A Novel Wave-Propagation Approach For Fully Conservative Eulerian Multi-Material Simulation K....
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Transcript of A Novel Wave-Propagation Approach For Fully Conservative Eulerian Multi-Material Simulation K....
A Novel Wave-Propagation Approach For Fully Conservative Eulerian Multi-Material SimulationK. 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. 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 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
• 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
• 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.
• 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
• 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 Extension
• The algorithm for updating becomes:
1. Compute RP solutions at t=0 for all regular and embedded interfaces.
2. Initialize interface-adjacent states to initial cell values for all mixed cells.
3. While there are waves interacting with an embedded interface:
a) Identify next wave to interact
b) Update interface adjacent states to star states of most recent interface RP
c) Add incoming wave jump to relevant state, e.g. :=
d) Solve resultant intermediate RP between , and store.
4. Update solution using contributions from all waves from all RPs.
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
• Snapshots of the results of both tests at :
• Total mass conservation errors are of the order of machine accuracy:
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
• 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
• Mass conservation:
• Momentum conservation:
• Energy conservation:
• Total deformation:
• Plastic deformation:
• Work hardening:
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 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 & Drikakis 2010
• 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.
One-Dimensional Strength Results
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
• 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, 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.