A Novel Wave-Propagation Approach For Fully Conservative Eulerian Multi-Material Simulation K. Nordin-Bates Lab. for Scientific Computing, Cavendish Lab., University of Cambridge With thanks to AWE for funding!

Outline Motivation Brief introduction to Cartesian cut-cell approaches LeVeque & Shyue’s Front-tracking Wave Propagation Method Extension to a fully conservative multi-material algorithm Examples in 1D – Fluid and strength. Extension to two dimensions Fluid examples in 2D Conclusions and Next steps

Motivation We’re interested in simulating high-speed solid-solid and solid-fluid interaction involving large deformations. Our current approach employs a level set method for representation of interfaces coupled with a deformation gradient formulation for elastic-plastic strength. This appears to give reasonable results for many situations. However, there are some drawbacks to the approach: Spatial accuracy at the boundary is limited since the precise location of the interface within cells is lost. The method is not mass or energy conservative (even if the level set is updated in a conservative manner). Problems may occur at concave interfaces since single ghost cells attempt to satisfy multiple boundary conditions.

Cut-Cell Meshes and their Challenges We are therefore also investigating the use of Cartesian cut-cell meshes for the simulation of such configurations. In such methods the material interface (or boundary) cuts through a regular underlying mesh, resulting in a single layer of irregular cells adjacent to the interface. The primary challenge associated with solving hyperbolic systems on such meshes using standard explicit methods is a time-step limit of the order of the cell volume (and these volumes may be arbitrarily small)

Some Existing Cut-Cell Approaches Various approaches have been developed to overcome this ‘small cell problem’: Cell merging: e.g. Clarke, Salas & Hassan 1986 Flux redistribution / stabilisation schemes: e.g. Colella et al Rotated Grid / h-Box scheme: Berger et al We ideally want a method that : Is stable at a time-step determined by regular cells. Copes with moving interfaces. Can handle arbitrary no. of materials and interfaces in a cell Works in multi-dimensions and preserves symmetry

Leveque & Shyue Front Tracking Method LeVeque & Shyue 1996 introduced a method for the simulation of problems in which an embedded front is tracked explicitly in parallel with a solution on a regular mesh They proposed a ‘large time-step’ scheme in which the propagation of waves from each interface into multiple target volumes is considered. However, this scheme as originally constructed is not fully conservative for multi-material simulation, since waves from cell interfaces cross the embedded interface. (Diagram taken from Leveque & Shyue 1996)

1D Wave Propagation Method

Front Tracking Version of WPM

Through the use of a multi-material RP solver at the tracked front, the basic mechanism may be extended in a natural way to cope with multiple materials. (The details of multi-material RP solvers are skipped here.) To make the approach conservative within each material, we need to avoid waves crossing the embedded interface: We achieve this here by identifying the arrival time of the incoming wave with the interface and posing an intermediate RP at this point. This is repeated for each wave arriving at the interface Fully Conservative Multi-Material Extension

Some comments

Fully Conservative Multi-Material Extension

One-Dimensional Fluid Examples

One-Dimensional Fluid Example: Single Material

Second Order Extension

One dimensional Fluid Example: Multi-material Barrier x=0Barrier x=1

One-Dimensional Strength Examples

Strength test comments

One-Dimensional Strength Examples

One-Dimensional Strength Results

Conservation for non-Roe-type solvers

2D Unsplit Wave Propagation Method For now, we consider the extension of the method to 2D for a single material with rigid interfaces. Consider a single regular cell interface in 2D: solving the Riemann problem normal to the interface gives a family of waves emitted from the interface: Dimensionally unsplit WPM accounts for tangential propagation of these waves by decomposing each of them using the tangential flux eigenstructure:

2D Unsplit WPM with Interfaces Essentially, we obtain a collection of parallelograms and update the solution based on the wave jumps and the cells overlapped by the parallelograms. We can do the same at interior interfaces (using a multi-material RP solver for the normal direction)

2D Wave-Interface Interaction As in 1D, waves from regular interfaces may cross the interior interface within a time-step. This impact is determined by simple geometric test and an ‘impact area’ identified. Impact time may be taken as the average (i.e. time at which centre is hit). At this point, we pose a new interface RP (for entire interior interface) and propagate resultant waves.

2D Results We demonstrate the method in action with a very simple example problem of a Mach 1.49 shock in air hitting a ‘double’ wedge at 55°.

Conclusions & Next Steps Demonstrated a novel multi-interaction version of the Wave Propagation Method incorporating material interface tracking giving mass conservation in each material individually (to machine accuracy) in 1D. The approach has been extended to 2D and again shows mass conservation for a single material with static embedded boundaries. Additionally have preliminary results with moving interfaces in 2D (not presented) – not yet fully conservative. Further research: Rigorous comparison of accuracy and expense as compared to alternative cut-cell approaches. Investigate use of approximate multi-material Riemann solvers. Further investigating of moving boundary conservation in 2D. 3D!

Constant Interface Velocity Modification While the original approach is functional in 1D, varying of interface velocity within a time-step is impractical in multi-dimensions. We therefore propose a modification in which the interface velocity remains constant within each time-step. This velocity is decided by the RP solution at the beginning of the time-step, which is solved in the normal way. Intermediate interactions then present a modified interface problem, in which we no longer require pressures to match at the interface, but instead require it to match the prescribed velocity.