Software

I. Front tracking
A. Robust but first order tracking
With Front tracking, our aim is to achieve enhanced resolution of interfaces between fluids and maintain this resolution throughout the simulation. This allows us to minimize numerical mass diffusion associated with steep concentration or thermal gradients. We see improved results even for coarser grids, since the discontinuity that normally leads to the smearing in coarser grids is dynamically tracked, preventing one fluid from numerically interacting across the interface with the other.

We accomplish this goal by creating a separate, lower dimensional mesh, along the front (interface) which separates the fluids, for example following an iso-temperature contour within a sharp thermal gradient. The front then separates fluids with distinct physical characteristics. During each time step we propagate the front's mesh to its new location based on the surrounding interior states. The front points are propagated in a normal direction, utilizing a Riemann problem fomulation to resolve the different velocities obtained via one sided extrapolation from interior states to the front location.

When interfacial tangling occurs, we utilize algorithms to untangle and redistribute points on the front mesh, allowing for a logical reconstruction of a tangled interface. For more information on this process and the FronTier Front Tracking algorithm, we refer the reader to [ GliGroZha99b]. [ DuFixGli05]. [ BoLiuGli10].

The front mesh is overlayed on a normal eulerian grid. Where it intersects cells, we cut those cells into multiple pieces to ensure separation of the distinct fluids across an interface.

Discretized interface and cut cell construction. Tracked Front overlayed over Eulerian Grid, discretized at front points.

Display of cut cells created when an interface cuts through a cell.
When updating interior states on a cut cell, ghost cells constructed from states defined in neighboring cells on the same side of the front allow for a 1-sided update, consisting only of states from those cells on the same side of the discontinuity interface.

This construction allows us to control numerical diffusion, especially with complicated interfaces, as arise in hydro instabilities in ICF processes.

B. Higher order tracking
C. Large eddy simulations
The purpose of subgrid models is to capture the effect of the unresolved scales (below the grid level) on those that are resolved. We start with the interpretation of the numerical solution values as representing grid cell averaged quantities. Conventionally, in the derivation of Reynolds averaged Navier-Stokes equations, there is a convolution average by a positive weight function (the filter), and the equations are derived for the filtered quantities. We omit this filter step, and use the cell averaged quantities directly. The average of the nonlinear terms introduces new unknowns into the equations, for which new equations, called closures are required. These show up as flux type terms, and are the origin of the turbulent transport. The closure term must be modeled; often a solution gradient is used for a turbulent flux, with an undetermined coefficient. With dynamic subgrid models [ GerPioMoi91] [ MoiSquCab91] [ Ma06] the otherwise missing coefficient is determined by the numerical solution itself, and will vary with space-time according to local flow conditions. The determination is achieved by consideration of the closure on the current grid and on a once coarser grid, and by an asymptotic assumption that the model coefficient is a mesh convergent or asymptotically mesh independent quantity, so that a common value can be used between the two grid levels considered. The advantage of cell averages as opposed to a filter is the absence of loss of spatial resolution and of solution fluctuations associated with the filter.
the otherwise missing coefficient is determined by the numerical solution itself, and will vary with space-time according to local flow conditions. The determination is achieved by consideration of the closure on the current grid and on a once coarser grid, and by an asymptotic assumption that the model coefficient is a mesh convergent or asymptotically mesh independent quantity, so that a common value can be used between the two grid levels considered. The advantage of cell averages as opposed to a filter is the absence of loss of spatial resolution and of solution fluctuations associated with the filter.
II. Supported physical problems and associated software
A. Compressible hydrodynamics
B. Incompressible hydrodynamics
C. Combustion
D. Cavitation and phase transitions
E. MHD
F. Brittle fracture
G. Fluid-structure interactions

Tools

I. w*
To assess stochastic convergence, we have built a tool which reads output files, and with user specified supercells, will assemble all data in a supercedll to define a CDF. For comparing two stochastic simulations, either to assess mesh convergence or the results of variation of some parameter, this tool will take the CDFs defined on supercells, and form the difference of the CDF and its L_1 norm. These L_1 norms are then averaged over a larger domain to get a space-time-solution space norm convergence of fluctuations. See W* Manual .
II. API Application programming interface
To link Front Tracking to a user code, is under construction. See API Manual . Currently, links are under development to FLASH, a high energy density physics code from the U. Chicago, and WRF, an atmospheric modeling code supported by NCAR. The API comes in two stages, passive and active tracking. This division reflects the two primarey functions of the API, to provide two way communication between the Front Tracking code and a client code. With communicatoin from the clent to FronTeir only, we enable passivel tracking; in this case interfaces, once initialized, are moved by the client code but the computation of the client code is not modified by this coupling. With two way communication, the client code has revised algorithms for advection of solution discontinuity surfaces or of solution iso-surfaces representing steep concentration gradients.

All functions in the API are designated as belonging to the client or to FronTier. Functions that need to access data structures of the client are logically regarded as belonging to the client. For these the API has some reference implementations, but cannot support the full range of user codes, and a user will need to adapt one of the reference implementations to their own situation. Functions which do not access client data structures belong logically to FronTier.

A. Passive tracking
The main client function is a velocity function, which will return a fluid velocity at a general (non-mesh) point.

The main FronTier function is a point propagate function, wiich, using the client velocity, will move the front points.

B. Active tracking
The main FronTier function is a front assessment function. For a given stencil (as part of the client finite difference code), FronTier will decide if the stencil crosses the front or not. If it does the stencil is marked for later "repair" of the client solution.

A second FronTier function is to fill the stencil with new states, based on the idea of ghost cell extrapolation. This function is based on extrapolation, and will use stencil values from the clients "to be repaired" list of stencils.

The main client function is its normal solver or integration finite difference operator. An additional client function will loop over all "to be repaired" stencils and update them using the modified states located within the stencil.

III. Subgrid terms for large eddy simulations
We derive a formal expansion (convergence assuming K41) in polynomials in first difference operators acting on solution primitive variables. The expansion, a modification of the renormalization group fixed point expansion, is in units of the refined mesh spacing on which the difference operators are defined. Based on an asymptotic assumption, these same expressions can be transformed to presently resolved meshes, leading to a type of dynamic determination of the subgrid terms. This step, similar to currently used dynamic SGS models, differs in one key respect. Not only is the coefficient of the model determined from theory, the actual model itself is determined from theory, with a systematic expansion for higher order corrections, or for assessement of approximation error for any finite level of expansion. See Theory (add link) and (refs).