// Copyright (c) 2017-2026, 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 /// libCEED QFunctions for diffusion operator example for a scalar field on the sphere using PETSc #include #ifndef CEED_RUNNING_JIT_PASS #include #endif // ----------------------------------------------------------------------------- // 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; } // -----------------------------------------------------------------------------