1 static char help[] = "Transient nonlinear driven cavity in 2d.\n\ 2 \n\ 3 The 2D driven cavity problem is solved in a velocity-vorticity formulation.\n\ 4 The flow can be driven with the lid or with buoyancy or both:\n\ 5 -lidvelocity <lid>, where <lid> = dimensionless velocity of lid\n\ 6 -grashof <gr>, where <gr> = dimensionless temperature gradent\n\ 7 -prandtl <pr>, where <pr> = dimensionless thermal/momentum diffusity ratio\n\ 8 -contours : draw contour plots of solution\n\n"; 9 /* 10 See src/snes/tutorials/ex19.c for the steady-state version. 11 12 There used to be a SNES example (src/snes/tutorials/ex27.c) that 13 implemented this algorithm without using TS and was used for the numerical 14 results in the paper 15 16 Todd S. Coffey and C. T. Kelley and David E. Keyes, Pseudotransient 17 Continuation and Differential-Algebraic Equations, 2003. 18 19 That example was removed because it used obsolete interfaces, but the 20 algorithms from the paper can be reproduced using this example. 21 22 Note: The paper describes the algorithm as being linearly implicit but the 23 numerical results were created using nonlinearly implicit Euler. The 24 algorithm as described (linearly implicit) is more efficient and is the 25 default when using TSPSEUDO. If you want to reproduce the numerical 26 results from the paper, you'll have to change the SNES to converge the 27 nonlinear solve (e.g., -snes_type newtonls). The DAE versus ODE variants 28 are controlled using the -parabolic option. 29 30 Comment preserved from snes/tutorials/ex27.c, since removed: 31 32 [H]owever Figure 3.1 was generated with a slightly different algorithm 33 (see targets runex27 and runex27_p) than described in the paper. In 34 particular, the described algorithm is linearly implicit, advancing to 35 the next step after one Newton step, so that the steady state residual 36 is always used, but the figure was generated by converging each step to 37 a relative tolerance of 1.e-3. On the example problem, setting 38 -snes_type ksponly has only minor impact on number of steps, but 39 significantly reduces the required number of linear solves. 40 41 See also https://lists.mcs.anl.gov/pipermail/petsc-dev/2010-March/002362.html 42 */ 43 44 /* ------------------------------------------------------------------------ 45 46 We thank David E. Keyes for contributing the driven cavity discretization 47 within this example code. 48 49 This problem is modeled by the partial differential equation system 50 51 - Lap(U) - Grad_y(Omega) = 0 52 - Lap(V) + Grad_x(Omega) = 0 53 Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0 54 T_t - Lap(T) + PR*Div([U*T,V*T]) = 0 55 56 in the unit square, which is uniformly discretized in each of x and 57 y in this simple encoding. 58 59 No-slip, rigid-wall Dirichlet conditions are used for [U,V]. 60 Dirichlet conditions are used for Omega, based on the definition of 61 vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each 62 constant coordinate boundary, the tangential derivative is zero. 63 Dirichlet conditions are used for T on the left and right walls, 64 and insulation homogeneous Neumann conditions are used for T on 65 the top and bottom walls. 66 67 A finite difference approximation with the usual 5-point stencil 68 is used to discretize the boundary value problem to obtain a 69 nonlinear system of equations. Upwinding is used for the divergence 70 (convective) terms and central for the gradient (source) terms. 71 72 The Jacobian can be either 73 * formed via finite differencing using coloring (the default), or 74 * applied matrix-free via the option -snes_mf 75 (for larger grid problems this variant may not converge 76 without a preconditioner due to ill-conditioning). 77 78 ------------------------------------------------------------------------- */ 79 80 /* 81 Include "petscdmda.h" so that we can use distributed arrays (DMDAs). 82 Include "petscts.h" so that we can use TS solvers. Note that this 83 file automatically includes: 84 petscsys.h - base PETSc routines petscvec.h - vectors 85 petscmat.h - matrices 86 petscis.h - index sets petscksp.h - Krylov subspace methods 87 petscviewer.h - viewers petscpc.h - preconditioners 88 petscksp.h - linear solvers petscsnes.h - nonlinear solvers 89 */ 90 #include <petscts.h> 91 #include <petscdm.h> 92 #include <petscdmda.h> 93 94 /* 95 User-defined routines and data structures 96 */ 97 typedef struct { 98 PetscScalar u, v, omega, temp; 99 } Field; 100 101 PetscErrorCode FormIFunctionLocal(DMDALocalInfo *, PetscReal, Field **, Field **, Field **, void *); 102 103 typedef struct { 104 PetscReal lidvelocity, prandtl, grashof; /* physical parameters */ 105 PetscBool parabolic; /* allow a transient term corresponding roughly to artificial compressibility */ 106 PetscReal cfl_initial; /* CFL for first time step */ 107 } AppCtx; 108 109 PetscErrorCode FormInitialSolution(TS, Vec, AppCtx *); 110 111 int main(int argc, char **argv) 112 { 113 AppCtx user; /* user-defined work context */ 114 PetscInt mx, my, steps; 115 TS ts; 116 DM da; 117 Vec X; 118 PetscReal ftime; 119 TSConvergedReason reason; 120 121 PetscFunctionBeginUser; 122 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 123 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); 124 PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 4, 4, PETSC_DECIDE, PETSC_DECIDE, 4, 1, 0, 0, &da)); 125 PetscCall(DMSetFromOptions(da)); 126 PetscCall(DMSetUp(da)); 127 PetscCall(TSSetDM(ts, (DM)da)); 128 129 PetscCall(DMDAGetInfo(da, 0, &mx, &my, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE)); 130 /* 131 Problem parameters (velocity of lid, prandtl, and grashof numbers) 132 */ 133 user.lidvelocity = 1.0 / (mx * my); 134 user.prandtl = 1.0; 135 user.grashof = 1.0; 136 user.parabolic = PETSC_FALSE; 137 user.cfl_initial = 50.; 138 139 PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Driven cavity/natural convection options", ""); 140 PetscCall(PetscOptionsReal("-lidvelocity", "Lid velocity, related to Reynolds number", "", user.lidvelocity, &user.lidvelocity, NULL)); 141 PetscCall(PetscOptionsReal("-prandtl", "Ratio of viscous to thermal diffusivity", "", user.prandtl, &user.prandtl, NULL)); 142 PetscCall(PetscOptionsReal("-grashof", "Ratio of buoyant to viscous forces", "", user.grashof, &user.grashof, NULL)); 143 PetscCall(PetscOptionsBool("-parabolic", "Relax incompressibility to make the system parabolic instead of differential-algebraic", "", user.parabolic, &user.parabolic, NULL)); 144 PetscCall(PetscOptionsReal("-cfl_initial", "Advective CFL for the first time step", "", user.cfl_initial, &user.cfl_initial, NULL)); 145 PetscOptionsEnd(); 146 147 PetscCall(DMDASetFieldName(da, 0, "x-velocity")); 148 PetscCall(DMDASetFieldName(da, 1, "y-velocity")); 149 PetscCall(DMDASetFieldName(da, 2, "Omega")); 150 PetscCall(DMDASetFieldName(da, 3, "temperature")); 151 152 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 153 Create user context, set problem data, create vector data structures. 154 Also, compute the initial guess. 155 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 156 157 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 158 Create time integration context 159 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 160 PetscCall(DMSetApplicationContext(da, &user)); 161 PetscCall(DMDATSSetIFunctionLocal(da, INSERT_VALUES, (DMDATSIFunctionLocalFn *)FormIFunctionLocal, &user)); 162 PetscCall(TSSetMaxSteps(ts, 10000)); 163 PetscCall(TSSetMaxTime(ts, 1e12)); 164 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); 165 PetscCall(TSSetTimeStep(ts, user.cfl_initial / (user.lidvelocity * mx))); 166 PetscCall(TSSetFromOptions(ts)); 167 168 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%" PetscInt_FMT "x%" PetscInt_FMT " grid, lid velocity = %g, prandtl # = %g, grashof # = %g\n", mx, my, (double)user.lidvelocity, (double)user.prandtl, (double)user.grashof)); 169 170 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 171 Solve the nonlinear system 172 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 173 174 PetscCall(DMCreateGlobalVector(da, &X)); 175 PetscCall(FormInitialSolution(ts, X, &user)); 176 177 PetscCall(TSSolve(ts, X)); 178 PetscCall(TSGetSolveTime(ts, &ftime)); 179 PetscCall(TSGetStepNumber(ts, &steps)); 180 PetscCall(TSGetConvergedReason(ts, &reason)); 181 182 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%s at time %g after %" PetscInt_FMT " steps\n", TSConvergedReasons[reason], (double)ftime, steps)); 183 184 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 185 Free work space. All PETSc objects should be destroyed when they 186 are no longer needed. 187 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 188 PetscCall(VecDestroy(&X)); 189 PetscCall(DMDestroy(&da)); 190 PetscCall(TSDestroy(&ts)); 191 192 PetscCall(PetscFinalize()); 193 return 0; 194 } 195 196 /* ------------------------------------------------------------------- */ 197 198 /* 199 FormInitialSolution - Forms initial approximation. 200 201 Input Parameters: 202 user - user-defined application context 203 X - vector 204 205 Output Parameter: 206 X - vector 207 */ 208 PetscErrorCode FormInitialSolution(TS ts, Vec X, AppCtx *user) 209 { 210 DM da; 211 PetscInt i, j, mx, xs, ys, xm, ym; 212 PetscReal grashof, dx; 213 Field **x; 214 215 PetscFunctionBeginUser; 216 grashof = user->grashof; 217 PetscCall(TSGetDM(ts, &da)); 218 PetscCall(DMDAGetInfo(da, 0, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)); 219 dx = 1.0 / (mx - 1); 220 221 /* 222 Get local grid boundaries (for 2-dimensional DMDA): 223 xs, ys - starting grid indices (no ghost points) 224 xm, ym - widths of local grid (no ghost points) 225 */ 226 PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL)); 227 228 /* 229 Get a pointer to vector data. 230 - For default PETSc vectors, VecGetArray() returns a pointer to 231 the data array. Otherwise, the routine is implementation dependent. 232 - You MUST call VecRestoreArray() when you no longer need access to 233 the array. 234 */ 235 PetscCall(DMDAVecGetArray(da, X, &x)); 236 237 /* 238 Compute initial guess over the locally owned part of the grid 239 Initial condition is motionless fluid and equilibrium temperature 240 */ 241 for (j = ys; j < ys + ym; j++) { 242 for (i = xs; i < xs + xm; i++) { 243 x[j][i].u = 0.0; 244 x[j][i].v = 0.0; 245 x[j][i].omega = 0.0; 246 x[j][i].temp = (grashof > 0) * i * dx; 247 } 248 } 249 250 /* 251 Restore vector 252 */ 253 PetscCall(DMDAVecRestoreArray(da, X, &x)); 254 PetscFunctionReturn(PETSC_SUCCESS); 255 } 256 257 PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info, PetscReal ptime, Field **x, Field **xdot, Field **f, void *ptr) 258 { 259 AppCtx *user = (AppCtx *)ptr; 260 PetscInt xints, xinte, yints, yinte, i, j; 261 PetscReal hx, hy, dhx, dhy, hxdhy, hydhx; 262 PetscReal grashof, prandtl, lid; 263 PetscScalar u, udot, uxx, uyy, vx, vy, avx, avy, vxp, vxm, vyp, vym; 264 265 PetscFunctionBeginUser; 266 grashof = user->grashof; 267 prandtl = user->prandtl; 268 lid = user->lidvelocity; 269 270 /* 271 Define mesh intervals ratios for uniform grid. 272 273 Note: FD formulae below are normalized by multiplying through by 274 local volume element (i.e. hx*hy) to obtain coefficients O(1) in two dimensions. 275 276 */ 277 dhx = (PetscReal)(info->mx - 1); 278 dhy = (PetscReal)(info->my - 1); 279 hx = 1.0 / dhx; 280 hy = 1.0 / dhy; 281 hxdhy = hx * dhy; 282 hydhx = hy * dhx; 283 284 xints = info->xs; 285 xinte = info->xs + info->xm; 286 yints = info->ys; 287 yinte = info->ys + info->ym; 288 289 /* Test whether we are on the bottom edge of the global array */ 290 if (yints == 0) { 291 j = 0; 292 yints = yints + 1; 293 /* bottom edge */ 294 for (i = info->xs; i < info->xs + info->xm; i++) { 295 f[j][i].u = x[j][i].u; 296 f[j][i].v = x[j][i].v; 297 f[j][i].omega = x[j][i].omega + (x[j + 1][i].u - x[j][i].u) * dhy; 298 f[j][i].temp = x[j][i].temp - x[j + 1][i].temp; 299 } 300 } 301 302 /* Test whether we are on the top edge of the global array */ 303 if (yinte == info->my) { 304 j = info->my - 1; 305 yinte = yinte - 1; 306 /* top edge */ 307 for (i = info->xs; i < info->xs + info->xm; i++) { 308 f[j][i].u = x[j][i].u - lid; 309 f[j][i].v = x[j][i].v; 310 f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j - 1][i].u) * dhy; 311 f[j][i].temp = x[j][i].temp - x[j - 1][i].temp; 312 } 313 } 314 315 /* Test whether we are on the left edge of the global array */ 316 if (xints == 0) { 317 i = 0; 318 xints = xints + 1; 319 /* left edge */ 320 for (j = info->ys; j < info->ys + info->ym; j++) { 321 f[j][i].u = x[j][i].u; 322 f[j][i].v = x[j][i].v; 323 f[j][i].omega = x[j][i].omega - (x[j][i + 1].v - x[j][i].v) * dhx; 324 f[j][i].temp = x[j][i].temp; 325 } 326 } 327 328 /* Test whether we are on the right edge of the global array */ 329 if (xinte == info->mx) { 330 i = info->mx - 1; 331 xinte = xinte - 1; 332 /* right edge */ 333 for (j = info->ys; j < info->ys + info->ym; j++) { 334 f[j][i].u = x[j][i].u; 335 f[j][i].v = x[j][i].v; 336 f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i - 1].v) * dhx; 337 f[j][i].temp = x[j][i].temp - (PetscReal)(grashof > 0); 338 } 339 } 340 341 /* Compute over the interior points */ 342 for (j = yints; j < yinte; j++) { 343 for (i = xints; i < xinte; i++) { 344 /* 345 convective coefficients for upwinding 346 */ 347 vx = x[j][i].u; 348 avx = PetscAbsScalar(vx); 349 vxp = .5 * (vx + avx); 350 vxm = .5 * (vx - avx); 351 vy = x[j][i].v; 352 avy = PetscAbsScalar(vy); 353 vyp = .5 * (vy + avy); 354 vym = .5 * (vy - avy); 355 356 /* U velocity */ 357 u = x[j][i].u; 358 udot = user->parabolic ? xdot[j][i].u : 0.; 359 uxx = (2.0 * u - x[j][i - 1].u - x[j][i + 1].u) * hydhx; 360 uyy = (2.0 * u - x[j - 1][i].u - x[j + 1][i].u) * hxdhy; 361 f[j][i].u = udot + uxx + uyy - .5 * (x[j + 1][i].omega - x[j - 1][i].omega) * hx; 362 363 /* V velocity */ 364 u = x[j][i].v; 365 udot = user->parabolic ? xdot[j][i].v : 0.; 366 uxx = (2.0 * u - x[j][i - 1].v - x[j][i + 1].v) * hydhx; 367 uyy = (2.0 * u - x[j - 1][i].v - x[j + 1][i].v) * hxdhy; 368 f[j][i].v = udot + uxx + uyy + .5 * (x[j][i + 1].omega - x[j][i - 1].omega) * hy; 369 370 /* Omega */ 371 u = x[j][i].omega; 372 uxx = (2.0 * u - x[j][i - 1].omega - x[j][i + 1].omega) * hydhx; 373 uyy = (2.0 * u - x[j - 1][i].omega - x[j + 1][i].omega) * hxdhy; 374 f[j][i].omega = (xdot[j][i].omega + uxx + uyy + (vxp * (u - x[j][i - 1].omega) + vxm * (x[j][i + 1].omega - u)) * hy + (vyp * (u - x[j - 1][i].omega) + vym * (x[j + 1][i].omega - u)) * hx - .5 * grashof * (x[j][i + 1].temp - x[j][i - 1].temp) * hy); 375 376 /* Temperature */ 377 u = x[j][i].temp; 378 uxx = (2.0 * u - x[j][i - 1].temp - x[j][i + 1].temp) * hydhx; 379 uyy = (2.0 * u - x[j - 1][i].temp - x[j + 1][i].temp) * hxdhy; 380 f[j][i].temp = (xdot[j][i].temp + uxx + uyy + prandtl * ((vxp * (u - x[j][i - 1].temp) + vxm * (x[j][i + 1].temp - u)) * hy + (vyp * (u - x[j - 1][i].temp) + vym * (x[j + 1][i].temp - u)) * hx)); 381 } 382 } 383 384 /* 385 Flop count (multiply-adds are counted as 2 operations) 386 */ 387 PetscCall(PetscLogFlops(84.0 * info->ym * info->xm)); 388 PetscFunctionReturn(PETSC_SUCCESS); 389 } 390 391 /*TEST 392 393 test: 394 args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03d.vts' 395 requires: !complex !single 396 397 test: 398 suffix: 1_steps 399 args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_run_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03d.vts' 400 requires: !complex !single 401 402 test: 403 suffix: 2 404 nsize: 4 405 args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03d.vts' -ts_monitor_solution_vtk_interval -1 406 requires: !complex !single 407 408 test: 409 suffix: 3 410 nsize: 4 411 args: -da_refine 2 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 -pc_type none -ts_type beuler -ts_monitor -snes_monitor_short -snes_type aspin -da_overlap 4 -npc_sub_ksp_type preonly -npc_sub_pc_type lu 412 requires: !complex !single 413 414 test: 415 suffix: 4 416 nsize: 2 417 args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 418 requires: !complex !single 419 420 test: 421 suffix: asm 422 nsize: 4 423 args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 424 requires: !complex !single 425 426 TEST*/ 427