# Presentation: 376449

## Front Tracking

**Tutorial Lectures by James Glimm**
**with thanks to the Front Tracking team**
***S. Dutta, E. George, J. Grove, H. Jin, Y. Kang, M.-N. Kim,T. Lee, X.-L. Li, T. Liu, X.-F. Liu, A. Marchese, W. Oh, A. Pamgemanan, R. Samulyak, D. S. Sharp, Z. Xu, Y. Yu,Y. Zhang, M. Zhao, N. Zhao***
**at BNL, LANL, Univ. Stony Brook**

## Outline of Presentation

**Overview**
- Basic idea of Front Tracking
- Advantages and disadvantages of Front Tracking
- Modular software design
- Use and availability of software
**Technical Description**
- Geometry: interfaces and the description of free surface
- Grid free and grid based formulations
- Physics: fronts and the propagation of states and points
**Advanced Topics**
- Conservative and nonconservative formulations
**Ongoing Research**

## Part I: Overview -- The Basic Ideas

- Front is a lower dimensional grid, moving through the volume filling grid
- Key ideas are
- the geometrical description of the front
- the algorithm to propagate it
- the modification of the finite difference stencils which cross the front so that the stencils see only states on one side of the front
- Front tracking is the ultimate ALE code as it is pure Eulerian except for a lower dimensional Lagrangian surface grid
- Beyond ALE: front tracking has built in slide surfaces for interfaces (shear discontinuity allowed)

## Conservative Equations for Front Tracking

- Hyperbolic:

- Elliptic:

- Track discontinuities in U

- Track discontinuities in

- Parabolic:

- Mixed: any or all of the above in different
- subsystems of equations

## Schematic for Front Tracking

**Schematic for Front Tracking**

- R

- R

## Front Tracking:Advantages and Disadvantages

- Advantages:
- Often gives the best solutions on coarser grids compared to other methods for problems with important discontinuity interfaces
- Solves interface problems not solvable by other methods
- Disadvantages:
- For shocks: too complex relative to benefits
- Not well suited to diffused or spread out fronts
- Software complexity implies learning period for use

## Three Examples

- Code comparison and grid convergence study for spherical implosions and explosions: shock passage through an interface (the spherical Richtmyer-Meshkov problem)
- Code comparisons for a single mode accelerated interface (the planar 2D Rayleigh-Taylor problem)
- Code comparison for 3D steady acceleration of a density discontinuity interface (the planar Rayleigh-Taylor problem)

## Tracked (left) and untracked (right) spherical implosions, 200x200 grids

## Comparison of L1 error for contact for tracked and untracked simulations

## Single mode Rayleigh-Taylor instability comparison (20 cells across): Frontier, Tracked TVD, TVD

## Comparison (40 cells across);FronTier, TVD Tracked, TVD

## Fluid Mixing Simulation

**Early time FronTier simulation of****	****	****Late time FronTier simulation of a**
**3D RT mixing layer.****	****	****	****	****3D RT mixing layer.**

##   Comparison of Simulation, Theory, Experiments

- 			**penetration distance of light fluid into heavy **

**0.05 -- 0.077 (Experiment)**
**0.05 -- 0.06   (Theory)**
**0.07              (Simulation - tracked)**
**0.035            (Simulation - TVD untracked)**
**0.06              (Simulation - TVD untracked, diffusion renormalized)**

## FronTier and TVD Simulations without / with diffusion remormalization

## Density at Z = const. Cross section.

**Density at Z = const. Cross section.**
**Comparison of FronTier (left) and TVD (right)**

## Modular Code Design

**Interface library**: Describes geometry of an interface. This is at the level of nonmanifold geometry, meaning that the interface surfaces can intersect on curves, which can meet at points.
- Typical support routines: make_object, print, read_print, copy, delete, modify where object = point, bond, curve, triangle, surface, interface
- Higher level support routines: test for intersections, find side or closest interface point or component from a general position in space, glue pieces for parallel communication

## Modular Code Design

**Front Library**: Describes Interface with physical states (at this level, a unit of storage)
- Typical support routines: Propagation of interface points; re-meshing of interface points
**Hyp Library**: Assembles stencils for explicit solution of hyperbolic equations.
**Gas**: Provides Riemann solvers to Front and finite difference stencil operators to Hyp.
**EOS**: Contains constitutive laws to close equations

- Reference: J. Glimm, J. Grove, X.-L. Li, K-M. Shyue, Q. Zhang,
- Y. Zeng. “*Three Dimnsionslal Front Tracking.”* SISC **19**
- (1998), 703-707

## Software Availability

- Interface -- geometrical routines
**Freely available**
- Hyperbolic tracking -- finite difference for fronts and for interior states near fronts
**Available by request**
**Plans to make this portion into freely available library**
- Elliptic and parabolic tracking -- finite elements for elliptic operators with discontinuous coefficients
**Available by request**
- Physics libraries -- Gas, MHD, Solid, Porous media

- http://www.ams.sunysb.edu/~FronTier.Ftmain.html

## Part II: Technical Description

**Geometry**: interfaces and the description of free surfaces
- Grid free and grid based formulations
- Conservative, higher order formulation
- Interface operations and support
- Untangle, remesh
**Dynamics**: fronts and the propagation of states and points
- Local and nonlocal Riemann solvers
- Interior difference solvers near a tracked front
**Parallel communication** (and AMR)
- repatch pieces of fronts after parallel communication

## II.1: GeometryThe front separates space into connected components, each with possibly different physics

## Interface data structures

**Coords**: (pointer to) three numbers in a 3D space
**Point**: coords, left state, right state
**Bond**: point for each end, and pointers to next, prev bond
**Node**: beginning/end of curve. This is a curve with a list of
- incoming and outgoing curves
**Curve**: doubly linked list of bonds, starting and ending at a node,
- pointers to first/last bond, start/end node
- Tri: three points for vertices and pointers to neighbors

## Interface data structures

**Surface**: defined by bounding curves and by linked tris
**Hypersurface element**: tri in 3D, bond in 2D
**Hypersurface**: surface in 3D, curve in 2D. Left/right
- component index
**Interface**: has all of the above

## Elementary Interface Operations

- For each object (POINT, BOND, CURVE, TRI, SURFACE, INTERFACE):
- allocate, install, copy, print, read_print, delete, next  (iterators)
- For parallel communication:
- communicate in blocks
- reset all pointers in communicated blocks
- reconnect interface patches communicated near edges or over buffer zones at edges

## Advanced Interface Operations

- Test for intersections
- Resolve intersections (untangle)
- Locate relative to interface
- which side, or connected component
- closest interface point
- Determine crossings of interface with lines
- to define finite difference stencil
- to define grid based reconstruction
- Precomputation (hash tables) for efficiency

## Grid free vs. Grid based

**Grid free**: interface and interior (volume) grid share a comon length scale but are otherwise unrelated
**Grid based**: the interface is directly tied to the volume grid.
- The interface is defined by its intersections with the grid cell edges.
- It is assumed that each grid cell edge has at most one intersection with the interface.
- In the interior of the cell, the interface is reconstructed from its cell edge crossings.

## Grid free vs. Grid based

**Grid free**
- can be more accurate
- is less robust
**Grid based**
- highly robust
- Lorensen and Cline. “*Marching Cubes”.* Computer Graphics, **21** (1987), 163-169.
**Hybrid**: alternate grid free and grid based at some frequency
- robust since grid based is used as a backup
- improved interface description
- best of three algorithms

## The 3D interface: Grid free

**The 3D interface: Grid free**

## Grid-based interface reconstruction:14 nonisomorphic cases in 3D

**Grid-based interface reconstruction:****14 nonisomorphic cases in 3D**

## Grid-free interface evolution

**Grid-free interface evolution**

## Grid-free surface

**Grid-free surface**

## Grid-based surface

**Grid-based surface**

## References for Interface Construction

- J. Glimm and O. McBryan, *“A Computational Model for *
*Interfaces”,* Adv. Appl. Math. **6** (1985), 422-435
- J. Glimm, J. W. Grove, X.-L. Li, K.-M. Shyue, Q. Zhang
- snd Y. Zeng, *“Three Dimensional Front Tracking”,* SIAM
- J. Sci. Comp. **19** (1998), 703-727.
- J. Glimm, J. W. Grove, X.-L. Li and D. Tan, *“A Robust*
*Computational Algorithm for Dymanic Interface Tracking in*
*Three Dimensions”,* SIAM J. Sci. Comp. **21** (2000), 2240-
- 2256.
- J. Glimm, J. W. Grove, X. L.Li and D. C. Tan “*Robust Computational Algorithms for Dymanic Interface Tracking in Three Dimaneions”, *SIAM J. Sci. Comp. **21** (2000) 2240-2256.

## Redistribution of points on a curve will ensure equal spacing and provide some smoothing

## Grid free redistribute in 3D is based on elementary triangle operations

## 3D redistribution improves the triangle quality for the surface

## Grid based redistribution

- The grid based algorithm is automatically redistributed
- every step.
- The algorithm is based on reconstruction of the interface
- from the crossings of the interface with the grid cell edges.
- The reconstruction can be viewed as a special type of
- redistribution.

## Tangles:Self-intersecting curves and surfaces

## Typical 2D interface tangles, resolved within a time step

## Grid-free untangling (3D)

**Grid-free untangling (3D)**

## Grid free untangle

- 1. Test all triangle pairs for intersections (use hash table)
- 2. Find (cross) bonds defined by intersecting trinagles
- 3. Link cross bonds to form cross curves
- 4. Install cross curves into interface, cutting surfaces along
- line of intersection
- 5. Test for and remove unphysical surfaces
- 6. Remove unneeded cross curves
- 2D algorithm: just the analogues of 1+2+6 needed

## The idea of grid-based untangling

**The idea of grid-based untangling**

## Grid-based topological correctionThe same construction works for 3D. Untangle is an elementary step for grid based tracking.

**Grid-based topological correction****The same construction works for 3D. Untangle is an elementary step for grid based**** tracking.**

## Grid based reconstruction (includes redistribute and untangle)

- 1 Determine the crossings of the interface with the cell block
- edges
- 2. Determine the components at the corners of the cell block.
- This is done, starting with a point not swept by the interface
- (thus with the same component as the previous time step), and
- followed by a walk through all mesh block squares. Double
- crossings and other unphysical crossings are eliminated at this
- step.
- 3. Reconstruct the interface, using the one of the 14
-  nonisomorphic templates which matches the give cell
- 4. Check for and resolve any possible inconsistency at each
- cell face

## Grid based matching at cell faces

- On cell faces, the interface is also grid based, in the
- sense that it is determined by reconstruction with the
- intersections of the interface with the edges of that face.
- Thus two cells with a common face share a boundary
- with common data (edge intersections) and a common
- solution (the reconstruction). Thus the interface is
- consistent across adjacent cells after reconstruction
- (it is watertight).
- Exception: 4 edge crossings for a single face allow a nonunique
- reconstruction, and an explicit watertight patch is needed.
- Uniqueness of reconstruction matching is important for
- parallel communication.

## II.2: DynamicsFlow Chart for Front Tracking

## Propagation of Points and States

**Front points** (each point has a left and right side state)
- Normal propagate: solution of a nonlocal Riemann problem in one dimension
- Tangential propagate: project surface onto tangent plane and apply finite differences there
**Interior points**
- Many different finite difference methods supported. No modification in case the stencil does not cross the front
- Use of ghost cells to reconstruct stencil states in case the stencil crosses the front

## Algorithms for differencing with multi-components defined by fronts

- The only routine which sees multicomponents is the
- Riemann solver. The Riemann solver will accept left
- and right states describing possibly different physics
- (*e.g.* a different Equation of State). Its solution defines
- the coupling or boundary conditions between these two
- regimes.
- All other routines (finite differences, interpolation, finite
- elements, tangential front update) see only states from
- one component at a time.
- In this way there is no numerical mass diffusion across a
- front.

## The interface motion is split into normal and tangential steps

## Three steps in the normal propagation algorithm

- Determine states at new point via
- characteristic equations.
- Solve RP at new point to get left
- and right states.
-  Repropagate point using average of t0
-  and t 0+    t velocities to achieve higher
- order accuracy

## A Solution Function

- Evaluation of the solution at the foot of a backwards
- characteristic will fall at an arbitrary point relative to
- the grid.
- A solution function is provided to evaluate the solution
- at an arbitrary point.
- It is based on interpolation from (regular) grid data points
- using bilinear interpolation and from front points using
- linear interpolation on triangles

## Interpolation grid used to define solution function: Interpolation of states from a single component only

## Tangential propagation

- Tangential propagation applies to front states.
- States on each side of the front are propagated
- separately, by a conventional finite difference algorithm.
- Motion of points is optional. Propagation (normal
- and tangential) can occur in any Galilean frame,
- as the equations are frame invariant. Choice of frame
- affects the tangential component of point propagation.
- Tangential motion is an isomorphism of the interface,
- and has no dynamical significance.

## Grid base front propagation

- The front propagation algorithm will yield a general
- interface even if it starts from a grid based one.
- For grid based front  propagation, the final step in the
- propagation algorithm is to reconstruct the propagated
- front to be grid based.
- As indicated before, we determine the intersections of’
- the front with the grid cell edges and use these
- intersections to give a new grid based propagated front.

## Propagation of interior states

**The problem**: irregular stencils which cross the front.
**The solution**: ghost cells and extrapolation.
- This method is nonconservative.
- Locally conservative tracking requires a space time
- tracked grid.
- The locally conservative construction gains one and
-  potentially two additional orders of accuracy.

## Interior states (continued)

- For efficiency, the interior solution consists of two
- passes.
- The first pass ignores the front and solves for all points,
- regular or irregular in a uniform manner.
- This pass is vectorized.
- The second pass returns to the cells with an irregular
- stencil and solves taking the front into account, in effect
- overwriting the answer of the first pass for those cells.

## Stencil states for the ghost cell method: extrapolate states across the interface

**Stencil states for the ghost cell method: extrapolate states across the interface**

## Ghost cell extrapolation copies a state on a curve to a ghost cell regular stencil point. The completed stencil always has states from a single component

## Ghost Cells

- For stencils to the left, left side states are
- extrapolated to right to fill states at the
- locations where they are needed on the
- right.
- Similarly on the right.
- Thus the finite difference scheme sees only
- states from a single side.

## The ghost cell extrapolation method

- Original reference
- J. Glimm, D. Marchesin, O. McBryan. *“A Numerical Method for Two Phase Flow with an Unstable Interface.”* J. Comp. Phys. **39** (1981), 179-200
- Used without attribution by Fedkiw *et al.*
- R. Fedkiw, T. Aslam, B. Merriman, S. Osher. *“A Non-Oscillatory Eulerian Approach to Interfaces in Multiphase Flow.”* J. Comp. Phys. **152** (1999), 452-492.

## Parallel Communication

- For interior states, use ghost cells. Communicate after
- interior sweeps
- For the front: cut a patch to extend beyond ghost cell region.
- Communicate patch. Install patch in image domain.
- Installation requires a matching condition, defined by
- floating point comparison of points, with redundancy
- through use of the coordinates of the (2 or 3) points
- defining a bond or triangle.
- Grid based matching depends on cell face data, and is easier.

## General Reference for Front Tracking: geometry and dynamics

- J. Glimm, J. W. Grove and Y. Zhang, *“Interface Tracking for Axisymmetric Flows”,* SIAM J. SciComp **24** (2002), 208-236.

## Part III: Introduction to Advanced Topics

- Conservative tracking
- conservation
- higher order accuracy
- simpler numerical methods; all difficulty transferred to the space time geometry
- Parabolic and elliptic problems with free surfaces
- Free surface MHD
- Porous media with sharp fronts
- Navier-Stokes with two fluids (distinct viscosities)

## Locally Conservative Tracking

- Ghost cells are not conservative
- Ghost cells are locally zero order accurate, as is the case with other finite difference methods at discontinuities
- A locally conservative higher order method requires a space time interface
- All difficulty is transferred to the geometrical issues of interface construction
- Finite differencing is standard

## Locally Conservative Front Tracking

**Lax-Wendroff theorem[1960]:** A conservative consistent scheme “converges” to a function u, the limit u is a weak solution.

- The Rankine-Hugoniot condition:

**The Key**: Utilization of the _**dynamic flux**_, which not only satisfies Rankine-Hugoniot condition but also gives equal numerical flux on both sides of cell boundary.

## Conservation laws in Integral Form

- Figure 1: The changes of volume V inside the flow

- For the right hand side, omitting the higher order term, dividing both sides by        and taking the limit of                     , we have

## Define dynamic flux with moving boundary:

- For a fixed cell such as a rectangular cell in an Eulerian grid,

*** ******Conservation laws in Integral Form***

- Define _**dynamic flux**__** **_with moving boundary:

- Difference function near boundary

## The conservation property

- due to the Rankine-Hugoniot relations for the
- conservation law. Thus the method is conservative.

## 1D Conservative Front Tracking Geometry

**Two cases**** **
**Fronts do not cross the cell center in one time step.**
**Fronts do cross the cell center in one time step.**

**New cell average      and         :**

## 4-way comparison: exact vs. untracked (x), conserv. tracked (o), nonconserv. tracked (+)

## L1 convergence order for shock interacting with rarefaction wave (having smooth edges): comparison of conservative and nonconservative tracking

- Grid          Conservative  Nonconservative

## Accuracy Order ofLocally Conservative Tracking

- 1D locally conservative algorithm is also locally 2nd order accurate at the tracked front
- Front propagation is 2nd order accurate in position. Uses a predictor corrector.
- 2D locally conservative algorithm is 1st order accurate at tracked front
- Implementation is1st order accurate at present.
- 2nd order requires curvature corrections in 2D
- 2D space time grid uses 3D grid based interface

## Major Steps in 2D Algorithm

- Propagate 2D spatial interface
- Connect old, new 2D grids to form 3D space time grid, and reconstruct this to be grid based
- Merge cells with small tops to ensure CFL stability
- Conservative differencing in 3D space time cells using dynamic flux, and piecewise linear state reconstruction

## 2D Space time interface

**2D Space time interface**

## Irregular cells cut by 2D space time interface

**Irregular cells cut by 2D space time interface**

## Irregular volume grid after merger of small cells

- After cell merger

**Irregular volume grid after merger of small cells**

## Conservative tracking: single mode Richtmyer-Meshkov instability, 40 cells across

## Conservative tracking (40 cells) vs. Nonconservative tracking, 40, 80, 160 cells 

- C 40

- NC 40

- NC 80

- NC 160

## Comparison of growth rates:40, 160 cell Cons. Tracked and 160 noncons. Tracked are similar; 40 cell Noncons. Tracked has slower growth

## Conservative Front Tracking

- J. Glimm, X. L. Li, and Y.-J. Liu, *“Conservative Front Tracking with Improved Accuracy”,* Siam J. Num. Analys. Submitted (2003).
- J. Glimm, X.-L. Li and Y.-J. Liu, *“Conservative Front Tracking in Higher Space Dimensions”, *Transactions of Nanjing University of Aeronautics and Astronautics **18**, Suppl. 1-15.
- J. Glimm, X.-L. Li, Y.-J. Liu and N. Zhao, “*Conservative Front Tracking and Level Set Algorithms”,* Proc. Nat. Acad. Sci. **98** (2001) 14196-14201.

## Parabolic and Elliptic Problems:Discontinuous Coefficients in the Elliptic Operator

- Multiple applications (transport properties with discontinuities in the materials)
- Shift grid lines or surfaces to the discontinuity interface
- Preserve well conditioned mesh elements for numerical stability
- Rectangular index structure desirable but not essential, for fast solvers

## Point-Shifted Triangular Grid

- 1. Irregular
-     Rectangular Grid

- Density Function

## Point-Shifted Triangular Grid

- 2. Point-Shifting

- a. intersections

## Point-Shifted Triangular Grid

- 2. Point-Shifting

- a. intersections

- b. redistribution

- 3. Error Handling

- local mesh refinement

## Point-Shifted Triangular Grid

- 2. Point-Shifting

- a. intersections

- b. redistribution

## Point-Shifted Triangular Grid

- 2. Point-Shifting

- a. intersections

- b. redistribution

- c. shift grid nodes

## Point-Shifted Triangular Grid

- 3. Triangulation

## 3D Construction of Surface Constrained Grid for Elliptic Finite Element Solver

- Find intersections of triangle edges with the grid block surfaces; add new points to the triangle there
- Split resulting polygons to get triangles again
- Collapse small triangles, remove some points
- Record all grid block surface and volume diagonals enforced by surface
- Add new grid lines if topology is too complex to resolve
- Shift grid points to interface or vica versa
- Tetrahedralize (breadth first)
- Introduce Steiner points if needed (rarely)

## Simulation ResultsConformity

## Part IV: Ongoing Research

- Algorithms
- Locally Conservative tracking
- Automatic Mesh Refinement
- Applications: Engineering and physics
- Axisymmetric spherical flows
- Laser accelerated targets
- Jet breakup and spray
- Late stage fluid mixing
- Packaging and usability
- Uniform calling interface (TSTT)
- Merge with other code frameworks (Overature)
- Library formulation

## Automatic Mesh Refinement for FT

- Merge with Overature (LLNL) code; acquire AMR from Overature.
- Patch based AMR as with M. Berger
- Assume that the front occurs on the finest grid level only
- Assume that each patch lies in a single parallel processor domain

## AMR: A shock-contact interactionInitial data

## Level 4 AMRAfter shock passage through contact

## Applications of FronTier-Gas

- Acceleration driven mixing
- E. George, J. Glimm, X.-L. Li, A. Marchese and Z. L. Xu *“A comparison of  Experimental, Theoretical, and Numerical Simulation of Rayleigh-Taylor Mixing Rates”,* Proc. National Academy of Sci. **99** (2002) 2587-2592
- R. L.Holmes, B. Fryxell, M. Gittings, J. W. Grove, G. Dimonte, M. Schneider, D. H. Sharp, A. Velikovich, R. P. Weaver, and Q. Zhang *“Richtmyer-Meshkov Instability Growth: Experiment, Simulation, and Theory”,* J. Fluid  Mech. **389** (1999) 55-79.
- S. Dutta,, E. George, J. Glimm, X. L. Li, A. Marchese, Z. L. Xu, Y. M. Zhang, J. W. Grove and D. H. Sharp, *“Numerical Methods for the Determination of Mioxing”, *Laser and Particle Beams, submitted (2003).

## Applications of FronTier-gas

- Breakup of a diesel jet into spray
- J. Glimm, X.-L. Li, W. Oh, A. Marchese, M.-N. Kim, R. Samulyak and C. Tzanos, *“Jet breakup and spray formation in a diesel engine”,* Proceedings of Second MIT conference on Computational Flluid and Solid Mechanics, 2003.
- Laser Induced Fluid Mixing
-  R. P. Drake, H. F. Robey, O. A. Hurricane, B. A. Remington, J. Knauer, J. Glimm, Y. Zhang, D. Arnett, D. D. Ryutov, J. O. Kane, K. S. Budil and J. W. Grove, *“Experiments to produce a hydrodynamically unstable spherical divergine system of relevance to instabilities in supernovae”,* Astrophysics Journal **546** (2002), 896-906.
- Axisymmetric Fluid Flows
- J. Glimm, J. W. Grove, Y. Zhang and S. Dutta *“Numerical Study of Axisymmetric Richtmyer-Meshkov Instability and Azimuthal Effect on Spherical Mixing”, *J. Stat. Phys. **107** (2002) 241-260.
- Target and Detector Design for High Energy Particle Accelerator
- R. Samulyak, *“Numerical Simulation of hydro- and magnetohydrodynamic processes in the Muon Collider target”,* Lecture Notes in Computer Science, 2002 (submitted).
- R. Samulyak, L. Lu, J. Glimm, X. L. Li, and P. Spentzouris,*“Numerical Simulation of PMT Implosion Effects in MiniBooNE”*, BNL Technical Report, 2003.

## Diesel injection: 

- Fuel injector (liquid-gas EOS)

**Diesel injection: **
**Four time steps in jet breakup**

## Pressure vs. Density (EOS): The phase change EOS is one of several difficulties in this problem

## FronTier Simulation of 

***FronTier*** **Simulation** **of**
**NLUF** 2 **Experiment**

- CHGe capsule surrounded by CRF foam. The RM instability is driven by strong shock of Mach number 300 by the Omega laser

## Comparison of the FronTier and CALE 

- Comparison of the *FronTier* and CALE
- Simulations with Experiment

## Shock imploding randomly perturbed initial contact surface (light Imploding heavy)

## Application: Cracking of PMT detector.Simulation of accident at Super K detector

## Other Extensions of FronTier

- FronTier-res
**P. Daripa, J. Glimm, W. B.Lindquist and O. McBryan, *****“Ploymer floods: A case study of nonlinear wave analysis and of instability control in tertiary oil recovery”,***** Siam J. Appl. Math. 48 (1988) 353-373**
- FronTier-MHD
**R. Samulyak, *****“Numerical Simulation of hydro- and magnetohydrodynamic processes in the Muon Collider target”,***** Lecture Notes in Computer Science, 2002 (submitted).**
- FronTier-solid
- F. Wang, J. Glimm, J. W. Grove, B. Plohr and D. H. Sharp, *“A conservative Eulertian Numerical Scheme for Elasto-Plasticity and Application to Plate Impace Problems”, *Impact Comput. Sci. Engrg **5** (1993) 285-308.*.*
- FronTier-mphase
- J. Glimm, H. Jin, M. Laforest, and F. Tangerman, *“A*
*two pressure numerical model of two fluid mixtures”,* J. Multiscale Modeling and Simulation. Accepted for publication.:

## MHD: Pure hydo energy deposition into jet. Successive time steps in instability development

## MHD: Energy deposition into jet with increasing strength of magnetic field

## Packaging and usability of FronTier

- Plans for an externally callable library
- Merger with other codes (Overature) and library systems underway
- J. Glimm, J. Grove, X. L. Li, Y. Liu, and Z. Xu “*Unstructured grids in 3D and 4D for a time dependent interface in front tracking with improved accuracy”, *Proceedings of the 8th International Conference on Numerical Grid Generation in Computational Field Simulations, June 2-6, 2002, Honolulu Hawaii,

## Related Lectures at this Conference

- MS24, Monday Feb 10, 4:15-4:40PM, Garden Room F.  High resolution algorithms for fluid mixing. J. Glimm, M. Kim, X. LI, A. Marchese, Z. Xu, and N. Zhao.
-  MS 41,Tuesday Feb. 11, 3:15-3:40 PM, Mission Ballroom B, Uncertainty Quantification for Numerical Simulaitons, J. Glimm
- MS 51 Wednesday Feb 12, 10:30-10:55, Regency Ballroom C, Simplifying the Front Tracking Method to Track Complex Interfaces in High Dimensions, X. Li.
- MS 75 Thursday Feb 13, Regency Ballroom C. Error Distribution Models for Strong Shock Interactions, J. Grove.

**Thank you for your attention**