generalizedIncompressibleBoundaryTemplates.h

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#ifndef GENERALIZED_INCOMPRESSIBLE_BOUNDARY_TEMPLATES_H
#define GENERALIZED_INCOMPRESSIBLE_BOUNDARY_TEMPLATES_H

#include "generalizedBoundaryDynamics.h"
#include "core/cell.h"
#include "core/dynamicsIdentifiers.h"
#include "latticeBoltzmann/indexTemplates.h"
#include "latticeBoltzmann/hermitePolynomialsTemplates.h"
#include <Eigen3/Core>
#include <Eigen3/LU>
#include <Eigen3/QR>
#include <Eigen3/Cholesky>
#include <Eigen3/SVD>
#include <Eigen3/Dense>


namespace plb {

template<typename T, template<typename U> class Descriptor>
struct generalizedIncomprBoundaryTemplates {
    
    static T equilibrium_ma2_over_rho(plint iPop, const Array<T,Descriptor<T>::d> &u, T uSqr) {
        T c_u = Descriptor<T>::c[iPop][0]*u[0];
        for (int iD=1; iD < Descriptor<T>::d; ++iD) {
            c_u += Descriptor<T>::c[iPop][iD]*u[iD];
        }
        return Descriptor<T>::t[iPop] * ( 
            (T)1 + Descriptor<T>::invCs2 * c_u +
            Descriptor<T>::invCs2/(T)2 * (
            Descriptor<T>::invCs2 * c_u*c_u - uSqr ) );
    }
    
    // f = w_i*rho*g_i+H2/(2*cs^4):PiNeq (rho and PiNeq unknowns)
    static void f_ma2_linear(plint iPop, const Array<T,Descriptor<T>::d> &u, 
                                     T uSqr, Eigen::RowVectorXd &a) {
        T eqOverRho = equilibrium_ma2_over_rho(iPop,u,uSqr);
        a[0] = eqOverRho;
        
        T factor = 0.5*Descriptor<T>::t[iPop]*Descriptor<T>::invCs2*Descriptor<T>::invCs2;
        Array<T,SymmetricTensor<T,Descriptor>::n> H2 = HermiteTemplate<T,Descriptor>::contractedOrder2(iPop);
        
        for (plint iPi=1; iPi<=SymmetricTensor<T,Descriptor>::n; ++iPi) a[iPi] = H2[iPi-1]*factor;
    }
    
    static void f_ma2_linear(plint iPop, const Array<T,Descriptor<T>::d> &u, T uSqr, Eigen::RowVectorXd &a,
                             T omega) {
        T eqOverRho = equilibrium_ma2_over_rho(iPop,u,uSqr);
        a(0) = eqOverRho;
        
        T factor = 0.5*Descriptor<T>::t[iPop]*Descriptor<T>::invCs2*Descriptor<T>::invCs2 * ((T)1-omega);
        Array<T,SymmetricTensor<T,Descriptor>::n> H2 = HermiteTemplate<T,Descriptor>::contractedOrder2(iPop);
        
        for (plint iPi=1; iPi<=SymmetricTensor<T,Descriptor>::n; ++iPi) a(iPi) = H2[iPi-1]*factor;
    }
        
    static void f_to_A_ma2_contrib(const std::vector<plint> kInd, const Array<T,Descriptor<T>::d> &u, 
                                     T uSqr, Eigen::MatrixXd &A) {
        for (pluint fInd = 0; fInd < kInd.size(); ++fInd) {
            plint iPop = kInd[fInd];
            Eigen::RowVectorXd lineA = Eigen::RowVectorXd::Zero(A.cols());
            generalizedIncomprBoundaryTemplates<T,Descriptor>::
                f_ma2_linear(iPop, u, uSqr, lineA);
            A.row(fInd) = lineA;
        }
    }
    
    static void f_to_b_contrib(Cell<T,Descriptor> &cell, const std::vector<plint> kInd, Eigen::VectorXd &b) {
        for (pluint fInd = 0; fInd < kInd.size(); ++fInd) {
            plint iPop = kInd[fInd];
            b[fInd] = fullF<T,Descriptor>(cell[iPop], iPop);
        }
    }
    
    static void from_macro_to_rho_j_pineq(const std::vector<T> &macro, T &rho, Array<T,Descriptor<T>::d> &j, 
                                          Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq) {
        rho = macro[0];
        for (plint iD = 0; iD < Descriptor<T>::d; ++iD) j[iD] = rho * (macro[1+iD]);
        for (plint iPi = 0; iPi < SymmetricTensor<T,Descriptor>::n; ++iPi) PiNeq[iPi] = macro[1+Descriptor<T>::d+iPi];
    }
    
    // generic tranformation methods
    static void fromRhoAndPiNeqToX(T rho, const Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq, Eigen::VectorXd &x) {
        x(0) = rho;
        for (plint iPi = 0; iPi < SymmetricTensor<T,Descriptor>::n; ++iPi) x(iPi+1) = PiNeq[iPi];
    }
    
    static void fromXtoRhoAndPiNeq(const Eigen::VectorXd &x, T &rho, Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq) {
        rho = x(0);
        for (plint iPi = 0; iPi < SymmetricTensor<T,Descriptor>::n; ++iPi) PiNeq[iPi] = x(iPi+1);
    }
    
    static void fromUandPiNeqToX(const Array<T,Descriptor<T>::d> &u, 
                                 const Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq, Eigen::VectorXd &x, plint dir) {
        x(0) = u[dir];
        for (plint iPi = 0; iPi < SymmetricTensor<T,Descriptor>::n; ++iPi) x(iPi+1) = PiNeq[iPi];
    }
    
    static void fromXtoUandPiNeq(const Eigen::VectorXd &x, Array<T,Descriptor<T>::d> &u, 
                                 Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq,plint dir) {
        u[dir] = x(0);
        for (plint iPi = 0; iPi < SymmetricTensor<T,Descriptor>::n; ++iPi) PiNeq[iPi] = x(iPi+1);
    }
    
    // creation of the over-determined (usually) linear system
//     static void createLinearSystem(const Cell<T,Descriptor>& cell, const Array<T,Descriptor<T>::d> &u,
//                                    const std::vector<plint> &missingIndices,
//                                    const std::vector<plint> &knownIndices,
//                                    Eigen::MatrixXd &A,Eigen::VectorXd &b) {
//         
//         T uSqr = VectorTemplate<T,Descriptor>::normSqr(u);
//         
//         plint systSizeX = SymmetricTensor<T,Descriptor>::n+1;
//         plint systSizeY = knownIndices.size()+1;
//         
//         // matrix of the system Ax=b
//         A = Eigen::MatrixXd::Zero(systSizeY,systSizeX);
//         // rhs of the equation Ax=b
//         b = Eigen::VectorXd::Zero(systSizeY);
//         
//         // f^k = A * x
//         // A = g, 1/(2c_s^4) H^2
//         // with g being feq/rho and H^2 the second order Hermite polynomial
//         for (pluint fInd = 0; fInd < knownIndices.size(); ++fInd) {
//             plint iPop = knownIndices[fInd];
//             Eigen::RowVectorXd lineA = Eigen::RowVectorXd::Zero(systSizeX);
//             computeMatrixRow(iPop, u, uSqr, lineA);
//             A.row(fInd) = lineA;
//         }
//         
//         T rhoTmp = T();
//         for (pluint kInd = 0; kInd < knownIndices.size(); ++kInd) {
//             plint iPop = knownIndices[kInd];
//             rhoTmp += fullF<T,Descriptor>(cell[iPop], iPop);
//             b(kInd) = fullF<T,Descriptor>(cell[iPop], iPop);
//         }
//         // rhoTtmp = sum_i->known f_i.
//         b(knownIndices.size()) = rhoTmp;
//         
//         // first row of the A matrix. imposing sum_i f_i = rho.
//         Eigen::RowVectorXd e0 = Eigen::RowVectorXd::Zero(systSizeX); 
//         e0(0) = 1.0;
//         
//         Eigen::RowVectorXd sumA = Eigen::RowVectorXd::Zero(systSizeX);
//         for (pluint fInd = 0; fInd < missingIndices.size(); ++fInd) {
//             plint iPop = missingIndices[fInd];
//             Eigen::RowVectorXd lineA = Eigen::RowVectorXd::Zero(systSizeX);
//             computeMatrixRow(iPop, u, uSqr, lineA);
//             for (plint iVec = 0; iVec < systSizeX; ++iVec) sumA(iVec) += lineA(iVec);
//         }
//         
//         A.row(knownIndices.size()) = e0-sumA;
//         
//     }
// 
//     static void solveLinearSystemEigen(const Cell<T,Descriptor>& cell, 
//                                       const Array<T,Descriptor<T>::d> &u,
//                                       const std::vector<plint> &missingIndices,
//                                       const std::vector<plint> &knownIndices,
//                                       T &rho, 
//                                       Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq) {
//         Eigen::MatrixXd A;
//         Eigen::VectorXd b;
//         
//         createLinearSystem(cell, u, missingIndices, knownIndices, A, b);
//         
//         Eigen::VectorXd x;
//         
//         Eigen::MatrixXd AT = A.transpose();
//         A = AT * A;
//         b = AT * b;
//         
//         #ifdef PLB_DEBUG
//         bool solutionExists = A.lu().solve(b,&x);   // using a LU factorization
//         PLB_ASSERT(solutionExists);
//         #else
//         A.lu().solve(b,&x);
//         #endif
// 
//         fromXtoRhoAndPiNeq(x,rho,PiNeq);
//     }
    
    // ========= Methods used for the density BCs ============== //
    static void compute_f_diff_u_dir_and_PiNeq(plint iPop, T rho, const Array<T,Descriptor<T>::d> &u, 
                                               Eigen::RowVectorXd &df, plint dir) {
        T tcs2 = Descriptor<T>::invCs2* Descriptor<T>::t[iPop];
        T factor = 0.5 * tcs2 * Descriptor<T>::invCs2;
        Array<T,SymmetricTensor<T,Descriptor>::n> H2 = HermiteTemplate<T,Descriptor>::contractedOrder2(iPop);
        
        T diffFeqUmissing = T();
        plint iPi = 0;
        for (plint iA = 0; iA < Descriptor<T>::d; ++iA) {
            for (plint iB = iA; iB < Descriptor<T>::d; ++iB) {
                if (iA == dir || iB == dir) {
                    if (iA == iB) {
                        diffFeqUmissing += (T)2*H2[iPi]*u[dir];
                    }
                    else if (iA != dir) {
                        diffFeqUmissing += H2[iPi]*u[iA];
                    }
                    else if (iB != dir) {
                        diffFeqUmissing += H2[iPi]*u[iB];
                    }
                }
                df(iPi+1) = factor*H2[iPi];
                ++iPi;
            }
        }
        diffFeqUmissing *= factor;
        diffFeqUmissing += Descriptor<T>::c[iPop][dir]*tcs2;
        diffFeqUmissing *= rho;
        
        df(0) = diffFeqUmissing;
    }
    
    static void computeDiffF(plint iPop, T rho, const Array<T,Descriptor<T>::d> &u, plint dir, Eigen::RowVectorXd &df) {
        T tcs2 = Descriptor<T>::invCs2* Descriptor<T>::t[iPop];
        T factor = 0.5 * tcs2 * Descriptor<T>::invCs2;
        Array<T,SymmetricTensor<T,Descriptor>::n> H2 = HermiteTemplate<T,Descriptor>::contractedOrder2(iPop);
        
        T diffFeqUmissing = T();
        plint iPi = 0;
        for (plint iA = 0; iA < Descriptor<T>::d; ++iA) {
            for (plint iB = iA; iB < Descriptor<T>::d; ++iB) {
                if (iA == dir || iB == dir) {
                    if (iA == iB) {
                        diffFeqUmissing += (T)2*H2[iPi]*u[dir];
                    }
                    else if (iA != dir) {
                        diffFeqUmissing += H2[iPi]*u[iA];
                    }
                    else if (iB != dir) {
                        diffFeqUmissing += H2[iPi]*u[iB];
                    }
                }
                df(iPi+1) = factor*H2[iPi];
                ++iPi;
            }
        }
        diffFeqUmissing *= factor;
        diffFeqUmissing += Descriptor<T>::c[iPop][dir]*tcs2;
        diffFeqUmissing *= rho;
        
        df(0) = diffFeqUmissing;
    }
    
    static void computeJacobian(T rho,
                                const Array<T,Descriptor<T>::d> &u,
                                const Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq,
                                const plint dir, // direction of the unknown velocity
                                const std::vector<plint> &knownIndices,
                                Eigen::MatrixXd &Jac) {
        plint systSizeX = SymmetricTensor<T,Descriptor>::n+1;
        
        Eigen::RowVectorXd df = Eigen::RowVectorXd::Zero(systSizeX);
        for (pluint iPop = 0; iPop < knownIndices.size(); ++iPop) {
            computeDiffF(knownIndices[iPop], rho, u, dir, df);
            Jac.row(iPop) = df;
        }
    }
    
    static void computeNonLinearFunction(const Cell<T,Descriptor>& cell, 
                                         T rho,
                                         const Array<T,Descriptor<T>::d> &u,
                                         const T uSqr,
                                         const Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq,
                                         const plint dir, // direction of the unknown velocity
                                         const std::vector<plint> &knownIndices,
                                         Eigen::VectorXd &f) {
        T rhoBar = Descriptor<T>::rhoBar(rho);
        T invRho = Descriptor<T>::invRho(rhoBar);
        Array<T,Descriptor<T>::d> j = rho * u;
        T jSqr = rho*rho*uSqr;
        for (pluint iPop = 0; iPop < knownIndices.size(); ++iPop) {
            f(iPop) = cell[knownIndices[iPop]] - (
                dynamicsTemplates<T,Descriptor>::bgk_ma2_equilibrium(knownIndices[iPop], rhoBar, invRho, j, jSqr) +
                offEquilibriumTemplates<T,Descriptor>::fromPiToFneq(knownIndices[iPop],PiNeq) );
        }
    }
    
    static bool converge(Eigen::VectorXd &x,
                         Eigen::VectorXd &dx,
                         T epsilon)
    {
        for (plint iPi = 0; iPi < x.rows(); ++iPi) {
            T res = (fabs(x[iPi]) > 1.0e-14 ? fabs(dx(iPi)/x(iPi)) : fabs(x(iPi)));
            
            if (res > epsilon) return false;
        }
        return true;
    }
    
    static void iterativelySolveSystem(const Cell<T,Descriptor>& cell, 
                                       T rho,
                                       Array<T,Descriptor<T>::d> &u,
                                       Array<T,SymmetricTensor<T,Descriptor>::n> &PiNeq,
                                       const int dir, // direction of the unknown velocity
                                       const std::vector<plint> &knownIndices,
                                       T epsilon) {
        // u and PiNeq contain the initial guess for the solution of the system
        plint maxT = 10000;
        
        plint systSizeX = SymmetricTensor<T,Descriptor>::n+1;
        plint systSizeY = knownIndices.size();
        
        Eigen::VectorXd f  = Eigen::VectorXd::Zero(systSizeY); // stores the non-linear function
        Eigen::VectorXd x  = Eigen::VectorXd::Zero(systSizeX); // stores the variables (u[dir] and PiNeq)
        Eigen::VectorXd dx = Eigen::VectorXd::Zero(systSizeX); // contains delta_u[dir], delta_PiNeq (the increments towards the solution)
        
        fromUandPiNeqToX(u,PiNeq,x,dir);
        Eigen::MatrixXd Jac  = Eigen::MatrixXd::Zero(systSizeY,systSizeX);
        Eigen::MatrixXd JacT = Jac.transpose();
        for (plint iT = 0; iT < maxT; ++iT) {
            T uSqr = VectorTemplate<T,Descriptor>::normSqr(u);
            computeNonLinearFunction(cell,rho,u,uSqr,PiNeq,dir, knownIndices,f);
//             std::cout << iT << " " << f << std::endl << std::endl;
            computeJacobian(rho, u, PiNeq, dir, knownIndices, Jac);
            JacT = Jac.transpose();
            
            Eigen::MatrixXd JacSqr = JacT * Jac;
            Eigen::VectorXd JacTf = JacT * f;
            
            #ifdef PLB_DEBUG
//             bool solutionExists = JacSqr.lu().solve(JacTf,&dx);   // using a LU factorization
//             PLB_ASSERT(solutionExists);
            dx = JacSqr.fullPivLu().solve(JacTf);
            T relError = (JacSqr*dx - JacTf).norm() / JacTf.norm();
            PLB_ASSERT(relError < 1.0e-12);
            #else
            dx = JacSqr.fullPivLu().solve(JacTf);
//             JacSqr.lu().solve(JacTf,&dx);
//             dx = JacSqr.fullPivLu().solve(JacTf);
            #endif
            
            T stepMult = (T)1; // step size (step mult can only be <= 1 (usually = 1).
            x += stepMult*dx;  // increment solution
            fromXtoUandPiNeq(x,u,PiNeq,dir);
            if (converge(x,dx,epsilon)) {
//                 pcout << "Converged after " << iT << " iterations." << std::endl;
//                 std::cout << x << std::endl << std::endl;
                break;
            }
        }
//         pcout << "NEVER CONVERGED!!!." << std::endl;
    }

};  // struct generalizedIncomprBoundaryTemplates

}  // namespace plb

#endif  // GENERALIZED_BOUNDARY_DYNAMICS_HH