1 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. 12 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 16 17 Todd S. Coffey and C. T. Kelley and David E. Keyes, Pseudotransient 18 Continuation and Differential-Algebraic Equations, 2003. 19 20 That example was removed because it used obsolete interfaces, but the 21 algorithms from the paper can be reproduced using this example. 22 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. 30 31 Comment preserved from snes/tutorials/ex27.c, since removed: 32 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. 41 42 See also https://lists.mcs.anl.gov/pipermail/petsc-dev/2010-March/002362.html 43 */ 44 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 /* ------------------------------------------------------------------------ 53 54 We thank David E. Keyes for contributing the driven cavity discretization 55 within this example code. 56 57 This problem is modeled by the partial differential equation system 58 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 63 64 in the unit square, which is uniformly discretized in each of x and 65 y in this simple encoding. 66 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. 74 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. 79 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). 85 86 ------------------------------------------------------------------------- */ 87 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> 101 102 /* 103 User-defined routines and data structures 104 */ 105 typedef struct { 106 PetscScalar u,v,omega,temp; 107 } Field; 108 109 PetscErrorCode FormIFunctionLocal(DMDALocalInfo*,PetscReal,Field**,Field**,Field**,void*); 110 111 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; 116 117 PetscErrorCode FormInitialSolution(TS,Vec,AppCtx*); 118 119 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; 129 130 ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr; 131 CHKERRQ(TSCreate(PETSC_COMM_WORLD,&ts)); 132 CHKERRQ(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 CHKERRQ(DMSetFromOptions(da)); 134 CHKERRQ(DMSetUp(da)); 135 CHKERRQ(TSSetDM(ts,(DM)da)); 136 137 ierr = 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);CHKERRQ(ierr); 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.; 147 148 ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Driven cavity/natural convection options","");CHKERRQ(ierr); 149 CHKERRQ(PetscOptionsReal("-lidvelocity","Lid velocity, related to Reynolds number","",user.lidvelocity,&user.lidvelocity,NULL)); 150 CHKERRQ(PetscOptionsReal("-prandtl","Ratio of viscous to thermal diffusivity","",user.prandtl,&user.prandtl,NULL)); 151 CHKERRQ(PetscOptionsReal("-grashof","Ratio of bouyant to viscous forces","",user.grashof,&user.grashof,NULL)); 152 CHKERRQ(PetscOptionsBool("-parabolic","Relax incompressibility to make the system parabolic instead of differential-algebraic","",user.parabolic,&user.parabolic,NULL)); 153 CHKERRQ(PetscOptionsReal("-cfl_initial","Advective CFL for the first time step","",user.cfl_initial,&user.cfl_initial,NULL)); 154 ierr = PetscOptionsEnd();CHKERRQ(ierr); 155 156 CHKERRQ(DMDASetFieldName(da,0,"x-velocity")); 157 CHKERRQ(DMDASetFieldName(da,1,"y-velocity")); 158 CHKERRQ(DMDASetFieldName(da,2,"Omega")); 159 CHKERRQ(DMDASetFieldName(da,3,"temperature")); 160 161 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 162 Create user context, set problem data, create vector data structures. 163 Also, compute the initial guess. 164 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 165 166 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 167 Create time integration context 168 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 169 CHKERRQ(DMSetApplicationContext(da,&user)); 170 CHKERRQ(DMDATSSetIFunctionLocal(da,INSERT_VALUES,(DMDATSIFunctionLocal)FormIFunctionLocal,&user)); 171 CHKERRQ(TSSetMaxSteps(ts,10000)); 172 CHKERRQ(TSSetMaxTime(ts,1e12)); 173 CHKERRQ(TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER)); 174 CHKERRQ(TSSetTimeStep(ts,user.cfl_initial/(user.lidvelocity*mx))); 175 CHKERRQ(TSSetFromOptions(ts)); 176 177 CHKERRQ(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)); 178 179 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 180 Solve the nonlinear system 181 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 182 183 CHKERRQ(DMCreateGlobalVector(da,&X)); 184 CHKERRQ(FormInitialSolution(ts,X,&user)); 185 186 CHKERRQ(TSSolve(ts,X)); 187 CHKERRQ(TSGetSolveTime(ts,&ftime)); 188 CHKERRQ(TSGetStepNumber(ts,&steps)); 189 CHKERRQ(TSGetConvergedReason(ts,&reason)); 190 191 CHKERRQ(PetscPrintf(PETSC_COMM_WORLD,"%s at time %g after %D steps\n",TSConvergedReasons[reason],(double)ftime,steps)); 192 193 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 194 Free work space. All PETSc objects should be destroyed when they 195 are no longer needed. 196 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 197 CHKERRQ(VecDestroy(&X)); 198 CHKERRQ(DMDestroy(&da)); 199 CHKERRQ(TSDestroy(&ts)); 200 201 ierr = PetscFinalize(); 202 return ierr; 203 } 204 205 /* ------------------------------------------------------------------- */ 206 207 /* 208 FormInitialSolution - Forms initial approximation. 209 210 Input Parameters: 211 user - user-defined application context 212 X - vector 213 214 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; 223 224 grashof = user->grashof; 225 CHKERRQ(TSGetDM(ts,&da)); 226 CHKERRQ(DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0)); 227 dx = 1.0/(mx-1); 228 229 /* 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 CHKERRQ(DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL)); 235 236 /* 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 CHKERRQ(DMDAVecGetArray(da,X,&x)); 244 245 /* 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 } 257 258 /* 259 Restore vector 260 */ 261 CHKERRQ(DMDAVecRestoreArray(da,X,&x)); 262 return 0; 263 } 264 265 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; 272 273 PetscFunctionBeginUser; 274 grashof = user->grashof; 275 prandtl = user->prandtl; 276 lid = user->lidvelocity; 277 278 /* 279 Define mesh intervals ratios for uniform grid. 280 281 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. 283 284 */ 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; 288 289 xints = info->xs; xinte = info->xs+info->xm; yints = info->ys; yinte = info->ys+info->ym; 290 291 /* 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 } 303 304 /* 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 } 316 317 /* 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 } 329 330 /* 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 } 342 343 /* Compute over the interior points */ 344 for (j=yints; j<yinte; j++) { 345 for (i=xints; i<xinte; i++) { 346 347 /* 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); 354 355 /* 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; 361 362 /* 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; 368 369 /* 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); 379 380 /* 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 } 391 392 /* 393 Flop count (multiply-adds are counted as 2 operations) 394 */ 395 CHKERRQ(PetscLogFlops(84.0*info->ym*info->xm)); 396 PetscFunctionReturn(0); 397 } 398 399 /*TEST 400 401 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 404 405 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 410 411 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 416 417 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 422 423 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 428 429 TEST*/ 430