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;
123: PetscInitialize(&argc, &argv, (char *)0, help);
124: TSCreate(PETSC_COMM_WORLD, &ts);
125: 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: DMSetFromOptions(da);
127: DMSetUp(da);
128: TSSetDM(ts, (DM)da);
130: 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: PetscOptionsReal("-lidvelocity", "Lid velocity, related to Reynolds number", "", user.lidvelocity, &user.lidvelocity, NULL);
142: PetscOptionsReal("-prandtl", "Ratio of viscous to thermal diffusivity", "", user.prandtl, &user.prandtl, NULL);
143: PetscOptionsReal("-grashof", "Ratio of buoyant to viscous forces", "", user.grashof, &user.grashof, NULL);
144: PetscOptionsBool("-parabolic", "Relax incompressibility to make the system parabolic instead of differential-algebraic", "", user.parabolic, &user.parabolic, NULL);
145: PetscOptionsReal("-cfl_initial", "Advective CFL for the first time step", "", user.cfl_initial, &user.cfl_initial, NULL);
146: PetscOptionsEnd();
148: DMDASetFieldName(da, 0, "x-velocity");
149: DMDASetFieldName(da, 1, "y-velocity");
150: DMDASetFieldName(da, 2, "Omega");
151: 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: DMSetApplicationContext(da, &user);
162: DMDATSSetIFunctionLocal(da, INSERT_VALUES, (DMDATSIFunctionLocal)FormIFunctionLocal, &user);
163: TSSetMaxSteps(ts, 10000);
164: TSSetMaxTime(ts, 1e12);
165: TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER);
166: TSSetTimeStep(ts, user.cfl_initial / (user.lidvelocity * mx));
167: TSSetFromOptions(ts);
169: 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: DMCreateGlobalVector(da, &X);
176: FormInitialSolution(ts, X, &user);
178: TSSolve(ts, X);
179: TSGetSolveTime(ts, &ftime);
180: TSGetStepNumber(ts, &steps);
181: TSGetConvergedReason(ts, &reason);
183: 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: VecDestroy(&X);
190: DMDestroy(&da);
191: TSDestroy(&ts);
193: 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: grashof = user->grashof;
217: TSGetDM(ts, &da);
218: DMDAGetInfo(da, 0, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0);
219: dx = 1.0 / (mx - 1);
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: DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL);
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: DMDAVecGetArray(da, X, &x);
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: }
250: /*
251: Restore vector
252: */
253: DMDAVecRestoreArray(da, X, &x);
254: return 0;
255: }
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;
266: grashof = user->grashof;
267: prandtl = user->prandtl;
268: lid = user->lidvelocity;
270: /*
271: Define mesh intervals ratios for uniform grid.
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.
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;
284: xints = info->xs;
285: xinte = info->xs + info->xm;
286: yints = info->ys;
287: yinte = info->ys + info->ym;
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: }
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: }
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: }
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: }
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);
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;
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;
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);
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: }
384: /*
385: Flop count (multiply-adds are counted as 2 operations)
386: */
387: PetscLogFlops(84.0 * info->ym * info->xm);
388: return 0;
389: }
391: /*TEST
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
397: test:
398: suffix: 2
399: nsize: 4
400: 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'
401: requires: !complex !single
403: test:
404: suffix: 3
405: nsize: 4
406: 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
407: requires: !complex !single
409: test:
410: suffix: 4
411: nsize: 2
412: args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
413: requires: !complex !single
415: test:
416: suffix: asm
417: nsize: 4
418: args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
419: requires: !complex !single
421: TEST*/