Actual source code: ex26.c

petsc-3.5.1 2014-08-06
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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:       See src/snes/examples/tutorials/ex27.c for a version written specifically
 14:       for pseudo-transient continuation, without using TS.
 15: */

 17: /*T
 18:    Concepts: TS^solving a system of nonlinear equations (parallel multicomponent example);
 19:    Concepts: DMDA^using distributed arrays;
 20:    Concepts: TS^multicomponent
 21:    Concepts: TS^differential-algebraic equation
 22:    Processors: n
 23: T*/
 24: /* ------------------------------------------------------------------------

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

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

 31:         - Lap(U) - Grad_y(Omega) = 0
 32:         - Lap(V) + Grad_x(Omega) = 0
 33:         Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0
 34:         T_t - Lap(T) + PR*Div([U*T,V*T]) = 0

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

 39:     No-slip, rigid-wall Dirichlet conditions are used for [U,V].
 40:     Dirichlet conditions are used for Omega, based on the definition of
 41:     vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each
 42:     constant coordinate boundary, the tangential derivative is zero.
 43:     Dirichlet conditions are used for T on the left and right walls,
 44:     and insulation homogeneous Neumann conditions are used for T on
 45:     the top and bottom walls.

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

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

 58:   ------------------------------------------------------------------------- */

 60: /*
 61:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 62:    Include "petscts.h" so that we can use TS solvers.  Note that this
 63:    file automatically includes:
 64:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 65:      petscmat.h - matrices
 66:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 67:      petscviewer.h - viewers               petscpc.h  - preconditioners
 68:      petscksp.h   - linear solvers         petscsnes.h - nonlinear solvers
 69: */
 70: #include <petscts.h>
 71: #include <petscdm.h>
 72: #include <petscdmda.h>

 74: /*
 75:    User-defined routines and data structures
 76: */
 77: typedef struct {
 78:   PetscScalar u,v,omega,temp;
 79: } Field;

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

 83: typedef struct {
 84:   PassiveReal lidvelocity,prandtl,grashof;   /* physical parameters */
 85:   PetscBool   parabolic;                     /* allow a transient term corresponding roughly to artificial compressibility */
 86:   PetscReal   cfl_initial;                   /* CFL for first time step */
 87: } AppCtx;

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

 93: int main(int argc,char **argv)
 94: {
 95:   AppCtx            user;             /* user-defined work context */
 96:   PetscInt          mx,my,steps;
 97:   PetscErrorCode    ierr;
 98:   TS                ts;
 99:   DM                da;
100:   Vec               X;
101:   PetscReal         ftime;
102:   TSConvergedReason reason;

104:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return(1);

106:   TSCreate(PETSC_COMM_WORLD,&ts);

108:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,-4,-4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);
109:   TSSetDM(ts,(DM)da);

111:   DMDAGetInfo(da,0,&mx,&my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
112:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
113:   /*
114:      Problem parameters (velocity of lid, prandtl, and grashof numbers)
115:   */
116:   user.lidvelocity = 1.0/(mx*my);
117:   user.prandtl     = 1.0;
118:   user.grashof     = 1.0;
119:   user.parabolic   = PETSC_FALSE;
120:   user.cfl_initial = 50.;

122:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Driven cavity/natural convection options","");
123:   PetscOptionsReal("-lidvelocity","Lid velocity, related to Reynolds number","",user.lidvelocity,&user.lidvelocity,NULL);
124:   PetscOptionsReal("-prandtl","Ratio of viscous to thermal diffusivity","",user.prandtl,&user.prandtl,NULL);
125:   PetscOptionsReal("-grashof","Ratio of bouyant to viscous forces","",user.grashof,&user.grashof,NULL);
126:   PetscOptionsBool("-parabolic","Relax incompressibility to make the system parabolic instead of differential-algebraic","",user.parabolic,&user.parabolic,NULL);
127:   PetscOptionsReal("-cfl_initial","Advective CFL for the first time step","",user.cfl_initial,&user.cfl_initial,NULL);
128:   PetscOptionsEnd();

130:   DMDASetFieldName(da,0,"x-velocity");
131:   DMDASetFieldName(da,1,"y-velocity");
132:   DMDASetFieldName(da,2,"Omega");
133:   DMDASetFieldName(da,3,"temperature");

135:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136:      Create user context, set problem data, create vector data structures.
137:      Also, compute the initial guess.
138:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

140:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
141:      Create time integration context
142:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
143:   DMSetApplicationContext(da,&user);
144:   DMDATSSetIFunctionLocal(da,INSERT_VALUES,(DMDATSIFunctionLocal)FormIFunctionLocal,&user);
145:   TSSetDuration(ts,10000,1e12);
146:   TSSetInitialTimeStep(ts,0.0,user.cfl_initial/(user.lidvelocity*mx));
147:   TSSetFromOptions(ts);

149:   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);


152:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153:      Solve the nonlinear system
154:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

156:   DMCreateGlobalVector(da,&X);
157:   FormInitialSolution(ts,X,&user);

159:   TSSolve(ts,X);
160:   TSGetSolveTime(ts,&ftime);
161:   TSGetTimeStepNumber(ts,&steps);
162:   TSGetConvergedReason(ts,&reason);

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

166:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
167:      Free work space.  All PETSc objects should be destroyed when they
168:      are no longer needed.
169:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
170:   VecDestroy(&X);
171:   DMDestroy(&da);
172:   TSDestroy(&ts);

174:   PetscFinalize();
175:   return 0;
176: }

178: /* ------------------------------------------------------------------- */


183: /*
184:    FormInitialSolution - Forms initial approximation.

186:    Input Parameters:
187:    user - user-defined application context
188:    X - vector

190:    Output Parameter:
191:    X - vector
192:  */
193: PetscErrorCode FormInitialSolution(TS ts,Vec X,AppCtx *user)
194: {
195:   DM             da;
196:   PetscInt       i,j,mx,xs,ys,xm,ym;
198:   PetscReal      grashof,dx;
199:   Field          **x;

201:   grashof = user->grashof;
202:   TSGetDM(ts,&da);
203:   DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0);
204:   dx      = 1.0/(mx-1);

206:   /*
207:      Get local grid boundaries (for 2-dimensional DMDA):
208:        xs, ys   - starting grid indices (no ghost points)
209:        xm, ym   - widths of local grid (no ghost points)
210:   */
211:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

213:   /*
214:      Get a pointer to vector data.
215:        - For default PETSc vectors, VecGetArray() returns a pointer to
216:          the data array.  Otherwise, the routine is implementation dependent.
217:        - You MUST call VecRestoreArray() when you no longer need access to
218:          the array.
219:   */
220:   DMDAVecGetArray(da,X,&x);

222:   /*
223:      Compute initial guess over the locally owned part of the grid
224:      Initial condition is motionless fluid and equilibrium temperature
225:   */
226:   for (j=ys; j<ys+ym; j++) {
227:     for (i=xs; i<xs+xm; i++) {
228:       x[j][i].u     = 0.0;
229:       x[j][i].v     = 0.0;
230:       x[j][i].omega = 0.0;
231:       x[j][i].temp  = (grashof>0)*i*dx;
232:     }
233:   }

235:   /*
236:      Restore vector
237:   */
238:   DMDAVecRestoreArray(da,X,&x);
239:   return 0;
240: }

244: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info,PetscReal ptime,Field **x,Field **xdot,Field **f,void *ptr)
245: {
246:   AppCtx         *user = (AppCtx*)ptr;
248:   PetscInt       xints,xinte,yints,yinte,i,j;
249:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
250:   PetscReal      grashof,prandtl,lid;
251:   PetscScalar    u,udot,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;

254:   grashof = user->grashof;
255:   prandtl = user->prandtl;
256:   lid     = user->lidvelocity;

258:   /*
259:      Define mesh intervals ratios for uniform grid.

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


265:   */
266:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
267:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
268:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

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

272:   /* Test whether we are on the bottom edge of the global array */
273:   if (yints == 0) {
274:     j     = 0;
275:     yints = yints + 1;
276:     /* bottom edge */
277:     for (i=info->xs; i<info->xs+info->xm; i++) {
278:       f[j][i].u     = x[j][i].u;
279:       f[j][i].v     = x[j][i].v;
280:       f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
281:       f[j][i].temp  = x[j][i].temp-x[j+1][i].temp;
282:     }
283:   }

285:   /* Test whether we are on the top edge of the global array */
286:   if (yinte == info->my) {
287:     j     = info->my - 1;
288:     yinte = yinte - 1;
289:     /* top edge */
290:     for (i=info->xs; i<info->xs+info->xm; i++) {
291:       f[j][i].u     = x[j][i].u - lid;
292:       f[j][i].v     = x[j][i].v;
293:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy;
294:       f[j][i].temp  = x[j][i].temp-x[j-1][i].temp;
295:     }
296:   }

298:   /* Test whether we are on the left edge of the global array */
299:   if (xints == 0) {
300:     i     = 0;
301:     xints = xints + 1;
302:     /* left edge */
303:     for (j=info->ys; j<info->ys+info->ym; j++) {
304:       f[j][i].u     = x[j][i].u;
305:       f[j][i].v     = x[j][i].v;
306:       f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
307:       f[j][i].temp  = x[j][i].temp;
308:     }
309:   }

311:   /* Test whether we are on the right edge of the global array */
312:   if (xinte == info->mx) {
313:     i     = info->mx - 1;
314:     xinte = xinte - 1;
315:     /* right edge */
316:     for (j=info->ys; j<info->ys+info->ym; j++) {
317:       f[j][i].u     = x[j][i].u;
318:       f[j][i].v     = x[j][i].v;
319:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx;
320:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof>0);
321:     }
322:   }

324:   /* Compute over the interior points */
325:   for (j=yints; j<yinte; j++) {
326:     for (i=xints; i<xinte; i++) {

328:       /*
329:         convective coefficients for upwinding
330:       */
331:       vx  = x[j][i].u; avx = PetscAbsScalar(vx);
332:       vxp = .5*(vx+avx); vxm = .5*(vx-avx);
333:       vy  = x[j][i].v; avy = PetscAbsScalar(vy);
334:       vyp = .5*(vy+avy); vym = .5*(vy-avy);

336:       /* U velocity */
337:       u         = x[j][i].u;
338:       udot      = user->parabolic ? xdot[j][i].u : 0.;
339:       uxx       = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
340:       uyy       = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
341:       f[j][i].u = udot + uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx;

343:       /* V velocity */
344:       u         = x[j][i].v;
345:       udot      = user->parabolic ? xdot[j][i].v : 0.;
346:       uxx       = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
347:       uyy       = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
348:       f[j][i].v = udot + uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy;

350:       /* Omega */
351:       u             = x[j][i].omega;
352:       uxx           = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
353:       uyy           = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
354:       f[j][i].omega = (xdot[j][i].omega + uxx + uyy
355:                        + (vxp*(u - x[j][i-1].omega)
356:                           + vxm*(x[j][i+1].omega - u)) * hy
357:                        + (vyp*(u - x[j-1][i].omega)
358:                           + vym*(x[j+1][i].omega - u)) * hx
359:                        - .5 * grashof * (x[j][i+1].temp - x[j][i-1].temp) * hy);

361:       /* Temperature */
362:       u            = x[j][i].temp;
363:       uxx          = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
364:       uyy          = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
365:       f[j][i].temp =  (xdot[j][i].temp + uxx + uyy
366:                        + prandtl * ((vxp*(u - x[j][i-1].temp)
367:                                      + vxm*(x[j][i+1].temp - u)) * hy
368:                                     + (vyp*(u - x[j-1][i].temp)
369:                                        + vym*(x[j+1][i].temp - u)) * hx));
370:     }
371:   }

373:   /*
374:      Flop count (multiply-adds are counted as 2 operations)
375:   */
376:   PetscLogFlops(84.0*info->ym*info->xm);
377:   return(0);
378: }