xref: /libCEED/examples/solids/qfunctions/linear.h (revision 3d8e882215d238700cdceb37404f76ca7fa24eaa)

// 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
/// Linear elasticity for solid mechanics example using PETSc

#ifndef ELAS_LINEAR_H
#define ELAS_LINEAR_H

#include <math.h>

#ifndef PHYSICS_STRUCT
#define PHYSICS_STRUCT
typedef struct Physics_private *Physics;
struct Physics_private {
  CeedScalar   nu;      // Poisson's ratio
  CeedScalar   E;       // Young's Modulus
};
#endif

// -----------------------------------------------------------------------------
// Residual evaluation for linear elasticity
// -----------------------------------------------------------------------------
CEED_QFUNCTION(ElasLinearF)(void *ctx, CeedInt Q, const CeedScalar *const *in,
                            CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*ug)[3][CEED_Q_VLA] = (const CeedScalar(*)[3][CEED_Q_VLA])in[0],
                   (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[1];

  // Outputs
  CeedScalar (*dvdX)[3][CEED_Q_VLA] = (CeedScalar(*)[3][CEED_Q_VLA])out[0];
             // grad_u not used for linear elasticity
             // (*grad_u)[3][CEED_Q_VLA] = (CeedScalar(*)[3][CEED_Q_VLA])out[1];
  // *INDENT-ON*

  // Context
  const Physics context = (Physics)ctx;
  const CeedScalar E  = context->E;
  const CeedScalar nu = context->nu;

  // Quadrature Point Loop
  CeedPragmaSIMD
  for (CeedInt i=0; i<Q; i++) {
    // Read spatial derivatives of u
    // *INDENT-OFF*
    const CeedScalar du[3][3]   = {{ug[0][0][i],
                                    ug[1][0][i],
                                    ug[2][0][i]},
                                   {ug[0][1][i],
                                    ug[1][1][i],
                                    ug[2][1][i]},
                                   {ug[0][2][i],
                                    ug[1][2][i],
                                    ug[2][2][i]}
                                  };
    // -- Qdata
    const CeedScalar wdetJ      =   q_data[0][i];
    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*

    // Compute grad_u
    //   dXdx = (dx/dX)^(-1)
    // Apply dXdx to du = grad_u
    CeedScalar grad_u[3][3];
    for (CeedInt j = 0; j < 3; j++)     // Component
      for (CeedInt k = 0; k < 3; k++) { // Derivative
        grad_u[j][k] = 0;
        for (CeedInt m = 0; m < 3; m++)
          grad_u[j][k] += dXdx[m][k] * du[j][m];
      }

    // Compute Strain : e (epsilon)
    // e = 1/2 (grad u + (grad u)^T)

    // *INDENT-OFF*
    const CeedScalar e[3][3] = {{(grad_u[0][0] + grad_u[0][0])/2.,
                                 (grad_u[0][1] + grad_u[1][0])/2.,
                                 (grad_u[0][2] + grad_u[2][0])/2.},
                                {(grad_u[1][0] + grad_u[0][1])/2.,
                                 (grad_u[1][1] + grad_u[1][1])/2.,
                                 (grad_u[1][2] + grad_u[2][1])/2.},
                                {(grad_u[2][0] + grad_u[0][2])/2.,
                                 (grad_u[2][1] + grad_u[1][2])/2.,
                                 (grad_u[2][2] + grad_u[2][2])/2.}
                               };
    // *INDENT-ON*

    //
    // Formulation uses Voigt notation:
    //  stress (sigma)      strain (epsilon)
    //
    //    [sigma00]             [e00]
    //    [sigma11]             [e11]
    //    [sigma22]   =  S   *  [e22]
    //    [sigma12]             [e12]
    //    [sigma02]             [e02]
    //    [sigma01]             [e01]
    //
    //        where
    //                         [1-nu   nu    nu                                    ]
    //                         [ nu   1-nu   nu                                    ]
    //                         [ nu    nu   1-nu                                   ]
    // S = E/((1+nu)*(1-2*nu)) [                  (1-2*nu)/2                       ]
    //                         [                             (1-2*nu)/2            ]
    //                         [                                        (1-2*nu)/2 ]

    // Above Voigt Notation is placed in a 3x3 matrix:
    const CeedScalar ss      =  E / ((1 + nu)*(1 - 2*nu));
    const CeedScalar sigma00 = ss*((1 - nu)*e[0][0] + nu*e[1][1] + nu*e[2][2]),
                     sigma11 = ss*(nu*e[0][0] + (1 - nu)*e[1][1] + nu*e[2][2]),
                     sigma22 = ss*(nu*e[0][0] + nu*e[1][1] + (1 - nu)*e[2][2]),
                     sigma12 = ss*(1 - 2*nu)*e[1][2]*0.5,
                     sigma02 = ss*(1 - 2*nu)*e[0][2]*0.5,
                     sigma01 = ss*(1 - 2*nu)*e[0][1]*0.5;
    // *INDENT-OFF*
    const CeedScalar sigma[3][3] = {{sigma00, sigma01, sigma02},
                                    {sigma01, sigma11, sigma12},
                                    {sigma02, sigma12, sigma22}
                                   };
    // *INDENT-ON*

    // Apply dXdx^T and weight to sigma
    for (CeedInt j = 0; j < 3; j++)     // Component
      for (CeedInt k = 0; k < 3; k++) { // Derivative
        dvdX[k][j][i] = 0;
        for (CeedInt m = 0; m < 3; m++)
          dvdX[k][j][i] += dXdx[k][m] * sigma[j][m] * wdetJ;
      }

  } // End of Quadrature Point Loop

  return 0;
}

// -----------------------------------------------------------------------------
// Jacobian evaluation for linear elasticity
// -----------------------------------------------------------------------------
CEED_QFUNCTION(ElasLineardF)(void *ctx, CeedInt Q, const CeedScalar *const *in,
                             CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*deltaug)[3][CEED_Q_VLA] = (const CeedScalar(*)[3][CEED_Q_VLA])in[0],
                   (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[1];
                   // grad_u not used for linear elasticity
                   // (*grad_u)[3][Q] = (CeedScalar(*)[3][Q])in[2];

  // Outputs
  CeedScalar (*deltadvdX)[3][CEED_Q_VLA] = (CeedScalar(*)[3][CEED_Q_VLA])out[0];
  // *INDENT-ON*

  // Context
  const Physics context = (Physics)ctx;
  const CeedScalar E  = context->E;
  const CeedScalar nu = context->nu;

  // Quadrature Point Loop
  CeedPragmaSIMD
  for (CeedInt i=0; i<Q; i++) {
    // Read spatial derivatives of u
    // *INDENT-OFF*
    const CeedScalar deltadu[3][3] = {{deltaug[0][0][i],
                                       deltaug[1][0][i],
                                       deltaug[2][0][i]},
                                      {deltaug[0][1][i],
                                       deltaug[1][1][i],
                                       deltaug[2][1][i]},
                                      {deltaug[0][2][i],
                                       deltaug[1][2][i],
                                       deltaug[2][2][i]}
                                     };
    // -- Qdata
    const CeedScalar wdetJ      =      q_data[0][i];
    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*

    // Compute graddeltau
    //   dXdx = (dx/dX)^(-1)
    // Apply dXdx to deltadu = graddeltau
    CeedScalar graddeltau[3][3];
    for (CeedInt j = 0; j < 3; j++)     // Component
      for (CeedInt k = 0; k < 3; k++) { // Derivative
        graddeltau[j][k] = 0;
        for (CeedInt m = 0; m < 3; m++)
          graddeltau[j][k] += dXdx[m][k] * deltadu[j][m];
      }

    // Compute Strain : e (epsilon)
    // e = 1/2 (grad u + (grad u)^T)
    // *INDENT-OFF*
    const CeedScalar de[3][3] = {{(graddeltau[0][0] + graddeltau[0][0])/2.,
                                  (graddeltau[0][1] + graddeltau[1][0])/2.,
                                  (graddeltau[0][2] + graddeltau[2][0])/2.},
                                 {(graddeltau[1][0] + graddeltau[0][1])/2.,
                                  (graddeltau[1][1] + graddeltau[1][1])/2.,
                                  (graddeltau[1][2] + graddeltau[2][1])/2.},
                                 {(graddeltau[2][0] + graddeltau[0][2])/2.,
                                  (graddeltau[2][1] + graddeltau[1][2])/2.,
                                  (graddeltau[2][2] + graddeltau[2][2])/2.}
                                };

    // *INDENT-ON*
    // Formulation uses Voigt notation:
    //  stress (sigma)      strain (epsilon)
    //
    //    [dsigma00]             [de00]
    //    [dsigma11]             [de11]
    //    [dsigma22]   =  S   *  [de22]
    //    [dsigma12]             [de12]
    //    [dsigma02]             [de02]
    //    [dsigma01]             [de01]
    //
    //        where
    //                         [1-nu   nu    nu                                    ]
    //                         [ nu   1-nu   nu                                    ]
    //                         [ nu    nu   1-nu                                   ]
    // S = E/((1+nu)*(1-2*nu)) [                  (1-2*nu)/2                       ]
    //                         [                             (1-2*nu)/2            ]
    //                         [                                        (1-2*nu)/2 ]

    // Above Voigt Notation is placed in a 3x3 matrix:
    const CeedScalar ss      =  E / ((1 + nu)*(1 - 2*nu));
    const CeedScalar dsigma00 = ss*((1 - nu)*de[0][0]+nu*de[1][1]+nu*de[2][2]),
                     dsigma11 = ss*(nu*de[0][0]+(1 - nu)*de[1][1]+nu*de[2][2]),
                     dsigma22 = ss*(nu*de[0][0]+nu*de[1][1]+(1 - nu)*de[2][2]),
                     dsigma12 = ss*(1 - 2*nu)*de[1][2] / 2,
                     dsigma02 = ss*(1 - 2*nu)*de[0][2] / 2,
                     dsigma01 = ss*(1 - 2*nu)*de[0][1] / 2;
    // *INDENT-OFF*
    const CeedScalar dsigma[3][3] = {{dsigma00, dsigma01, dsigma02},
                                     {dsigma01, dsigma11, dsigma12},
                                     {dsigma02, dsigma12, dsigma22}
                                    };
    // *INDENT-ON*

    // Apply dXdx^T and weight
    for (CeedInt j = 0; j < 3; j++)     // Component
      for (CeedInt k = 0; k < 3 ; k++) { // Derivative
        deltadvdX[k][j][i] = 0;
        for (CeedInt m = 0; m < 3; m++)
          deltadvdX[k][j][i] += dXdx[k][m] * dsigma[j][m] * wdetJ;
      }

  } // End of Quadrature Point Loop

  return 0;
}

// -----------------------------------------------------------------------------
// Strain energy computation for linear elasticity
// -----------------------------------------------------------------------------
CEED_QFUNCTION(ElasLinearEnergy)(void *ctx, CeedInt Q,
                                 const CeedScalar *const *in,
                                 CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*ug)[3][CEED_Q_VLA] = (const CeedScalar(*)[3][CEED_Q_VLA])in[0],
                   (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[1];

  // Outputs
  CeedScalar (*energy) = (CeedScalar(*))out[0];
  // *INDENT-ON*

  // Context
  const Physics context = (Physics)ctx;
  const CeedScalar E  = context->E;
  const CeedScalar nu = context->nu;

  // Constants
  const CeedScalar TwoMu = E / (1 + nu);
  const CeedScalar mu = TwoMu / 2;
  const CeedScalar Kbulk = E / (3*(1 - 2*nu)); // Bulk Modulus
  const CeedScalar lambda = (3*Kbulk - TwoMu) / 3;

  // Quadrature Point Loop
  CeedPragmaSIMD
  for (CeedInt i=0; i<Q; i++) {
    // Read spatial derivatives of u
    // *INDENT-OFF*
    const CeedScalar du[3][3]   = {{ug[0][0][i],
                                    ug[1][0][i],
                                    ug[2][0][i]},
                                   {ug[0][1][i],
                                    ug[1][1][i],
                                    ug[2][1][i]},
                                   {ug[0][2][i],
                                    ug[1][2][i],
                                    ug[2][2][i]}
                                  };
    // -- Qdata
    const CeedScalar wdetJ      =   q_data[0][i];
    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*

    // Compute grad_u
    //   dXdx = (dx/dX)^(-1)
    // Apply dXdx to du = grad_u
    CeedScalar grad_u[3][3];
    for (CeedInt j = 0; j < 3; j++)     // Component
      for (CeedInt k = 0; k < 3; k++) { // Derivative
        grad_u[j][k] = 0;
        for (CeedInt m = 0; m < 3; m++)
          grad_u[j][k] += dXdx[m][k] * du[j][m];
      }

    // Compute Strain : e (epsilon)
    // e = 1/2 (grad u + (grad u)^T)

    // *INDENT-OFF*
    const CeedScalar e[3][3] = {{(grad_u[0][0] + grad_u[0][0])/2.,
                                 (grad_u[0][1] + grad_u[1][0])/2.,
                                 (grad_u[0][2] + grad_u[2][0])/2.},
                                {(grad_u[1][0] + grad_u[0][1])/2.,
                                 (grad_u[1][1] + grad_u[1][1])/2.,
                                 (grad_u[1][2] + grad_u[2][1])/2.},
                                {(grad_u[2][0] + grad_u[0][2])/2.,
                                 (grad_u[2][1] + grad_u[1][2])/2.,
                                 (grad_u[2][2] + grad_u[2][2])/2.}
                               };
    // *INDENT-ON*

    // Strain energy
    const CeedScalar strain_vol = e[0][0] + e[1][1] + e[2][2];
    energy[i] = (lambda*strain_vol*strain_vol/2. + strain_vol*mu +
                 (e[0][1]*e[0][1]+e[0][2]*e[0][2]+e[1][2]*e[1][2])*2*mu)*wdetJ;

  } // End of Quadrature Point Loop

  return 0;
}

// -----------------------------------------------------------------------------
// Nodal diagnostic quantities for linear elasticity
// -----------------------------------------------------------------------------
CEED_QFUNCTION(ElasLinearDiagnostic)(void *ctx, CeedInt Q,
                                     const CeedScalar *const *in,
                                     CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*u)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0],
                   (*ug)[3][CEED_Q_VLA] = (const CeedScalar(*)[3][CEED_Q_VLA])in[1],
                   (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[2];

  // Outputs
  CeedScalar (*diagnostic)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0];
  // *INDENT-ON*

  // Context
  const Physics context = (Physics)ctx;
  const CeedScalar E  = context->E;
  const CeedScalar nu = context->nu;

  // Constants
  const CeedScalar TwoMu = E / (1 + nu);
  const CeedScalar mu = TwoMu / 2;
  const CeedScalar Kbulk = E / (3*(1 - 2*nu)); // Bulk Modulus
  const CeedScalar lambda = (3*Kbulk - TwoMu) / 3;

  // Quadrature Point Loop
  CeedPragmaSIMD
  for (CeedInt i=0; i<Q; i++) {
    // Read spatial derivatives of u
    // *INDENT-OFF*
    const CeedScalar du[3][3]   = {{ug[0][0][i],
                                    ug[1][0][i],
                                    ug[2][0][i]},
                                   {ug[0][1][i],
                                    ug[1][1][i],
                                    ug[2][1][i]},
                                   {ug[0][2][i],
                                    ug[1][2][i],
                                    ug[2][2][i]}
                                  };
    // -- Qdata
    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*

    // Compute grad_u
    //   dXdx = (dx/dX)^(-1)
    // Apply dXdx to du = grad_u
    CeedScalar grad_u[3][3];
    for (CeedInt j = 0; j < 3; j++)     // Component
      for (CeedInt k = 0; k < 3; k++) { // Derivative
        grad_u[j][k] = 0;
        for (CeedInt m = 0; m < 3; m++)
          grad_u[j][k] += dXdx[m][k] * du[j][m];
      }

    // Compute Strain : e (epsilon)
    // e = 1/2 (grad u + (grad u)^T)

    // *INDENT-OFF*
    const CeedScalar e[3][3] = {{(grad_u[0][0] + grad_u[0][0])/2.,
                                 (grad_u[0][1] + grad_u[1][0])/2.,
                                 (grad_u[0][2] + grad_u[2][0])/2.},
                                {(grad_u[1][0] + grad_u[0][1])/2.,
                                 (grad_u[1][1] + grad_u[1][1])/2.,
                                 (grad_u[1][2] + grad_u[2][1])/2.},
                                {(grad_u[2][0] + grad_u[0][2])/2.,
                                 (grad_u[2][1] + grad_u[1][2])/2.,
                                 (grad_u[2][2] + grad_u[2][2])/2.}
                               };
    // *INDENT-ON*

    // Displacement
    diagnostic[0][i] = u[0][i];
    diagnostic[1][i] = u[1][i];
    diagnostic[2][i] = u[2][i];

    // Pressure
    const CeedScalar strain_vol = e[0][0] + e[1][1] + e[2][2];
    diagnostic[3][i] = -lambda*strain_vol;

    // Stress tensor invariants
    diagnostic[4][i] = strain_vol;
    diagnostic[5][i] = 0.;
    for (CeedInt j = 0; j < 3; j++)
      for (CeedInt m = 0; m < 3; m++)
        diagnostic[5][i] += e[j][m] * e[m][j];
    diagnostic[6][i] = 1 + strain_vol;

    // Strain energy
    diagnostic[7][i] = (lambda*strain_vol*strain_vol/2. + strain_vol*mu +
                        (e[0][1]*e[0][1]+e[0][2]*e[0][2]+e[1][2]*e[1][2])*2*mu);

  } // End of Quadrature Point Loop

  return 0;
}
// -----------------------------------------------------------------------------

#endif //End of ELAS_LINEAR_H