1 // SPDX-FileCopyrightText: Copyright (c) 2017-2024, HONEE contributors. 2 // SPDX-License-Identifier: Apache-2.0 OR BSD-2-Clause 3 #pragma once 4 5 #include <ceed.h> 6 #include <math.h> 7 8 #ifndef M_PI 9 #define M_PI 3.14159265358979323846 10 #endif 11 12 CEED_QFUNCTION_HELPER CeedScalar Max(CeedScalar a, CeedScalar b) { return a < b ? b : a; } 13 CEED_QFUNCTION_HELPER CeedScalar Min(CeedScalar a, CeedScalar b) { return a < b ? a : b; } 14 15 CEED_QFUNCTION_HELPER void SwapScalar(CeedScalar *a, CeedScalar *b) { 16 CeedScalar temp = *a; 17 *a = *b; 18 *b = temp; 19 } 20 21 CEED_QFUNCTION_HELPER CeedScalar Square(CeedScalar x) { return x * x; } 22 CEED_QFUNCTION_HELPER CeedScalar Cube(CeedScalar x) { return x * x * x; } 23 24 // @brief Scale vector of length N by scalar alpha 25 CEED_QFUNCTION_HELPER void ScaleN(CeedScalar *u, const CeedScalar alpha, const CeedInt N) { 26 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) u[i] *= alpha; 27 } 28 29 // @brief Set vector of length N to a value alpha 30 CEED_QFUNCTION_HELPER void SetValueN(CeedScalar *u, const CeedScalar alpha, const CeedInt N) { 31 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) u[i] = alpha; 32 } 33 34 // @brief Copy N elements from x to y 35 CEED_QFUNCTION_HELPER void CopyN(const CeedScalar *x, CeedScalar *y, const CeedInt N) { CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) y[i] = x[i]; } 36 37 // @brief Copy 3x3 matrix from A to B 38 CEED_QFUNCTION_HELPER void CopyMat3(const CeedScalar A[3][3], CeedScalar B[3][3]) { CopyN((const CeedScalar *)A, (CeedScalar *)B, 9); } 39 40 // @brief Dot product of vectors with N elements 41 CEED_QFUNCTION_HELPER CeedScalar DotN(const CeedScalar *u, const CeedScalar *v, const CeedInt N) { 42 CeedScalar output = 0; 43 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) output += u[i] * v[i]; 44 return output; 45 } 46 47 // @brief Dot product of 3 element vectors 48 CEED_QFUNCTION_HELPER CeedScalar Dot3(const CeedScalar *u, const CeedScalar *v) { return u[0] * v[0] + u[1] * v[1] + u[2] * v[2]; } 49 50 // @brief \ell^2 norm of 3 element vectors 51 CEED_QFUNCTION_HELPER CeedScalar Norm3(const CeedScalar *u) { return sqrt(u[0] * u[0] + u[1] * u[1] + u[2] * u[2]); } 52 53 // @brief Cross product of vectors with 3 elements 54 CEED_QFUNCTION_HELPER void Cross3(const CeedScalar u[3], const CeedScalar v[3], CeedScalar w[3]) { 55 w[0] = (u[1] * v[2]) - (u[2] * v[1]); 56 w[1] = (u[2] * v[0]) - (u[0] * v[2]); 57 w[2] = (u[0] * v[1]) - (u[1] * v[0]); 58 } 59 60 // @brief Curl of vector given its gradient 61 CEED_QFUNCTION_HELPER void Curl3(const CeedScalar gradient[3][3], CeedScalar v[3]) { 62 v[0] = gradient[2][1] - gradient[1][2]; 63 v[1] = gradient[0][2] - gradient[2][0]; 64 v[2] = gradient[1][0] - gradient[0][1]; 65 } 66 67 // @brief Matrix vector product, b = Ax + b. A is NxM, x is M, b is N 68 CEED_QFUNCTION_HELPER void MatVecNM(const CeedScalar *A, const CeedScalar *x, const CeedInt N, const CeedInt M, const CeedTransposeMode transpose_A, 69 CeedScalar *b) { 70 switch (transpose_A) { 71 case CEED_NOTRANSPOSE: 72 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) b[i] += DotN(&A[i * M], x, M); 73 break; 74 case CEED_TRANSPOSE: 75 CeedPragmaSIMD for (CeedInt i = 0; i < M; i++) { CeedPragmaSIMD for (CeedInt j = 0; j < N; j++) b[i] += A[j * M + i] * x[j]; } 76 break; 77 } 78 } 79 80 // @brief 3x3 Matrix vector product b = Ax + b. 81 CEED_QFUNCTION_HELPER void MatVec3(const CeedScalar A[3][3], const CeedScalar x[3], const CeedTransposeMode transpose_A, CeedScalar b[3]) { 82 MatVecNM((const CeedScalar *)A, (const CeedScalar *)x, 3, 3, transpose_A, (CeedScalar *)b); 83 } 84 85 // @brief Matrix-Matrix product, B = DA + B, where D is diagonal. 86 // @details A is NxM, D is diagonal NxN, represented by a vector of length N, and B is NxM. Optionally, A may be transposed. 87 CEED_QFUNCTION_HELPER void MatDiagNM(const CeedScalar *A, const CeedScalar *D, const CeedInt N, const CeedInt M, const CeedTransposeMode transpose_A, 88 CeedScalar *B) { 89 switch (transpose_A) { 90 case CEED_NOTRANSPOSE: 91 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) { CeedPragmaSIMD for (CeedInt j = 0; j < M; j++) B[i * M + j] += D[i] * A[i * M + j]; } 92 break; 93 case CEED_TRANSPOSE: 94 CeedPragmaSIMD for (CeedInt i = 0; i < M; i++) { CeedPragmaSIMD for (CeedInt j = 0; j < N; j++) B[i * N + j] += D[i] * A[j * M + i]; } 95 break; 96 } 97 } 98 99 // @brief 3x3 Matrix-Matrix product, B = DA + B, where D is diagonal. 100 // @details Optionally, A may be transposed. 101 CEED_QFUNCTION_HELPER void MatDiag3(const CeedScalar A[3][3], const CeedScalar D[3], const CeedTransposeMode transpose_A, CeedScalar B[3][3]) { 102 MatDiagNM((const CeedScalar *)A, (const CeedScalar *)D, 3, 3, transpose_A, (CeedScalar *)B); 103 } 104 // @brief NxN Matrix-Matrix product, C = AB + C 105 CEED_QFUNCTION_HELPER void MatMatN(const CeedScalar *A, const CeedScalar *B, const CeedInt N, const CeedTransposeMode transpose_A, 106 const CeedTransposeMode transpose_B, CeedScalar *C) { 107 switch (transpose_A) { 108 case CEED_NOTRANSPOSE: 109 switch (transpose_B) { 110 case CEED_NOTRANSPOSE: 111 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) { 112 CeedPragmaSIMD for (CeedInt j = 0; j < N; j++) { 113 CeedPragmaSIMD for (CeedInt k = 0; k < N; k++) C[i * N + j] += A[i * N + k] * B[k * N + j]; 114 } 115 } 116 break; 117 case CEED_TRANSPOSE: 118 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) { 119 CeedPragmaSIMD for (CeedInt j = 0; j < N; j++) { 120 CeedPragmaSIMD for (CeedInt k = 0; k < N; k++) C[i * N + j] += A[i * N + k] * B[j * N + k]; 121 } 122 } 123 break; 124 } 125 break; 126 case CEED_TRANSPOSE: 127 switch (transpose_B) { 128 case CEED_NOTRANSPOSE: 129 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) { 130 CeedPragmaSIMD for (CeedInt j = 0; j < N; j++) { 131 CeedPragmaSIMD for (CeedInt k = 0; k < N; k++) C[i * N + j] += A[k * N + i] * B[k * N + j]; 132 } 133 } 134 break; 135 case CEED_TRANSPOSE: 136 CeedPragmaSIMD for (CeedInt i = 0; i < N; i++) { 137 CeedPragmaSIMD for (CeedInt j = 0; j < N; j++) { 138 CeedPragmaSIMD for (CeedInt k = 0; k < N; k++) C[i * N + j] += A[k * N + i] * B[j * N + k]; 139 } 140 } 141 break; 142 } 143 break; 144 } 145 } 146 147 // @brief 3x3 Matrix-Matrix product, C = AB + C 148 CEED_QFUNCTION_HELPER void MatMat3(const CeedScalar A[3][3], const CeedScalar B[3][3], const CeedTransposeMode transpose_A, 149 const CeedTransposeMode transpose_B, CeedScalar C[3][3]) { 150 MatMatN((const CeedScalar *)A, (const CeedScalar *)B, 3, transpose_A, transpose_B, (CeedScalar *)C); 151 } 152 153 /** 154 @brief MxN Matrix-Matrix product, C = AB + C 155 156 C is NxM, A is NxP, B is PxM 157 158 @param[in] mat_A Row-major matrix `A` 159 @param[in] mat_B Row-major matrix `B` 160 @param[out] mat_C Row-major output matrix `C` 161 @param[in] N Number of rows of `C` 162 @param[in] M Number of columns of `C` 163 @param[in] P Number of columns of `A`/rows of `B` 164 **/ 165 CEED_QFUNCTION_HELPER void MatMatNM(const CeedScalar *mat_A, const CeedScalar *mat_B, CeedScalar *mat_C, CeedInt N, CeedInt M, CeedInt P) { 166 for (CeedInt i = 0; i < N; i++) { 167 for (CeedInt j = 0; j < M; j++) { 168 for (CeedInt k = 0; k < P; k++) mat_C[i * M + j] += mat_A[i * P + k] * mat_B[k * M + j]; 169 } 170 } 171 } 172 173 // @brief Unpack Kelvin-Mandel notation symmetric tensor into full tensor 174 CEED_QFUNCTION_HELPER void KMUnpack(const CeedScalar v[6], CeedScalar A[3][3]) { 175 const CeedScalar weight = 1 / sqrt(2.); 176 A[0][0] = v[0]; 177 A[1][1] = v[1]; 178 A[2][2] = v[2]; 179 A[2][1] = A[1][2] = weight * v[3]; 180 A[2][0] = A[0][2] = weight * v[4]; 181 A[1][0] = A[0][1] = weight * v[5]; 182 } 183 184 // @brief Pack full tensor into Kelvin-Mandel notation symmetric tensor 185 CEED_QFUNCTION_HELPER void KMPack(const CeedScalar A[3][3], CeedScalar v[6]) { 186 const CeedScalar weight = sqrt(2.); 187 v[0] = A[0][0]; 188 v[1] = A[1][1]; 189 v[2] = A[2][2]; 190 v[3] = A[2][1] * weight; 191 v[4] = A[2][0] * weight; 192 v[5] = A[1][0] * weight; 193 } 194 195 // @brief Calculate metric tensor from mapping, g_{ij} = xi_{k,i} xi_{k,j} = dXdx^T dXdx 196 CEED_QFUNCTION_HELPER void KMMetricTensor(const CeedScalar dXdx[3][3], CeedScalar km_g_ij[6]) { 197 CeedScalar g_ij[3][3] = {{0.}}; 198 MatMat3(dXdx, dXdx, CEED_TRANSPOSE, CEED_NOTRANSPOSE, g_ij); 199 KMPack(g_ij, km_g_ij); 200 } 201 202 // @brief Linear ramp evaluation 203 CEED_QFUNCTION_HELPER CeedScalar LinearRampCoefficient(CeedScalar amplitude, CeedScalar length, CeedScalar start, CeedScalar x) { 204 if (x < start) { 205 return amplitude; 206 } else if (x < start + length) { 207 return amplitude * ((x - start) * (-1 / length) + 1); 208 } else { 209 return 0; 210 } 211 } 212 213 /** 214 @brief Pack stored values at quadrature point 215 216 @param[in] Q Number of quadrature points 217 @param[in] i Current quadrature point 218 @param[in] start Starting index to store components 219 @param[in] num_comp Number of components to store 220 @param[in] values_at_qpnt Local values for quadrature point i 221 @param[out] stored Stored values 222 223 @return An error code: 0 - success, otherwise - failure 224 **/ 225 CEED_QFUNCTION_HELPER int StoredValuesPack(CeedInt Q, CeedInt i, CeedInt start, CeedInt num_comp, const CeedScalar *values_at_qpnt, 226 CeedScalar *stored) { 227 for (CeedInt j = 0; j < num_comp; j++) stored[(start + j) * Q + i] = values_at_qpnt[j]; 228 229 return CEED_ERROR_SUCCESS; 230 } 231 232 /** 233 @brief Unpack stored values at quadrature point 234 235 @param[in] Q Number of quadrature points 236 @param[in] i Current quadrature point 237 @param[in] start Starting index to store components 238 @param[in] num_comp Number of components to store 239 @param[in] stored Stored values 240 @param[out] values_at_qpnt Local values for quadrature point i 241 242 @return An error code: 0 - success, otherwise - failure 243 **/ 244 CEED_QFUNCTION_HELPER int StoredValuesUnpack(CeedInt Q, CeedInt i, CeedInt start, CeedInt num_comp, const CeedScalar *stored, 245 CeedScalar *values_at_qpnt) { 246 for (CeedInt j = 0; j < num_comp; j++) values_at_qpnt[j] = stored[(start + j) * Q + i]; 247 248 return CEED_ERROR_SUCCESS; 249 } 250 251 /** 252 @brief Unpack N-D element q_data at quadrature point 253 254 @param[in] dim Dimension of the element 255 @param[in] Q Number of quadrature points 256 @param[in] i Current quadrature point 257 @param[in] q_data Pointer to q_data (generated by `setupgeo.h:Setup`) 258 @param[out] wdetJ Quadrature weight times determinant of the mapping Jacobian, or `NULL` 259 @param[out] dXdx Inverse of the mapping Jacobian (shape [dim][dim]), or `NULL` 260 261 @return An error code: 0 - success, otherwise - failure 262 **/ 263 CEED_QFUNCTION_HELPER int QdataUnpack_ND(CeedInt dim, CeedInt Q, CeedInt i, const CeedScalar *q_data, CeedScalar *wdetJ, CeedScalar *dXdx) { 264 switch (dim) { 265 case 2: 266 if (wdetJ) StoredValuesUnpack(Q, i, 0, 1, q_data, wdetJ); 267 if (dXdx) StoredValuesUnpack(Q, i, 1, 4, q_data, dXdx); 268 break; 269 case 3: 270 if (wdetJ) StoredValuesUnpack(Q, i, 0, 1, q_data, wdetJ); 271 if (dXdx) StoredValuesUnpack(Q, i, 1, 9, q_data, dXdx); 272 break; 273 } 274 return CEED_ERROR_SUCCESS; 275 } 276 277 /** 278 @brief Unpack boundary element q_data for N-D problem at quadrature point 279 280 @param[in] dim Dimension of the element 281 @param[in] Q Number of quadrature points 282 @param[in] i Current quadrature point 283 @param[in] q_data Pointer to q_data (generated by `setupgeo.h:SetupBoundary`) 284 @param[out] wdetJ Quadrature weight times determinant of the mapping Jacobian, or `NULL` 285 @param[out] dXdx Inverse of the mapping Jacobian (shape [dim - 1][dim]), or `NULL` 286 @param[out] normal Components of the normal vector (shape [dim]), or `NULL` 287 288 @return An error code: 0 - success, otherwise - failure 289 **/ 290 CEED_QFUNCTION_HELPER int QdataBoundaryUnpack_ND(CeedInt dim, CeedInt Q, CeedInt i, const CeedScalar *q_data, CeedScalar *wdetJ, CeedScalar *dXdx, 291 CeedScalar *normal) { 292 switch (dim) { 293 case 2: 294 if (wdetJ) StoredValuesUnpack(Q, i, 0, 1, q_data, wdetJ); 295 if (normal) StoredValuesUnpack(Q, i, 1, 2, q_data, normal); 296 break; 297 case 3: 298 if (wdetJ) StoredValuesUnpack(Q, i, 0, 1, q_data, wdetJ); 299 if (normal) StoredValuesUnpack(Q, i, 1, 3, q_data, normal); 300 if (dXdx) StoredValuesUnpack(Q, i, 4, 6, q_data, (CeedScalar *)dXdx); 301 break; 302 } 303 return CEED_ERROR_SUCCESS; 304 } 305 306 /** 307 @brief Unpack 3D element q_data at quadrature point 308 309 @param[in] Q Number of quadrature points 310 @param[in] i Current quadrature point 311 @param[in] q_data Pointer to q_data (generated by `setupgeo.h:Setup`) 312 @param[out] wdetJ Quadrature weight times determinant of the mapping Jacobian 313 @param[out] dXdx Inverse of the mapping Jacobian (shape [3][3]) 314 315 @return An error code: 0 - success, otherwise - failure 316 **/ 317 CEED_QFUNCTION_HELPER int QdataUnpack_3D(CeedInt Q, CeedInt i, const CeedScalar *q_data, CeedScalar *wdetJ, CeedScalar dXdx[3][3]) { 318 return QdataUnpack_ND(3, Q, i, q_data, wdetJ, (CeedScalar *)dXdx); 319 } 320 321 /** 322 @brief Unpack boundary element q_data for 3D problem at quadrature point 323 324 @param[in] Q Number of quadrature points 325 @param[in] i Current quadrature point 326 @param[in] q_data Pointer to q_data (generated by `setupgeo.h:SetupBoundary`) 327 @param[out] wdetJ Quadrature weight times determinant of the mapping Jacobian, or `NULL` 328 @param[out] dXdx Inverse of the mapping Jacobian (shape [2][3]), or `NULL` 329 @param[out] normal Components of the normal vector (shape [3]), or `NULL` 330 331 @return An error code: 0 - success, otherwise - failure 332 **/ 333 CEED_QFUNCTION_HELPER int QdataBoundaryUnpack_3D(CeedInt Q, CeedInt i, const CeedScalar *q_data, CeedScalar *wdetJ, CeedScalar dXdx[2][3], 334 CeedScalar normal[3]) { 335 return QdataBoundaryUnpack_ND(3, Q, i, q_data, wdetJ, (CeedScalar *)dXdx, normal); 336 } 337 338 /** 339 @brief Unpack boundary element q_data for 3D problem at quadrature point 340 341 @param[in] Q Number of quadrature points 342 @param[in] i Current quadrature point 343 @param[in] q_data Pointer to q_data (generated by `setupgeo.h:SetupBoundary`) 344 @param[out] wdetJ Quadrature weight times determinant of the mapping Jacobian, or `NULL` 345 @param[out] dXdx Inverse of the mapping Jacobian (shape [3][3]), or `NULL` 346 @param[out] normal Components of the normal vector (shape [3]), or `NULL` 347 348 @return An error code: 0 - success, otherwise - failure 349 **/ 350 CEED_QFUNCTION_HELPER int QdataBoundaryGradientUnpack_3D(CeedInt Q, CeedInt i, const CeedScalar *q_data, CeedScalar *wdetJ, CeedScalar dXdx[3][3], 351 CeedScalar normal[3]) { 352 if (wdetJ) StoredValuesUnpack(Q, i, 0, 1, q_data, wdetJ); 353 if (dXdx) StoredValuesUnpack(Q, i, 1, 9, q_data, (CeedScalar *)dXdx); 354 if (normal) StoredValuesUnpack(Q, i, 10, 3, q_data, normal); 355 return CEED_ERROR_SUCCESS; 356 } 357 358 /** 359 @brief Unpack 2D element q_data at quadrature point 360 361 @param[in] Q Number of quadrature points 362 @param[in] i Current quadrature point 363 @param[in] q_data Pointer to q_data (generated by `setupgeo.h:Setup`) 364 @param[out] wdetJ Quadrature weight times determinant of the mapping Jacobian 365 @param[out] dXdx Inverse of the mapping Jacobian (shape [2][2]) 366 367 @return An error code: 0 - success, otherwise - failure 368 **/ 369 CEED_QFUNCTION_HELPER int QdataUnpack_2D(CeedInt Q, CeedInt i, const CeedScalar *q_data, CeedScalar *wdetJ, CeedScalar dXdx[2][2]) { 370 QdataUnpack_ND(2, Q, i, q_data, wdetJ, (CeedScalar *)dXdx); 371 return CEED_ERROR_SUCCESS; 372 } 373 374 /** 375 @brief Unpack boundary element q_data for 2D problem at quadrature point 376 377 @param[in] Q Number of quadrature points 378 @param[in] i Current quadrature point 379 @param[in] q_data Pointer to q_data (generated by `setupgeo.h:SetupBoundary2d`) 380 @param[out] wdetJ Quadrature weight times determinant of the mapping Jacobian, or `NULL` 381 @param[out] normal Components of the normal vector (shape [2]), or `NULL` 382 383 @return An error code: 0 - success, otherwise - failure 384 **/ 385 CEED_QFUNCTION_HELPER int QdataBoundaryUnpack_2D(CeedInt Q, CeedInt i, const CeedScalar *q_data, CeedScalar *wdetJ, CeedScalar normal[2]) { 386 QdataBoundaryUnpack_ND(3, Q, i, q_data, wdetJ, NULL, normal); 387 return CEED_ERROR_SUCCESS; 388 } 389 390 /** 391 @brief Unpack `CEED_EVAL_GRAD` QF input into quadrature-point local array 392 393 @param[in] Q Number of quadrature points 394 @param[in] i Current quadrature point 395 @param[in] num_comp Number of components of the input 396 @param[in] dim Topological dimension of the element (ie. number of derivative terms per component) 397 @param[in] grad QF gradient input, shape `[dim][num_comp][Q]` 398 @param[out] local Gradient array at quadrature point Q, shape `[num_comp][dim]` 399 **/ 400 CEED_QFUNCTION_HELPER void GradUnpackN(CeedInt Q, CeedInt i, CeedInt num_comp, CeedInt dim, const CeedScalar *grad, CeedScalar *local) { 401 for (CeedInt d = 0; d < dim; d++) { 402 for (CeedInt c = 0; c < num_comp; c++) { 403 local[dim * c + d] = grad[(Q * num_comp) * d + Q * c + i]; 404 } 405 } 406 } 407 408 /** 409 @brief Unpack `CEED_EVAL_GRAD` QF input into quadrature-point local array for 3D elements 410 411 @param[in] Q Number of quadrature points 412 @param[in] i Current quadrature point 413 @param[in] num_comp Number of components of the input 414 @param[in] grad QF gradient input, shape `[dim][num_comp][Q]` 415 @param[out] local Gradient array at quadrature point Q, shape `[num_comp][dim]` 416 **/ 417 CEED_QFUNCTION_HELPER void GradUnpack3(CeedInt Q, CeedInt i, CeedInt num_comp, const CeedScalar *grad, CeedScalar (*local)[3]) { 418 GradUnpackN(Q, i, num_comp, 3, grad, (CeedScalar *)local); 419 } 420 421 /** 422 @brief Calculate divergence from reference gradient 423 424 Given gradient array G_{ij} and inverse element mapping X_{ij}, then the divergence is 425 426 G_{ij} X{ji} 427 428 @param[in] grad_qn Gradient array, orientation [vector component][gradient direction] 429 @param[in] dXdx Inverse of the mapping Jacobian (shape [dim][dim]) 430 @param[in] dim Dimension of the problem 431 @param[out] divergence The divergence 432 **/ 433 CEED_QFUNCTION_HELPER void DivergenceND(const CeedScalar *grad_qn, const CeedScalar *dXdx, const CeedInt dim, CeedScalar *divergence) { 434 for (CeedInt i = 0; i < dim; i++) { 435 for (CeedInt j = 0; j < dim; j++) { 436 *divergence += grad_qn[i * dim + j] * dXdx[j * dim + i]; 437 } 438 } 439 } 440 441 /** 442 @brief Calculate divergence from reference gradient for 3D problem 443 444 Given gradient array G_{ij} and inverse element mapping X_{ij}, then the divergence is 445 446 G_{ij} X{ji} 447 448 @param[in] grad_qn Gradient array, orientation [vector component][gradient direction] 449 @param[in] dXdx Inverse of the mapping Jacobian (shape [3][3]) 450 @param[out] divergence The divergence 451 **/ 452 CEED_QFUNCTION_HELPER void Divergence3D(const CeedScalar grad_qn[3][3], const CeedScalar dXdx[3][3], CeedScalar *divergence) { 453 DivergenceND((const CeedScalar *)grad_qn, (const CeedScalar *)dXdx, 3, divergence); 454 } 455