Actual source code: ex26.c

petsc-master 2019-11-22
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  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/examples/tutorials/ex19.c for the steady-state version.

 13:       There used to be a SNES example (src/snes/examples/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/examples/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);if (ierr) return ierr;
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);


180:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
181:      Solve the nonlinear system
182:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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

202:   PetscFinalize();
203:   return ierr;
204: }

206: /* ------------------------------------------------------------------- */


209: /*
210:    FormInitialSolution - Forms initial approximation.

212:    Input Parameters:
213:    user - user-defined application context
214:    X - vector

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

227:   grashof = user->grashof;
228:   TSGetDM(ts,&da);
229:   DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0);
230:   dx      = 1.0/(mx-1);

232:   /*
233:      Get local grid boundaries (for 2-dimensional DMDA):
234:        xs, ys   - starting grid indices (no ghost points)
235:        xm, ym   - widths of local grid (no ghost points)
236:   */
237:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

239:   /*
240:      Get a pointer to vector data.
241:        - For default PETSc vectors, VecGetArray() returns a pointer to
242:          the data array.  Otherwise, the routine is implementation dependent.
243:        - You MUST call VecRestoreArray() when you no longer need access to
244:          the array.
245:   */
246:   DMDAVecGetArray(da,X,&x);

248:   /*
249:      Compute initial guess over the locally owned part of the grid
250:      Initial condition is motionless fluid and equilibrium temperature
251:   */
252:   for (j=ys; j<ys+ym; j++) {
253:     for (i=xs; i<xs+xm; i++) {
254:       x[j][i].u     = 0.0;
255:       x[j][i].v     = 0.0;
256:       x[j][i].omega = 0.0;
257:       x[j][i].temp  = (grashof>0)*i*dx;
258:     }
259:   }

261:   /*
262:      Restore vector
263:   */
264:   DMDAVecRestoreArray(da,X,&x);
265:   return 0;
266: }

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

278:   grashof = user->grashof;
279:   prandtl = user->prandtl;
280:   lid     = user->lidvelocity;

282:   /*
283:      Define mesh intervals ratios for uniform grid.

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


289:   */
290:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
291:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
292:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

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

296:   /* Test whether we are on the bottom edge of the global array */
297:   if (yints == 0) {
298:     j     = 0;
299:     yints = yints + 1;
300:     /* bottom edge */
301:     for (i=info->xs; i<info->xs+info->xm; i++) {
302:       f[j][i].u     = x[j][i].u;
303:       f[j][i].v     = x[j][i].v;
304:       f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
305:       f[j][i].temp  = x[j][i].temp-x[j+1][i].temp;
306:     }
307:   }

309:   /* Test whether we are on the top edge of the global array */
310:   if (yinte == info->my) {
311:     j     = info->my - 1;
312:     yinte = yinte - 1;
313:     /* top edge */
314:     for (i=info->xs; i<info->xs+info->xm; i++) {
315:       f[j][i].u     = x[j][i].u - lid;
316:       f[j][i].v     = x[j][i].v;
317:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy;
318:       f[j][i].temp  = x[j][i].temp-x[j-1][i].temp;
319:     }
320:   }

322:   /* Test whether we are on the left edge of the global array */
323:   if (xints == 0) {
324:     i     = 0;
325:     xints = xints + 1;
326:     /* left edge */
327:     for (j=info->ys; j<info->ys+info->ym; j++) {
328:       f[j][i].u     = x[j][i].u;
329:       f[j][i].v     = x[j][i].v;
330:       f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
331:       f[j][i].temp  = x[j][i].temp;
332:     }
333:   }

335:   /* Test whether we are on the right edge of the global array */
336:   if (xinte == info->mx) {
337:     i     = info->mx - 1;
338:     xinte = xinte - 1;
339:     /* right edge */
340:     for (j=info->ys; j<info->ys+info->ym; j++) {
341:       f[j][i].u     = x[j][i].u;
342:       f[j][i].v     = x[j][i].v;
343:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx;
344:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof>0);
345:     }
346:   }

348:   /* Compute over the interior points */
349:   for (j=yints; j<yinte; j++) {
350:     for (i=xints; i<xinte; i++) {

352:       /*
353:         convective coefficients for upwinding
354:       */
355:       vx  = x[j][i].u; avx = PetscAbsScalar(vx);
356:       vxp = .5*(vx+avx); vxm = .5*(vx-avx);
357:       vy  = x[j][i].v; avy = PetscAbsScalar(vy);
358:       vyp = .5*(vy+avy); vym = .5*(vy-avy);

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

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

374:       /* Omega */
375:       u             = x[j][i].omega;
376:       uxx           = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
377:       uyy           = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
378:       f[j][i].omega = (xdot[j][i].omega + uxx + uyy
379:                        + (vxp*(u - x[j][i-1].omega)
380:                           + vxm*(x[j][i+1].omega - u)) * hy
381:                        + (vyp*(u - x[j-1][i].omega)
382:                           + vym*(x[j+1][i].omega - u)) * hx
383:                        - .5 * grashof * (x[j][i+1].temp - x[j][i-1].temp) * hy);

385:       /* Temperature */
386:       u            = x[j][i].temp;
387:       uxx          = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
388:       uyy          = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
389:       f[j][i].temp =  (xdot[j][i].temp + uxx + uyy
390:                        + prandtl * ((vxp*(u - x[j][i-1].temp)
391:                                      + vxm*(x[j][i+1].temp - u)) * hy
392:                                     + (vyp*(u - x[j-1][i].temp)
393:                                        + vym*(x[j+1][i].temp - u)) * hx));
394:     }
395:   }

397:   /*
398:      Flop count (multiply-adds are counted as 2 operations)
399:   */
400:   PetscLogFlops(84.0*info->ym*info->xm);
401:   return(0);
402: }

404: /*TEST

406:     test:
407:       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'
408:       requires: !complex !single

410:     test:
411:       suffix: 2
412:       nsize: 4
413:       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'
414:       requires: !complex !single

416:     test:
417:       suffix: 3
418:       nsize: 4
419:       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
420:       requires: !complex !single

422:     test:
423:       suffix: 4
424:       nsize: 2
425:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
426:       requires: !complex !single

428:     test:
429:       suffix: asm
430:       nsize: 4
431:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
432:       requires: !complex !single

434: TEST*/