ex18.hpp

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//                  MFEM Example 18 - Serial/Parallel Shared Code

#include "mfem.hpp"

using namespace std;
using namespace mfem;

// Problem definition
extern int problem;

// Maximum characteristic speed (updated by integrators)
extern double max_char_speed;

extern const int num_equation;
extern const double specific_heat_ratio;
extern const double gas_constant;

// Time-dependent operator for the right-hand side of the ODE representing the
// DG weak form.
class FE_Evolution : public TimeDependentOperator
{
private:
   const int dim;

   FiniteElementSpace &vfes;
   Operator &A;
   SparseMatrix &Aflux;
   DenseTensor Me_inv;

   mutable Vector state;
   mutable DenseMatrix f;
   mutable DenseTensor flux;
   mutable Vector z;

   void GetFlux(const DenseMatrix &state, DenseTensor &flux) const;

public:
   FE_Evolution(FiniteElementSpace &vfes_,
                Operator &A_, SparseMatrix &Aflux_);

   virtual void Mult(const Vector &x, Vector &y) const;

   virtual ~FE_Evolution() { }
};

// Implements a simple Rusanov flux
class RiemannSolver
{
private:
   Vector flux1;
   Vector flux2;

public:
   RiemannSolver();
   double Eval(const Vector &state1, const Vector &state2,
               const Vector &nor, Vector &flux);
};


// Constant (in time) mixed bilinear form multiplying the flux grid function.
// The form is (vec(v), grad(w)) where the trial space = vector L2 space (mesh
// dim) and test space = scalar L2 space.
class DomainIntegrator : public BilinearFormIntegrator
{
private:
   Vector shape;
   DenseMatrix flux;
   DenseMatrix dshapedr;
   DenseMatrix dshapedx;

public:
   DomainIntegrator(const int dim);

   virtual void AssembleElementMatrix2(const FiniteElement &trial_fe,
                                       const FiniteElement &test_fe,
                                       ElementTransformation &Tr,
                                       DenseMatrix &elmat);
};

// Interior face term: <F.n(u),[w]>
class FaceIntegrator : public NonlinearFormIntegrator
{
private:
   RiemannSolver rsolver;
   Vector shape1;
   Vector shape2;
   Vector funval1;
   Vector funval2;
   Vector nor;
   Vector fluxN;

public:
   FaceIntegrator(RiemannSolver &rsolver_, const int dim);

   virtual void AssembleFaceVector(const FiniteElement &el1,
                                   const FiniteElement &el2,
                                   FaceElementTransformations &Tr,
                                   const Vector &elfun, Vector &elvect);
};

// Implementation of class FE_Evolution
FE_Evolution::FE_Evolution(FiniteElementSpace &vfes_,
                           Operator &A_, SparseMatrix &Aflux_)
   : TimeDependentOperator(A_.Height()),
     dim(vfes_.GetFE(0)->GetDim()),
     vfes(vfes_),
     A(A_),
     Aflux(Aflux_),
     Me_inv(vfes.GetFE(0)->GetDof(), vfes.GetFE(0)->GetDof(), vfes.GetNE()),
     state(num_equation),
     f(num_equation, dim),
     flux(vfes.GetNDofs(), dim, num_equation),
     z(A.Height())
{
   // Standard local assembly and inversion for energy mass matrices.
   const int dof = vfes.GetFE(0)->GetDof();
   DenseMatrix Me(dof);
   DenseMatrixInverse inv(&Me);
   MassIntegrator mi;
   for (int i = 0; i < vfes.GetNE(); i++)
   {
      mi.AssembleElementMatrix(*vfes.GetFE(i), *vfes.GetElementTransformation(i), Me);
      inv.Factor();
      inv.GetInverseMatrix(Me_inv(i));
   }
}

void FE_Evolution::Mult(const Vector &x, Vector &y) const
{
   // 0. Reset wavespeed computation before operator application.
   max_char_speed = 0.;

   // 1. Create the vector z with the face terms -<F.n(u), [w]>.
   A.Mult(x, z);

   // 2. Add the element terms.
   // i.  computing the flux approximately as a grid function by interpolating
   //     at the solution nodes.
   // ii. multiplying this grid function by a (constant) mixed bilinear form for
   //     each of the num_equation, computing (F(u), grad(w)) for each equation.

   DenseMatrix xmat(x.GetData(), vfes.GetNDofs(), num_equation);
   GetFlux(xmat, flux);

   for (int k = 0; k < num_equation; k++)
   {
      Vector fk(flux(k).GetData(), dim * vfes.GetNDofs());
      Vector zk(z.GetData() + k * vfes.GetNDofs(), vfes.GetNDofs());
      Aflux.AddMult(fk, zk);
   }

   // 3. Multiply element-wise by the inverse mass matrices.
   Vector zval;
   Array<int> vdofs;
   const int dof = vfes.GetFE(0)->GetDof();
   DenseMatrix zmat, ymat(dof, num_equation);

   for (int i = 0; i < vfes.GetNE(); i++)
   {
      // Return the vdofs ordered byNODES
      vfes.GetElementVDofs(i, vdofs);
      z.GetSubVector(vdofs, zval);
      zmat.UseExternalData(zval.GetData(), dof, num_equation);
      mfem::Mult(Me_inv(i), zmat, ymat);
      y.SetSubVector(vdofs, ymat.GetData());
   }
}

// Physicality check (at end)
bool StateIsPhysical(const Vector &state, const int dim);

// Pressure (EOS) computation
inline double ComputePressure(const Vector &state, int dim)
{
   const double den = state(0);
   const Vector den_vel(state.GetData() + 1, dim);
   const double den_energy = state(1 + dim);

   double den_vel2 = 0;
   for (int d = 0; d < dim; d++) { den_vel2 += den_vel(d) * den_vel(d); }
   den_vel2 /= den;

   return (specific_heat_ratio - 1.0) * (den_energy - 0.5 * den_vel2);
}

// Compute the vector flux F(u)
void ComputeFlux(const Vector &state, int dim, DenseMatrix &flux)
{
   const double den = state(0);
   const Vector den_vel(state.GetData() + 1, dim);
   const double den_energy = state(1 + dim);

   MFEM_ASSERT(StateIsPhysical(state, dim), "");

   const double pres = ComputePressure(state, dim);

   for (int d = 0; d < dim; d++)
   {
      flux(0, d) = den_vel(d);
      for (int i = 0; i < dim; i++)
      {
         flux(1+i, d) = den_vel(i) * den_vel(d) / den;
      }
      flux(1+d, d) += pres;
   }

   const double H = (den_energy + pres) / den;
   for (int d = 0; d < dim; d++)
   {
      flux(1+dim, d) = den_vel(d) * H;
   }
}

// Compute the scalar F(u).n
void ComputeFluxDotN(const Vector &state, const Vector &nor,
                     Vector &fluxN)
{
   // NOTE: nor in general is not a unit normal
   const int dim = nor.Size();
   const double den = state(0);
   const Vector den_vel(state.GetData() + 1, dim);
   const double den_energy = state(1 + dim);

   MFEM_ASSERT(StateIsPhysical(state, dim), "");

   const double pres = ComputePressure(state, dim);

   double den_velN = 0;
   for (int d = 0; d < dim; d++) { den_velN += den_vel(d) * nor(d); }

   fluxN(0) = den_velN;
   for (int d = 0; d < dim; d++)
   {
      fluxN(1+d) = den_velN * den_vel(d) / den + pres * nor(d);
   }

   const double H = (den_energy + pres) / den;
   fluxN(1 + dim) = den_velN * H;
}

// Compute the maximum characteristic speed.
inline double ComputeMaxCharSpeed(const Vector &state, const int dim)
{
   const double den = state(0);
   const Vector den_vel(state.GetData() + 1, dim);

   double den_vel2 = 0;
   for (int d = 0; d < dim; d++) { den_vel2 += den_vel(d) * den_vel(d); }
   den_vel2 /= den;

   const double pres = ComputePressure(state, dim);
   const double sound = sqrt(specific_heat_ratio * pres / den);
   const double vel = sqrt(den_vel2 / den);

   return vel + sound;
}

// Compute the flux at solution nodes.
void FE_Evolution::GetFlux(const DenseMatrix &x, DenseTensor &flux) const
{
   const int dof = flux.SizeI();
   const int dim = flux.SizeJ();

   for (int i = 0; i < dof; i++)
   {
      for (int k = 0; k < num_equation; k++) { state(k) = x(i, k); }
      ComputeFlux(state, dim, f);

      for (int d = 0; d < dim; d++)
      {
         for (int k = 0; k < num_equation; k++)
         {
            flux(i, d, k) = f(k, d);
         }
      }

      // Update max char speed
      const double mcs = ComputeMaxCharSpeed(state, dim);
      if (mcs > max_char_speed) { max_char_speed = mcs; }
   }
}

// Implementation of class RiemannSolver
RiemannSolver::RiemannSolver() :
   flux1(num_equation),
   flux2(num_equation) { }

double RiemannSolver::Eval(const Vector &state1, const Vector &state2,
                           const Vector &nor, Vector &flux)
{
   // NOTE: nor in general is not a unit normal
   const int dim = nor.Size();

   MFEM_ASSERT(StateIsPhysical(state1, dim), "");
   MFEM_ASSERT(StateIsPhysical(state2, dim), "");

   const double maxE1 = ComputeMaxCharSpeed(state1, dim);
   const double maxE2 = ComputeMaxCharSpeed(state2, dim);

   const double maxE = max(maxE1, maxE2);

   ComputeFluxDotN(state1, nor, flux1);
   ComputeFluxDotN(state2, nor, flux2);

   double normag = 0;
   for (int i = 0; i < dim; i++)
   {
      normag += nor(i) * nor(i);
   }
   normag = sqrt(normag);

   for (int i = 0; i < num_equation; i++)
   {
      flux(i) = 0.5 * (flux1(i) + flux2(i))
                - 0.5 * maxE * (state2(i) - state1(i)) * normag;
   }

   return maxE;
}

// Implementation of class DomainIntegrator
DomainIntegrator::DomainIntegrator(const int dim) : flux(num_equation, dim) { }

void DomainIntegrator::AssembleElementMatrix2(const FiniteElement &trial_fe,
                                              const FiniteElement &test_fe,
                                              ElementTransformation &Tr,
                                              DenseMatrix &elmat)
{
   // Assemble the form (vec(v), grad(w))

   // Trial space = vector L2 space (mesh dim)
   // Test space  = scalar L2 space

   const int dof_trial = trial_fe.GetDof();
   const int dof_test = test_fe.GetDof();
   const int dim = trial_fe.GetDim();

   shape.SetSize(dof_trial);
   dshapedr.SetSize(dof_test, dim);
   dshapedx.SetSize(dof_test, dim);

   elmat.SetSize(dof_test, dof_trial * dim);
   elmat = 0.0;

   const int maxorder = max(trial_fe.GetOrder(), test_fe.GetOrder());
   const int intorder = 2 * maxorder;
   const IntegrationRule *ir = &IntRules.Get(trial_fe.GetGeomType(), intorder);

   for (int i = 0; i < ir->GetNPoints(); i++)
   {
      const IntegrationPoint &ip = ir->IntPoint(i);

      // Calculate the shape functions
      trial_fe.CalcShape(ip, shape);
      shape *= ip.weight;

      // Compute the physical gradients of the test functions
      Tr.SetIntPoint(&ip);
      test_fe.CalcDShape(ip, dshapedr);
      Mult(dshapedr, Tr.AdjugateJacobian(), dshapedx);

      for (int d = 0; d < dim; d++)
      {
         for (int j = 0; j < dof_test; j++)
         {
            for (int k = 0; k < dof_trial; k++)
            {
               elmat(j, k + d * dof_trial) += shape(k) * dshapedx(j, d);
            }
         }
      }
   }
}

// Implementation of class FaceIntegrator
FaceIntegrator::FaceIntegrator(RiemannSolver &rsolver_, const int dim) :
   rsolver(rsolver_),
   funval1(num_equation),
   funval2(num_equation),
   nor(dim),
   fluxN(num_equation) { }

void FaceIntegrator::AssembleFaceVector(const FiniteElement &el1,
                                        const FiniteElement &el2,
                                        FaceElementTransformations &Tr,
                                        const Vector &elfun, Vector &elvect)
{
   // Compute the term <F.n(u),[w]> on the interior faces.
   const int dof1 = el1.GetDof();
   const int dof2 = el2.GetDof();

   shape1.SetSize(dof1);
   shape2.SetSize(dof2);

   elvect.SetSize((dof1 + dof2) * num_equation);
   elvect = 0.0;

   DenseMatrix elfun1_mat(elfun.GetData(), dof1, num_equation);
   DenseMatrix elfun2_mat(elfun.GetData() + dof1 * num_equation, dof2,
                          num_equation);

   DenseMatrix elvect1_mat(elvect.GetData(), dof1, num_equation);
   DenseMatrix elvect2_mat(elvect.GetData() + dof1 * num_equation, dof2,
                           num_equation);

   // Integration order calculation from DGTraceIntegrator
   int intorder;
   if (Tr.Elem2No >= 0)
      intorder = (min(Tr.Elem1->OrderW(), Tr.Elem2->OrderW()) +
                  2*max(el1.GetOrder(), el2.GetOrder()));
   else
   {
      intorder = Tr.Elem1->OrderW() + 2*el1.GetOrder();
   }
   if (el1.Space() == FunctionSpace::Pk)
   {
      intorder++;
   }
   const IntegrationRule *ir = &IntRules.Get(Tr.GetGeometryType(), intorder);

   for (int i = 0; i < ir->GetNPoints(); i++)
   {
      const IntegrationPoint &ip = ir->IntPoint(i);

      Tr.SetAllIntPoints(&ip); // set face and element int. points

      // Calculate basis functions on both elements at the face
      el1.CalcShape(Tr.GetElement1IntPoint(), shape1);
      el2.CalcShape(Tr.GetElement2IntPoint(), shape2);

      // Interpolate elfun at the point
      elfun1_mat.MultTranspose(shape1, funval1);
      elfun2_mat.MultTranspose(shape2, funval2);

      // Get the normal vector and the flux on the face
      CalcOrtho(Tr.Jacobian(), nor);
      const double mcs = rsolver.Eval(funval1, funval2, nor, fluxN);

      // Update max char speed
      if (mcs > max_char_speed) { max_char_speed = mcs; }

      fluxN *= ip.weight;
      for (int k = 0; k < num_equation; k++)
      {
         for (int s = 0; s < dof1; s++)
         {
            elvect1_mat(s, k) -= fluxN(k) * shape1(s);
         }
         for (int s = 0; s < dof2; s++)
         {
            elvect2_mat(s, k) += fluxN(k) * shape2(s);
         }
      }
   }
}

// Check that the state is physical - enabled in debug mode
bool StateIsPhysical(const Vector &state, const int dim)
{
   const double den = state(0);
   const Vector den_vel(state.GetData() + 1, dim);
   const double den_energy = state(1 + dim);

   if (den < 0)
   {
      cout << "Negative density: ";
      for (int i = 0; i < state.Size(); i++)
      {
         cout << state(i) << " ";
      }
      cout << endl;
      return false;
   }
   if (den_energy <= 0)
   {
      cout << "Negative energy: ";
      for (int i = 0; i < state.Size(); i++)
      {
         cout << state(i) << " ";
      }
      cout << endl;
      return false;
   }

   double den_vel2 = 0;
   for (int i = 0; i < dim; i++) { den_vel2 += den_vel(i) * den_vel(i); }
   den_vel2 /= den;

   const double pres = (specific_heat_ratio - 1.0) * (den_energy - 0.5 * den_vel2);

   if (pres <= 0)
   {
      cout << "Negative pressure: " << pres << ", state: ";
      for (int i = 0; i < state.Size(); i++)
      {
         cout << state(i) << " ";
      }
      cout << endl;
      return false;
   }
   return true;
}

// Initial condition
void InitialCondition(const Vector &x, Vector &y)
{
   MFEM_ASSERT(x.Size() == 2, "");

   double radius = 0, Minf = 0, beta = 0;
   if (problem == 1)
   {
      // "Fast vortex"
      radius = 0.2;
      Minf = 0.5;
      beta = 1. / 5.;
   }
   else if (problem == 2)
   {
      // "Slow vortex"
      radius = 0.2;
      Minf = 0.05;
      beta = 1. / 50.;
   }
   else
   {
      mfem_error("Cannot recognize problem."
                 "Options are: 1 - fast vortex, 2 - slow vortex");
   }

   const double xc = 0.0, yc = 0.0;

   // Nice units
   const double vel_inf = 1.;
   const double den_inf = 1.;

   // Derive remainder of background state from this and Minf
   const double pres_inf = (den_inf / specific_heat_ratio) * (vel_inf / Minf) *
                           (vel_inf / Minf);
   const double temp_inf = pres_inf / (den_inf * gas_constant);

   double r2rad = 0.0;
   r2rad += (x(0) - xc) * (x(0) - xc);
   r2rad += (x(1) - yc) * (x(1) - yc);
   r2rad /= (radius * radius);

   const double shrinv1 = 1.0 / (specific_heat_ratio - 1.);

   const double velX = vel_inf * (1 - beta * (x(1) - yc) / radius * exp(
                                     -0.5 * r2rad));
   const double velY = vel_inf * beta * (x(0) - xc) / radius * exp(-0.5 * r2rad);
   const double vel2 = velX * velX + velY * velY;

   const double specific_heat = gas_constant * specific_heat_ratio * shrinv1;
   const double temp = temp_inf - 0.5 * (vel_inf * beta) *
                       (vel_inf * beta) / specific_heat * exp(-r2rad);

   const double den = den_inf * pow(temp/temp_inf, shrinv1);
   const double pres = den * gas_constant * temp;
   const double energy = shrinv1 * pres / den + 0.5 * vel2;

   y(0) = den;
   y(1) = den * velX;
   y(2) = den * velY;
   y(3) = den * energy;
}