Actual source code: ex26.c


  2: static char help[] = "Transient nonlinear driven cavity in 2d.\n\
  3:   \n\
  4: The 2D driven cavity problem is solved in a velocity-vorticity formulation.\n\
  5: The flow can be driven with the lid or with buoyancy or both:\n\
  6:   -lidvelocity <lid>, where <lid> = dimensionless velocity of lid\n\
  7:   -grashof <gr>, where <gr> = dimensionless temperature gradent\n\
  8:   -prandtl <pr>, where <pr> = dimensionless thermal/momentum diffusity ratio\n\
  9:   -contours : draw contour plots of solution\n\n";
 10: /*
 11:       See src/snes/tutorials/ex19.c for the steady-state version.

 13:       There used to be a SNES example (src/snes/tutorials/ex27.c) that
 14:       implemented this algorithm without using TS and was used for the numerical
 15:       results in the paper

 17:         Todd S. Coffey and C. T. Kelley and David E. Keyes, Pseudotransient
 18:         Continuation and Differential-Algebraic Equations, 2003.

 20:       That example was removed because it used obsolete interfaces, but the
 21:       algorithms from the paper can be reproduced using this example.

 23:       Note: The paper describes the algorithm as being linearly implicit but the
 24:       numerical results were created using nonlinearly implicit Euler.  The
 25:       algorithm as described (linearly implicit) is more efficient and is the
 26:       default when using TSPSEUDO.  If you want to reproduce the numerical
 27:       results from the paper, you'll have to change the SNES to converge the
 28:       nonlinear solve (e.g., -snes_type newtonls).  The DAE versus ODE variants
 29:       are controlled using the -parabolic option.

 31:       Comment preserved from snes/tutorials/ex27.c, since removed:

 33:         [H]owever Figure 3.1 was generated with a slightly different algorithm
 34:         (see targets runex27 and runex27_p) than described in the paper.  In
 35:         particular, the described algorithm is linearly implicit, advancing to
 36:         the next step after one Newton step, so that the steady state residual
 37:         is always used, but the figure was generated by converging each step to
 38:         a relative tolerance of 1.e-3.  On the example problem, setting
 39:         -snes_type ksponly has only minor impact on number of steps, but
 40:         significantly reduces the required number of linear solves.

 42:       See also https://lists.mcs.anl.gov/pipermail/petsc-dev/2010-March/002362.html
 43: */

 45: /* ------------------------------------------------------------------------

 47:     We thank David E. Keyes for contributing the driven cavity discretization
 48:     within this example code.

 50:     This problem is modeled by the partial differential equation system

 52:         - Lap(U) - Grad_y(Omega) = 0
 53:         - Lap(V) + Grad_x(Omega) = 0
 54:         Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0
 55:         T_t - Lap(T) + PR*Div([U*T,V*T]) = 0

 57:     in the unit square, which is uniformly discretized in each of x and
 58:     y in this simple encoding.

 60:     No-slip, rigid-wall Dirichlet conditions are used for [U,V].
 61:     Dirichlet conditions are used for Omega, based on the definition of
 62:     vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each
 63:     constant coordinate boundary, the tangential derivative is zero.
 64:     Dirichlet conditions are used for T on the left and right walls,
 65:     and insulation homogeneous Neumann conditions are used for T on
 66:     the top and bottom walls.

 68:     A finite difference approximation with the usual 5-point stencil
 69:     is used to discretize the boundary value problem to obtain a
 70:     nonlinear system of equations.  Upwinding is used for the divergence
 71:     (convective) terms and central for the gradient (source) terms.

 73:     The Jacobian can be either
 74:       * formed via finite differencing using coloring (the default), or
 75:       * applied matrix-free via the option -snes_mf
 76:         (for larger grid problems this variant may not converge
 77:         without a preconditioner due to ill-conditioning).

 79:   ------------------------------------------------------------------------- */

 81: /*
 82:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 83:    Include "petscts.h" so that we can use TS solvers.  Note that this
 84:    file automatically includes:
 85:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 86:      petscmat.h - matrices
 87:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 88:      petscviewer.h - viewers               petscpc.h  - preconditioners
 89:      petscksp.h   - linear solvers         petscsnes.h - nonlinear solvers
 90: */
 91: #include <petscts.h>
 92: #include <petscdm.h>
 93: #include <petscdmda.h>

 95: /*
 96:    User-defined routines and data structures
 97: */
 98: typedef struct {
 99:   PetscScalar u, v, omega, temp;
100: } Field;

102: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *, PetscReal, Field **, Field **, Field **, void *);

104: typedef struct {
105:   PetscReal lidvelocity, prandtl, grashof; /* physical parameters */
106:   PetscBool parabolic;                     /* allow a transient term corresponding roughly to artificial compressibility */
107:   PetscReal cfl_initial;                   /* CFL for first time step */
108: } AppCtx;

110: PetscErrorCode FormInitialSolution(TS, Vec, AppCtx *);

112: int main(int argc, char **argv)
113: {
114:   AppCtx            user; /* user-defined work context */
115:   PetscInt          mx, my, steps;
116:   TS                ts;
117:   DM                da;
118:   Vec               X;
119:   PetscReal         ftime;
120:   TSConvergedReason reason;

122:   PetscFunctionBeginUser;
123:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
124:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
125:   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));
126:   PetscCall(DMSetFromOptions(da));
127:   PetscCall(DMSetUp(da));
128:   PetscCall(TSSetDM(ts, (DM)da));

130:   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));
131:   /*
132:      Problem parameters (velocity of lid, prandtl, and grashof numbers)
133:   */
134:   user.lidvelocity = 1.0 / (mx * my);
135:   user.prandtl     = 1.0;
136:   user.grashof     = 1.0;
137:   user.parabolic   = PETSC_FALSE;
138:   user.cfl_initial = 50.;

140:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Driven cavity/natural convection options", "");
141:   PetscCall(PetscOptionsReal("-lidvelocity", "Lid velocity, related to Reynolds number", "", user.lidvelocity, &user.lidvelocity, NULL));
142:   PetscCall(PetscOptionsReal("-prandtl", "Ratio of viscous to thermal diffusivity", "", user.prandtl, &user.prandtl, NULL));
143:   PetscCall(PetscOptionsReal("-grashof", "Ratio of buoyant to viscous forces", "", user.grashof, &user.grashof, NULL));
144:   PetscCall(PetscOptionsBool("-parabolic", "Relax incompressibility to make the system parabolic instead of differential-algebraic", "", user.parabolic, &user.parabolic, NULL));
145:   PetscCall(PetscOptionsReal("-cfl_initial", "Advective CFL for the first time step", "", user.cfl_initial, &user.cfl_initial, NULL));
146:   PetscOptionsEnd();

148:   PetscCall(DMDASetFieldName(da, 0, "x-velocity"));
149:   PetscCall(DMDASetFieldName(da, 1, "y-velocity"));
150:   PetscCall(DMDASetFieldName(da, 2, "Omega"));
151:   PetscCall(DMDASetFieldName(da, 3, "temperature"));

153:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
154:      Create user context, set problem data, create vector data structures.
155:      Also, compute the initial guess.
156:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

158:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
159:      Create time integration context
160:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
161:   PetscCall(DMSetApplicationContext(da, &user));
162:   PetscCall(DMDATSSetIFunctionLocal(da, INSERT_VALUES, (DMDATSIFunctionLocal)FormIFunctionLocal, &user));
163:   PetscCall(TSSetMaxSteps(ts, 10000));
164:   PetscCall(TSSetMaxTime(ts, 1e12));
165:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
166:   PetscCall(TSSetTimeStep(ts, user.cfl_initial / (user.lidvelocity * mx)));
167:   PetscCall(TSSetFromOptions(ts));

169:   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));

171:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
172:      Solve the nonlinear system
173:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

175:   PetscCall(DMCreateGlobalVector(da, &X));
176:   PetscCall(FormInitialSolution(ts, X, &user));

178:   PetscCall(TSSolve(ts, X));
179:   PetscCall(TSGetSolveTime(ts, &ftime));
180:   PetscCall(TSGetStepNumber(ts, &steps));
181:   PetscCall(TSGetConvergedReason(ts, &reason));

183:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%s at time %g after %" PetscInt_FMT " steps\n", TSConvergedReasons[reason], (double)ftime, steps));

185:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
186:      Free work space.  All PETSc objects should be destroyed when they
187:      are no longer needed.
188:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
189:   PetscCall(VecDestroy(&X));
190:   PetscCall(DMDestroy(&da));
191:   PetscCall(TSDestroy(&ts));

193:   PetscCall(PetscFinalize());
194:   return 0;
195: }

197: /* ------------------------------------------------------------------- */

199: /*
200:    FormInitialSolution - Forms initial approximation.

202:    Input Parameters:
203:    user - user-defined application context
204:    X - vector

206:    Output Parameter:
207:    X - vector
208:  */
209: PetscErrorCode FormInitialSolution(TS ts, Vec X, AppCtx *user)
210: {
211:   DM        da;
212:   PetscInt  i, j, mx, xs, ys, xm, ym;
213:   PetscReal grashof, dx;
214:   Field   **x;

216:   PetscFunctionBeginUser;
217:   grashof = user->grashof;
218:   PetscCall(TSGetDM(ts, &da));
219:   PetscCall(DMDAGetInfo(da, 0, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
220:   dx = 1.0 / (mx - 1);

222:   /*
223:      Get local grid boundaries (for 2-dimensional DMDA):
224:        xs, ys   - starting grid indices (no ghost points)
225:        xm, ym   - widths of local grid (no ghost points)
226:   */
227:   PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

229:   /*
230:      Get a pointer to vector data.
231:        - For default PETSc vectors, VecGetArray() returns a pointer to
232:          the data array.  Otherwise, the routine is implementation dependent.
233:        - You MUST call VecRestoreArray() when you no longer need access to
234:          the array.
235:   */
236:   PetscCall(DMDAVecGetArray(da, X, &x));

238:   /*
239:      Compute initial guess over the locally owned part of the grid
240:      Initial condition is motionless fluid and equilibrium temperature
241:   */
242:   for (j = ys; j < ys + ym; j++) {
243:     for (i = xs; i < xs + xm; i++) {
244:       x[j][i].u     = 0.0;
245:       x[j][i].v     = 0.0;
246:       x[j][i].omega = 0.0;
247:       x[j][i].temp  = (grashof > 0) * i * dx;
248:     }
249:   }

251:   /*
252:      Restore vector
253:   */
254:   PetscCall(DMDAVecRestoreArray(da, X, &x));
255:   PetscFunctionReturn(PETSC_SUCCESS);
256: }

258: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info, PetscReal ptime, Field **x, Field **xdot, Field **f, void *ptr)
259: {
260:   AppCtx     *user = (AppCtx *)ptr;
261:   PetscInt    xints, xinte, yints, yinte, i, j;
262:   PetscReal   hx, hy, dhx, dhy, hxdhy, hydhx;
263:   PetscReal   grashof, prandtl, lid;
264:   PetscScalar u, udot, uxx, uyy, vx, vy, avx, avy, vxp, vxm, vyp, vym;

266:   PetscFunctionBeginUser;
267:   grashof = user->grashof;
268:   prandtl = user->prandtl;
269:   lid     = user->lidvelocity;

271:   /*
272:      Define mesh intervals ratios for uniform grid.

274:      Note: FD formulae below are normalized by multiplying through by
275:      local volume element (i.e. hx*hy) to obtain coefficients O(1) in two dimensions.

277:   */
278:   dhx   = (PetscReal)(info->mx - 1);
279:   dhy   = (PetscReal)(info->my - 1);
280:   hx    = 1.0 / dhx;
281:   hy    = 1.0 / dhy;
282:   hxdhy = hx * dhy;
283:   hydhx = hy * dhx;

285:   xints = info->xs;
286:   xinte = info->xs + info->xm;
287:   yints = info->ys;
288:   yinte = info->ys + info->ym;

290:   /* Test whether we are on the bottom edge of the global array */
291:   if (yints == 0) {
292:     j     = 0;
293:     yints = yints + 1;
294:     /* bottom edge */
295:     for (i = info->xs; i < info->xs + info->xm; i++) {
296:       f[j][i].u     = x[j][i].u;
297:       f[j][i].v     = x[j][i].v;
298:       f[j][i].omega = x[j][i].omega + (x[j + 1][i].u - x[j][i].u) * dhy;
299:       f[j][i].temp  = x[j][i].temp - x[j + 1][i].temp;
300:     }
301:   }

303:   /* Test whether we are on the top edge of the global array */
304:   if (yinte == info->my) {
305:     j     = info->my - 1;
306:     yinte = yinte - 1;
307:     /* top edge */
308:     for (i = info->xs; i < info->xs + info->xm; i++) {
309:       f[j][i].u     = x[j][i].u - lid;
310:       f[j][i].v     = x[j][i].v;
311:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j - 1][i].u) * dhy;
312:       f[j][i].temp  = x[j][i].temp - x[j - 1][i].temp;
313:     }
314:   }

316:   /* Test whether we are on the left edge of the global array */
317:   if (xints == 0) {
318:     i     = 0;
319:     xints = xints + 1;
320:     /* left edge */
321:     for (j = info->ys; j < info->ys + info->ym; j++) {
322:       f[j][i].u     = x[j][i].u;
323:       f[j][i].v     = x[j][i].v;
324:       f[j][i].omega = x[j][i].omega - (x[j][i + 1].v - x[j][i].v) * dhx;
325:       f[j][i].temp  = x[j][i].temp;
326:     }
327:   }

329:   /* Test whether we are on the right edge of the global array */
330:   if (xinte == info->mx) {
331:     i     = info->mx - 1;
332:     xinte = xinte - 1;
333:     /* right edge */
334:     for (j = info->ys; j < info->ys + info->ym; j++) {
335:       f[j][i].u     = x[j][i].u;
336:       f[j][i].v     = x[j][i].v;
337:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i - 1].v) * dhx;
338:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof > 0);
339:     }
340:   }

342:   /* Compute over the interior points */
343:   for (j = yints; j < yinte; j++) {
344:     for (i = xints; i < xinte; i++) {
345:       /*
346:         convective coefficients for upwinding
347:       */
348:       vx  = x[j][i].u;
349:       avx = PetscAbsScalar(vx);
350:       vxp = .5 * (vx + avx);
351:       vxm = .5 * (vx - avx);
352:       vy  = x[j][i].v;
353:       avy = PetscAbsScalar(vy);
354:       vyp = .5 * (vy + avy);
355:       vym = .5 * (vy - avy);

357:       /* U velocity */
358:       u         = x[j][i].u;
359:       udot      = user->parabolic ? xdot[j][i].u : 0.;
360:       uxx       = (2.0 * u - x[j][i - 1].u - x[j][i + 1].u) * hydhx;
361:       uyy       = (2.0 * u - x[j - 1][i].u - x[j + 1][i].u) * hxdhy;
362:       f[j][i].u = udot + uxx + uyy - .5 * (x[j + 1][i].omega - x[j - 1][i].omega) * hx;

364:       /* V velocity */
365:       u         = x[j][i].v;
366:       udot      = user->parabolic ? xdot[j][i].v : 0.;
367:       uxx       = (2.0 * u - x[j][i - 1].v - x[j][i + 1].v) * hydhx;
368:       uyy       = (2.0 * u - x[j - 1][i].v - x[j + 1][i].v) * hxdhy;
369:       f[j][i].v = udot + uxx + uyy + .5 * (x[j][i + 1].omega - x[j][i - 1].omega) * hy;

371:       /* Omega */
372:       u   = x[j][i].omega;
373:       uxx = (2.0 * u - x[j][i - 1].omega - x[j][i + 1].omega) * hydhx;
374:       uyy = (2.0 * u - x[j - 1][i].omega - x[j + 1][i].omega) * hxdhy;
375:       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);

377:       /* Temperature */
378:       u            = x[j][i].temp;
379:       uxx          = (2.0 * u - x[j][i - 1].temp - x[j][i + 1].temp) * hydhx;
380:       uyy          = (2.0 * u - x[j - 1][i].temp - x[j + 1][i].temp) * hxdhy;
381:       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));
382:     }
383:   }

385:   /*
386:      Flop count (multiply-adds are counted as 2 operations)
387:   */
388:   PetscCall(PetscLogFlops(84.0 * info->ym * info->xm));
389:   PetscFunctionReturn(PETSC_SUCCESS);
390: }

392: /*TEST

394:     test:
395:       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'
396:       requires: !complex !single

398:     test:
399:       suffix: 2
400:       nsize: 4
401:       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'
402:       requires: !complex !single

404:     test:
405:       suffix: 3
406:       nsize: 4
407:       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
408:       requires: !complex !single

410:     test:
411:       suffix: 4
412:       nsize: 2
413:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
414:       requires: !complex !single

416:     test:
417:       suffix: asm
418:       nsize: 4
419:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
420:       requires: !complex !single

422: TEST*/