1 // Copyright (c) 2017-2022, 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 /// Euler traveling vortex initial condition and operator for Navier-Stokes 10 /// example using PETSc 11 12 // Model from: 13 // On the Order of Accuracy and Numerical Performance of Two Classes of 14 // Finite Volume WENO Schemes, Zhang, Zhang, and Shu (2011). 15 16 #ifndef eulervortex_h 17 #define eulervortex_h 18 19 #include <math.h> 20 #include <ceed.h> 21 22 #ifndef M_PI 23 #define M_PI 3.14159265358979323846 24 #endif 25 26 #ifndef euler_context_struct 27 #define euler_context_struct 28 typedef struct EulerContext_ *EulerContext; 29 struct EulerContext_ { 30 CeedScalar center[3]; 31 CeedScalar curr_time; 32 CeedScalar vortex_strength; 33 CeedScalar c_tau; 34 CeedScalar mean_velocity[3]; 35 bool implicit; 36 int euler_test; 37 int stabilization; // See StabilizationType: 0=none, 1=SU, 2=SUPG 38 }; 39 #endif 40 41 // ***************************************************************************** 42 // This function sets the initial conditions 43 // 44 // Temperature: 45 // T = 1 - (gamma - 1) vortex_strength**2 exp(1 - r**2) / (8 gamma pi**2) 46 // Density: 47 // rho = (T/S_vortex)^(1 / (gamma - 1)) 48 // Pressure: 49 // P = rho * T 50 // Velocity: 51 // ui = 1 + vortex_strength exp((1 - r**2)/2.) [yc - y, x - xc] / (2 pi) 52 // r = sqrt( (x - xc)**2 + (y - yc)**2 ) 53 // Velocity/Momentum Density: 54 // Ui = rho ui 55 // Total Energy: 56 // E = P / (gamma - 1) + rho (u u)/2 57 // 58 // Constants: 59 // cv , Specific heat, constant volume 60 // cp , Specific heat, constant pressure 61 // vortex_strength , Strength of vortex 62 // center , Location of bubble center 63 // gamma = cp / cv, Specific heat ratio 64 // 65 // ***************************************************************************** 66 67 // ***************************************************************************** 68 // This helper function provides support for the exact, time-dependent solution 69 // (currently not implemented) and IC formulation for Euler traveling vortex 70 // ***************************************************************************** 71 CEED_QFUNCTION_HELPER int Exact_Euler(CeedInt dim, CeedScalar time, 72 const CeedScalar X[], CeedInt Nf, CeedScalar q[], 73 void *ctx) { 74 // Context 75 const EulerContext context = (EulerContext)ctx; 76 const CeedScalar vortex_strength = context->vortex_strength; 77 const CeedScalar *center = context->center; // Center of the domain 78 const CeedScalar *mean_velocity = context->mean_velocity; 79 80 // Setup 81 const CeedScalar gamma = 1.4; 82 const CeedScalar cv = 2.5; 83 const CeedScalar R = 1.; 84 const CeedScalar x = X[0], y = X[1]; // Coordinates 85 // Vortex center 86 const CeedScalar xc = center[0] + mean_velocity[0] * time; 87 const CeedScalar yc = center[1] + mean_velocity[1] * time; 88 89 const CeedScalar x0 = x - xc; 90 const CeedScalar y0 = y - yc; 91 const CeedScalar r = sqrt( x0*x0 + y0*y0 ); 92 const CeedScalar C = vortex_strength * exp((1. - r*r)/2.) / (2. * M_PI); 93 const CeedScalar delta_T = - (gamma - 1.) * vortex_strength * vortex_strength * 94 exp(1 - r*r) / (8. * gamma * M_PI * M_PI); 95 const CeedScalar S_vortex = 1; // no perturbation in the entropy P / rho^gamma 96 const CeedScalar S_bubble = (gamma - 1.) * vortex_strength * vortex_strength / 97 (8.*gamma*M_PI*M_PI); 98 CeedScalar rho, P, T, E, u[3] = {0.}; 99 100 // Initial Conditions 101 switch (context->euler_test) { 102 case 0: // Traveling vortex 103 T = 1 + delta_T; 104 // P = rho * T 105 // P = S * rho^gamma 106 // Solve for rho, then substitute for P 107 rho = pow(T/S_vortex, 1 / (gamma - 1.)); 108 P = rho * T; 109 u[0] = mean_velocity[0] - C*y0; 110 u[1] = mean_velocity[1] + C*x0; 111 112 // Assign exact solution 113 q[0] = rho; 114 q[1] = rho * u[0]; 115 q[2] = rho * u[1]; 116 q[3] = rho * u[2]; 117 q[4] = P / (gamma - 1.) + rho * (u[0]*u[0] + u[1]*u[1]) / 2.; 118 break; 119 case 1: // Constant zero velocity, density constant, total energy constant 120 rho = 1.; 121 E = 2.; 122 123 // Assign exact solution 124 q[0] = rho; 125 q[1] = rho * u[0]; 126 q[2] = rho * u[1]; 127 q[3] = rho * u[2]; 128 q[4] = E; 129 break; 130 case 2: // Constant nonzero velocity, density constant, total energy constant 131 rho = 1.; 132 E = 2.; 133 u[0] = mean_velocity[0]; 134 u[1] = mean_velocity[1]; 135 136 // Assign exact solution 137 q[0] = rho; 138 q[1] = rho * u[0]; 139 q[2] = rho * u[1]; 140 q[3] = rho * u[2]; 141 q[4] = E; 142 break; 143 case 3: // Velocity zero, pressure constant 144 // (so density and internal energy will be non-constant), 145 // but the velocity should stay zero and the bubble won't diffuse 146 // (for Euler, where there is no thermal conductivity) 147 P = 1.; 148 T = 1. - S_bubble * exp(1. - r*r); 149 rho = P / (R*T); 150 151 // Assign exact solution 152 q[0] = rho; 153 q[1] = rho * u[0]; 154 q[2] = rho * u[1]; 155 q[3] = rho * u[2]; 156 q[4] = rho * (cv * T + (u[0]*u[0] + u[1]*u[1])/2.); 157 break; 158 case 4: // Constant nonzero velocity, pressure constant 159 // (so density and internal energy will be non-constant), 160 // it should be transported across the domain, but velocity stays constant 161 P = 1.; 162 T = 1. - S_bubble * exp(1. - r*r); 163 rho = P / (R*T); 164 u[0] = mean_velocity[0]; 165 u[1] = mean_velocity[1]; 166 167 // Assign exact solution 168 q[0] = rho; 169 q[1] = rho * u[0]; 170 q[2] = rho * u[1]; 171 q[3] = rho * u[2]; 172 q[4] = rho * (cv * T + (u[0]*u[0] + u[1]*u[1])/2.); 173 break; 174 case 5: // non-smooth thermal bubble - cylinder 175 P = 1.; 176 T = 1. - (r < 1. ? S_bubble : 0.); 177 rho = P / (R*T); 178 u[0] = mean_velocity[0]; 179 u[1] = mean_velocity[1]; 180 181 // Assign exact solution 182 q[0] = rho; 183 q[1] = rho * u[0]; 184 q[2] = rho * u[1]; 185 q[3] = rho * u[2]; 186 q[4] = rho * (cv * T + (u[0]*u[0] + u[1]*u[1])/2.); 187 break; 188 } 189 // Return 190 return 0; 191 } 192 193 // ***************************************************************************** 194 // Helper function for computing flux Jacobian 195 // ***************************************************************************** 196 CEED_QFUNCTION_HELPER void ConvectiveFluxJacobian_Euler(CeedScalar dF[3][5][5], 197 const CeedScalar rho, const CeedScalar u[3], const CeedScalar E, 198 const CeedScalar gamma) { 199 CeedScalar u_sq = u[0]*u[0] + u[1]*u[1] + u[2]*u[2]; // Velocity square 200 for (CeedInt i=0; i<3; i++) { // Jacobian matrices for 3 directions 201 for (CeedInt j=0; j<3; j++) { // Rows of each Jacobian matrix 202 dF[i][j+1][0] = ((i==j) ? ((gamma-1.)*(u_sq/2.)) : 0.) - u[i]*u[j]; 203 for (CeedInt k=0; k<3; k++) { // Columns of each Jacobian matrix 204 dF[i][0][k+1] = ((i==k) ? 1. : 0.); 205 dF[i][j+1][k+1] = ((j==k) ? u[i] : 0.) + 206 ((i==k) ? u[j] : 0.) - 207 ((i==j) ? u[k] : 0.) * (gamma-1.); 208 dF[i][4][k+1] = ((i==k) ? (E*gamma/rho - (gamma-1.)*u_sq/2.) : 0.) - 209 (gamma-1.)*u[i]*u[k]; 210 } 211 dF[i][j+1][4] = ((i==j) ? (gamma-1.) : 0.); 212 } 213 dF[i][4][0] = u[i] * ((gamma-1.)*u_sq - E*gamma/rho); 214 dF[i][4][4] = u[i] * gamma; 215 } 216 } 217 218 // ***************************************************************************** 219 // Helper function for computing Tau elements (stabilization constant) 220 // Model from: 221 // Stabilized Methods for Compressible Flows, Hughes et al 2010 222 // 223 // Spatial criterion #2 - Tau is a 3x3 diagonal matrix 224 // Tau[i] = c_tau h[i] Xi(Pe) / rho(A[i]) (no sum) 225 // 226 // Where 227 // c_tau = stabilization constant (0.5 is reported as "optimal") 228 // h[i] = 2 length(dxdX[i]) 229 // Pe = Peclet number ( Pe = sqrt(u u) / dot(dXdx,u) diffusivity ) 230 // Xi(Pe) = coth Pe - 1. / Pe (1. at large local Peclet number ) 231 // rho(A[i]) = spectral radius of the convective flux Jacobian i, 232 // wave speed in direction i 233 // ***************************************************************************** 234 CEED_QFUNCTION_HELPER void Tau_spatial(CeedScalar Tau_x[3], 235 const CeedScalar dXdx[3][3], const CeedScalar u[3], 236 const CeedScalar sound_speed, const CeedScalar c_tau) { 237 for (int i=0; i<3; i++) { 238 // length of element in direction i 239 CeedScalar h = 2 / sqrt(dXdx[0][i]*dXdx[0][i] + dXdx[1][i]*dXdx[1][i] + 240 dXdx[2][i]*dXdx[2][i]); 241 // fastest wave in direction i 242 CeedScalar fastest_wave = fabs(u[i]) + sound_speed; 243 Tau_x[i] = c_tau * h / fastest_wave; 244 } 245 } 246 247 // ***************************************************************************** 248 // This QFunction sets the initial conditions for Euler traveling vortex 249 // ***************************************************************************** 250 CEED_QFUNCTION(ICsEuler)(void *ctx, CeedInt Q, 251 const CeedScalar *const *in, CeedScalar *const *out) { 252 // Inputs 253 const CeedScalar (*X)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0]; 254 255 // Outputs 256 CeedScalar (*q0)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0]; 257 const EulerContext context = (EulerContext)ctx; 258 259 CeedPragmaSIMD 260 // Quadrature Point Loop 261 for (CeedInt i=0; i<Q; i++) { 262 const CeedScalar x[] = {X[0][i], X[1][i], X[2][i]}; 263 CeedScalar q[5] = {0.}; 264 265 Exact_Euler(3, context->curr_time, x, 5, q, ctx); 266 267 for (CeedInt j=0; j<5; j++) 268 q0[j][i] = q[j]; 269 } // End of Quadrature Point Loop 270 271 // Return 272 return 0; 273 } 274 275 // ***************************************************************************** 276 // This QFunction implements the following formulation of Euler equations 277 // with explicit time stepping method 278 // 279 // This is 3D Euler for compressible gas dynamics in conservation 280 // form with state variables of density, momentum density, and total 281 // energy density. 282 // 283 // State Variables: q = ( rho, U1, U2, U3, E ) 284 // rho - Mass Density 285 // Ui - Momentum Density, Ui = rho ui 286 // E - Total Energy Density, E = P / (gamma - 1) + rho (u u)/2 287 // 288 // Euler Equations: 289 // drho/dt + div( U ) = 0 290 // dU/dt + div( rho (u x u) + P I3 ) = 0 291 // dE/dt + div( (E + P) u ) = 0 292 // 293 // Equation of State: 294 // P = (gamma - 1) (E - rho (u u) / 2) 295 // 296 // Constants: 297 // cv , Specific heat, constant volume 298 // cp , Specific heat, constant pressure 299 // g , Gravity 300 // gamma = cp / cv, Specific heat ratio 301 // ***************************************************************************** 302 CEED_QFUNCTION(Euler)(void *ctx, CeedInt Q, 303 const CeedScalar *const *in, CeedScalar *const *out) { 304 // *INDENT-OFF* 305 // Inputs 306 const CeedScalar (*q)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0], 307 (*dq)[5][CEED_Q_VLA] = (const CeedScalar(*)[5][CEED_Q_VLA])in[1], 308 (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[2]; 309 // Outputs 310 CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0], 311 (*dv)[5][CEED_Q_VLA] = (CeedScalar(*)[5][CEED_Q_VLA])out[1]; 312 313 EulerContext context = (EulerContext)ctx; 314 const CeedScalar c_tau = context->c_tau; 315 const CeedScalar gamma = 1.4; 316 317 CeedPragmaSIMD 318 // Quadrature Point Loop 319 for (CeedInt i=0; i<Q; i++) { 320 // *INDENT-OFF* 321 // Setup 322 // -- Interp in 323 const CeedScalar rho = q[0][i]; 324 const CeedScalar u[3] = {q[1][i] / rho, 325 q[2][i] / rho, 326 q[3][i] / rho 327 }; 328 const CeedScalar E = q[4][i]; 329 const CeedScalar drho[3] = {dq[0][0][i], 330 dq[1][0][i], 331 dq[2][0][i] 332 }; 333 const CeedScalar dU[3][3] = {{dq[0][1][i], 334 dq[1][1][i], 335 dq[2][1][i]}, 336 {dq[0][2][i], 337 dq[1][2][i], 338 dq[2][2][i]}, 339 {dq[0][3][i], 340 dq[1][3][i], 341 dq[2][3][i]} 342 }; 343 const CeedScalar dE[3] = {dq[0][4][i], 344 dq[1][4][i], 345 dq[2][4][i] 346 }; 347 // -- Interp-to-Interp q_data 348 const CeedScalar wdetJ = q_data[0][i]; 349 // -- Interp-to-Grad q_data 350 // ---- Inverse of change of coordinate matrix: X_i,j 351 // *INDENT-OFF* 352 const CeedScalar dXdx[3][3] = {{q_data[1][i], 353 q_data[2][i], 354 q_data[3][i]}, 355 {q_data[4][i], 356 q_data[5][i], 357 q_data[6][i]}, 358 {q_data[7][i], 359 q_data[8][i], 360 q_data[9][i]} 361 }; 362 // *INDENT-ON* 363 // dU/dx 364 CeedScalar drhodx[3] = {0.}; 365 CeedScalar dEdx[3] = {0.}; 366 CeedScalar dUdx[3][3] = {{0.}}; 367 CeedScalar dXdxdXdxT[3][3] = {{0.}}; 368 for (int j=0; j<3; j++) { 369 for (int k=0; k<3; k++) { 370 drhodx[j] += drho[k] * dXdx[k][j]; 371 dEdx[j] += dE[k] * dXdx[k][j]; 372 for (int l=0; l<3; l++) { 373 dUdx[j][k] += dU[j][l] * dXdx[l][k]; 374 dXdxdXdxT[j][k] += dXdx[j][l]*dXdx[k][l]; //dXdx_j,k * dXdx_k,j 375 } 376 } 377 } 378 // Pressure 379 const CeedScalar 380 E_kinetic = 0.5 * rho * (u[0]*u[0] + u[1]*u[1] + u[2]*u[2]), 381 E_internal = E - E_kinetic, 382 P = E_internal * (gamma - 1.); // P = pressure 383 384 // The Physics 385 // Zero v and dv so all future terms can safely sum into it 386 for (int j=0; j<5; j++) { 387 v[j][i] = 0.; 388 for (int k=0; k<3; k++) 389 dv[k][j][i] = 0.; 390 } 391 392 // -- Density 393 // ---- u rho 394 for (int j=0; j<3; j++) 395 dv[j][0][i] += wdetJ*(rho*u[0]*dXdx[j][0] + rho*u[1]*dXdx[j][1] + 396 rho*u[2]*dXdx[j][2]); 397 // -- Momentum 398 // ---- rho (u x u) + P I3 399 for (int j=0; j<3; j++) 400 for (int k=0; k<3; k++) 401 dv[k][j+1][i] += wdetJ*((rho*u[j]*u[0] + (j==0?P:0.))*dXdx[k][0] + 402 (rho*u[j]*u[1] + (j==1?P:0.))*dXdx[k][1] + 403 (rho*u[j]*u[2] + (j==2?P:0.))*dXdx[k][2]); 404 // -- Total Energy Density 405 // ---- (E + P) u 406 for (int j=0; j<3; j++) 407 dv[j][4][i] += wdetJ * (E + P) * (u[0]*dXdx[j][0] + u[1]*dXdx[j][1] + 408 u[2]*dXdx[j][2]); 409 410 // --Stabilization terms 411 // ---- jacob_F_conv[3][5][5] = dF(convective)/dq at each direction 412 CeedScalar jacob_F_conv[3][5][5] = {{{0.}}}; 413 ConvectiveFluxJacobian_Euler(jacob_F_conv, rho, u, E, gamma); 414 415 // ---- dqdx collects drhodx, dUdx and dEdx in one vector 416 CeedScalar dqdx[5][3]; 417 for (int j=0; j<3; j++) { 418 dqdx[0][j] = drhodx[j]; 419 dqdx[4][j] = dEdx[j]; 420 for (int k=0; k<3; k++) 421 dqdx[k+1][j] = dUdx[k][j]; 422 } 423 424 // ---- strong_conv = dF/dq * dq/dx (Strong convection) 425 CeedScalar strong_conv[5] = {0.}; 426 for (int j=0; j<3; j++) 427 for (int k=0; k<5; k++) 428 for (int l=0; l<5; l++) 429 strong_conv[k] += jacob_F_conv[j][k][l] * dqdx[l][j]; 430 431 // Stabilization 432 // -- Tau elements 433 const CeedScalar sound_speed = sqrt(gamma * P / rho); 434 CeedScalar Tau_x[3] = {0.}; 435 Tau_spatial(Tau_x, dXdx, u, sound_speed, c_tau); 436 437 // -- Stabilization method: none or SU 438 CeedScalar stab[5][3] = {{0.}}; 439 switch (context->stabilization) { 440 case 0: // Galerkin 441 break; 442 case 1: // SU 443 for (int j=0; j<3; j++) 444 for (int k=0; k<5; k++) 445 for (int l=0; l<5; l++) 446 stab[k][j] += jacob_F_conv[j][k][l] * Tau_x[j] * strong_conv[l]; 447 448 for (int j=0; j<5; j++) 449 for (int k=0; k<3; k++) 450 dv[k][j][i] -= wdetJ*(stab[j][0] * dXdx[k][0] + 451 stab[j][1] * dXdx[k][1] + 452 stab[j][2] * dXdx[k][2]); 453 break; 454 case 2: // SUPG is not implemented for explicit scheme 455 break; 456 } 457 458 } // End Quadrature Point Loop 459 460 // Return 461 return 0; 462 } 463 464 // ***************************************************************************** 465 // This QFunction implements the Euler equations with (mentioned above) 466 // with implicit time stepping method 467 // 468 // ***************************************************************************** 469 CEED_QFUNCTION(IFunction_Euler)(void *ctx, CeedInt Q, 470 const CeedScalar *const *in, CeedScalar *const *out) { 471 // *INDENT-OFF* 472 // Inputs 473 const CeedScalar (*q)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0], 474 (*dq)[5][CEED_Q_VLA] = (const CeedScalar(*)[5][CEED_Q_VLA])in[1], 475 (*q_dot)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[2], 476 (*q_data)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[3]; 477 // Outputs 478 CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0], 479 (*dv)[5][CEED_Q_VLA] = (CeedScalar(*)[5][CEED_Q_VLA])out[1]; 480 481 EulerContext context = (EulerContext)ctx; 482 const CeedScalar c_tau = context->c_tau; 483 const CeedScalar gamma = 1.4; 484 485 CeedPragmaSIMD 486 // Quadrature Point Loop 487 for (CeedInt i=0; i<Q; i++) { 488 // *INDENT-OFF* 489 // Setup 490 // -- Interp in 491 const CeedScalar rho = q[0][i]; 492 const CeedScalar u[3] = {q[1][i] / rho, 493 q[2][i] / rho, 494 q[3][i] / rho 495 }; 496 const CeedScalar E = q[4][i]; 497 const CeedScalar drho[3] = {dq[0][0][i], 498 dq[1][0][i], 499 dq[2][0][i] 500 }; 501 const CeedScalar dU[3][3] = {{dq[0][1][i], 502 dq[1][1][i], 503 dq[2][1][i]}, 504 {dq[0][2][i], 505 dq[1][2][i], 506 dq[2][2][i]}, 507 {dq[0][3][i], 508 dq[1][3][i], 509 dq[2][3][i]} 510 }; 511 const CeedScalar dE[3] = {dq[0][4][i], 512 dq[1][4][i], 513 dq[2][4][i] 514 }; 515 // -- Interp-to-Interp q_data 516 const CeedScalar wdetJ = q_data[0][i]; 517 // -- Interp-to-Grad q_data 518 // ---- Inverse of change of coordinate matrix: X_i,j 519 // *INDENT-OFF* 520 const CeedScalar dXdx[3][3] = {{q_data[1][i], 521 q_data[2][i], 522 q_data[3][i]}, 523 {q_data[4][i], 524 q_data[5][i], 525 q_data[6][i]}, 526 {q_data[7][i], 527 q_data[8][i], 528 q_data[9][i]} 529 }; 530 // *INDENT-ON* 531 // dU/dx 532 CeedScalar drhodx[3] = {0.}; 533 CeedScalar dEdx[3] = {0.}; 534 CeedScalar dUdx[3][3] = {{0.}}; 535 CeedScalar dXdxdXdxT[3][3] = {{0.}}; 536 for (int j=0; j<3; j++) { 537 for (int k=0; k<3; k++) { 538 drhodx[j] += drho[k] * dXdx[k][j]; 539 dEdx[j] += dE[k] * dXdx[k][j]; 540 for (int l=0; l<3; l++) { 541 dUdx[j][k] += dU[j][l] * dXdx[l][k]; 542 dXdxdXdxT[j][k] += dXdx[j][l]*dXdx[k][l]; //dXdx_j,k * dXdx_k,j 543 } 544 } 545 } 546 const CeedScalar 547 E_kinetic = 0.5 * rho * (u[0]*u[0] + u[1]*u[1] + u[2]*u[2]), 548 E_internal = E - E_kinetic, 549 P = E_internal * (gamma - 1.); // P = pressure 550 551 // The Physics 552 // Zero v and dv so all future terms can safely sum into it 553 for (int j=0; j<5; j++) { 554 v[j][i] = 0.; 555 for (int k=0; k<3; k++) 556 dv[k][j][i] = 0.; 557 } 558 //-----mass matrix 559 for (int j=0; j<5; j++) 560 v[j][i] += wdetJ*q_dot[j][i]; 561 562 // -- Density 563 // ---- u rho 564 for (int j=0; j<3; j++) 565 dv[j][0][i] -= wdetJ*(rho*u[0]*dXdx[j][0] + rho*u[1]*dXdx[j][1] + 566 rho*u[2]*dXdx[j][2]); 567 // -- Momentum 568 // ---- rho (u x u) + P I3 569 for (int j=0; j<3; j++) 570 for (int k=0; k<3; k++) 571 dv[k][j+1][i] -= wdetJ*((rho*u[j]*u[0] + (j==0?P:0.))*dXdx[k][0] + 572 (rho*u[j]*u[1] + (j==1?P:0.))*dXdx[k][1] + 573 (rho*u[j]*u[2] + (j==2?P:0.))*dXdx[k][2]); 574 // -- Total Energy Density 575 // ---- (E + P) u 576 for (int j=0; j<3; j++) 577 dv[j][4][i] -= wdetJ * (E + P) * (u[0]*dXdx[j][0] + u[1]*dXdx[j][1] + 578 u[2]*dXdx[j][2]); 579 580 // -- Stabilization terms 581 // ---- jacob_F_conv[3][5][5] = dF(convective)/dq at each direction 582 CeedScalar jacob_F_conv[3][5][5] = {{{0.}}}; 583 ConvectiveFluxJacobian_Euler(jacob_F_conv, rho, u, E, gamma); 584 585 // ---- dqdx collects drhodx, dUdx and dEdx in one vector 586 CeedScalar dqdx[5][3]; 587 for (int j=0; j<3; j++) { 588 dqdx[0][j] = drhodx[j]; 589 dqdx[4][j] = dEdx[j]; 590 for (int k=0; k<3; k++) 591 dqdx[k+1][j] = dUdx[k][j]; 592 } 593 594 // ---- strong_conv = dF/dq * dq/dx (Strong convection) 595 CeedScalar strong_conv[5] = {0.}; 596 for (int j=0; j<3; j++) 597 for (int k=0; k<5; k++) 598 for (int l=0; l<5; l++) 599 strong_conv[k] += jacob_F_conv[j][k][l] * dqdx[l][j]; 600 601 // ---- Strong residual 602 CeedScalar strong_res[5]; 603 for (int j=0; j<5; j++) 604 strong_res[j] = q_dot[j][i] + strong_conv[j]; 605 606 // Stabilization 607 // -- Tau elements 608 const CeedScalar sound_speed = sqrt(gamma * P / rho); 609 CeedScalar Tau_x[3] = {0.}; 610 Tau_spatial(Tau_x, dXdx, u, sound_speed, c_tau); 611 612 // -- Stabilization method: none, SU, or SUPG 613 CeedScalar stab[5][3] = {{0.}}; 614 switch (context->stabilization) { 615 case 0: // Galerkin 616 break; 617 case 1: // SU 618 for (int j=0; j<3; j++) 619 for (int k=0; k<5; k++) 620 for (int l=0; l<5; l++) 621 stab[k][j] += jacob_F_conv[j][k][l] * Tau_x[j] * strong_conv[l]; 622 623 for (int j=0; j<5; j++) 624 for (int k=0; k<3; k++) 625 dv[k][j][i] += wdetJ*(stab[j][0] * dXdx[k][0] + 626 stab[j][1] * dXdx[k][1] + 627 stab[j][2] * dXdx[k][2]); 628 break; 629 case 2: // SUPG 630 for (int j=0; j<3; j++) 631 for (int k=0; k<5; k++) 632 for (int l=0; l<5; l++) 633 stab[k][j] = jacob_F_conv[j][k][l] * Tau_x[j] * strong_res[l]; 634 635 for (int j=0; j<5; j++) 636 for (int k=0; k<3; k++) 637 dv[k][j][i] += wdetJ*(stab[j][0] * dXdx[k][0] + 638 stab[j][1] * dXdx[k][1] + 639 stab[j][2] * dXdx[k][2]); 640 break; 641 } 642 } // End Quadrature Point Loop 643 644 // Return 645 return 0; 646 } 647 // ***************************************************************************** 648 // This QFunction sets the inflow boundary conditions for 649 // the traveling vortex problem. 650 // 651 // Prescribed T_inlet and P_inlet are converted to conservative variables 652 // and applied weakly. 653 // 654 // ***************************************************************************** 655 CEED_QFUNCTION(TravelingVortex_Inflow)(void *ctx, CeedInt Q, 656 const CeedScalar *const *in, 657 CeedScalar *const *out) { 658 // *INDENT-OFF* 659 // Inputs 660 const CeedScalar (*q_data_sur)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[1]; 661 // Outputs 662 CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0]; 663 // *INDENT-ON* 664 EulerContext context = (EulerContext)ctx; 665 const int euler_test = context->euler_test; 666 const bool implicit = context->implicit; 667 CeedScalar *mean_velocity = context->mean_velocity; 668 const CeedScalar cv = 2.5; 669 const CeedScalar R = 1.; 670 CeedScalar T_inlet; 671 CeedScalar P_inlet; 672 673 // For test cases 1 and 3 the background velocity is zero 674 if (euler_test == 1 || euler_test == 3) 675 for (CeedInt i=0; i<3; i++) mean_velocity[i] = 0.; 676 677 // For test cases 1 and 2, T_inlet = T_inlet = 0.4 678 if (euler_test == 1 || euler_test == 2) T_inlet = P_inlet = .4; 679 else T_inlet = P_inlet = 1.; 680 681 CeedPragmaSIMD 682 // Quadrature Point Loop 683 for (CeedInt i=0; i<Q; i++) { 684 // Setup 685 // -- Interp-to-Interp q_data 686 // For explicit mode, the surface integral is on the RHS of ODE q_dot = f(q). 687 // For implicit mode, it gets pulled to the LHS of implicit ODE/DAE g(q_dot, q). 688 // We can effect this by swapping the sign on this weight 689 const CeedScalar wdetJb = (implicit ? -1. : 1.) * q_data_sur[0][i]; 690 // ---- Normal vect 691 const CeedScalar norm[3] = {q_data_sur[1][i], 692 q_data_sur[2][i], 693 q_data_sur[3][i] 694 }; 695 696 // face_normal = Normal vector of the face 697 const CeedScalar face_normal = norm[0]*mean_velocity[0] + 698 norm[1]*mean_velocity[1] + 699 norm[2]*mean_velocity[2]; 700 // The Physics 701 // Zero v so all future terms can safely sum into it 702 for (int j=0; j<5; j++) v[j][i] = 0.; 703 704 // Implementing in/outflow BCs 705 if (face_normal > 0) { 706 } else { // inflow 707 const CeedScalar rho_inlet = P_inlet/(R*T_inlet); 708 const CeedScalar E_kinetic_inlet = (mean_velocity[0]*mean_velocity[0] + 709 mean_velocity[1]*mean_velocity[1]) / 2.; 710 // incoming total energy 711 const CeedScalar E_inlet = rho_inlet * (cv * T_inlet + E_kinetic_inlet); 712 713 // The Physics 714 // -- Density 715 v[0][i] -= wdetJb * rho_inlet * face_normal; 716 717 // -- Momentum 718 for (int j=0; j<3; j++) 719 v[j+1][i] -= wdetJb *(rho_inlet * face_normal * mean_velocity[j] + 720 norm[j] * P_inlet); 721 722 // -- Total Energy Density 723 v[4][i] -= wdetJb * face_normal * (E_inlet + P_inlet); 724 } 725 726 } // End Quadrature Point Loop 727 return 0; 728 } 729 730 // ***************************************************************************** 731 // This QFunction sets the outflow boundary conditions for 732 // the Euler solver. 733 // 734 // Outflow BCs: 735 // The validity of the weak form of the governing equations is 736 // extended to the outflow. 737 // 738 // ***************************************************************************** 739 CEED_QFUNCTION(Euler_Outflow)(void *ctx, CeedInt Q, 740 const CeedScalar *const *in, 741 CeedScalar *const *out) { 742 // *INDENT-OFF* 743 // Inputs 744 const CeedScalar (*q)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0], 745 (*q_data_sur)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[1]; 746 // Outputs 747 CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0]; 748 // *INDENT-ON* 749 EulerContext context = (EulerContext)ctx; 750 const bool implicit = context->implicit; 751 CeedScalar *mean_velocity = context->mean_velocity; 752 753 const CeedScalar gamma = 1.4; 754 755 CeedPragmaSIMD 756 // Quadrature Point Loop 757 for (CeedInt i=0; i<Q; i++) { 758 // Setup 759 // -- Interp in 760 const CeedScalar rho = q[0][i]; 761 const CeedScalar u[3] = {q[1][i] / rho, 762 q[2][i] / rho, 763 q[3][i] / rho 764 }; 765 const CeedScalar E = q[4][i]; 766 767 // -- Interp-to-Interp q_data 768 // For explicit mode, the surface integral is on the RHS of ODE q_dot = f(q). 769 // For implicit mode, it gets pulled to the LHS of implicit ODE/DAE g(q_dot, q). 770 // We can effect this by swapping the sign on this weight 771 const CeedScalar wdetJb = (implicit ? -1. : 1.) * q_data_sur[0][i]; 772 // ---- Normal vectors 773 const CeedScalar norm[3] = {q_data_sur[1][i], 774 q_data_sur[2][i], 775 q_data_sur[3][i] 776 }; 777 778 // face_normal = Normal vector of the face 779 const CeedScalar face_normal = norm[0]*mean_velocity[0] + 780 norm[1]*mean_velocity[1] + 781 norm[2]*mean_velocity[2]; 782 // The Physics 783 // Zero v so all future terms can safely sum into it 784 for (int j=0; j<5; j++) v[j][i] = 0; 785 786 // Implementing in/outflow BCs 787 if (face_normal > 0) { // outflow 788 const CeedScalar E_kinetic = (u[0]*u[0] + u[1]*u[1]) / 2.; 789 const CeedScalar P = (E - E_kinetic * rho) * (gamma - 1.); // pressure 790 const CeedScalar u_normal = norm[0]*u[0] + norm[1]*u[1] + 791 norm[2]*u[2]; // Normal velocity 792 // The Physics 793 // -- Density 794 v[0][i] -= wdetJb * rho * u_normal; 795 796 // -- Momentum 797 for (int j=0; j<3; j++) 798 v[j+1][i] -= wdetJb *(rho * u_normal * u[j] + norm[j] * P); 799 800 // -- Total Energy Density 801 v[4][i] -= wdetJb * u_normal * (E + P); 802 } 803 } // End Quadrature Point Loop 804 return 0; 805 } 806 807 // ***************************************************************************** 808 809 #endif // eulervortex_h 810