fluids/qfunctions/advection.h

// Copyright (c) 2017-2022, Lawrence Livermore National Security, LLC and other CEED contributors.
// All Rights Reserved. See the top-level LICENSE and NOTICE files for details.
//
// SPDX-License-Identifier: BSD-2-Clause
//
// This file is part of CEED:  http://github.com/ceed

/// @file
/// Advection initial condition and operator for Navier-Stokes example using PETSc

#ifndef advection_h
#define advection_h

#include <math.h>

typedef struct SetupContext_ *SetupContext;
struct SetupContext_ {
  CeedScalar rc;
  CeedScalar lx;
  CeedScalar ly;
  CeedScalar lz;
  CeedScalar wind[3];
  CeedScalar time;
  int wind_type;              // See WindType: 0=ROTATION, 1=TRANSLATION
  int bubble_type;            // See BubbleType: 0=SPHERE, 1=CYLINDER
  int bubble_continuity_type; // See BubbleContinuityType: 0=SMOOTH, 1=BACK_SHARP 2=THICK
};

typedef struct AdvectionContext_ *AdvectionContext;
struct AdvectionContext_ {
  CeedScalar CtauS;
  CeedScalar strong_form;
  CeedScalar E_wind;
  bool implicit;
  int stabilization; // See StabilizationType: 0=none, 1=SU, 2=SUPG
};

CEED_QFUNCTION_HELPER CeedScalar Square(CeedScalar x) { return x*x; }

// *****************************************************************************
// This QFunction sets the initial conditions and the boundary conditions
//   for two test cases: ROTATION and TRANSLATION
//
// -- ROTATION (default)
//      Initial Conditions:
//        Mass Density:
//          Constant mass density of 1.0
//        Momentum Density:
//          Rotational field in x,y
//        Energy Density:
//          Maximum of 1. x0 decreasing linearly to 0. as radial distance
//            increases to (1.-r/rc), then 0. everywhere else
//
//      Boundary Conditions:
//        Mass Density:
//          0.0 flux
//        Momentum Density:
//          0.0
//        Energy Density:
//          0.0 flux
//
// -- TRANSLATION
//      Initial Conditions:
//        Mass Density:
//          Constant mass density of 1.0
//        Momentum Density:
//           Constant rectilinear field in x,y
//        Energy Density:
//          Maximum of 1. x0 decreasing linearly to 0. as radial distance
//            increases to (1.-r/rc), then 0. everywhere else
//
//      Boundary Conditions:
//        Mass Density:
//          0.0 flux
//        Momentum Density:
//          0.0
//        Energy Density:
//          Inflow BCs:
//            E = E_wind
//          Outflow BCs:
//            E = E(boundary)
//          Both In/Outflow BCs for E are applied weakly in the
//            QFunction "Advection_Sur"
//
// *****************************************************************************

// *****************************************************************************
// This helper function provides support for the exact, time-dependent solution
//   (currently not implemented) and IC formulation for 3D advection
// *****************************************************************************
CEED_QFUNCTION_HELPER int Exact_Advection(CeedInt dim, CeedScalar time,
    const CeedScalar X[], CeedInt Nf, CeedScalar q[], void *ctx) {
  const SetupContext context = (SetupContext)ctx;
  const CeedScalar rc    = context->rc;
  const CeedScalar lx    = context->lx;
  const CeedScalar ly    = context->ly;
  const CeedScalar lz    = context->lz;
  const CeedScalar *wind = context->wind;

  // Setup
  const CeedScalar x0[3] = {0.25*lx, 0.5*ly, 0.5*lz};
  const CeedScalar center[3] = {0.5*lx, 0.5*ly, 0.5*lz};

  // -- Coordinates
  const CeedScalar x = X[0];
  const CeedScalar y = X[1];
  const CeedScalar z = X[2];

  // -- Energy
  CeedScalar r = 0.;
  switch (context->bubble_type) {
  //  original sphere
  case 0: { // (dim=3)
    r = sqrt(Square(x - x0[0]) +
             Square(y - x0[1]) +
             Square(z - x0[2]));
  } break;
  // cylinder (needs periodicity to work properly)
  case 1: { // (dim=2)
    r = sqrt(Square(x - x0[0]) + Square(y - x0[1]));
  } break;
  }

  // Initial Conditions
  switch (context->wind_type) {
  case 0:    // Rotation
    q[0] = 1.;
    q[1] = -(y - center[1]);
    q[2] =  (x - center[0]);
    q[3] = 0;
    break;
  case 1:    // Translation
    q[0] = 1.;
    q[1] = wind[0];
    q[2] = wind[1];
    q[3] = wind[2];
    break;
  }

  switch (context->bubble_continuity_type) {
  // original continuous, smooth shape
  case 0: {
    q[4] = r <= rc ? (1.-r/rc) : 0.;
  } break;
  // discontinuous, sharp back half shape
  case 1: {
    q[4] = ((r <= rc) && (y<center[1])) ? (1.-r/rc) : 0.;
  } break;
  // attempt to define a finite thickness that will get resolved under grid refinement
  case 2: {
    q[4] = ((r <= rc)
            && (y<center[1])) ? (1.-r/rc)*fmin(1.0,(center[1]-y)/1.25) : 0.;
  } break;
  }
  return 0;
}

// *****************************************************************************
// This QFunction sets the initial conditions for 3D advection
// *****************************************************************************
CEED_QFUNCTION(ICsAdvection)(void *ctx, CeedInt Q,
                             const CeedScalar *const *in,
                             CeedScalar *const *out) {
  // Inputs
  const CeedScalar (*X)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0];
  // Outputs
  CeedScalar (*q0)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0];

  CeedPragmaSIMD
  // Quadrature Point Loop
  for (CeedInt i=0; i<Q; i++) {
    const CeedScalar x[] = {X[0][i], X[1][i], X[2][i]};
    CeedScalar q[5] = {0.};

    Exact_Advection(3, 0., x, 5, q, ctx);
    for (CeedInt j=0; j<5; j++) q0[j][i] = q[j];
  } // End of Quadrature Point Loop

  // Return
  return 0;
}

// *****************************************************************************
// This QFunction implements the following formulation of the advection equation
//
// This is 3D advection given in two formulations based upon the weak form.
//
// State Variables: q = ( rho, U1, U2, U3, E )
//   rho - Mass Density
//   Ui  - Momentum Density    ,  Ui = rho ui
//   E   - Total Energy Density
//
// Advection Equation:
//   dE/dt + div( E u ) = 0
//
// *****************************************************************************
CEED_QFUNCTION(Advection)(void *ctx, CeedInt Q,
                          const CeedScalar *const *in, CeedScalar *const *out) {
  // Inputs
  // *INDENT-OFF*
  const CeedScalar (*q)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0],
                   (*dq)[5][CEED_Q_VLA] = (const CeedScalar(*)[5][CEED_Q_VLA])in[1],
                   (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[2];

  // Outputs
  CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0],
             (*dv)[5][CEED_Q_VLA] = (CeedScalar(*)[5][CEED_Q_VLA])out[1];
  // *INDENT-ON*

  // Context
  AdvectionContext context = (AdvectionContext)ctx;
  const CeedScalar CtauS       = context->CtauS;
  const CeedScalar strong_form = context->strong_form;

  CeedPragmaSIMD
  // Quadrature Point Loop
  for (CeedInt i=0; i<Q; i++) {
    // Setup
    // -- Interp in
    const CeedScalar rho        =    q[0][i];
    const CeedScalar u[3]       =   {q[1][i] / rho,
                                     q[2][i] / rho,
                                     q[3][i] / rho
                                    };
    const CeedScalar E          =    q[4][i];
    // -- Grad in
    const CeedScalar drho[3]    =   {dq[0][0][i],
                                     dq[1][0][i],
                                     dq[2][0][i]
                                    };
    // *INDENT-OFF*
    const CeedScalar du[3][3]   = {{(dq[0][1][i] - drho[0]*u[0]) / rho,
                                    (dq[1][1][i] - drho[1]*u[0]) / rho,
                                    (dq[2][1][i] - drho[2]*u[0]) / rho},
                                   {(dq[0][2][i] - drho[0]*u[1]) / rho,
                                    (dq[1][2][i] - drho[1]*u[1]) / rho,
                                    (dq[2][2][i] - drho[2]*u[1]) / rho},
                                   {(dq[0][3][i] - drho[0]*u[2]) / rho,
                                    (dq[1][3][i] - drho[1]*u[2]) / rho,
                                    (dq[2][3][i] - drho[2]*u[2]) / rho}
                                  };
    // *INDENT-ON*
    const CeedScalar dE[3]      =   {dq[0][4][i],
                                     dq[1][4][i],
                                     dq[2][4][i]
                                    };
    // -- Interp-to-Interp q_data
    const CeedScalar wdetJ      =    q_data[0][i];
    // -- Interp-to-Grad q_data
    // ---- Inverse of change of coordinate matrix: X_i,j
    // *INDENT-OFF*
    const CeedScalar dXdx[3][3] =  {{q_data[1][i],
                                     q_data[2][i],
                                     q_data[3][i]},
                                    {q_data[4][i],
                                     q_data[5][i],
                                     q_data[6][i]},
                                    {q_data[7][i],
                                     q_data[8][i],
                                     q_data[9][i]}
                                   };
    // *INDENT-ON*
    // The Physics
    // Note with the order that du was filled and the order that dXdx was filled
    //   du[j][k]= du_j / dX_K    (note cap K to be clear this is u_{j,xi_k})
    //   dXdx[k][j] = dX_K / dx_j
    //   X_K=Kth reference element coordinate (note cap X and K instead of xi_k}
    //   x_j and u_j are jth  physical position and velocity components

    // No Change in density or momentum
    for (CeedInt f=0; f<4; f++) {
      for (CeedInt j=0; j<3; j++)
        dv[j][f][i] = 0;
      v[f][i] = 0;
    }

    // -- Total Energy
    // Evaluate the strong form using div(E u) = u . grad(E) + E div(u)
    // or in index notation: (u_j E)_{,j} = u_j E_j + E u_{j,j}
    CeedScalar div_u = 0, u_dot_grad_E = 0;
    for (CeedInt j=0; j<3; j++) {
      CeedScalar dEdx_j = 0;
      for (CeedInt k=0; k<3; k++) {
        div_u += du[j][k] * dXdx[k][j]; // u_{j,j} = u_{j,K} X_{K,j}
        dEdx_j += dE[k] * dXdx[k][j];
      }
      u_dot_grad_E += u[j] * dEdx_j;
    }
    CeedScalar strong_conv = E*div_u + u_dot_grad_E;

    // Weak Galerkin convection term: dv \cdot (E u)
    for (CeedInt j=0; j<3; j++)
      dv[j][4][i] = (1 - strong_form) * wdetJ * E * (u[0]*dXdx[j][0] +
                    u[1]*dXdx[j][1] +
                    u[2]*dXdx[j][2]);
    v[4][i] = 0;

    // Strong Galerkin convection term: - v div(E u)
    v[4][i] = -strong_form * wdetJ * strong_conv;

    // Stabilization requires a measure of element transit time in the velocity
    //   field u.
    CeedScalar uX[3];
    for (CeedInt j=0; j<3;
         j++) uX[j] = dXdx[j][0]*u[0] + dXdx[j][1]*u[1] + dXdx[j][2]*u[2];
    const CeedScalar TauS = CtauS / sqrt(uX[0]*uX[0] + uX[1]*uX[1] + uX[2]*uX[2]);
    for (CeedInt j=0; j<3; j++)
      dv[j][4][i] -= wdetJ * TauS * strong_conv * uX[j];
  } // End Quadrature Point Loop

  return 0;
}

// *****************************************************************************
// This QFunction implements 3D (mentioned above) with
//   implicit time stepping method
//
// *****************************************************************************
CEED_QFUNCTION(IFunction_Advection)(void *ctx, CeedInt Q,
                                    const CeedScalar *const *in,
                                    CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*q)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0],
                   (*dq)[5][CEED_Q_VLA] = (const CeedScalar(*)[5][CEED_Q_VLA])in[1],
                   (*q_dot)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[2],
                   (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[3];
  // Outputs
  CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0],
             (*dv)[5][CEED_Q_VLA] = (CeedScalar(*)[5][CEED_Q_VLA])out[1];
  // *INDENT-ON*
  AdvectionContext context = (AdvectionContext)ctx;
  const CeedScalar CtauS       = context->CtauS;
  const CeedScalar strong_form = context->strong_form;

  CeedPragmaSIMD
  // Quadrature Point Loop
  for (CeedInt i=0; i<Q; i++) {
    // Setup
    // -- Interp in
    const CeedScalar rho        =    q[0][i];
    const CeedScalar u[3]       =   {q[1][i] / rho,
                                     q[2][i] / rho,
                                     q[3][i] / rho
                                    };
    const CeedScalar E          =    q[4][i];
    // -- Grad in
    const CeedScalar drho[3]    =   {dq[0][0][i],
                                     dq[1][0][i],
                                     dq[2][0][i]
                                    };
    // *INDENT-OFF*
    const CeedScalar du[3][3]   = {{(dq[0][1][i] - drho[0]*u[0]) / rho,
                                    (dq[1][1][i] - drho[1]*u[0]) / rho,
                                    (dq[2][1][i] - drho[2]*u[0]) / rho},
                                   {(dq[0][2][i] - drho[0]*u[1]) / rho,
                                    (dq[1][2][i] - drho[1]*u[1]) / rho,
                                    (dq[2][2][i] - drho[2]*u[1]) / rho},
                                   {(dq[0][3][i] - drho[0]*u[2]) / rho,
                                    (dq[1][3][i] - drho[1]*u[2]) / rho,
                                    (dq[2][3][i] - drho[2]*u[2]) / rho}
                                  };
    // *INDENT-ON*
    const CeedScalar dE[3]      =   {dq[0][4][i],
                                     dq[1][4][i],
                                     dq[2][4][i]
                                    };
    // -- Interp-to-Interp q_data
    const CeedScalar wdetJ      =    q_data[0][i];
    // -- Interp-to-Grad q_data
    // ---- Inverse of change of coordinate matrix: X_i,j
    // *INDENT-OFF*
    const CeedScalar dXdx[3][3] =  {{q_data[1][i],
                                     q_data[2][i],
                                     q_data[3][i]},
                                    {q_data[4][i],
                                     q_data[5][i],
                                     q_data[6][i]},
                                    {q_data[7][i],
                                     q_data[8][i],
                                     q_data[9][i]}
                                   };
    // *INDENT-ON*
    // The Physics
    // Note with the order that du was filled and the order that dXdx was filled
    //   du[j][k]= du_j / dX_K    (note cap K to be clear this is u_{j,xi_k} )
    //   dXdx[k][j] = dX_K / dx_j
    //   X_K=Kth reference element coordinate (note cap X and K instead of xi_k}
    //   x_j and u_j are jth  physical position and velocity components

    // No Change in density or momentum
    for (CeedInt f=0; f<4; f++) {
      for (CeedInt j=0; j<3; j++)
        dv[j][f][i] = 0;
      v[f][i] = wdetJ * q_dot[f][i]; //K Mass/transient term
    }

    // -- Total Energy
    // Evaluate the strong form using div(E u) = u . grad(E) + E div(u)
    //   or in index notation: (u_j E)_{,j} = u_j E_j + E u_{j,j}
    CeedScalar div_u = 0, u_dot_grad_E = 0;
    for (CeedInt j=0; j<3; j++) {
      CeedScalar dEdx_j = 0;
      for (CeedInt k=0; k<3; k++) {
        div_u += du[j][k] * dXdx[k][j]; // u_{j,j} = u_{j,K} X_{K,j}
        dEdx_j += dE[k] * dXdx[k][j];
      }
      u_dot_grad_E += u[j] * dEdx_j;
    }
    CeedScalar strong_conv = E*div_u + u_dot_grad_E;
    CeedScalar strong_res = q_dot[4][i] + strong_conv;

    v[4][i] = wdetJ * q_dot[4][i]; // transient part (ALWAYS)

    // Weak Galerkin convection term: -dv \cdot (E u)
    for (CeedInt j=0; j<3; j++)
      dv[j][4][i] = -wdetJ * (1 - strong_form) * E * (u[0]*dXdx[j][0] +
                    u[1]*dXdx[j][1] +
                    u[2]*dXdx[j][2]);

    // Strong Galerkin convection term: v div(E u)
    v[4][i] += wdetJ * strong_form * strong_conv;

    // Stabilization requires a measure of element transit time in the velocity
    //   field u.
    CeedScalar uX[3];
    for (CeedInt j=0; j<3;
         j++) uX[j] = dXdx[j][0]*u[0] + dXdx[j][1]*u[1] + dXdx[j][2]*u[2];
    const CeedScalar TauS = CtauS / sqrt(uX[0]*uX[0] + uX[1]*uX[1] + uX[2]*uX[2]);

    for (CeedInt j=0; j<3; j++)
      switch (context->stabilization) {
      case 0:
        break;
      case 1: dv[j][4][i] += wdetJ * TauS * strong_conv * uX[j];  //SU
        break;
      case 2: dv[j][4][i] += wdetJ * TauS * strong_res * uX[j];  //SUPG
        break;
      }
  } // End Quadrature Point Loop

  return 0;
}

// *****************************************************************************
// This QFunction implements consistent outflow and inflow BCs
//      for 3D advection
//
//  Inflow and outflow faces are determined based on sign(dot(wind, normal)):
//    sign(dot(wind, normal)) > 0 : outflow BCs
//    sign(dot(wind, normal)) < 0 : inflow BCs
//
//  Outflow BCs:
//    The validity of the weak form of the governing equations is extended
//    to the outflow and the current values of E are applied.
//
//  Inflow BCs:
//    A prescribed Total Energy (E_wind) is applied weakly.
//
// *****************************************************************************
CEED_QFUNCTION(Advection_InOutFlow)(void *ctx, CeedInt Q,
                                    const CeedScalar *const *in,
                                    CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*q)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0],
                   (*q_data_sur)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[1];
  // Outputs
  CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0];
  // *INDENT-ON*
  AdvectionContext context = (AdvectionContext)ctx;
  const CeedScalar E_wind      = context->E_wind;
  const CeedScalar strong_form = context->strong_form;
  const bool implicit          = context->implicit;

  CeedPragmaSIMD
  // Quadrature Point Loop
  for (CeedInt i=0; i<Q; i++) {
    // Setup
    // -- Interp in
    const CeedScalar rho        =    q[0][i];
    const CeedScalar u[3]       =   {q[1][i] / rho,
                                     q[2][i] / rho,
                                     q[3][i] / rho
                                    };
    const CeedScalar E          =    q[4][i];

    // -- Interp-to-Interp q_data
    // For explicit mode, the surface integral is on the RHS of ODE q_dot = f(q).
    // For implicit mode, it gets pulled to the LHS of implicit ODE/DAE g(q_dot, q).
    // We can effect this by swapping the sign on this weight
    const CeedScalar wdetJb     =   (implicit ? -1. : 1.) * q_data_sur[0][i];

    // ---- Normal vectors
    const CeedScalar norm[3]    =   {q_data_sur[1][i],
                                     q_data_sur[2][i],
                                     q_data_sur[3][i]
                                    };
    // Normal velocity
    const CeedScalar u_normal = norm[0]*u[0] + norm[1]*u[1] + norm[2]*u[2];

    // No Change in density or momentum
    for (CeedInt j=0; j<4; j++) {
      v[j][i] = 0;
    }
    // Implementing in/outflow BCs
    if (u_normal > 0) { // outflow
      v[4][i] = -(1 - strong_form) * wdetJb * E * u_normal;
    } else { // inflow
      v[4][i] = -(1 - strong_form) * wdetJb * E_wind * u_normal;
    }
  } // End Quadrature Point Loop
  return 0;
}
// *****************************************************************************

#endif // advection_h