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/types.h> 12 #ifndef CEED_RUNNING_JIT_PASS 13 #include <math.h> 14 #endif 15 16 // ----------------------------------------------------------------------------- 17 // This QFunction sets up the geometric factors required for integration and coordinate transformations when reference coordinates have a different 18 // dimension than the one of physical coordinates 19 // 20 // Reference (parent) 2D coordinates: X \in [-1, 1]^2 21 // 22 // Global 3D physical coordinates given by the mesh: xx \in [-R, R]^3 with R radius of the sphere 23 // 24 // Local 3D physical coordinates on the 2D manifold: x \in [-l, l]^3 with l half edge of the cube inscribed in the sphere 25 // 26 // Change of coordinates matrix computed by the library: 27 // (physical 3D coords relative to reference 2D coords) 28 // dxx_j/dX_i (indicial notation) [3 * 2] 29 // 30 // Change of coordinates x (on the 2D manifold) relative to xx (phyisical 3D): 31 // dx_i/dxx_j (indicial notation) [3 * 3] 32 // 33 // Change of coordinates x (on the 2D manifold) relative to X (reference 2D): 34 // (by chain rule) 35 // dx_i/dX_j [3 * 2] = dx_i/dxx_k [3 * 3] * dxx_k/dX_j [3 * 2] 36 // 37 // mod_J is given by the magnitude of the cross product of the columns of dx_i/dX_j 38 // 39 // The quadrature data is stored in the array q_data. 40 // 41 // We require the determinant of the Jacobian to properly compute integrals of the form: int( u v ) 42 // 43 // q_data[0]: mod_J * w 44 // 45 // 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 46 // form: int( gradv gradu ) 47 // 48 // dX_i/dx_j [2 * 3] = (dx_i/dX_j)+ = (dxdX^T dxdX)^(-1) dxdX 49 // 50 // and the product simplifies to yield the contravariant metric tensor 51 // 52 // g^{ij} = dX_i/dx_k dX_j/dx_k = (dxdX^T dxdX)^{-1} 53 // 54 // Stored: g^{ij} (in Voigt convention) in 55 // 56 // q_data[1:3]: [dXdxdXdxT00 dXdxdXdxT01] 57 // [dXdxdXdxT01 dXdxdXdxT11] 58 // ----------------------------------------------------------------------------- 59 CEED_QFUNCTION(SetupDiffGeo)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) { 60 const CeedScalar *X = in[0], *J = in[1], *w = in[2]; 61 CeedScalar *q_data = out[0]; 62 63 // Quadrature Point Loop 64 CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) { 65 // Read global Cartesian coordinates 66 const CeedScalar xx[3] = {X[i + 0 * Q], X[i + 1 * Q], X[i + 2 * Q]}; 67 68 // Read dxxdX Jacobian entries, stored as 69 // 0 3 70 // 1 4 71 // 2 5 72 const CeedScalar dxxdX[3][2] = { 73 {J[i + Q * 0], J[i + Q * 3]}, 74 {J[i + Q * 1], J[i + Q * 4]}, 75 {J[i + Q * 2], J[i + Q * 5]} 76 }; 77 78 // Setup 79 // x = xx (xx^T xx)^{-1/2} 80 // dx/dxx = I (xx^T xx)^{-1/2} - xx xx^T (xx^T xx)^{-3/2} 81 const CeedScalar mod_xx_sq = xx[0] * xx[0] + xx[1] * xx[1] + xx[2] * xx[2]; 82 CeedScalar xx_sq[3][3]; 83 for (int j = 0; j < 3; j++) { 84 for (int k = 0; k < 3; k++) xx_sq[j][k] = xx[j] * xx[k] / (sqrt(mod_xx_sq) * mod_xx_sq); 85 } 86 87 const CeedScalar dxdxx[3][3] = { 88 {1. / sqrt(mod_xx_sq) - xx_sq[0][0], -xx_sq[0][1], -xx_sq[0][2] }, 89 {-xx_sq[1][0], 1. / sqrt(mod_xx_sq) - xx_sq[1][1], -xx_sq[1][2] }, 90 {-xx_sq[2][0], -xx_sq[2][1], 1. / sqrt(mod_xx_sq) - xx_sq[2][2]} 91 }; 92 93 CeedScalar dxdX[3][2]; 94 for (int j = 0; j < 3; j++) { 95 for (int k = 0; k < 2; k++) { 96 dxdX[j][k] = 0; 97 for (int l = 0; l < 3; l++) dxdX[j][k] += dxdxx[j][l] * dxxdX[l][k]; 98 } 99 } 100 101 // J is given by the cross product of the columns of dxdX 102 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], 103 dxdX[0][0] * dxdX[1][1] - dxdX[1][0] * dxdX[0][1]}; 104 105 // Use the magnitude of J as our detJ (volume scaling factor) 106 const CeedScalar mod_J = sqrt(J[0] * J[0] + J[1] * J[1] + J[2] * J[2]); 107 108 // Interp-to-Interp q_data 109 q_data[i + Q * 0] = mod_J * w[i]; 110 111 // dxdX_k,j * dxdX_j,k 112 CeedScalar dxdXTdxdX[2][2]; 113 for (int j = 0; j < 2; j++) { 114 for (int k = 0; k < 2; k++) { 115 dxdXTdxdX[j][k] = 0; 116 for (int l = 0; l < 3; l++) dxdXTdxdX[j][k] += dxdX[l][j] * dxdX[l][k]; 117 } 118 } 119 120 const CeedScalar detdxdXTdxdX = dxdXTdxdX[0][0] * dxdXTdxdX[1][1] - dxdXTdxdX[1][0] * dxdXTdxdX[0][1]; 121 122 // Compute inverse of dxdXTdxdX, which is the 2x2 contravariant metric tensor g^{ij} 123 CeedScalar dxdXTdxdX_inv[2][2]; 124 dxdXTdxdX_inv[0][0] = dxdXTdxdX[1][1] / detdxdXTdxdX; 125 dxdXTdxdX_inv[0][1] = -dxdXTdxdX[0][1] / detdxdXTdxdX; 126 dxdXTdxdX_inv[1][0] = -dxdXTdxdX[1][0] / detdxdXTdxdX; 127 dxdXTdxdX_inv[1][1] = dxdXTdxdX[0][0] / detdxdXTdxdX; 128 129 // Stored in Voigt convention 130 q_data[i + Q * 1] = dxdXTdxdX_inv[0][0]; 131 q_data[i + Q * 2] = dxdXTdxdX_inv[1][1]; 132 q_data[i + Q * 3] = dxdXTdxdX_inv[0][1]; 133 } // End of Quadrature Point Loop 134 135 // Return 136 return 0; 137 } 138 139 // ----------------------------------------------------------------------------- 140 // This QFunction sets up the rhs and true solution for the problem 141 // ----------------------------------------------------------------------------- 142 CEED_QFUNCTION(SetupDiffRhs)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) { 143 // Inputs 144 const CeedScalar *X = in[0], *q_data = in[1]; 145 // Outputs 146 CeedScalar *true_soln = out[0], *rhs = out[1]; 147 148 // Context 149 const CeedScalar *context = (const CeedScalar *)ctx; 150 const CeedScalar R = context[0]; 151 152 // Quadrature Point Loop 153 CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) { 154 // Read global Cartesian coordinates 155 CeedScalar x = X[i + Q * 0], y = X[i + Q * 1], z = X[i + Q * 2]; 156 // Normalize quadrature point coordinates to sphere 157 CeedScalar rad = sqrt(x * x + y * y + z * z); 158 x *= R / rad; 159 y *= R / rad; 160 z *= R / rad; 161 // Compute latitude and longitude 162 const CeedScalar theta = asin(z / R); // latitude 163 const CeedScalar lambda = atan2(y, x); // longitude 164 165 true_soln[i + Q * 0] = sin(lambda) * cos(theta); 166 167 rhs[i + Q * 0] = q_data[i + Q * 0] * 2 * sin(lambda) * cos(theta) / (R * R); 168 } // End of Quadrature Point Loop 169 170 return 0; 171 } 172 173 // ----------------------------------------------------------------------------- 174 // This QFunction applies the diffusion operator for a scalar field. 175 // 176 // Inputs: 177 // ug - Input vector gradient at quadrature points 178 // q_data - Geometric factors 179 // 180 // Output: 181 // vg - Output vector (test functions) gradient at quadrature points 182 // ----------------------------------------------------------------------------- 183 CEED_QFUNCTION(Diff)(void *ctx, CeedInt Q, const CeedScalar *const *in, CeedScalar *const *out) { 184 // Inputs 185 const CeedScalar *ug = in[0], *q_data = in[1]; 186 // Outputs 187 CeedScalar *vg = out[0]; 188 189 // Quadrature Point Loop 190 CeedPragmaSIMD for (CeedInt i = 0; i < Q; i++) { 191 // Read spatial derivatives of u 192 const CeedScalar du[2] = {ug[i + Q * 0], ug[i + Q * 1]}; 193 // Read q_data 194 const CeedScalar w_det_J = q_data[i + Q * 0]; 195 // -- Grad-to-Grad q_data 196 // ---- dXdx_j,k * dXdx_k,j 197 const CeedScalar dXdxdXdx_T[2][2] = { 198 {q_data[i + Q * 1], q_data[i + Q * 3]}, 199 {q_data[i + Q * 3], q_data[i + Q * 2]} 200 }; 201 202 for (int j = 0; j < 2; j++) { // j = direction of vg 203 vg[i + j * Q] = w_det_J * (du[0] * dXdxdXdx_T[0][j] + du[1] * dXdxdXdx_T[1][j]); 204 } 205 } // End of Quadrature Point Loop 206 207 return 0; 208 } 209 // ----------------------------------------------------------------------------- 210