ncpflash.hh

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#ifndef DUMUX_NCP_FLASH_HH
#define DUMUX_NCP_FLASH_HH
 
#include <dune/common/fvector.hh>
#include <dune/common/fmatrix.hh>
 
#include <dumux/common/exceptions.hh>
#include <dumux/material/fluidmatrixinteractions/mp/mpadapter.hh>
 
namespace Dumux {
 
template <class Scalar, class FluidSystem>
class NcpFlash
{
    enum { numPhases = FluidSystem::numPhases };
    enum { numComponents = FluidSystem::numComponents };
 
    using ParameterCache = typename FluidSystem::ParameterCache;
 
    static constexpr int numEq = numPhases*(numComponents + 1);
 
    using Matrix = Dune::FieldMatrix<Scalar, numEq, numEq>;
    using Vector = Dune::FieldVector<Scalar, numEq>;
 
public:
    using ComponentVector = Dune::FieldVector<Scalar, numComponents>;
 
    explicit NcpFlash(int wettingPhaseIdx = 0)
    : wettingPhaseIdx_(wettingPhaseIdx)
    {}
 
    template <class FluidState>
    void guessInitial(FluidState &fluidState,
                      ParameterCache &paramCache,
                      const ComponentVector &globalMolarities)
    {
        // the sum of all molarities
        Scalar sumMoles = 0;
        for (int compIdx = 0; compIdx < numComponents; ++compIdx)
            sumMoles += globalMolarities[compIdx];
 
        for (int phaseIdx = 0; phaseIdx < numPhases; ++ phaseIdx) {
            // composition
            for (int compIdx = 0; compIdx < numComponents; ++ compIdx)
                fluidState.setMoleFraction(phaseIdx,
                                           compIdx,
                                           globalMolarities[compIdx]/sumMoles);
 
            // pressure. use atmospheric pressure as initial guess
            fluidState.setPressure(phaseIdx, 1.0135e5);
 
            // saturation. assume all fluids to be equally distributed
            fluidState.setSaturation(phaseIdx, 1.0/numPhases);
        }
 
        // set the fugacity coefficients of all components in all phases
        paramCache.updateAll(fluidState);
        for (int phaseIdx = 0; phaseIdx < numPhases; ++ phaseIdx) {
            for (int compIdx = 0; compIdx < numComponents; ++ compIdx) {
                Scalar phi = FluidSystem::fugacityCoefficient(fluidState, paramCache, phaseIdx, compIdx);
                fluidState.setFugacityCoefficient(phaseIdx, compIdx, phi);
            }
        }
    }
 
    template <class MaterialLaw, class FluidState>
    void solve(FluidState &fluidState,
               ParameterCache &paramCache,
               const MaterialLaw& material,
               const ComponentVector &globalMolarities)
    {
        // Newton method
 
        // Jacobian matrix
        Matrix J;
        // solution, i.e. phase composition
        Vector deltaX;
        // right hand side
        Vector b;
 
        // make the fluid state consistent with the fluid system.
        completeFluidState_(fluidState, paramCache, material);
 
        /*
        std::cout << "--------------------\n";
        std::cout << "globalMolarities: ";
        for (int compIdx = 0; compIdx < numComponents; ++ compIdx)
            std::cout << globalMolarities[compIdx] << " ";
        std::cout << "\n";
        */
 
        const int nMax = 50; // <- maximum number of newton iterations
        for (int nIdx = 0; nIdx < nMax; ++nIdx) {
            // calculate Jacobian matrix and right hand side
            linearize_(J, b, fluidState, paramCache, material, globalMolarities);
 
            // Solve J*x = b
            deltaX = 0;
 
            try {
 
                // simple preconditioning of the linear system to reduce condition number
                int eqIdx = 0;
                for (int compIdx = 0; compIdx < numComponents; ++compIdx)
                {
                    for (int phaseIdx = 1; phaseIdx < numPhases; ++phaseIdx)
                    {
                        J[eqIdx] *= 10e-7; // roughly the magnitude of the fugacities
                        b[eqIdx] *= 10e-7;
                        ++eqIdx;
                    }
                }
 
                J.solve(deltaX, b);
 
            }
            catch (Dune::FMatrixError &e)
            {
                /*
                printFluidState_(fluidState);
                std::cout << "error: " << e << "\n";
                std::cout << "b: " << b << "\n";
                std::cout << "J: " << J << "\n";
                */
 
                throw NumericalProblem(e.what());
            }
 
            /*
            printFluidState_(fluidState);
            std::cout << "J:\n";
            for (int i = 0; i < numEq; ++i) {
                for (int j = 0; j < numEq; ++j) {
                    std::ostringstream os;
                    os << J[i][j];
 
                    std::string s(os.str());
                    do {
                        s += " ";
                    } while (s.size() < 20);
                    std::cout << s;
                }
                std::cout << "\n";
            }
 
            std::cout << "deltaX: " << deltaX << "\n";
            std::cout << "---------------\n";
            */
 
            // update the fluid quantities.
            Scalar relError = update_(fluidState, paramCache, material, deltaX);
 
            if (relError < 1e-9)
                return;
        }
 
        /*
        printFluidState_(fluidState);
        std::cout << "globalMolarities: ";
        for (int compIdx = 0; compIdx < numComponents; ++ compIdx)
            std::cout << globalMolarities[compIdx] << " ";
        std::cout << "\n";
        */
 
        DUNE_THROW(NumericalProblem,
                   "Flash calculation failed."
                   " {c_alpha^kappa} = {" << globalMolarities << "}, T = "
                   << fluidState.temperature(/*phaseIdx=*/0));
    }
 
 
protected:
    template <class FluidState>
    void printFluidState_(const FluidState &fs)
    {
        std::cout << "saturations: ";
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx)
            std::cout << fs.saturation(phaseIdx) << " ";
        std::cout << "\n";
 
        std::cout << "pressures: ";
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx)
            std::cout << fs.pressure(phaseIdx) << " ";
        std::cout << "\n";
 
        std::cout << "densities: ";
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx)
            std::cout << fs.density(phaseIdx) << " ";
        std::cout << "\n";
 
        std::cout << "molar densities: ";
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx)
            std::cout << fs.molarDensity(phaseIdx) << " ";
        std::cout << "\n";
 
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
            std::cout << "composition " << FluidSystem::phaseName(phaseIdx) << "Phase: ";
            for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
                std::cout << fs.moleFraction(phaseIdx, compIdx) << " ";
            }
            std::cout << "\n";
        }
 
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
            std::cout << "fugacities " << FluidSystem::phaseName(phaseIdx) << "Phase: ";
            for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
                std::cout << fs.fugacity(phaseIdx, compIdx) << " ";
            }
            std::cout << "\n";
        }
 
        std::cout << "global component molarities: ";
        for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
            Scalar sum = 0;
            for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
                sum += fs.saturation(phaseIdx)*fs.molarity(phaseIdx, compIdx);
            }
            std::cout << sum << " ";
        }
        std::cout << "\n";
    }
 
    template <class MaterialLaw, class FluidState>
    void linearize_(Matrix &J,
                    Vector &b,
                    FluidState &fluidState,
                    ParameterCache &paramCache,
                    const MaterialLaw& material,
                    const ComponentVector &globalMolarities)
    {
        FluidState origFluidState(fluidState);
        ParameterCache origParamCache(paramCache);
 
        Vector tmp;
 
        // reset jacobian
        J = 0;
 
        calculateDefect_(b, fluidState, fluidState, globalMolarities);
 
        // assemble jacobian matrix
        for (int pvIdx = 0; pvIdx < numEq; ++ pvIdx) {
            // approximately calculate partial derivatives of the
            // fugacity defect of all components in regard to the mole
            // fraction of the i-th component. This is done via
            // forward differences
 
            // deviate the mole fraction of the i-th component
            Scalar x_i = getQuantity_(fluidState, pvIdx);
            const Scalar eps = 1e-8/quantityWeight_(fluidState, pvIdx);
            setQuantity_(fluidState, paramCache, material, pvIdx, x_i + eps);
 
            // compute derivative of the defect
            calculateDefect_(tmp, origFluidState, fluidState, globalMolarities);
            tmp -= b;
            tmp /= eps;
 
            // store derivative in jacobian matrix
            for (int eqIdx = 0; eqIdx < numEq; ++eqIdx)
                J[eqIdx][pvIdx] = tmp[eqIdx];
 
            // fluid state and parameter cache to their original values
            fluidState = origFluidState;
            paramCache = origParamCache;
 
            // end forward differences
        }
    }
 
    template <class FluidState>
    void calculateDefect_(Vector &b,
                          const FluidState &fluidStateEval,
                          const FluidState &fluidState,
                          const ComponentVector &globalMolarities)
    {
        int eqIdx = 0;
 
        // fugacity of any component must be equal in all phases
        for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
            for (int phaseIdx = 1; phaseIdx < numPhases; ++phaseIdx) {
                b[eqIdx] =
                    fluidState.fugacity(/*phaseIdx=*/0, compIdx) -
                    fluidState.fugacity(phaseIdx, compIdx);
                ++eqIdx;
            }
        }
 
        assert(eqIdx == numComponents*(numPhases - 1));
 
        // the fact saturations must sum up to 1 is included explicitly!
 
        // capillary pressures are explicitly included!
 
        // global molarities are given
        for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
            b[eqIdx] = 0.0;
            for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
                b[eqIdx] +=
                    fluidState.saturation(phaseIdx)
                    * fluidState.molarity(phaseIdx, compIdx);
            }
 
            b[eqIdx] -= globalMolarities[compIdx];
            ++eqIdx;
        }
 
        // model assumptions (-> non-linear complementarity functions)
        // must be adhered
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
            Scalar sumMoleFracEval = 0.0;
            for (int compIdx = 0; compIdx < numComponents; ++compIdx)
                sumMoleFracEval += fluidStateEval.moleFraction(phaseIdx, compIdx);
 
            if (1.0 - sumMoleFracEval > fluidStateEval.saturation(phaseIdx)) {
                b[eqIdx] = fluidState.saturation(phaseIdx);
            }
            else {
                Scalar sumMoleFrac = 0.0;
                for (int compIdx = 0; compIdx < numComponents; ++compIdx)
                    sumMoleFrac += fluidState.moleFraction(phaseIdx, compIdx);
                b[eqIdx] = 1.0 - sumMoleFrac;
            }
 
            ++eqIdx;
        }
    }
 
    template <class MaterialLaw, class FluidState>
    Scalar update_(FluidState &fluidState,
                   ParameterCache &paramCache,
                   const MaterialLaw& material,
                   const Vector &deltaX)
    {
        Scalar relError = 0;
        for (int pvIdx = 0; pvIdx < numEq; ++ pvIdx) {
            Scalar tmp = getQuantity_(fluidState, pvIdx);
            Scalar delta = deltaX[pvIdx];
            using std::max;
            using std::abs;
            using std::min;
            relError = max(relError, abs(delta)*quantityWeight_(fluidState, pvIdx));
 
            if (isSaturationIdx_(pvIdx)) {
                // dampen to at most 20% change in saturation per
                // iteration
                delta = min(0.2, max(-0.2, delta));
            }
            else if (isMoleFracIdx_(pvIdx)) {
                // dampen to at most 15% change in mole fraction per
                // iteration
                delta = min(0.15, max(-0.15, delta));
            }
            else if (isPressureIdx_(pvIdx)) {
                // dampen to at most 15% change in pressure per
                // iteration
                delta = min(0.15*fluidState.pressure(0), max(-0.15*fluidState.pressure(0), delta));
            }
 
            setQuantityRaw_(fluidState, pvIdx, tmp - delta);
        }
 
        // make sure all saturations, pressures and mole fractions are non-negative
        Scalar sumSat = 0;
        for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
            Scalar value = fluidState.saturation(phaseIdx);
            if (value < -0.05) {
                value = -0.05;
                fluidState.setSaturation(phaseIdx, value);
            }
            sumSat += value;
 
            value = fluidState.pressure(phaseIdx);
            if (value < 0)
                fluidState.setPressure(phaseIdx, 0.0);
 
            bool allMoleFractionsAreZero = true;
            for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
                value = fluidState.moleFraction(phaseIdx, compIdx);
                if (value > 0)
                    allMoleFractionsAreZero = false;
                else if (value < 0)
                    fluidState.setMoleFraction(phaseIdx, compIdx, 0.0);
            }
 
            // set at least one mole fraction to a positive value
            // to prevent breakdown of the linear solver in completeFluidState_
            if (allMoleFractionsAreZero)
                fluidState.setMoleFraction(phaseIdx, /*compIdx=*/0, 1e-10);
        }
 
        // last saturation
        if (sumSat > 1.05) {
            for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
                Scalar value = fluidState.saturation(phaseIdx)/(0.95*sumSat);
                fluidState.setSaturation(phaseIdx, value);
            }
        }
 
        completeFluidState_(fluidState, paramCache, material);
 
        return relError;
    }
 
    template <class MaterialLaw, class FluidState>
    void completeFluidState_(FluidState &fluidState,
                             ParameterCache &paramCache,
                             const MaterialLaw& material)
    {
        // calculate the saturation of the last phase as a function of
        // the other saturations
        Scalar sumSat = 0.0;
        for (int phaseIdx = 0; phaseIdx < numPhases - 1; ++phaseIdx)
            sumSat += fluidState.saturation(phaseIdx);
        fluidState.setSaturation(/*phaseIdx=*/numPhases - 1, 1.0 - sumSat);
 
        // update the pressures using the material law (saturations
        // and first pressure are already set because it is implicitly
        // solved for.)
        const auto pc = material.capillaryPressures(fluidState, wettingPhaseIdx_);
        for (int phaseIdx = 1; phaseIdx < numPhases; ++phaseIdx)
            fluidState.setPressure(phaseIdx,
                                   fluidState.pressure(0)
                                   + (pc[phaseIdx] - pc[0]));
 
        // update the parameter cache
        paramCache.updateAll(fluidState, /*except=*/ParameterCache::Temperature);
 
        // update all densities and fugacity coefficients
        for (int phaseIdx = 0; phaseIdx < numPhases; ++ phaseIdx) {
 
            Scalar rho = FluidSystem::density(fluidState, paramCache, phaseIdx);
            Scalar rhoMolar = FluidSystem::molarDensity(fluidState, paramCache, phaseIdx);
            fluidState.setDensity(phaseIdx, rho);
            fluidState.setMolarDensity(phaseIdx, rhoMolar);
 
            for (int compIdx = 0; compIdx < numComponents; ++ compIdx) {
                Scalar phi = FluidSystem::fugacityCoefficient( fluidState, paramCache, phaseIdx, compIdx);
                fluidState.setFugacityCoefficient(phaseIdx, compIdx, phi);
            }
        }
    }
 
    bool isPressureIdx_(int pvIdx)
    { return pvIdx == 0; }
 
    bool isSaturationIdx_(int pvIdx)
    { return 1 <= pvIdx && pvIdx < numPhases; }
 
    bool isMoleFracIdx_(int pvIdx)
    { return numPhases <= pvIdx && pvIdx < numPhases + numPhases*numComponents; }
 
    // retrieves a quantity from the fluid state
    template <class FluidState>
    Scalar getQuantity_(const FluidState &fs, int pvIdx)
    {
        assert(pvIdx < numEq);
 
        // first pressure
        if (pvIdx < 1) {
            int phaseIdx = 0;
            return fs.pressure(phaseIdx);
        }
        // first M - 1 saturations
        else if (pvIdx < numPhases) {
            int phaseIdx = pvIdx - 1;
            return fs.saturation(phaseIdx);
        }
        // mole fractions
        else // if (pvIdx < numPhases + numPhases*numComponents)
        {
            int phaseIdx = (pvIdx - numPhases)/numComponents;
            int compIdx = (pvIdx - numPhases)%numComponents;
            return fs.moleFraction(phaseIdx, compIdx);
        }
    }
 
    // set a quantity in the fluid state
    template <class MaterialLaw, class FluidState>
    void setQuantity_(FluidState &fs,
                      ParameterCache &paramCache,
                      const MaterialLaw& material,
                      int pvIdx,
                      Scalar value)
    {
        assert(0 <= pvIdx && pvIdx < numEq);
 
        if (pvIdx < 1) { // <- first pressure
            Scalar delta = value - fs.pressure(0);
 
            // set all pressures. here we assume that the capillary
            // pressure does not depend on absolute pressure.
            for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx)
                fs.setPressure(phaseIdx, fs.pressure(phaseIdx) + delta);
            paramCache.updateAllPressures(fs);
 
            // update all densities and fugacity coefficients
            for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
                Scalar rho = FluidSystem::density(fs, paramCache, phaseIdx);
                Scalar rhoMolar = FluidSystem::molarDensity(fs, paramCache, phaseIdx);
                fs.setDensity(phaseIdx, rho);
                fs.setMolarDensity(phaseIdx, rhoMolar);
 
                for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
                    Scalar phi = FluidSystem::fugacityCoefficient(fs, paramCache, phaseIdx, compIdx);
                    fs.setFugacityCoefficient(phaseIdx, compIdx, phi);
                }
            }
        }
        else if (pvIdx < numPhases) { // <- first M - 1 saturations
            Scalar delta = value - fs.saturation(/*phaseIdx=*/pvIdx - 1);
            fs.setSaturation(/*phaseIdx=*/pvIdx - 1, value);
 
            // set last saturation (-> minus the change of the
            // satuation of the other phase)
            fs.setSaturation(/*phaseIdx=*/numPhases - 1,
                             fs.saturation(numPhases - 1) - delta);
 
            // update all fluid pressures using the capillary pressure
            // law
            const auto pc = material.capillaryPressures(fs, wettingPhaseIdx_);
            for (int phaseIdx = 1; phaseIdx < numPhases; ++phaseIdx)
                fs.setPressure(phaseIdx,
                               fs.pressure(0)
                               + (pc[phaseIdx] - pc[0]));
            paramCache.updateAllPressures(fs);
 
            // update all densities and fugacity coefficients
            for (int phaseIdx = 0; phaseIdx < numPhases; ++phaseIdx) {
                Scalar rho = FluidSystem::density(fs, paramCache, phaseIdx);
                Scalar rhoMolar = FluidSystem::molarDensity(fs, paramCache, phaseIdx);
                fs.setDensity(phaseIdx, rho);
                fs.setMolarDensity(phaseIdx, rhoMolar);
 
                for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
                    Scalar phi = FluidSystem::fugacityCoefficient(fs, paramCache, phaseIdx, compIdx);
                    fs.setFugacityCoefficient(phaseIdx, compIdx, phi);
                }
            }
        }
        else if (pvIdx < numPhases + numPhases*numComponents) // <- mole fractions
        {
            int phaseIdx = (pvIdx - numPhases)/numComponents;
            int compIdx = (pvIdx - numPhases)%numComponents;
 
            fs.setMoleFraction(phaseIdx, compIdx, value);
            paramCache.updateSingleMoleFraction(fs, phaseIdx, compIdx);
 
            // update the density of the phase
            Scalar rho = FluidSystem::density(fs, paramCache, phaseIdx);
            Scalar rhoMolar = FluidSystem::molarDensity(fs, paramCache, phaseIdx);
            fs.setDensity(phaseIdx, rho);
            fs.setMolarDensity(phaseIdx, rhoMolar);
 
            // if the phase's fugacity coefficients are composition
            // dependent, update them as well.
            if (!FluidSystem::isIdealMixture(phaseIdx)) {
                for (int compIdx2 = 0; compIdx2 < numComponents; ++compIdx2) {
                    Scalar phi = FluidSystem::fugacityCoefficient(fs, paramCache, phaseIdx, compIdx2);
                    fs.setFugacityCoefficient(phaseIdx, compIdx2, phi);
                }
            }
        }
        else {
            assert(false);
        }
    }
 
    // set a quantity in the fluid state
    template <class FluidState>
    void setQuantityRaw_(FluidState &fs, int pvIdx, Scalar value)
    {
        assert(pvIdx < numEq);
 
        // first pressure
        if (pvIdx < 1) {
            int phaseIdx = 0;
            fs.setPressure(phaseIdx, value);
        }
        // first M - 1 saturations
        else if (pvIdx < numPhases) {
            int phaseIdx = pvIdx - 1;
            fs.setSaturation(phaseIdx, value);
        }
        // mole fractions
        else // if (pvIdx < numPhases + numPhases*numComponents)
        {
            int phaseIdx = (pvIdx - numPhases)/numComponents;
            int compIdx = (pvIdx - numPhases)%numComponents;
            fs.setMoleFraction(phaseIdx, compIdx, value);
        }
    }
 
    template <class FluidState>
    Scalar quantityWeight_(const FluidState &fs, int pvIdx)
    {
        // first pressure
        if (pvIdx < 1)
            return 1.0/1e5;
        // first M - 1 saturations
        else if (pvIdx < numPhases)
            return 1.0;
        // mole fractions
        else // if (pvIdx < numPhases + numPhases*numComponents)
            return 1.0;
    }
 
private:
    int wettingPhaseIdx_ = 0; 
};
 
} // end namespace Dumux
 
#endif