qfunctions/bps/bp3sphere.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 element and spectral
// element discretizations for exascale applications. For more information and
// source code availability see http://github.com/ceed
//
// The CEED research is supported by the Exascale Computing Project 17-SC-20-SC,
// a collaborative effort of two U.S. Department of Energy organizations (Office
// of Science and the National Nuclear Security Administration) responsible for
// the planning and preparation of a capable exascale ecosystem, including
// software, applications, hardware, advanced system engineering and early
// testbed platforms, in support of the nation's exascale computing imperative.

/// @file
/// libCEED QFunctions for diffusion operator example for a scalar field on the sphere using PETSc

#ifndef bp3sphere_h
#define bp3sphere_h

#include <ceed.h>
#include <math.h>

// -----------------------------------------------------------------------------
// This QFunction sets up the geometric factors required for integration and
//   coordinate transformations when reference coordinates have a different
//   dimension than the one of physical coordinates
//
// Reference (parent) 2D coordinates: X \in [-1, 1]^2
//
// Global 3D physical coordinates given by the mesh: xx \in [-R, R]^3
//   with R radius of the sphere
//
// Local 3D physical coordinates on the 2D manifold: x \in [-l, l]^3
//   with l half edge of the cube inscribed in the sphere
//
// Change of coordinates matrix computed by the library:
//   (physical 3D coords relative to reference 2D coords)
//   dxx_j/dX_i (indicial notation) [3 * 2]
//
// Change of coordinates x (on the 2D manifold) relative to xx (phyisical 3D):
//   dx_i/dxx_j (indicial notation) [3 * 3]
//
// Change of coordinates x (on the 2D manifold) relative to X (reference 2D):
//   (by chain rule)
//   dx_i/dX_j [3 * 2] = dx_i/dxx_k [3 * 3] * dxx_k/dX_j [3 * 2]
//
// mod_J is given by the magnitude of the cross product of the columns of dx_i/dX_j
//
// The quadrature data is stored in the array q_data.
//
// We require the determinant of the Jacobian to properly compute integrals of
//   the form: int( u v )
//
// q_data[0]: mod_J * w
//
// We use the Moore–Penrose (left) pseudoinverse of dx_i/dX_j, to compute dX_i/dx_j (and its transpose),
//   needed to properly compute integrals of the form: int( gradv gradu )
//
// dX_i/dx_j [2 * 3] = (dx_i/dX_j)+ = (dxdX^T dxdX)^(-1) dxdX
//
// and the product simplifies to yield the contravariant metric tensor
//
// g^{ij} = dX_i/dx_k dX_j/dx_k = (dxdX^T dxdX)^{-1}
//
// Stored: g^{ij} (in Voigt convention) in
//
//   q_data[1:3]: [dXdxdXdxT00 dXdxdXdxT01]
//               [dXdxdXdxT01 dXdxdXdxT11]
// -----------------------------------------------------------------------------
CEED_QFUNCTION(SetupDiffGeo)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) {
  const CeedScalar *X = in[0], *J = in[1], *w = in[2];
  CeedScalar       *q_data = out[0];

  // Quadrature Point Loop
  CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) {
    // Read global Cartesian coordinates
    const CeedScalar xx[3] = {X[i + 0 * Q], X[i + 1 * Q], X[i + 2 * Q]};

    // Read dxxdX Jacobian entries, stored as
    // 0 3
    // 1 4
    // 2 5
    const CeedScalar dxxdX[3][2] = {
        {J[i + Q * 0], J[i + Q * 3]},
        {J[i + Q * 1], J[i + Q * 4]},
        {J[i + Q * 2], J[i + Q * 5]}
    };

    // Setup
    // x = xx (xx^T xx)^{-1/2}
    // dx/dxx = I (xx^T xx)^{-1/2} - xx xx^T (xx^T xx)^{-3/2}
    const CeedScalar mod_xx_sq = xx[0] * xx[0] + xx[1] * xx[1] + xx[2] * xx[2];
    CeedScalar       xx_sq[3][3];
    for (int j = 0; j < 3; j++) {
      for (int k = 0; k < 3; k++) xx_sq[j][k] = xx[j] * xx[k] / (sqrt(mod_xx_sq) * mod_xx_sq);
    }

    const CeedScalar dxdxx[3][3] = {
        {1. / sqrt(mod_xx_sq) - xx_sq[0][0], -xx_sq[0][1],                       -xx_sq[0][2]                      },
        {-xx_sq[1][0],                       1. / sqrt(mod_xx_sq) - xx_sq[1][1], -xx_sq[1][2]                      },
        {-xx_sq[2][0],                       -xx_sq[2][1],                       1. / sqrt(mod_xx_sq) - xx_sq[2][2]}
    };

    CeedScalar dxdX[3][2];
    for (int j = 0; j < 3; j++) {
      for (int k = 0; k < 2; k++) {
        dxdX[j][k] = 0;
        for (int l = 0; l < 3; l++) dxdX[j][k] += dxdxx[j][l] * dxxdX[l][k];
      }
    }

    // J is given by the cross product of the columns of dxdX
    const CeedScalar J[3] = {dxdX[1][0] * dxdX[2][1] - dxdX[2][0] * dxdX[1][1], dxdX[2][0] * dxdX[0][1] - dxdX[0][0] * dxdX[2][1],
                             dxdX[0][0] * dxdX[1][1] - dxdX[1][0] * dxdX[0][1]};

    // Use the magnitude of J as our detJ (volume scaling factor)
    const CeedScalar mod_J = sqrt(J[0] * J[0] + J[1] * J[1] + J[2] * J[2]);

    // Interp-to-Interp q_data
    q_data[i + Q * 0] = mod_J * w[i];

    // dxdX_k,j * dxdX_j,k
    CeedScalar dxdXTdxdX[2][2];
    for (int j = 0; j < 2; j++) {
      for (int k = 0; k < 2; k++) {
        dxdXTdxdX[j][k] = 0;
        for (int l = 0; l < 3; l++) dxdXTdxdX[j][k] += dxdX[l][j] * dxdX[l][k];
      }
    }

    const CeedScalar detdxdXTdxdX = dxdXTdxdX[0][0] * dxdXTdxdX[1][1] - dxdXTdxdX[1][0] * dxdXTdxdX[0][1];

    // Compute inverse of dxdXTdxdX, which is the 2x2 contravariant metric tensor g^{ij}
    CeedScalar dxdXTdxdX_inv[2][2];
    dxdXTdxdX_inv[0][0] = dxdXTdxdX[1][1] / detdxdXTdxdX;
    dxdXTdxdX_inv[0][1] = -dxdXTdxdX[0][1] / detdxdXTdxdX;
    dxdXTdxdX_inv[1][0] = -dxdXTdxdX[1][0] / detdxdXTdxdX;
    dxdXTdxdX_inv[1][1] = dxdXTdxdX[0][0] / detdxdXTdxdX;

    // Stored in Voigt convention
    q_data[i + Q * 1] = dxdXTdxdX_inv[0][0];
    q_data[i + Q * 2] = dxdXTdxdX_inv[1][1];
    q_data[i + Q * 3] = dxdXTdxdX_inv[0][1];
  }  // End of Quadrature Point Loop

  // Return
  return 0;
}

// -----------------------------------------------------------------------------
// This QFunction sets up the rhs and true solution for the problem
// -----------------------------------------------------------------------------
CEED_QFUNCTION(SetupDiffRhs)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) {
  // Inputs
  const CeedScalar *X = in[0], *q_data = in[1];
  // Outputs
  CeedScalar *true_soln = out[0], *rhs = out[1];

  // Context
  const CeedScalar *context = (const CeedScalar *)ctx;
  const CeedScalar  R       = context[0];

  // Quadrature Point Loop
  CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) {
    // Read global Cartesian coordinates
    CeedScalar x = X[i + Q * 0], y = X[i + Q * 1], z = X[i + Q * 2];
    // Normalize quadrature point coordinates to sphere
    CeedScalar rad = sqrt(x * x + y * y + z * z);
    x *= R / rad;
    y *= R / rad;
    z *= R / rad;
    // Compute latitude and longitude
    const CeedScalar theta  = asin(z / R);  // latitude
    const CeedScalar lambda = atan2(y, x);  // longitude

    true_soln[i + Q * 0] = sin(lambda) * cos(theta);

    rhs[i + Q * 0] = q_data[i + Q * 0] * 2 * sin(lambda) * cos(theta) / (R * R);
  }  // End of Quadrature Point Loop

  return 0;
}

// -----------------------------------------------------------------------------
// This QFunction applies the diffusion operator for a scalar field.
//
// Inputs:
//   ug     - Input vector gradient at quadrature points
//   q_data  - Geometric factors
//
// Output:
//   vg     - Output vector (test functions) gradient at quadrature points
//
// -----------------------------------------------------------------------------
CEED_QFUNCTION(Diff)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) {
  // Inputs
  const CeedScalar *ug = in[0], *q_data = in[1];
  // Outputs
  CeedScalar *vg = out[0];

  // Quadrature Point Loop
  CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) {
    // Read spatial derivatives of u
    const CeedScalar du[2] = {ug[i + Q * 0], ug[i + Q * 1]};
    // Read q_data
    const CeedScalar w_det_J = q_data[i + Q * 0];
    // -- Grad-to-Grad q_data
    // ---- dXdx_j,k * dXdx_k,j
    const CeedScalar dXdxdXdx_T[2][2] = {
        {q_data[i + Q * 1], q_data[i + Q * 3]},
        {q_data[i + Q * 3], q_data[i + Q * 2]}
    };

    for (int j = 0; j < 2; j++) {  // j = direction of vg
      vg[i + j * Q] = w_det_J * (du[0] * dXdxdXdx_T[0][j] + du[1] * dXdxdXdx_T[1][j]);
    }
  }  // End of Quadrature Point Loop

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

#endif  // bp3sphere_h