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 bouyancy 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: /*T
 46:    Concepts: TS^solving a system of nonlinear equations (parallel multicomponent example);
 47:    Concepts: DMDA^using distributed arrays;
 48:    Concepts: TS^multicomponent
 49:    Concepts: TS^differential-algebraic equation
 50:    Processors: n
 51: T*/
 52: /* ------------------------------------------------------------------------

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

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

 59:         - Lap(U) - Grad_y(Omega) = 0
 60:         - Lap(V) + Grad_x(Omega) = 0
 61:         Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0
 62:         T_t - Lap(T) + PR*Div([U*T,V*T]) = 0

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

 67:     No-slip, rigid-wall Dirichlet conditions are used for [U,V].
 68:     Dirichlet conditions are used for Omega, based on the definition of
 69:     vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each
 70:     constant coordinate boundary, the tangential derivative is zero.
 71:     Dirichlet conditions are used for T on the left and right walls,
 72:     and insulation homogeneous Neumann conditions are used for T on
 73:     the top and bottom walls.

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

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

 86:   ------------------------------------------------------------------------- */

 88: /*
 89:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 90:    Include "petscts.h" so that we can use TS solvers.  Note that this
 91:    file automatically includes:
 92:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 93:      petscmat.h - matrices
 94:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 95:      petscviewer.h - viewers               petscpc.h  - preconditioners
 96:      petscksp.h   - linear solvers         petscsnes.h - nonlinear solvers
 97: */
 98: #include <petscts.h>
 99: #include <petscdm.h>
100: #include <petscdmda.h>

102: /*
103:    User-defined routines and data structures
104: */
105: typedef struct {
106:   PetscScalar u,v,omega,temp;
107: } Field;

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

111: typedef struct {
112:   PetscReal   lidvelocity,prandtl,grashof;   /* physical parameters */
113:   PetscBool   parabolic;                     /* allow a transient term corresponding roughly to artificial compressibility */
114:   PetscReal   cfl_initial;                   /* CFL for first time step */
115: } AppCtx;

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

119: int main(int argc,char **argv)
120: {
121:   AppCtx            user;             /* user-defined work context */
122:   PetscInt          mx,my,steps;
123:   PetscErrorCode    ierr;
124:   TS                ts;
125:   DM                da;
126:   Vec               X;
127:   PetscReal         ftime;
128:   TSConvergedReason reason;

130:   PetscInitialize(&argc,&argv,(char*)0,help);
131:   TSCreate(PETSC_COMM_WORLD,&ts);
132:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);
133:   DMSetFromOptions(da);
134:   DMSetUp(da);
135:   TSSetDM(ts,(DM)da);

137:   DMDAGetInfo(da,0,&mx,&my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
138:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
139:   /*
140:      Problem parameters (velocity of lid, prandtl, and grashof numbers)
141:   */
142:   user.lidvelocity = 1.0/(mx*my);
143:   user.prandtl     = 1.0;
144:   user.grashof     = 1.0;
145:   user.parabolic   = PETSC_FALSE;
146:   user.cfl_initial = 50.;

148:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Driven cavity/natural convection options","");
149:   PetscOptionsReal("-lidvelocity","Lid velocity, related to Reynolds number","",user.lidvelocity,&user.lidvelocity,NULL);
150:   PetscOptionsReal("-prandtl","Ratio of viscous to thermal diffusivity","",user.prandtl,&user.prandtl,NULL);
151:   PetscOptionsReal("-grashof","Ratio of bouyant to viscous forces","",user.grashof,&user.grashof,NULL);
152:   PetscOptionsBool("-parabolic","Relax incompressibility to make the system parabolic instead of differential-algebraic","",user.parabolic,&user.parabolic,NULL);
153:   PetscOptionsReal("-cfl_initial","Advective CFL for the first time step","",user.cfl_initial,&user.cfl_initial,NULL);
154:   PetscOptionsEnd();

156:   DMDASetFieldName(da,0,"x-velocity");
157:   DMDASetFieldName(da,1,"y-velocity");
158:   DMDASetFieldName(da,2,"Omega");
159:   DMDASetFieldName(da,3,"temperature");

161:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
162:      Create user context, set problem data, create vector data structures.
163:      Also, compute the initial guess.
164:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

166:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
167:      Create time integration context
168:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
169:   DMSetApplicationContext(da,&user);
170:   DMDATSSetIFunctionLocal(da,INSERT_VALUES,(DMDATSIFunctionLocal)FormIFunctionLocal,&user);
171:   TSSetMaxSteps(ts,10000);
172:   TSSetMaxTime(ts,1e12);
173:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
174:   TSSetTimeStep(ts,user.cfl_initial/(user.lidvelocity*mx));
175:   TSSetFromOptions(ts);

177:   PetscPrintf(PETSC_COMM_WORLD,"%Dx%D grid, lid velocity = %g, prandtl # = %g, grashof # = %g\n",mx,my,(double)user.lidvelocity,(double)user.prandtl,(double)user.grashof);

179:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
180:      Solve the nonlinear system
181:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

183:   DMCreateGlobalVector(da,&X);
184:   FormInitialSolution(ts,X,&user);

186:   TSSolve(ts,X);
187:   TSGetSolveTime(ts,&ftime);
188:   TSGetStepNumber(ts,&steps);
189:   TSGetConvergedReason(ts,&reason);

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

193:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194:      Free work space.  All PETSc objects should be destroyed when they
195:      are no longer needed.
196:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
197:   VecDestroy(&X);
198:   DMDestroy(&da);
199:   TSDestroy(&ts);

201:   PetscFinalize();
202:   return 0;
203: }

205: /* ------------------------------------------------------------------- */

207: /*
208:    FormInitialSolution - Forms initial approximation.

210:    Input Parameters:
211:    user - user-defined application context
212:    X - vector

214:    Output Parameter:
215:    X - vector
216:  */
217: PetscErrorCode FormInitialSolution(TS ts,Vec X,AppCtx *user)
218: {
219:   DM             da;
220:   PetscInt       i,j,mx,xs,ys,xm,ym;
221:   PetscReal      grashof,dx;
222:   Field          **x;

224:   grashof = user->grashof;
225:   TSGetDM(ts,&da);
226:   DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0);
227:   dx      = 1.0/(mx-1);

229:   /*
230:      Get local grid boundaries (for 2-dimensional DMDA):
231:        xs, ys   - starting grid indices (no ghost points)
232:        xm, ym   - widths of local grid (no ghost points)
233:   */
234:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

236:   /*
237:      Get a pointer to vector data.
238:        - For default PETSc vectors, VecGetArray() returns a pointer to
239:          the data array.  Otherwise, the routine is implementation dependent.
240:        - You MUST call VecRestoreArray() when you no longer need access to
241:          the array.
242:   */
243:   DMDAVecGetArray(da,X,&x);

245:   /*
246:      Compute initial guess over the locally owned part of the grid
247:      Initial condition is motionless fluid and equilibrium temperature
248:   */
249:   for (j=ys; j<ys+ym; j++) {
250:     for (i=xs; i<xs+xm; i++) {
251:       x[j][i].u     = 0.0;
252:       x[j][i].v     = 0.0;
253:       x[j][i].omega = 0.0;
254:       x[j][i].temp  = (grashof>0)*i*dx;
255:     }
256:   }

258:   /*
259:      Restore vector
260:   */
261:   DMDAVecRestoreArray(da,X,&x);
262:   return 0;
263: }

265: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info,PetscReal ptime,Field **x,Field **xdot,Field **f,void *ptr)
266: {
267:   AppCtx         *user = (AppCtx*)ptr;
268:   PetscInt       xints,xinte,yints,yinte,i,j;
269:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
270:   PetscReal      grashof,prandtl,lid;
271:   PetscScalar    u,udot,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;

274:   grashof = user->grashof;
275:   prandtl = user->prandtl;
276:   lid     = user->lidvelocity;

278:   /*
279:      Define mesh intervals ratios for uniform grid.

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

284:   */
285:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
286:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
287:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

289:   xints = info->xs; xinte = info->xs+info->xm; yints = info->ys; yinte = info->ys+info->ym;

291:   /* Test whether we are on the bottom edge of the global array */
292:   if (yints == 0) {
293:     j     = 0;
294:     yints = yints + 1;
295:     /* bottom edge */
296:     for (i=info->xs; i<info->xs+info->xm; i++) {
297:       f[j][i].u     = x[j][i].u;
298:       f[j][i].v     = x[j][i].v;
299:       f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
300:       f[j][i].temp  = x[j][i].temp-x[j+1][i].temp;
301:     }
302:   }

304:   /* Test whether we are on the top edge of the global array */
305:   if (yinte == info->my) {
306:     j     = info->my - 1;
307:     yinte = yinte - 1;
308:     /* top edge */
309:     for (i=info->xs; i<info->xs+info->xm; i++) {
310:       f[j][i].u     = x[j][i].u - lid;
311:       f[j][i].v     = x[j][i].v;
312:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy;
313:       f[j][i].temp  = x[j][i].temp-x[j-1][i].temp;
314:     }
315:   }

317:   /* Test whether we are on the left edge of the global array */
318:   if (xints == 0) {
319:     i     = 0;
320:     xints = xints + 1;
321:     /* left edge */
322:     for (j=info->ys; j<info->ys+info->ym; j++) {
323:       f[j][i].u     = x[j][i].u;
324:       f[j][i].v     = x[j][i].v;
325:       f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
326:       f[j][i].temp  = x[j][i].temp;
327:     }
328:   }

330:   /* Test whether we are on the right edge of the global array */
331:   if (xinte == info->mx) {
332:     i     = info->mx - 1;
333:     xinte = xinte - 1;
334:     /* right edge */
335:     for (j=info->ys; j<info->ys+info->ym; j++) {
336:       f[j][i].u     = x[j][i].u;
337:       f[j][i].v     = x[j][i].v;
338:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx;
339:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof>0);
340:     }
341:   }

343:   /* Compute over the interior points */
344:   for (j=yints; j<yinte; j++) {
345:     for (i=xints; i<xinte; i++) {

347:       /*
348:         convective coefficients for upwinding
349:       */
350:       vx  = x[j][i].u; avx = PetscAbsScalar(vx);
351:       vxp = .5*(vx+avx); vxm = .5*(vx-avx);
352:       vy  = x[j][i].v; avy = PetscAbsScalar(vy);
353:       vyp = .5*(vy+avy); vym = .5*(vy-avy);

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

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

369:       /* Omega */
370:       u             = x[j][i].omega;
371:       uxx           = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
372:       uyy           = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
373:       f[j][i].omega = (xdot[j][i].omega + uxx + uyy
374:                        + (vxp*(u - x[j][i-1].omega)
375:                           + vxm*(x[j][i+1].omega - u)) * hy
376:                        + (vyp*(u - x[j-1][i].omega)
377:                           + vym*(x[j+1][i].omega - u)) * hx
378:                        - .5 * grashof * (x[j][i+1].temp - x[j][i-1].temp) * hy);

380:       /* Temperature */
381:       u            = x[j][i].temp;
382:       uxx          = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
383:       uyy          = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
384:       f[j][i].temp =  (xdot[j][i].temp + uxx + uyy
385:                        + prandtl * ((vxp*(u - x[j][i-1].temp)
386:                                      + vxm*(x[j][i+1].temp - u)) * hy
387:                                     + (vyp*(u - x[j-1][i].temp)
388:                                        + vym*(x[j+1][i].temp - u)) * hx));
389:     }
390:   }

392:   /*
393:      Flop count (multiply-adds are counted as 2 operations)
394:   */
395:   PetscLogFlops(84.0*info->ym*info->xm);
396:   return 0;
397: }

399: /*TEST

401:     test:
402:       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'
403:       requires: !complex !single

405:     test:
406:       suffix: 2
407:       nsize: 4
408:       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'
409:       requires: !complex !single

411:     test:
412:       suffix: 3
413:       nsize: 4
414:       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
415:       requires: !complex !single

417:     test:
418:       suffix: 4
419:       nsize: 2
420:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
421:       requires: !complex !single

423:     test:
424:       suffix: asm
425:       nsize: 4
426:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
427:       requires: !complex !single

429: TEST*/