Mixed Cell Closures

In the single-material Euler equations, mass and energy are typically evolved and the EOS is called to provide a pressure for the momentum equations. When transitioning to a multi-material approach, a single velocity is typically used and the Euler equations are solved with respect to the bulk fluid motion. In this case, the pressure contribution to momentum isn’t well-defined and in principle each material could have its own pressure contribution to material motion. Furthermore, the paritioning of volume and energy between the materials in the flow is not well-defined either.

As a result, a multi-material closure rule is needed to determine both how to compute the pressure response of the flow and how to partition the volume and energy between the individual materials. In this situation, one must decide how to compute thermodynamic quantities in that cell for each material.

Governing Equations and Assumptions

In a general sense the mixed material closure rule takes the form

\[\begin{split}P_i = F(\rho, \epsilon, \mu_1, ..., \mu_i, ..., \mu_{N-1}) \\ \rho_i = G(\rho, \epsilon, \mu_1, ..., \mu_i, ..., \mu_{N-1}) \\ \epsilon_i = H(\rho, \epsilon, \mu_1, ..., \mu_i, ..., \mu_{N-1}) \\ T_i = J(\rho, \epsilon, \mu_1, ..., \mu_i, ..., \mu_{N-1})\end{split}\]

where each material has its own density, \(\rho_i\), and specific internal energy, \(e_i\), as well as in principle its own pressure and temperature. Each will be a function of the bulk density, \(\rho\), the bulk specific internal energy, \(\epsilon\), and the mass fractions, \(\mu_j\), of all other materials present.

For convenience of minimizing input arguments though, the solvers in singularity-eos are typically posed in a different way that can sometimes be unintuitive.

For some finite total volume \(V\), each material occupies some fraction of that volume given by the volume fraction \(f_i\) such that

\[\sum_{i=0}^{N - 1} f_i = f_\mathrm{tot},\]

where \(f_\mathrm{tot}\) is the total fraction of the total volume being considered (in principle, different closure models can be used for different sets of materials). To consider the entire volume, \(f_\mathrm{tot}\) can simply be set to one.

The average density, \(\overline{\rho}_i\), (i.e. mass per total volume) for a material in the total volume is

\[\overline{\rho}_i = \rho_i f_i,\]

where \(\rho_i\) is the physical density (i.e. material mass per material volume). It is important to note here that while the densities, \(\rho_i\), and the volume fractions, \(f_i\), will vary as the closure model is applied, the average densities, \(\overline{\rho}_i\), will all remain constant, motiviating their internal use in the closure solvers. The total density (mass of participating materials per total volume) is then

\[\rho = \sum_{i=0}^{N - 1} \overline{\rho}_i = \sum_{i=0}^{N-1} \rho_i f_i\]

Similarly the energy can be summed in a similar way so that

\[u = \rho \epsilon = \sum_{i = 0}^{N - 1} \rho_i \epsilon_i = \sum_{i = 0}^{N - 1} u_i\]

where \(u\) is the total internal energy density (internal energy per unit volume). Similarly, \(u_i\) is analagous to \(\overline{\rho}_i\) in that it is the internal energy for a material averaged over the entire control volume.

Internally, the closer models in singularity-eos operate on \(f_i\), \(\overline{\rho}_i\), and \(u_i\) as well as their total counterparts. This is different than the forms stated at the beginning of this section so that in essence the PTE solver has the form

\[\begin{split}P_i = F(\epsilon, f_\mathrm{tot}, f_1, ..., f_i, ..., f_{N-1}, \rho_1, ..., \rho_i, ..., \rho_{N-1}) \\ \rho_i = G(\epsilon, f_\mathrm{tot}, f_1, ..., f_i, ..., f_{N-1}, \rho_1, ..., \rho_i, ..., \rho_{N-1}) \\ \epsilon_i = H(\epsilon, f_\mathrm{tot}, f_1, ..., f_i, ..., f_{N-1}, \rho_1, ..., \rho_i, ..., \rho_{N-1}) \\ T_i = J(\epsilon, f_\mathrm{tot}, f_1, ..., f_i, ..., f_{N-1}, \rho_1, ..., \rho_i, ..., \rho_{N-1})\end{split}\]

The important nuance here is that the volume fractions and densities are both inputs and outputs in the current singularity-eos formulation of the closure models. From physics perspective this can be confusing, but from a code perspective this limits the number of variables that need to be passed to the PTE solver and provides a convenient way to specify an initial guess for the closure state.

Warning

The volume fractions and material densities must be consistent so that

\[\rho = \sum_{i=0}^{N - 1} \overline{\rho}_i = \sum_{i=0}^{N-1} \rho_i f_i\]

Note

Since mass fraction information is encoded in the specification of the component densities, \(\rho_i\), and component volume fractions, \(f_i\), they must be consistent so that

\[\rho_i = \frac{\mu_i \rho}{f_i}\]

\[\sum\limits_{i=0}^{N-1} f_i = f_\mathrm{tot}.\]

For most practical applications, a previous PTE state for the current masses (i.e. from a previous timestep in a Lagrangian frame) or an appropriate prediction of the new PTE state (i.e. from advected values in an Eulerian frame) is usually available. This is usually the preferred input for the volume fractions and densities provided that they are consistent with the current mass fractions in the control volume.

When a previous state is not available, an assuption must be made for how volume is partitioned between the materials. The simplest (but perhaps not the most effective) assumption is that volume is equipartitioned such that

\[f_i = \frac{1}{N}.\]

It is important to note though that this may not be sufficient in many cases. A better guess just use the mass fractions so that

\[f_i = \mu_i = \frac{\overline{\rho}_i}{\rho},\]

but this is really only valid when the materials have similar compressibilities. A further improvement could be made by weighting the mass fractions by the material bulk moduli to reflect the relative compressibilities.

Pressure-Temperature Equilibrium

At present, singularity-eos focuses on several methods for finding a PTE solution, i.e. one where the pressures and temperatures of the individual materials are all the same. The methods presented differ in what they treat as independent variables, and thus what precise system of equations they solve. However they all share the above mathematical formulation.

In essence, the PTE equations can be posed as two residual equations:

\[f_\mathrm{tot} - \sum\limits_{i=0}^{N-1} f_i = \sum\limits_{i=0}^{N-1} f_i^*(x_i^*, y_i^*) - f_i(x_i, y_i)\]

\[u_\mathrm{tot} - \sum\limits_{i=0}^{N-1} u_i = \sum\limits_{i=0}^{N-1} u_i^*(x_i^*, y_i^*) - u_i(x_i, y_i)\]

where the superscript \(^*\) denotes the variables at the PTE state, \(f\) corresponds to the volume fractions, and \(u\) to the energy density (see the previous section for more information). In these equations, \(x\) and \(y\) represent some choice of independent thermodynamic variables.

These are two non-linear residual equations that will need to be solved. In singularity-eos a Newton-Raphson method is used that first relies on Taylor-expanding the equations about the equilibrium state in order to cast the equations in terms of an update to the unknowns. The expansion about an equilibrium state described by \(f_i^*(\rho_i, y_i)\) and \(u_i^*(\rho_i, y_i)\) is

\[f_\mathrm{tot} - \sum\limits_{i=0}^{N-1} f_i(x_i, y_i) \approx \sum\limits_{i=0}^{N-1} (x_i^* - x_i) \left(\frac{\partial f_i}{\partial x_i}\right)_{y_i} + \sum\limits_{i=0}^{N-1} (y_i^* - y_i) \left(\frac{\partial f_i}{\partial y_i}\right)_{x_i}\]

\[u - \sum\limits_{i=0}^{N-1} u_i(x_i, y_i) \approx \sum\limits_{i=0}^{N-1} (x_i^* - x_i) \left(\frac{\partial u_i}{\partial x_i}\right)_{y_i} + \sum\limits_{i=0}^{N-1} (y_i^* - y_i) \left(\frac{\partial u_i}{\partial y_i}\right)_{x_i},\]

providing a means to update the guess for the equilbrium state. Minor manipulations are needed to recast the derivatives in terms of accessible thermodynamic derivatives, and then these equations can be written in matrix form to solve for the unknown distance away from the equilibrum state. At each iteration of the Newton-Raphson solver, the derivatives are recomputed and a new update is found until some tolerance is reached. When a good initial guess is used (such as a previous PTE state), some algorithms may converge in relatively few iterations.

The choice of \(x\) and \(y\) is discussed below, but crucially it determines the number of equations and unknowns needed to specify the system. For example, if pressure, \(P\), and temperature, \(T\), are chosen, then the subscripts are eliminated since we seek a solution where all materials have the same temperature and pressure. In this formulation, there are two equations and two unkowns, but due to the difficulty of inverting an equation of state to be a function of pressure and temperature, singularity-eos does not have any PTE solvers that are designed to use pressure and temperature as independent variables.

Instead, all of the current PTE solvers in singularity-eos are cast in terms of volume fraction and some other independent variable. Using material volume fractions introduces \(N - 1\) additional unknowns since all but one of the volume fractions are independent from each other. The assumption of pressure equilibrium naturally leads to the addition of \(N - 1\) residual equations of the form

\[P_i(f_i, y_i) - P_j(f_j, y_j) = 0,\]

These can also be written as a Taylor expansion about the equilibrium state such that

\[\begin{split}P_i(f_i, y_j) - P_j(f_j, y_j) = (f^*_i - f_i) \left(\frac{\partial P_i}{\partial f_i}\right)_{y_i} + (y^*_i - y_i) \left(\frac{\partial P_i}{\partial y_i}\right)_{f_i} \\ - (f^*_j - f_j) \left(\frac{\partial P_j}{\partial f_j}\right)_{y_j} - (y^*_j - y_j) \left(\frac{\partial P_j}{\partial y_j}\right)_{f_j},\end{split}\]

where the equations are typically written such that \(j = i + 1\). Since the equlibrium pressure is the same for both materials, the term cancels out and the material pressures are left.

Formulating the closure equations in terms of volume fractions instead of densities has the benefit of allowing the volume constraint to be written in terms of just the volume fractions:

\[f_\mathrm{tot} - \sum\limits_{i=0}^{N-1} f_i = \sum\limits_{i=0}^{N-1} (f_i^* - f_i).\]

The EOS only returns derivatives in terms of density though, so a the density derivatives must be transformed to volume fraction derivatives via

\[\left(\frac{\partial Q}{\partial f_i}\right)_X = - \frac{\rho_i^2}{\rho}\left(\frac{\partial Q}{\partial \rho_i}\right)_X,\]

were \(Q\) and \(X\) are arbitrary thermodynamic variables. At this point, there are \(N + 1\) equations and unknowns in the PTE sover. The choice of the second independent variable is discussed below and has implications for both the number of additional unknowns and the stability of the method.

The Density-Energy Formulation

One choice is to treat volume fractions and material energies as independent quantities, but the material energies provide \(N - 1\) additional unknowns. The additional degrees of freedom are satisfied by requiring that the material temperatures be equal. As a result, we add \(N - 1\) residual equations of the form

\[T_i(\rho_i, \epsilon_i) - T_j(\rho_j, \epsilon_j) = 0.\]

Again Taylor expanding about the equilibirum state, this results in a set of equations of the form

\[\begin{split}T_i(f_i, \epsilon_i) - T_j(f_j, \epsilon_j) = (f^*_i - f_i) \left(\frac{\partial T_i}{\partial f_i}\right)_{\epsilon_i} + (\epsilon^*_i - \epsilon_i) \ \left(\frac{\partial T_i}{\partial \epsilon_i}\right)_{f_i} \\ - (f^*_j - f_j) \left(\frac{\partial T_j}{\partial f_j}\right)_{\epsilon_j} - (\epsilon^*_j - \epsilon_j) \left(\frac{\partial T_j}{\partial \epsilon_j}\right)_{f_j}\end{split}\]

Here there are a total number of \(2N\) equations and unknowns, which results in a fairly large matrix to invert when many materials are present in a cell. Further, the density-energy derivatives may require inversion of any EOS with density and temperature as the natural variables. In the case of tabular EOS, an iterative inversion step may be required to find the density-energy state by iterating on temperature; there may also be a loss of accuracy in the derivatives depending on how they are calculated.

In general, the density-temperature formulation of the PTE solver seems to be more stable and performant and is usually preferrred to this formulation.

In the code this is referred to as the PTESolverRhoU.

The Density-Temperature Formulation

Another choice is to treat the temperature as an independent variable, requiring no additional equations. The energy residual equation then takes the form

\[u - \sum\limits_{i=0}^{N-1} u_i(f_i, T) \approx \sum\limits_{i=0}^{N-1} (f_i^* - f_i) \left(\frac{\partial u_i}{\partial f_i}\right)_{T} + (T^* - T)\sum\limits_{i=0}^{N-1} \left(\frac{\partial u_i}{\partial T}\right)_{f_i},\]

where the temperature difference can be factored out of the sum since it doesn’t depend on material index.

In the code this is referred to as the PTESolverRhoT.

Fixed Pressure or Temperature

For initialization, the energy of a mixed material region is usually unknown while the density, mass fractions, and either temperature or pressure are known. To find the energy, a PTE solve is required, but with the added constraint of the fixed pressure or temperature.

Fixed temperature

When the temperature and total density are known, the equilibrium pressure and the component densities are unknown. This requires a total of \(N\) equations and unknowns. Since the total energy is unknown, it can be dropped from the contraints leaving just the \(N - 1\) pressure equality equations and the volume fraction sum constraint. The pressure residuals can then be simplified to be

\[P_i(f_i, T) - P_j(f_j, T) = (f^*_i - f_i) \left(\frac{\partial P_i}{\partial f_i}\right)_{T} - (f^*_j - f_j) \left(\frac{\partial P_j}{\partial f_j}\right)_{T}\]

In the code this is referred to as the PTESolverFixedT.

Fixed pressure

When the pressure and total density are known, the procedure is slightly more complicated. Since the pressure is known but the independent variables are density and temperature, there are \(N + 1\) unknowns for the component densities and the unknown equilibrium temperature.

Once again, the energy constraint is dropped since the energy is unknown, but since the equilibrium pressure is a specified quantity, the pressure residual equations must be modified to take the form

\[P_i^*(f^*_i, T) - P_i(f_i, T) = (f^*_i - f_i) \left(\frac{\partial P_i}{\partial f_i}\right)_{T} - (T^* - T) \left(\frac{\partial P_i}{\partial T}\right)_{f_i}.\]

Note that this results in \(N\) equations for each of the individual material pressures.

In the code this is referred to as the PTESolverFixedP.

Using the Pressure-Temperature Equilibrium Solver

The PTE machinery is implemented in the singularity-es/closure/mixed_cell_models.hpp header. It is entirely header only.

There are several moving parts. First, one must allocate scratch space used by the solver. There are helper routines for providing the needed scratch space, wich will tell you how many bytes per mixed cell are required. For example:

int PTESolverRhoTRequiredScratch(const int nmat);

and

int PTESolverRhoURequiredScratch(const int nmat);

provide the number of real numbers (i.e., either float or double) required for a single cell given a number of materials in equilibriun for either the RhoT or RhoU solver. The equivalent functions

size_t PTESolverRhoTRequiredScratchInBytes(const int nmat);

and

int PTESolverRhoURequiredScratchInBytes(const int nmat);

give the size in bytes needed to be allocated per cell given a number of materials nmat.

A solver in a given cell is initialized via a Solver object, either PTESolverRhoT or PTESolverRhoU. The constructor takes the number of materials, some set of total quantities required for the conservation constraints, and indexer objects for the equation of state, the independent and dependent variables, and the lambda objects for each equation of state, similar to the vector API for a given EOS. Here the indexers/vectors are not over cells, but materials.

Warning

It bears repeating: both the volume fractions and densities act as inputs and outputs. They are used to define the internal \(\overline {\rho}_i\) variables at the beginning of the PTE solve. The volume fractions and densities at the end of the PTE solve will represent those for the new PTE state. It’s important to note that \(\overline{\rho}_i\) remain constant throughout the calculation.

Warning

The PTE solvers require that all input densities and volume fractions are non-zero. As a result, nmat refers to the number of participating materials. The user is encouraged to wrap their data arrays using an Indexer concept where, for example, three paricipating PTE materials might be indexed as 5, 7, 20 in the material arrays. This requires overloading the square bracket operator to map from PTE idex to material index.

The constructor for the PTESolverRhoT is of the form

template <typename EOS_t, typename Real_t, typename Lambda_t>
PTESolverRhoT(const int nmat, EOS_t &&eos, const Real vfrac_tot, const Real sie_tot,
              Real_t &&rho, Real_t &&vfrac, Real_t &&sie, Real_t &&temp, Real_t &&press,
              Lambda_t &&lambda, Real *scratch, const Real Tguess = 0);

where nmat is the number of materials, eos is an indexer over equation of state objects, one per material, and vfrac_tot is a number \(\in (0,1]\) such that the sum over all volume fractions adds up to vfrac_tot. For a problem in which all materials participate in PTE, vfrac_tot_ should be 1. sie_tot is the total specific internal energy in the problem, rho is an indexer over densities, one per material. vfract is an indexer over volume fractions, one per material. sie is an indexer over temperatures, one per material. press is an indexer over pressures, one per material. lambda is an indexer over lambda arrays, one per material. scratch is a pointer to pre-allocated scratch memory, as described above. It is assumed enough scratch has been allocated. Finally, the optional argument Tguess allows for host codes to pass in an initial temperature guess for the solver. For more information on initial guesses, see the section below.

The constructor for the PTESolverRhoU has the same structure:

template <typename EOS_t, typename Real_t, typename Lambda_t>
PTESolverRhoU(const int nmat, const EOS_t &&eos, const Real vfrac_tot,
              const Real sie_tot, Real_t &&rho, Real_t &&vfrac, Real_t &&sie,
              Real_t &&temp, Real_t &&press, Lambda_t &&lambda, Real *scratch,
              const Real Tguess = 0);

Both constructors are callable on host or device. In gerneral, densities and internal energies are the required inputs. However, all indexer quantities are asusmed to be input/output, as the PTE solver may use unknowns, such as pressure and temperature, as initial guesses and may reset input quantities, such as material densities, to be thermodynamically consistent with the equilibrium solution.

Once a PTE solver has been constructed, one performs the solve with the PTESolver function, which takes a PTESolver object as input and returns a boolean status of either success or failure. For example:

auto method = PTESolverRhoT<decltype(eos), decltype(rho), decltype(lambda)>(NMAT, eos, 1.0, sie_tot, rho, vfrac, sie, temp, press, lambda, scratch);
bool success = PTESolver(method);

For an example of the PTE solver machinery in use, see the test_pte.cpp file in the tests directory.

Initial Guesses for PTE Solvers

As is always the case when solving systems of nonlinear equations, good initial guesses are important to ensure rapid convergence to the solution. For the PTE solvers, this means providing intial guesses for the material densities and the equilibrium temperature. For material densities, a good initial guess is often the previous value obtained from a prior call to the solver. singularity-eos does not provide any mechanism to cache these values from call to call, so it is up to the host code to provide these as input to the solvers. Note that the input values for the material densities and volume fractions are assumed to be consistent with the conserved cell-averaged material densities, or in other words, the produce of the input material densities, volume fractions, and cell volume should equal the amount of mass of each material in the cell. This consistency should be ensured for the input values or else the solvers will not provide correct answers.

For the temperature initial guess, one can similarly use a previous value for the cell. Alternatively, singularity-eos provides a function that can be used to provide an initial guess. This function takes the form

template <typename EOSIndexer, typename RealIndexer>
PORTABLE_INLINE_FUNCTION Real ApproxTemperatureFromRhoMatU(
  const int nmat, EOSIndexer &&eos, const Real u_tot, RealIndexer &&rho,
  RealIndexer &&vfrac, const Real Tguess = 0.0);

where nmat is the number of materials, eos is an indexer over equation of state objects, u_tot is the total material internal energy density (energy per unit volume), rho is an indexer over material density, vfrac is an indexer over material volume fractions, and the optional argument Tguess allows for callers to pass in a guess that could accelerate finding a solution. This function does a 1-D root find to find the temperature at which the material internal energies sum to the total. The root find does not have a tight tolerance – instead the hard-coded tolerance was selected to balance performance with the accuracy desired for an initial guess in a PTE solve. If a previous temperature value is unavailable or some other process may have significantly modified the temperature since it was last updated, this function can be quite effective.