1 // Copyright (c) 2017-2024, Lawrence Livermore National Security, LLC and other CEED contributors. 2 // All Rights Reserved. See the top-level LICENSE and NOTICE files for details. 3 // 4 // SPDX-License-Identifier: BSD-2-Clause 5 // 6 // This file is part of CEED: http://github.com/ceed 7 8 /// @file 9 /// libCEED QFunctions for diffusion operator example for a scalar field on the sphere using PETSc 10 11 #include <ceed.h> 12 #include <math.h> 13 14 // ----------------------------------------------------------------------------- 15 // This QFunction sets up the geometric factors required for integration and coordinate transformations when reference coordinates have a different 16 // dimension than the one of physical coordinates 17 // 18 // Reference (parent) 2D coordinates: X \in [-1, 1]^2 19 // 20 // Global 3D physical coordinates given by the mesh: xx \in [-R, R]^3 with R radius of the sphere 21 // 22 // Local 3D physical coordinates on the 2D manifold: x \in [-l, l]^3 with l half edge of the cube inscribed in the sphere 23 // 24 // Change of coordinates matrix computed by the library: 25 // (physical 3D coords relative to reference 2D coords) 26 // dxx_j/dX_i (indicial notation) [3 * 2] 27 // 28 // Change of coordinates x (on the 2D manifold) relative to xx (phyisical 3D): 29 // dx_i/dxx_j (indicial notation) [3 * 3] 30 // 31 // Change of coordinates x (on the 2D manifold) relative to X (reference 2D): 32 // (by chain rule) 33 // dx_i/dX_j [3 * 2] = dx_i/dxx_k [3 * 3] * dxx_k/dX_j [3 * 2] 34 // 35 // mod_J is given by the magnitude of the cross product of the columns of dx_i/dX_j 36 // 37 // The quadrature data is stored in the array q_data. 38 // 39 // We require the determinant of the Jacobian to properly compute integrals of the form: int( u v ) 40 // 41 // q_data[0]: mod_J * w 42 // 43 // 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 44 // form: int( gradv gradu ) 45 // 46 // dX_i/dx_j [2 * 3] = (dx_i/dX_j)+ = (dxdX^T dxdX)^(-1) dxdX 47 // 48 // and the product simplifies to yield the contravariant metric tensor 49 // 50 // g^{ij} = dX_i/dx_k dX_j/dx_k = (dxdX^T dxdX)^{-1} 51 // 52 // Stored: g^{ij} (in Voigt convention) in 53 // 54 // q_data[1:3]: [dXdxdXdxT00 dXdxdXdxT01] 55 // [dXdxdXdxT01 dXdxdXdxT11] 56 // ----------------------------------------------------------------------------- 57 CEED_QFUNCTION(SetupDiffGeo)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) { 58 const CeedScalar *X = in[0], *J = in[1], *w = in[2]; 59 CeedScalar *q_data = out[0]; 60 61 // Quadrature Point Loop 62 CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) { 63 // Read global Cartesian coordinates 64 const CeedScalar xx[3] = {X[i + 0 * Q], X[i + 1 * Q], X[i + 2 * Q]}; 65 66 // Read dxxdX Jacobian entries, stored as 67 // 0 3 68 // 1 4 69 // 2 5 70 const CeedScalar dxxdX[3][2] = { 71 {J[i + Q * 0], J[i + Q * 3]}, 72 {J[i + Q * 1], J[i + Q * 4]}, 73 {J[i + Q * 2], J[i + Q * 5]} 74 }; 75 76 // Setup 77 // x = xx (xx^T xx)^{-1/2} 78 // dx/dxx = I (xx^T xx)^{-1/2} - xx xx^T (xx^T xx)^{-3/2} 79 const CeedScalar mod_xx_sq = xx[0] * xx[0] + xx[1] * xx[1] + xx[2] * xx[2]; 80 CeedScalar xx_sq[3][3]; 81 for (int j = 0; j < 3; j++) { 82 for (int k = 0; k < 3; k++) xx_sq[j][k] = xx[j] * xx[k] / (sqrt(mod_xx_sq) * mod_xx_sq); 83 } 84 85 const CeedScalar dxdxx[3][3] = { 86 {1. / sqrt(mod_xx_sq) - xx_sq[0][0], -xx_sq[0][1], -xx_sq[0][2] }, 87 {-xx_sq[1][0], 1. / sqrt(mod_xx_sq) - xx_sq[1][1], -xx_sq[1][2] }, 88 {-xx_sq[2][0], -xx_sq[2][1], 1. / sqrt(mod_xx_sq) - xx_sq[2][2]} 89 }; 90 91 CeedScalar dxdX[3][2]; 92 for (int j = 0; j < 3; j++) { 93 for (int k = 0; k < 2; k++) { 94 dxdX[j][k] = 0; 95 for (int l = 0; l < 3; l++) dxdX[j][k] += dxdxx[j][l] * dxxdX[l][k]; 96 } 97 } 98 99 // J is given by the cross product of the columns of dxdX 100 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], 101 dxdX[0][0] * dxdX[1][1] - dxdX[1][0] * dxdX[0][1]}; 102 103 // Use the magnitude of J as our detJ (volume scaling factor) 104 const CeedScalar mod_J = sqrt(J[0] * J[0] + J[1] * J[1] + J[2] * J[2]); 105 106 // Interp-to-Interp q_data 107 q_data[i + Q * 0] = mod_J * w[i]; 108 109 // dxdX_k,j * dxdX_j,k 110 CeedScalar dxdXTdxdX[2][2]; 111 for (int j = 0; j < 2; j++) { 112 for (int k = 0; k < 2; k++) { 113 dxdXTdxdX[j][k] = 0; 114 for (int l = 0; l < 3; l++) dxdXTdxdX[j][k] += dxdX[l][j] * dxdX[l][k]; 115 } 116 } 117 118 const CeedScalar detdxdXTdxdX = dxdXTdxdX[0][0] * dxdXTdxdX[1][1] - dxdXTdxdX[1][0] * dxdXTdxdX[0][1]; 119 120 // Compute inverse of dxdXTdxdX, which is the 2x2 contravariant metric tensor g^{ij} 121 CeedScalar dxdXTdxdX_inv[2][2]; 122 dxdXTdxdX_inv[0][0] = dxdXTdxdX[1][1] / detdxdXTdxdX; 123 dxdXTdxdX_inv[0][1] = -dxdXTdxdX[0][1] / detdxdXTdxdX; 124 dxdXTdxdX_inv[1][0] = -dxdXTdxdX[1][0] / detdxdXTdxdX; 125 dxdXTdxdX_inv[1][1] = dxdXTdxdX[0][0] / detdxdXTdxdX; 126 127 // Stored in Voigt convention 128 q_data[i + Q * 1] = dxdXTdxdX_inv[0][0]; 129 q_data[i + Q * 2] = dxdXTdxdX_inv[1][1]; 130 q_data[i + Q * 3] = dxdXTdxdX_inv[0][1]; 131 } // End of Quadrature Point Loop 132 133 // Return 134 return 0; 135 } 136 137 // ----------------------------------------------------------------------------- 138 // This QFunction sets up the rhs and true solution for the problem 139 // ----------------------------------------------------------------------------- 140 CEED_QFUNCTION(SetupDiffRhs)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) { 141 // Inputs 142 const CeedScalar *X = in[0], *q_data = in[1]; 143 // Outputs 144 CeedScalar *true_soln = out[0], *rhs = out[1]; 145 146 // Context 147 const CeedScalar *context = (const CeedScalar *)ctx; 148 const CeedScalar R = context[0]; 149 150 // Quadrature Point Loop 151 CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) { 152 // Read global Cartesian coordinates 153 CeedScalar x = X[i + Q * 0], y = X[i + Q * 1], z = X[i + Q * 2]; 154 // Normalize quadrature point coordinates to sphere 155 CeedScalar rad = sqrt(x * x + y * y + z * z); 156 x *= R / rad; 157 y *= R / rad; 158 z *= R / rad; 159 // Compute latitude and longitude 160 const CeedScalar theta = asin(z / R); // latitude 161 const CeedScalar lambda = atan2(y, x); // longitude 162 163 true_soln[i + Q * 0] = sin(lambda) * cos(theta); 164 165 rhs[i + Q * 0] = q_data[i + Q * 0] * 2 * sin(lambda) * cos(theta) / (R * R); 166 } // End of Quadrature Point Loop 167 168 return 0; 169 } 170 171 // ----------------------------------------------------------------------------- 172 // This QFunction applies the diffusion operator for a scalar field. 173 // 174 // Inputs: 175 // ug - Input vector gradient at quadrature points 176 // q_data - Geometric factors 177 // 178 // Output: 179 // vg - Output vector (test functions) gradient at quadrature points 180 // ----------------------------------------------------------------------------- 181 CEED_QFUNCTION(Diff)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) { 182 // Inputs 183 const CeedScalar *ug = in[0], *q_data = in[1]; 184 // Outputs 185 CeedScalar *vg = out[0]; 186 187 // Quadrature Point Loop 188 CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) { 189 // Read spatial derivatives of u 190 const CeedScalar du[2] = {ug[i + Q * 0], ug[i + Q * 1]}; 191 // Read q_data 192 const CeedScalar w_det_J = q_data[i + Q * 0]; 193 // -- Grad-to-Grad q_data 194 // ---- dXdx_j,k * dXdx_k,j 195 const CeedScalar dXdxdXdx_T[2][2] = { 196 {q_data[i + Q * 1], q_data[i + Q * 3]}, 197 {q_data[i + Q * 3], q_data[i + Q * 2]} 198 }; 199 200 for (int j = 0; j < 2; j++) { // j = direction of vg 201 vg[i + j * Q] = w_det_J * (du[0] * dXdxdXdx_T[0][j] + du[1] * dXdxdXdx_T[1][j]); 202 } 203 } // End of Quadrature Point Loop 204 205 return 0; 206 } 207 // ----------------------------------------------------------------------------- 208