is perpendicular to the plane of weakness (or

strength), which for our application is the bedding plane or discrete interbedded layer. Tracking the orientation of material in a hybrid Lagrangian-Eulerian code, such as iSALE, is difficult, because material moves through a computational mesh that is fixed in space. Material history is not naturally recorded, as it would be in a Lagrangian approach (Collins et al. 2013). To track material movements and record its history, iSALE uses Lagrangian tracer particles. However, using these particles to recover bedding orientation in each cell is compromised by the need for a more-or-less even distribution of tracers across the domain. Instead, we implemented a cell-based scheme for tracking the orientation of originally flat layers based on a method described by Vitali and Benson (2012) for tracking the evolution of an initial spatial distribution of material properties.

A conventional approach for moving time-varying material properties through an Eulerian grid is to advect quantities of the material (e.g., mass, specific internal energy, etc.) between adjacent cells based on the volume swept out by the velocity vectors at the dividing face and local gradients in the advected variable. While this method guarantees conservation of the advected quantity, a disadvantage of the approach is that if the advected quantity varies non-linearly or non-smoothly in space, it will diffuse or ‘smear’ the field quantity (see Figure 1 in Vitali and Benson (2012) for a depiction of this effect). More accurate advection schemes can reduce this numerical diffusion problem, but not remove it.

For material quantities that do not vary in time and where conservation is not important, a device that allows a quantity to be advected without numerical diffusion is to associate the material quantity with its initial spatial coordinates and advect the initial coordinate field. As this field, by definition, varies linearly in space, it is advected with no numerical diffusion by even low-accuracy advection schemes (Van Leer 1977). The material quantity can then be recovered based on the advected initial coordinate field using interpolation. Vitali and Benson (2012) describe this approach in detail and use it to track the evolution of an initial distribution of flaws in a material. Tracking material orientation is a rather simpler task because this material property is directly related to the initial coordinate field, so that no interpolation is required.

Our approach is as follows. During initial mesh generation, an ‘initial coordinate mesh’ X is defined as the initial x- and y-coordinates of the cell centres, 𝑿 = 𝒙(𝑡 = 0). Each component of this cell-centred vector quantity is advected in the same way as other scalar fields, so that at each timestep the initial x and y location of the material in every cell is known. Assuming that the bedding plane orientation is initially perpendicular to the vertical coordinate y, its orientation at any subsequent timestep can be found from the gradient of the Y component of the advected initial coordinate field. To calculate grad Y, the values of the cell-centred initial coordinate field are first converted to vertex-centred coordinates, using linear interpolation. Then, a cell-centred grad Y is defined using the finite-difference approximation: Δ𝑌𝐶𝑖 = 𝑌_𝑉(𝑖, 𝑗 + 1) − 𝑌_𝑉(𝑖, 𝑗) Δ𝑦 Δ𝑌_𝐶𝑖+1 = 𝑌𝑉(𝑖 + 1, 𝑗 + 1) − 𝑌𝑉(𝑖 + 1, 𝑗) Δ𝑦 Δ𝑌_𝐶𝑗 = 𝑌𝑉(𝑖 + 1, 𝑗) − 𝑌𝑉(𝑖, 𝑗) Δ𝑥 Δ𝑌𝐶𝑗+1 = 𝑌_𝑉(𝑖 + 1, 𝑗 + 1) − 𝑌_𝑉(𝑖, 𝑗 + 1) Δ𝑥 ∇𝑌_𝐶(𝑖 +1 2, 𝑗 + 1 2) = ( 1 2[Δ𝑌𝐶𝑗+1+ Δ𝑌𝐶𝑗] , 1 2[Δ𝑌𝐶𝑖+1+ Δ𝑌𝐶𝑖]) (2.1)

The bedding angle is then computed using:

𝛾 = tan−1(∇𝑌𝐶𝑦

∇𝑌_𝐶𝑥) (2.2)

The gradient of the initial coordinate field is orthogonal to the layer orientation, so using 𝛾, we can apply the correct rotation (i.e., ±90°, depending on how the material was rotated previously) to find the bedding orientation, 𝜃. This process is outlined in Figure 2.1. The initial x- or y-coordinate field can be plotted in the same manner as any other field (e.g., density, pressure, etc.), allowing one to observe layer deformation without the need for

tracer particles (Fig. 2.2). Regardless of the method used to observe layer deformation, similar results are observed. Moments after impact (Fig. 2.2b), there is good agreement between the two methods. Towards the end of the simulation (Fig. 2.2c), there is some discrepancy in regions where multiple layers converge or mix together (i.e., in the centre of the crater where substantial collapse and uplift occur, or where layers overturn in the rim region). Tracers (which follow material throughout the simulation) do a better job at distinguishing layers in these regions, while the advected initial coordinate field tends to indicate an average of the initial depth of material in a given region of the mesh. However, material in the rim and central regions will likely be heavily damaged (and therefore, the strength of the material within these cells should be approximately constant and dependent entirely on friction), and the original anisotropic strength properties may not be preserved, so the discrepancies in these regions should not be a significant issue.

O

Figure 2.1: Illustration of the rotation scheme (see Equation 2.3 in 2.2.2)

implemented in the anisotropy model. Rotation is measured using vertex-centred initial y-coordinates, which are fluxed through the mesh in the same manner as other quantities (specific internal energy, mass, etc.). The stress tensor is rotated into the material reference frame, and then used to compute the Tsai-Hill yield criterion (see Equation 2.5 in 2.2.3).

Figure 2.2: Comparison of layer deformation using a contour plot of the initial y- coordinate field and massless tracer particles (white dots) at a) 0 s, b) 7 s, and c) 200 s after the start of simulation.