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

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a novel wave propagation approach for fully conservative eulerian multi material simulation

A Novel Wave-Propagation Approach For Fully Conservative Eulerian Multi-Material Simulation

K. Nordin-Bates

With thanks to AWE for funding!

Lab. for Scientific Computing, Cavendish Lab., University of Cambridge

  • 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
  • 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
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
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. 2006
  • Rotated Grid / h-Box scheme: Berger et al. 2003
  • 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 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
1D Wave Propagation Method
  • We begin by considering the original scheme of L&S in 1D for the system of conservation laws
  • At each interface we compute a Roe-type linearized Riemann Problem solution with wave speeds and state jumps across these - we have

, and

  • A conservative first order explicit update for the solution in cell is then given by
front tracking version of wpm
Front Tracking Version of WPM
  • This approach was extended to incorporate front-tracking. Suppose we have a point defining the front of interest laying in cell .
  • We store states in each portion of the cell, and can hence compute a RP between these, which gives waves propagating into each material as well as the speed of the front.
  • Note that waves from the front cross neighbouring cell interfaces, and vice-versa.
  • To obtain a stable update for the same time-step as the regular cells, we add the contributions of these waves to the multiple cells that they cross.
fully conservative multi material extension
Fully Conservative Multi-Material Extension
  • 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
some comments
Some comments
  • Won’t state full update formula here(!), but is built up from contribution due to waves.
  • Interacting waves impart a “full cell contribution” to the update of the relevant cut-cell states.
  • For example, in the example update to the right, the fastestwave from the interface contributes to the update of
  • …while the left-going wave from the first interaction contributes for the wave arrival timeand the corresponding RP solution
  • Note that if the interface crosses into a neighbouring cell within a time-step, we need to consider incoming waves from the edges of this cell too.
  • Also, note that if we’re only simulating a single material, the Roe-type Riemann solver used for the regular cell interfaces is insufficient for producing an interface solution.
fully conservative multi material extension1
Fully Conservative Multi-Material Extension
  • The algorithm for updating becomes:
    • Compute RP solutions at t=0 for all regular and embedded interfaces.
    • Initialize interface-adjacent states to initial cell values for all mixed cells.
    • While there are waves interacting with an embedded interface:
      • Identify next wave to interact
      • Update interface adjacent states to star states of most recent interface RP
      • Add incoming wave jump to relevant state, e.g. :=
      • Solve resultant intermediate RP between , and store.
    • Update solution using contributions from all waves from all RPs.
one dimensional fluid examples
One-Dimensional Fluid Examples
  • First examples demonstrate the scheme applied to single material Euler equations with ideal gas EoS.
  • (i.e. this essentially uses the scheme in a fully-conservative contact-front-tracking mode)
  • We consider a Sod-type problem with initial condition given by and with constant adiabatic constant .
  • Simulations are run at , with to , and we consider initial interface positions at and as illustrated below.
  • The exact solution consists of a right-travelling shock and contact and left-travelling rarefaction:
one dimensional fluid example single material
One-Dimensional Fluid Example: Single Material
  • Snapshots of the results of both tests at :
  • Total mass conservation errors are of the order of machine accuracy:
second order extension
Second Order Extension
  • The method is extended to second order accuracy in much the same way as the original WPM, with linear correction profiles added to each wave.
  • Taylor series analysis gives a correction
  • Modification is required on cut-cells.
  • We apply limiting to wave strengths toensure monotonicity.
  • Plot shows equivalent simulation to before, with 2nd order solver.
  • Conservation unaffected by correction.
one dimensional fluid example multi material
One dimensional Fluid Example: Multi-material
  • Now extend previous test to multi-material ideal gas case, with ,
  • Density snapshots at , run with in case with barrier at :
  • Relative mass error <1e15 for each component throughout in both cases

Barrier x=0

Barrier x=1

one dimensional strength examples
One-Dimensional Strength Examples
  • Mass conservation:
  • Momentum conservation:
  • Energy conservation:
  • Total deformation:
  • Plastic deformation:
  • Work hardening:
strength test comments
Strength test comments
  • Ignoring plastic source terms for now, this is a hyperbolic system with 22 waves (of which 16 are linearly degenerate of speed and 6 genuinely non-linear)
  • One longitudinal wave (p-wave) and 2 shearwaves (s-waves) in each direction.
  • Disclaimer: the linearized approximate Riemann solver used here does not satisfy Roe criterion, hence WPM is not conservative (even without interfaces!)
  • But, want to demonstrate stability of approach for systems involving more complex eigenstructures.
one dimensional strength examples1
One-Dimensional Strength Examples
  • We consider a purely elastic 1D impact problem in which pre-deformed copper and steel plates collide.
  • The problem is contrived such that the solution demonstrates a full family of waves
  • Initial conditions and material models are taken from first test-case of Barton & Drikakis2010
  • The problem is solved on a domain of length with regular cell size to time
  • The initial interface is located at and propagates through approx. 25 cells during the simulation.
  • ‘Stick’ interface conditions are used.
conservation for non roe type solvers
Conservation for non-Roe-type solvers
  • We can also consider how the approach may be made conservative for approximate Riemann solvers that do not satisfy the Roe criterion.
  • This requires considering the effect of the waves on the interface flux (rather than using them directly in the update).
  • For example, here we assemble a left interface flux
  • We may use these in a standard flux update.
  • This modification has an additional computational cost compared with the WPM approach, as the flux function must be evaluated multiple times per interface.
  • Results for fluid problems are qualitatively identical to WPM, but with exact conservation.
2d unsplit wave propagation method
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
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
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
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
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
Constant Interface Velocity Modification
  • While the original approach is functional in 1D,varyingof interface velocity within a time-step is impractical inmulti-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 beginningof 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.