xref: /petsc/src/snes/tutorials/ex30.c (revision 030f984af8d8bb4c203755d35bded3c05b3d83ce) !
1 static const char help[] = "Steady-state 2D subduction flow, pressure and temperature solver.\n\
2        The flow is driven by the subducting slab.\n\
3 ---------------------------------ex30 help---------------------------------\n\
4   -OPTION <DEFAULT> = (UNITS) DESCRIPTION.\n\n\
5   -width <320> = (km) width of domain.\n\
6   -depth <300> = (km) depth of domain.\n\
7   -slab_dip <45> = (degrees) dip angle of the slab (determines the grid aspect ratio).\n\
8   -lid_depth <35> = (km) depth of the static conductive lid.\n\
9   -fault_depth <35> = (km) depth of slab-wedge mechanical coupling\n\
10      (fault dept >= lid depth).\n\
11 \n\
12   -ni <82> = grid cells in x-direction. (nj adjusts to accommodate\n\
13       the slab dip & depth). DO NOT USE -da_grid_x option!!!\n\
14   -ivisc <3> = rheology option.\n\
15       0 --- constant viscosity.\n\
16       1 --- olivine diffusion creep rheology (T&P-dependent, newtonian).\n\
17       2 --- olivine dislocation creep rheology (T&P-dependent, non-newtonian).\n\
18       3 --- Full mantle rheology, combination of 1 & 2.\n\
19 \n\
20   -slab_velocity <5> = (cm/year) convergence rate of slab into subduction zone.\n\
21   -slab_age <50> = (million yrs) age of slab for thermal profile boundary condition.\n\
22   -lid_age <50> = (million yrs) age of lid for thermal profile boundary condition.\n\
23 \n\
24   FOR OTHER PARAMETER OPTIONS AND THEIR DEFAULT VALUES, see SetParams() in ex30.c.\n\
25 ---------------------------------ex30 help---------------------------------\n";
26 
27 /*F-----------------------------------------------------------------------
28 
29     This PETSc 2.2.0 example by Richard F. Katz
30     http://www.ldeo.columbia.edu/~katz/
31 
32     The problem is modeled by the partial differential equation system
33 
34 \begin{eqnarray}
35          -\nabla P + \nabla \cdot [\eta (\nabla v + \nabla v^T)] & = & 0  \\
36                                            \nabla \cdot v & = & 0   \\
37                     dT/dt + \nabla \cdot (vT) - 1/Pe \triangle^2(T) & = & 0  \\
38 \end{eqnarray}
39 
40  \begin{eqnarray}
41         \eta(T,Eps\_dot) &  = & \hbox{constant                        }    \hbox{if ivisc} ==0  \\
42                       &  = & \hbox{diffusion creep (T,P-dependent)    }     \hbox{if ivisc} ==1  \\
43                       &  = & \hbox{dislocation creep (T,P,v-dependent)}  \hbox{if ivisc} ==2  \\
44                       &  = & \hbox{mantle viscosity (difn and disl)   }  \hbox{if ivisc} ==3
45 \end{eqnarray}
46 
47     which is uniformly discretized on a staggered mesh:
48                       -------$w_{ij}$------
49                   $u_{i-1j}$    $P,T_{ij}$   $u_{ij}$
50                       ------$w_{ij-1}$-----
51 
52   ------------------------------------------------------------------------F*/
53 
54 #include <petscsnes.h>
55 #include <petscdm.h>
56 #include <petscdmda.h>
57 
58 #define VISC_CONST   0
59 #define VISC_DIFN    1
60 #define VISC_DISL    2
61 #define VISC_FULL    3
62 #define CELL_CENTER  0
63 #define CELL_CORNER  1
64 #define BC_ANALYTIC  0
65 #define BC_NOSTRESS  1
66 #define BC_EXPERMNT  2
67 #define ADVECT_FV    0
68 #define ADVECT_FROMM 1
69 #define PLATE_SLAB   0
70 #define PLATE_LID    1
71 #define EPS_ZERO     0.00000001
72 
73 typedef struct { /* holds the variables to be solved for */
74   PetscScalar u,w,p,T;
75 } Field;
76 
77 typedef struct { /* parameters needed to compute viscosity */
78   PetscReal A,n,Estar,Vstar;
79 } ViscParam;
80 
81 typedef struct { /* physical and miscelaneous parameters */
82   PetscReal width, depth, scaled_width, scaled_depth, peclet, potentialT;
83   PetscReal slab_dip, slab_age, slab_velocity, kappa, z_scale;
84   PetscReal c, d, sb, cb, skt, visc_cutoff, lid_age, eta0, continuation;
85   PetscReal L, V, lid_depth, fault_depth;
86   ViscParam diffusion, dislocation;
87   PetscInt  ivisc, adv_scheme, ibound, output_ivisc;
88   PetscBool quiet, param_test, output_to_file, pv_analytic;
89   PetscBool interrupted, stop_solve, toggle_kspmon, kspmon;
90   char      filename[PETSC_MAX_PATH_LEN];
91 } Parameter;
92 
93 typedef struct { /* grid parameters */
94   DMBoundaryType   bx,by;
95   DMDAStencilType  stencil;
96   PetscInt         corner,ni,nj,jlid,jfault,inose;
97   PetscInt         dof,stencil_width,mglevels;
98   PetscReal        dx,dz;
99 } GridInfo;
100 
101 typedef struct { /* application context */
102   Vec       x,Xguess;
103   Parameter *param;
104   GridInfo  *grid;
105 } AppCtx;
106 
107 /* Callback functions (static interface) */
108 extern PetscErrorCode FormFunctionLocal(DMDALocalInfo*,Field**,Field**,void*);
109 
110 /* Main routines */
111 extern PetscErrorCode SetParams(Parameter*, GridInfo*);
112 extern PetscErrorCode ReportParams(Parameter*, GridInfo*);
113 extern PetscErrorCode Initialize(DM);
114 extern PetscErrorCode UpdateSolution(SNES,AppCtx*, PetscInt*);
115 extern PetscErrorCode DoOutput(SNES,PetscInt);
116 
117 /* Post-processing & misc */
118 extern PetscErrorCode ViscosityField(DM,Vec,Vec);
119 extern PetscErrorCode StressField(DM);
120 extern PetscErrorCode SNESConverged_Interactive(SNES, PetscInt, PetscReal, PetscReal, PetscReal, SNESConvergedReason*, void*);
121 extern PetscErrorCode InteractiveHandler(int, void*);
122 
123 /*-----------------------------------------------------------------------*/
124 int main(int argc,char **argv)
125 /*-----------------------------------------------------------------------*/
126 {
127   SNES           snes;
128   AppCtx         *user;               /* user-defined work context */
129   Parameter      param;
130   GridInfo       grid;
131   PetscInt       nits;
132   PetscErrorCode ierr;
133   MPI_Comm       comm;
134   DM             da;
135 
136   ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
137   PetscOptionsSetValue(NULL,"-file","ex30_output");
138   PetscOptionsSetValue(NULL,"-snes_monitor_short",NULL);
139   PetscOptionsSetValue(NULL,"-snes_max_it","20");
140   PetscOptionsSetValue(NULL,"-ksp_max_it","1500");
141   PetscOptionsSetValue(NULL,"-ksp_gmres_restart","300");
142   PetscOptionsInsert(NULL,&argc,&argv,NULL);
143 
144   comm = PETSC_COMM_WORLD;
145 
146   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
147      Set up the problem parameters.
148      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
149   ierr = SetParams(&param,&grid);CHKERRQ(ierr);
150   ierr = ReportParams(&param,&grid);CHKERRQ(ierr);
151 
152   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
154   ierr = SNESCreate(comm,&snes);CHKERRQ(ierr);
155   ierr = DMDACreate2d(comm,grid.bx,grid.by,grid.stencil,grid.ni,grid.nj,PETSC_DECIDE,PETSC_DECIDE,grid.dof,grid.stencil_width,0,0,&da);CHKERRQ(ierr);
156   ierr = DMSetFromOptions(da);CHKERRQ(ierr);
157   ierr = DMSetUp(da);CHKERRQ(ierr);
158   ierr = SNESSetDM(snes,da);CHKERRQ(ierr);
159   ierr = DMDASetFieldName(da,0,"x-velocity");CHKERRQ(ierr);
160   ierr = DMDASetFieldName(da,1,"y-velocity");CHKERRQ(ierr);
161   ierr = DMDASetFieldName(da,2,"pressure");CHKERRQ(ierr);
162   ierr = DMDASetFieldName(da,3,"temperature");CHKERRQ(ierr);
163 
164   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
165      Create user context, set problem data, create vector data structures.
166      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
167   ierr        = PetscNew(&user);CHKERRQ(ierr);
168   user->param = &param;
169   user->grid  = &grid;
170   ierr        = DMSetApplicationContext(da,user);CHKERRQ(ierr);
171   ierr        = DMCreateGlobalVector(da,&(user->Xguess));CHKERRQ(ierr);
172 
173   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
174      Set up the SNES solver with callback functions.
175      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
176   ierr = DMDASNESSetFunctionLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionLocal,(void*)user);CHKERRQ(ierr);
177   ierr = SNESSetFromOptions(snes);CHKERRQ(ierr);
178 
179   ierr = SNESSetConvergenceTest(snes,SNESConverged_Interactive,(void*)user,NULL);CHKERRQ(ierr);
180   ierr = PetscPushSignalHandler(InteractiveHandler,(void*)user);CHKERRQ(ierr);
181 
182   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
183      Initialize and solve the nonlinear system
184      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
185   ierr = Initialize(da);CHKERRQ(ierr);
186   ierr = UpdateSolution(snes,user,&nits);CHKERRQ(ierr);
187 
188   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
189      Output variables.
190      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
191   ierr = DoOutput(snes,nits);CHKERRQ(ierr);
192 
193   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194      Free work space.
195      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
196   ierr = VecDestroy(&user->Xguess);CHKERRQ(ierr);
197   ierr = VecDestroy(&user->x);CHKERRQ(ierr);
198   ierr = PetscFree(user);CHKERRQ(ierr);
199   ierr = SNESDestroy(&snes);CHKERRQ(ierr);
200   ierr = DMDestroy(&da);CHKERRQ(ierr);
201   ierr = PetscPopSignalHandler();CHKERRQ(ierr);
202   ierr = PetscFinalize();
203   return ierr;
204 }
205 
206 /*=====================================================================
207   PETSc INTERACTION FUNCTIONS (initialize & call SNESSolve)
208   =====================================================================*/
209 
210 /*---------------------------------------------------------------------*/
211 /*  manages solve: adaptive continuation method  */
212 PetscErrorCode UpdateSolution(SNES snes, AppCtx *user, PetscInt *nits)
213 {
214   KSP                 ksp;
215   PC                  pc;
216   SNESConvergedReason reason = SNES_CONVERGED_ITERATING;
217   Parameter           *param   = user->param;
218   PetscReal           cont_incr=0.3;
219   PetscInt            its;
220   PetscErrorCode      ierr;
221   PetscBool           q = PETSC_FALSE;
222   DM                  dm;
223 
224   PetscFunctionBeginUser;
225   ierr = SNESGetDM(snes,&dm);CHKERRQ(ierr);
226   ierr = DMCreateGlobalVector(dm,&user->x);CHKERRQ(ierr);
227   ierr = SNESGetKSP(snes,&ksp);CHKERRQ(ierr);
228   ierr = KSPGetPC(ksp, &pc);CHKERRQ(ierr);
229   ierr = KSPSetComputeSingularValues(ksp, PETSC_TRUE);CHKERRQ(ierr);
230 
231   *nits=0;
232 
233   /* Isoviscous solve */
234   if (param->ivisc == VISC_CONST && !param->stop_solve) {
235     param->ivisc = VISC_CONST;
236 
237     ierr   = SNESSolve(snes,0,user->x);CHKERRQ(ierr);
238     ierr   = SNESGetConvergedReason(snes,&reason);CHKERRQ(ierr);
239     ierr   = SNESGetIterationNumber(snes,&its);CHKERRQ(ierr);
240     *nits += its;
241     ierr   = VecCopy(user->x,user->Xguess);CHKERRQ(ierr);
242     if (param->stop_solve) goto done;
243   }
244 
245   /* Olivine diffusion creep */
246   if (param->ivisc >= VISC_DIFN && !param->stop_solve) {
247     if (!q) {ierr = PetscPrintf(PETSC_COMM_WORLD,"Computing Variable Viscosity Solution\n");CHKERRQ(ierr);}
248 
249     /* continuation method on viscosity cutoff */
250     for (param->continuation=0.0;; param->continuation+=cont_incr) {
251       if (!q) {ierr = PetscPrintf(PETSC_COMM_WORLD," Continuation parameter = %g\n", (double)param->continuation);CHKERRQ(ierr);}
252 
253       /* solve the non-linear system */
254       ierr   = VecCopy(user->Xguess,user->x);CHKERRQ(ierr);
255       ierr   = SNESSolve(snes,0,user->x);CHKERRQ(ierr);
256       ierr   = SNESGetConvergedReason(snes,&reason);CHKERRQ(ierr);
257       ierr   = SNESGetIterationNumber(snes,&its);CHKERRQ(ierr);
258       *nits += its;
259       if (!q) {ierr = PetscPrintf(PETSC_COMM_WORLD," SNES iterations: %D, Cumulative: %D\n", its, *nits);CHKERRQ(ierr);}
260       if (param->stop_solve) goto done;
261 
262       if (reason<0) {
263         /* NOT converged */
264         cont_incr = -PetscAbsReal(cont_incr)/2.0;
265         if (PetscAbsReal(cont_incr)<0.01) goto done;
266 
267       } else {
268         /* converged */
269         ierr = VecCopy(user->x,user->Xguess);CHKERRQ(ierr);
270         if (param->continuation >= 1.0) goto done;
271         if (its<=3)      cont_incr = 0.30001;
272         else if (its<=8) cont_incr = 0.15001;
273         else             cont_incr = 0.10001;
274 
275         if (param->continuation+cont_incr > 1.0) cont_incr = 1.0 - param->continuation;
276       } /* endif reason<0 */
277     }
278   }
279 done:
280   if (param->stop_solve && !q) {ierr = PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: stopping solve.\n");CHKERRQ(ierr);}
281   if (reason<0 && !q) {ierr = PetscPrintf(PETSC_COMM_WORLD,"FAILED TO CONVERGE: stopping solve.\n");CHKERRQ(ierr);}
282   PetscFunctionReturn(0);
283 }
284 
285 /*=====================================================================
286   PHYSICS FUNCTIONS (compute the discrete residual)
287   =====================================================================*/
288 
289 /*---------------------------------------------------------------------*/
290 PETSC_STATIC_INLINE PetscScalar UInterp(Field **x, PetscInt i, PetscInt j)
291 /*---------------------------------------------------------------------*/
292 {
293   return 0.25*(x[j][i].u+x[j+1][i].u+x[j][i+1].u+x[j+1][i+1].u);
294 }
295 
296 /*---------------------------------------------------------------------*/
297 PETSC_STATIC_INLINE PetscScalar WInterp(Field **x, PetscInt i, PetscInt j)
298 /*---------------------------------------------------------------------*/
299 {
300   return 0.25*(x[j][i].w+x[j+1][i].w+x[j][i+1].w+x[j+1][i+1].w);
301 }
302 
303 /*---------------------------------------------------------------------*/
304 PETSC_STATIC_INLINE PetscScalar PInterp(Field **x, PetscInt i, PetscInt j)
305 /*---------------------------------------------------------------------*/
306 {
307   return 0.25*(x[j][i].p+x[j+1][i].p+x[j][i+1].p+x[j+1][i+1].p);
308 }
309 
310 /*---------------------------------------------------------------------*/
311 PETSC_STATIC_INLINE PetscScalar TInterp(Field **x, PetscInt i, PetscInt j)
312 /*---------------------------------------------------------------------*/
313 {
314   return 0.25*(x[j][i].T+x[j+1][i].T+x[j][i+1].T+x[j+1][i+1].T);
315 }
316 
317 /*---------------------------------------------------------------------*/
318 /*  isoviscous analytic solution for IC */
319 PETSC_STATIC_INLINE PetscScalar HorizVelocity(PetscInt i, PetscInt j, AppCtx *user)
320 /*---------------------------------------------------------------------*/
321 {
322   Parameter   *param = user->param;
323   GridInfo    *grid  = user->grid;
324   PetscScalar st, ct, th, c=param->c, d=param->d;
325   PetscReal   x, z,r;
326 
327   x  = (i - grid->jlid)*grid->dx;  z = (j - grid->jlid - 0.5)*grid->dz;
328   r  = PetscSqrtReal(x*x+z*z);
329   st = z/r;
330   ct = x/r;
331   th = PetscAtanReal(z/x);
332   return ct*(c*th*st+d*(st+th*ct)) + st*(c*(st-th*ct)+d*th*st);
333 }
334 
335 /*---------------------------------------------------------------------*/
336 /*  isoviscous analytic solution for IC */
337 PETSC_STATIC_INLINE PetscScalar VertVelocity(PetscInt i, PetscInt j, AppCtx *user)
338 /*---------------------------------------------------------------------*/
339 {
340   Parameter   *param = user->param;
341   GridInfo    *grid  = user->grid;
342   PetscScalar st, ct, th, c=param->c, d=param->d;
343   PetscReal   x, z, r;
344 
345   x = (i - grid->jlid - 0.5)*grid->dx;  z = (j - grid->jlid)*grid->dz;
346   r = PetscSqrtReal(x*x+z*z); st = z/r;  ct = x/r;  th = PetscAtanReal(z/x);
347   return st*(c*th*st+d*(st+th*ct)) - ct*(c*(st-th*ct)+d*th*st);
348 }
349 
350 /*---------------------------------------------------------------------*/
351 /*  isoviscous analytic solution for IC */
352 PETSC_STATIC_INLINE PetscScalar Pressure(PetscInt i, PetscInt j, AppCtx *user)
353 /*---------------------------------------------------------------------*/
354 {
355   Parameter   *param = user->param;
356   GridInfo    *grid  = user->grid;
357   PetscScalar x, z, r, st, ct, c=param->c, d=param->d;
358 
359   x = (i - grid->jlid - 0.5)*grid->dx;  z = (j - grid->jlid - 0.5)*grid->dz;
360   r = PetscSqrtReal(x*x+z*z);  st = z/r;  ct = x/r;
361   return (-2.0*(c*ct-d*st)/r);
362 }
363 
364 /*  computes the second invariant of the strain rate tensor */
365 PETSC_STATIC_INLINE PetscScalar CalcSecInv(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
366 /*---------------------------------------------------------------------*/
367 {
368   Parameter   *param = user->param;
369   GridInfo    *grid  = user->grid;
370   PetscInt    ilim   =grid->ni-1, jlim=grid->nj-1;
371   PetscScalar uN,uS,uE,uW,wN,wS,wE,wW;
372   PetscScalar eps11, eps12, eps22;
373 
374   if (i<j) return EPS_ZERO;
375   if (i==ilim) i--;
376   if (j==jlim) j--;
377 
378   if (ipos==CELL_CENTER) { /* on cell center */
379     if (j<=grid->jlid) return EPS_ZERO;
380 
381     uE = x[j][i].u; uW = x[j][i-1].u;
382     wN = x[j][i].w; wS = x[j-1][i].w;
383     wE = WInterp(x,i,j-1);
384     if (i==j) {
385       uN = param->cb; wW = param->sb;
386     } else {
387       uN = UInterp(x,i-1,j); wW = WInterp(x,i-1,j-1);
388     }
389 
390     if (j==grid->jlid+1) uS = 0.0;
391     else                 uS = UInterp(x,i-1,j-1);
392 
393   } else {       /* on CELL_CORNER */
394     if (j<grid->jlid) return EPS_ZERO;
395 
396     uN = x[j+1][i].u;  uS = x[j][i].u;
397     wE = x[j][i+1].w;  wW = x[j][i].w;
398     if (i==j) {
399       wN = param->sb;
400       uW = param->cb;
401     } else {
402       wN = WInterp(x,i,j);
403       uW = UInterp(x,i-1,j);
404     }
405 
406     if (j==grid->jlid) {
407       uE = 0.0;  uW = 0.0;
408       uS = -uN;
409       wS = -wN;
410     } else {
411       uE = UInterp(x,i,j);
412       wS = WInterp(x,i,j-1);
413     }
414   }
415 
416   eps11 = (uE-uW)/grid->dx;  eps22 = (wN-wS)/grid->dz;
417   eps12 = 0.5*((uN-uS)/grid->dz + (wE-wW)/grid->dx);
418 
419   return PetscSqrtReal(0.5*(eps11*eps11 + 2.0*eps12*eps12 + eps22*eps22));
420 }
421 
422 /*---------------------------------------------------------------------*/
423 /*  computes the shear viscosity */
424 PETSC_STATIC_INLINE PetscScalar Viscosity(PetscScalar T, PetscScalar eps, PetscScalar z, Parameter *param)
425 /*---------------------------------------------------------------------*/
426 {
427   PetscReal   result   =0.0;
428   ViscParam   difn     =param->diffusion, disl=param->dislocation;
429   PetscInt    iVisc    =param->ivisc;
430   PetscScalar eps_scale=param->V/(param->L*1000.0);
431   PetscScalar strain_power, v1, v2, P;
432   PetscScalar rho_g = 32340.0, R=8.3144;
433 
434   P = rho_g*(z*param->L*1000.0); /* Pa */
435 
436   if (iVisc==VISC_CONST) {
437     /* constant viscosity */
438     return 1.0;
439   } else if (iVisc==VISC_DIFN) {
440     /* diffusion creep rheology */
441     result = PetscRealPart((difn.A*PetscExpScalar((difn.Estar + P*difn.Vstar)/R/(T+273.0))/param->eta0));
442   } else if (iVisc==VISC_DISL) {
443     /* dislocation creep rheology */
444     strain_power = PetscPowScalar(eps*eps_scale, (1.0-disl.n)/disl.n);
445 
446     result = PetscRealPart(disl.A*PetscExpScalar((disl.Estar + P*disl.Vstar)/disl.n/R/(T+273.0))*strain_power/param->eta0);
447   } else if (iVisc==VISC_FULL) {
448     /* dislocation/diffusion creep rheology */
449     strain_power = PetscPowScalar(eps*eps_scale, (1.0-disl.n)/disl.n);
450 
451     v1 = difn.A*PetscExpScalar((difn.Estar + P*difn.Vstar)/R/(T+273.0))/param->eta0;
452     v2 = disl.A*PetscExpScalar((disl.Estar + P*disl.Vstar)/disl.n/R/(T+273.0))*strain_power/param->eta0;
453 
454     result = PetscRealPart(1.0/(1.0/v1 + 1.0/v2));
455   }
456 
457   /* max viscosity is param->eta0 */
458   result = PetscMin(result, 1.0);
459   /* min viscosity is param->visc_cutoff */
460   result = PetscMax(result, param->visc_cutoff);
461   /* continuation method */
462   result = PetscPowReal(result,param->continuation);
463   return result;
464 }
465 
466 /*---------------------------------------------------------------------*/
467 /*  computes the residual of the x-component of eqn (1) above */
468 PETSC_STATIC_INLINE PetscScalar XMomentumResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
469 /*---------------------------------------------------------------------*/
470 {
471   Parameter   *param=user->param;
472   GridInfo    *grid =user->grid;
473   PetscScalar dx    = grid->dx, dz=grid->dz;
474   PetscScalar etaN,etaS,etaE,etaW,epsN=0.0,epsS=0.0,epsE=0.0,epsW=0.0;
475   PetscScalar TE=0.0,TN=0.0,TS=0.0,TW=0.0, dPdx, residual, z_scale;
476   PetscScalar dudxW,dudxE,dudzN,dudzS,dwdxN,dwdxS;
477   PetscInt    jlim = grid->nj-1;
478 
479   z_scale = param->z_scale;
480 
481   if (param->ivisc==VISC_DIFN || param->ivisc>=VISC_DISL) { /* viscosity is T-dependent */
482     TS = param->potentialT * TInterp(x,i,j-1) * PetscExpScalar((j-1.0)*dz*z_scale);
483     if (j==jlim) TN = TS;
484     else         TN = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
485     TW = param->potentialT * x[j][i].T        * PetscExpScalar((j-0.5)*dz*z_scale);
486     TE = param->potentialT * x[j][i+1].T      * PetscExpScalar((j-0.5)*dz*z_scale);
487     if (param->ivisc>=VISC_DISL) { /* olivine dislocation creep */
488       epsN = CalcSecInv(x,i,j,  CELL_CORNER,user);
489       epsS = CalcSecInv(x,i,j-1,CELL_CORNER,user);
490       epsE = CalcSecInv(x,i+1,j,CELL_CENTER,user);
491       epsW = CalcSecInv(x,i,j,  CELL_CENTER,user);
492     }
493   }
494   etaN = Viscosity(TN,epsN,dz*(j+0.5),param);
495   etaS = Viscosity(TS,epsS,dz*(j-0.5),param);
496   etaW = Viscosity(TW,epsW,dz*j,param);
497   etaE = Viscosity(TE,epsE,dz*j,param);
498 
499   dPdx = (x[j][i+1].p - x[j][i].p)/dx;
500   if (j==jlim) dudzN = etaN * (x[j][i].w   - x[j][i+1].w)/dx;
501   else         dudzN = etaN * (x[j+1][i].u - x[j][i].u)  /dz;
502   dudzS = etaS * (x[j][i].u    - x[j-1][i].u)/dz;
503   dudxE = etaE * (x[j][i+1].u  - x[j][i].u)  /dx;
504   dudxW = etaW * (x[j][i].u    - x[j][i-1].u)/dx;
505 
506   residual = -dPdx                          /* X-MOMENTUM EQUATION*/
507              +(dudxE - dudxW)/dx
508              +(dudzN - dudzS)/dz;
509 
510   if (param->ivisc!=VISC_CONST) {
511     dwdxN = etaN * (x[j][i+1].w   - x[j][i].w)  /dx;
512     dwdxS = etaS * (x[j-1][i+1].w - x[j-1][i].w)/dx;
513 
514     residual += (dudxE - dudxW)/dx + (dwdxN - dwdxS)/dz;
515   }
516 
517   return residual;
518 }
519 
520 /*---------------------------------------------------------------------*/
521 /*  computes the residual of the z-component of eqn (1) above */
522 PETSC_STATIC_INLINE PetscScalar ZMomentumResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
523 /*---------------------------------------------------------------------*/
524 {
525   Parameter   *param=user->param;
526   GridInfo    *grid =user->grid;
527   PetscScalar dx    = grid->dx, dz=grid->dz;
528   PetscScalar etaN  =0.0,etaS=0.0,etaE=0.0,etaW=0.0,epsN=0.0,epsS=0.0,epsE=0.0,epsW=0.0;
529   PetscScalar TE    =0.0,TN=0.0,TS=0.0,TW=0.0, dPdz, residual,z_scale;
530   PetscScalar dudzE,dudzW,dwdxW,dwdxE,dwdzN,dwdzS;
531   PetscInt    ilim = grid->ni-1;
532 
533   /* geometric and other parameters */
534   z_scale = param->z_scale;
535 
536   /* viscosity */
537   if (param->ivisc==VISC_DIFN || param->ivisc>=VISC_DISL) { /* viscosity is T-dependent */
538     TN = param->potentialT * x[j+1][i].T      * PetscExpScalar((j+0.5)*dz*z_scale);
539     TS = param->potentialT * x[j][i].T        * PetscExpScalar((j-0.5)*dz*z_scale);
540     TW = param->potentialT * TInterp(x,i-1,j) * PetscExpScalar(j*dz*z_scale);
541     if (i==ilim) TE = TW;
542     else         TE = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
543     if (param->ivisc>=VISC_DISL) { /* olivine dislocation creep */
544       epsN = CalcSecInv(x,i,j+1,CELL_CENTER,user);
545       epsS = CalcSecInv(x,i,j,  CELL_CENTER,user);
546       epsE = CalcSecInv(x,i,j,  CELL_CORNER,user);
547       epsW = CalcSecInv(x,i-1,j,CELL_CORNER,user);
548     }
549   }
550   etaN = Viscosity(TN,epsN,dz*(j+1.0),param);
551   etaS = Viscosity(TS,epsS,dz*(j+0.0),param);
552   etaW = Viscosity(TW,epsW,dz*(j+0.5),param);
553   etaE = Viscosity(TE,epsE,dz*(j+0.5),param);
554 
555   dPdz  = (x[j+1][i].p - x[j][i].p)/dz;
556   dwdzN = etaN * (x[j+1][i].w - x[j][i].w)/dz;
557   dwdzS = etaS * (x[j][i].w - x[j-1][i].w)/dz;
558   if (i==ilim) dwdxE = etaE * (x[j][i].u   - x[j+1][i].u)/dz;
559   else         dwdxE = etaE * (x[j][i+1].w - x[j][i].w)  /dx;
560   dwdxW = 2.0*etaW * (x[j][i].w - x[j][i-1].w)/dx;
561 
562   /* Z-MOMENTUM */
563   residual = -dPdz                 /* constant viscosity terms */
564              +(dwdzN - dwdzS)/dz
565              +(dwdxE - dwdxW)/dx;
566 
567   if (param->ivisc!=VISC_CONST) {
568     dudzE = etaE * (x[j+1][i].u - x[j][i].u)/dz;
569     dudzW = etaW * (x[j+1][i-1].u - x[j][i-1].u)/dz;
570 
571     residual += (dwdzN - dwdzS)/dz + (dudzE - dudzW)/dx;
572   }
573 
574   return residual;
575 }
576 
577 /*---------------------------------------------------------------------*/
578 /*  computes the residual of eqn (2) above */
579 PETSC_STATIC_INLINE PetscScalar ContinuityResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
580 /*---------------------------------------------------------------------*/
581 {
582   GridInfo    *grid =user->grid;
583   PetscScalar uE,uW,wN,wS,dudx,dwdz;
584 
585   uW = x[j][i-1].u; uE = x[j][i].u; dudx = (uE - uW)/grid->dx;
586   wS = x[j-1][i].w; wN = x[j][i].w; dwdz = (wN - wS)/grid->dz;
587 
588   return dudx + dwdz;
589 }
590 
591 /*---------------------------------------------------------------------*/
592 /*  computes the residual of eqn (3) above */
593 PETSC_STATIC_INLINE PetscScalar EnergyResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
594 /*---------------------------------------------------------------------*/
595 {
596   Parameter   *param=user->param;
597   GridInfo    *grid =user->grid;
598   PetscScalar dx    = grid->dx, dz=grid->dz;
599   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, jlid=grid->jlid;
600   PetscScalar TE, TN, TS, TW, residual;
601   PetscScalar uE,uW,wN,wS;
602   PetscScalar fN,fS,fE,fW,dTdxW,dTdxE,dTdzN,dTdzS;
603 
604   dTdzN = (x[j+1][i].T - x[j][i].T)  /dz;
605   dTdzS = (x[j][i].T   - x[j-1][i].T)/dz;
606   dTdxE = (x[j][i+1].T - x[j][i].T)  /dx;
607   dTdxW = (x[j][i].T   - x[j][i-1].T)/dx;
608 
609   residual = ((dTdzN - dTdzS)/dz + /* diffusion term */
610               (dTdxE - dTdxW)/dx)*dx*dz/param->peclet;
611 
612   if (j<=jlid && i>=j) {
613     /* don't advect in the lid */
614     return residual;
615   } else if (i<j) {
616     /* beneath the slab sfc */
617     uW = uE = param->cb;
618     wS = wN = param->sb;
619   } else {
620     /* advect in the slab and wedge */
621     uW = x[j][i-1].u; uE = x[j][i].u;
622     wS = x[j-1][i].w; wN = x[j][i].w;
623   }
624 
625   if (param->adv_scheme==ADVECT_FV || i==ilim-1 || j==jlim-1 || i==1 || j==1) {
626     /* finite volume advection */
627     TS = (x[j][i].T + x[j-1][i].T)/2.0;
628     TN = (x[j][i].T + x[j+1][i].T)/2.0;
629     TE = (x[j][i].T + x[j][i+1].T)/2.0;
630     TW = (x[j][i].T + x[j][i-1].T)/2.0;
631     fN = wN*TN*dx; fS = wS*TS*dx;
632     fE = uE*TE*dz; fW = uW*TW*dz;
633 
634   } else {
635     /* Fromm advection scheme */
636     fE =     (uE *(-x[j][i+2].T + 5.0*(x[j][i+1].T+x[j][i].T)-x[j][i-1].T)/8.0
637               - PetscAbsScalar(uE)*(-x[j][i+2].T + 3.0*(x[j][i+1].T-x[j][i].T)+x[j][i-1].T)/8.0)*dz;
638     fW =     (uW *(-x[j][i+1].T + 5.0*(x[j][i].T+x[j][i-1].T)-x[j][i-2].T)/8.0
639               - PetscAbsScalar(uW)*(-x[j][i+1].T + 3.0*(x[j][i].T-x[j][i-1].T)+x[j][i-2].T)/8.0)*dz;
640     fN =     (wN *(-x[j+2][i].T + 5.0*(x[j+1][i].T+x[j][i].T)-x[j-1][i].T)/8.0
641               - PetscAbsScalar(wN)*(-x[j+2][i].T + 3.0*(x[j+1][i].T-x[j][i].T)+x[j-1][i].T)/8.0)*dx;
642     fS =     (wS *(-x[j+1][i].T + 5.0*(x[j][i].T+x[j-1][i].T)-x[j-2][i].T)/8.0
643               - PetscAbsScalar(wS)*(-x[j+1][i].T + 3.0*(x[j][i].T-x[j-1][i].T)+x[j-2][i].T)/8.0)*dx;
644   }
645 
646   residual -= (fE - fW + fN - fS);
647 
648   return residual;
649 }
650 
651 /*---------------------------------------------------------------------*/
652 /*  computes the shear stress---used on the boundaries */
653 PETSC_STATIC_INLINE PetscScalar ShearStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
654 /*---------------------------------------------------------------------*/
655 {
656   Parameter   *param=user->param;
657   GridInfo    *grid =user->grid;
658   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1;
659   PetscScalar uN, uS, wE, wW;
660 
661   if (j<=grid->jlid || i<j || i==ilim || j==jlim) return EPS_ZERO;
662 
663   if (ipos==CELL_CENTER) { /* on cell center */
664 
665     wE = WInterp(x,i,j-1);
666     if (i==j) {
667       wW = param->sb;
668       uN = param->cb;
669     } else {
670       wW = WInterp(x,i-1,j-1);
671       uN = UInterp(x,i-1,j);
672     }
673     if (j==grid->jlid+1) uS = 0.0;
674     else                 uS = UInterp(x,i-1,j-1);
675 
676   } else { /* on cell corner */
677 
678     uN = x[j+1][i].u;         uS = x[j][i].u;
679     wW = x[j][i].w;           wE = x[j][i+1].w;
680 
681   }
682 
683   return (uN-uS)/grid->dz + (wE-wW)/grid->dx;
684 }
685 
686 /*---------------------------------------------------------------------*/
687 /*  computes the normal stress---used on the boundaries */
688 PETSC_STATIC_INLINE PetscScalar XNormalStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
689 /*---------------------------------------------------------------------*/
690 {
691   Parameter   *param=user->param;
692   GridInfo    *grid =user->grid;
693   PetscScalar dx    = grid->dx, dz=grid->dz;
694   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, ivisc;
695   PetscScalar epsC  =0.0, etaC, TC, uE, uW, pC, z_scale;
696   if (i<j || j<=grid->jlid) return EPS_ZERO;
697 
698   ivisc=param->ivisc;  z_scale = param->z_scale;
699 
700   if (ipos==CELL_CENTER) { /* on cell center */
701 
702     TC = param->potentialT * x[j][i].T * PetscExpScalar((j-0.5)*dz*z_scale);
703     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CENTER,user);
704     etaC = Viscosity(TC,epsC,dz*j,param);
705 
706     uW = x[j][i-1].u;   uE = x[j][i].u;
707     pC = x[j][i].p;
708 
709   } else { /* on cell corner */
710     if (i==ilim || j==jlim) return EPS_ZERO;
711 
712     TC = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
713     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CORNER,user);
714     etaC = Viscosity(TC,epsC,dz*(j+0.5),param);
715 
716     if (i==j) uW = param->sb;
717     else      uW = UInterp(x,i-1,j);
718     uE = UInterp(x,i,j); pC = PInterp(x,i,j);
719   }
720 
721   return 2.0*etaC*(uE-uW)/dx - pC;
722 }
723 
724 /*---------------------------------------------------------------------*/
725 /*  computes the normal stress---used on the boundaries */
726 PETSC_STATIC_INLINE PetscScalar ZNormalStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
727 /*---------------------------------------------------------------------*/
728 {
729   Parameter   *param=user->param;
730   GridInfo    *grid =user->grid;
731   PetscScalar dz    =grid->dz;
732   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, ivisc;
733   PetscScalar epsC  =0.0, etaC, TC;
734   PetscScalar pC, wN, wS, z_scale;
735   if (i<j || j<=grid->jlid) return EPS_ZERO;
736 
737   ivisc=param->ivisc;  z_scale = param->z_scale;
738 
739   if (ipos==CELL_CENTER) { /* on cell center */
740 
741     TC = param->potentialT * x[j][i].T * PetscExpScalar((j-0.5)*dz*z_scale);
742     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CENTER,user);
743     etaC = Viscosity(TC,epsC,dz*j,param);
744     wN   = x[j][i].w; wS = x[j-1][i].w; pC = x[j][i].p;
745 
746   } else { /* on cell corner */
747     if ((i==ilim) || (j==jlim)) return EPS_ZERO;
748 
749     TC = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
750     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CORNER,user);
751     etaC = Viscosity(TC,epsC,dz*(j+0.5),param);
752     if (i==j) wN = param->sb;
753     else      wN = WInterp(x,i,j);
754     wS = WInterp(x,i,j-1); pC = PInterp(x,i,j);
755   }
756 
757   return 2.0*etaC*(wN-wS)/dz - pC;
758 }
759 
760 /*---------------------------------------------------------------------*/
761 
762 /*=====================================================================
763   INITIALIZATION, POST-PROCESSING AND OUTPUT FUNCTIONS
764   =====================================================================*/
765 
766 /*---------------------------------------------------------------------*/
767 /* initializes the problem parameters and checks for
768    command line changes */
769 PetscErrorCode SetParams(Parameter *param, GridInfo *grid)
770 /*---------------------------------------------------------------------*/
771 {
772   PetscErrorCode ierr, ierr_out=0;
773   PetscReal      SEC_PER_YR                    = 3600.00*24.00*365.2500;
774   PetscReal      alpha_g_on_cp_units_inverse_km=4.0e-5*9.8;
775 
776   /* domain geometry */
777   param->slab_dip    = 45.0;
778   param->width       = 320.0;                                              /* km */
779   param->depth       = 300.0;                                              /* km */
780   param->lid_depth   = 35.0;                                               /* km */
781   param->fault_depth = 35.0;                                               /* km */
782 
783   ierr = PetscOptionsGetReal(NULL,NULL,"-slab_dip",&(param->slab_dip),NULL);CHKERRQ(ierr);
784   ierr = PetscOptionsGetReal(NULL,NULL,"-width",&(param->width),NULL);CHKERRQ(ierr);
785   ierr = PetscOptionsGetReal(NULL,NULL,"-depth",&(param->depth),NULL);CHKERRQ(ierr);
786   ierr = PetscOptionsGetReal(NULL,NULL,"-lid_depth",&(param->lid_depth),NULL);CHKERRQ(ierr);
787   ierr = PetscOptionsGetReal(NULL,NULL,"-fault_depth",&(param->fault_depth),NULL);CHKERRQ(ierr);
788 
789   param->slab_dip = param->slab_dip*PETSC_PI/180.0;                    /* radians */
790 
791   /* grid information */
792   ierr     = PetscOptionsGetInt(NULL,NULL, "-jfault",&(grid->jfault),NULL);CHKERRQ(ierr);
793   grid->ni = 82;
794   ierr     = PetscOptionsGetInt(NULL,NULL, "-ni",&(grid->ni),NULL);CHKERRQ(ierr);
795 
796   grid->dx            = param->width/((PetscReal)(grid->ni-2));               /* km */
797   grid->dz            = grid->dx*PetscTanReal(param->slab_dip);               /* km */
798   grid->nj            = (PetscInt)(param->depth/grid->dz + 3.0);         /* gridpoints*/
799   param->depth        = grid->dz*(grid->nj-2);                             /* km */
800   grid->inose         = 0;                                          /* gridpoints*/
801   ierr                = PetscOptionsGetInt(NULL,NULL,"-inose",&(grid->inose),NULL);CHKERRQ(ierr);
802   grid->bx            = DM_BOUNDARY_NONE;
803   grid->by            = DM_BOUNDARY_NONE;
804   grid->stencil       = DMDA_STENCIL_BOX;
805   grid->dof           = 4;
806   grid->stencil_width = 2;
807   grid->mglevels      = 1;
808 
809   /* boundary conditions */
810   param->pv_analytic = PETSC_FALSE;
811   param->ibound      = BC_NOSTRESS;
812   ierr               = PetscOptionsGetInt(NULL,NULL,"-ibound",&(param->ibound),NULL);CHKERRQ(ierr);
813 
814   /* physical constants */
815   param->slab_velocity = 5.0;               /* cm/yr */
816   param->slab_age      = 50.0;              /* Ma */
817   param->lid_age       = 50.0;              /* Ma */
818   param->kappa         = 0.7272e-6;         /* m^2/sec */
819   param->potentialT    = 1300.0;            /* degrees C */
820 
821   ierr = PetscOptionsGetReal(NULL,NULL,"-slab_velocity",&(param->slab_velocity),NULL);CHKERRQ(ierr);
822   ierr = PetscOptionsGetReal(NULL,NULL,"-slab_age",&(param->slab_age),NULL);CHKERRQ(ierr);
823   ierr = PetscOptionsGetReal(NULL,NULL,"-lid_age",&(param->lid_age),NULL);CHKERRQ(ierr);
824   ierr = PetscOptionsGetReal(NULL,NULL,"-kappa",&(param->kappa),NULL);CHKERRQ(ierr);
825   ierr = PetscOptionsGetReal(NULL,NULL,"-potentialT",&(param->potentialT),NULL);CHKERRQ(ierr);
826 
827   /* viscosity */
828   param->ivisc        = 3;                  /* 0=isovisc, 1=difn creep, 2=disl creep, 3=full */
829   param->eta0         = 1e24;               /* Pa-s */
830   param->visc_cutoff  = 0.0;                /* factor of eta_0 */
831   param->continuation = 1.0;
832 
833   /* constants for diffusion creep */
834   param->diffusion.A     = 1.8e7;             /* Pa-s */
835   param->diffusion.n     = 1.0;               /* dim'less */
836   param->diffusion.Estar = 375e3;             /* J/mol */
837   param->diffusion.Vstar = 5e-6;              /* m^3/mol */
838 
839   /* constants for param->dislocationocation creep */
840   param->dislocation.A     = 2.8969e4;        /* Pa-s */
841   param->dislocation.n     = 3.5;             /* dim'less */
842   param->dislocation.Estar = 530e3;           /* J/mol */
843   param->dislocation.Vstar = 14e-6;           /* m^3/mol */
844 
845   ierr = PetscOptionsGetInt(NULL,NULL, "-ivisc",&(param->ivisc),NULL);CHKERRQ(ierr);
846   ierr = PetscOptionsGetReal(NULL,NULL,"-visc_cutoff",&(param->visc_cutoff),NULL);CHKERRQ(ierr);
847 
848   param->output_ivisc = param->ivisc;
849 
850   ierr = PetscOptionsGetInt(NULL,NULL,"-output_ivisc",&(param->output_ivisc),NULL);CHKERRQ(ierr);
851   ierr = PetscOptionsGetReal(NULL,NULL,"-vstar",&(param->dislocation.Vstar),NULL);CHKERRQ(ierr);
852 
853   /* output options */
854   param->quiet      = PETSC_FALSE;
855   param->param_test = PETSC_FALSE;
856 
857   ierr = PetscOptionsHasName(NULL,NULL,"-quiet",&(param->quiet));CHKERRQ(ierr);
858   ierr = PetscOptionsHasName(NULL,NULL,"-test",&(param->param_test));CHKERRQ(ierr);
859   ierr = PetscOptionsGetString(NULL,NULL,"-file",param->filename,sizeof(param->filename),&(param->output_to_file));CHKERRQ(ierr);
860 
861   /* advection */
862   param->adv_scheme = ADVECT_FROMM;       /* advection scheme: 0=finite vol, 1=Fromm */
863 
864   ierr = PetscOptionsGetInt(NULL,NULL,"-adv_scheme",&(param->adv_scheme),NULL);CHKERRQ(ierr);
865 
866   /* misc. flags */
867   param->stop_solve    = PETSC_FALSE;
868   param->interrupted   = PETSC_FALSE;
869   param->kspmon        = PETSC_FALSE;
870   param->toggle_kspmon = PETSC_FALSE;
871 
872   /* derived parameters for slab angle */
873   param->sb = PetscSinReal(param->slab_dip);
874   param->cb = PetscCosReal(param->slab_dip);
875   param->c  =  param->slab_dip*param->sb/(param->slab_dip*param->slab_dip-param->sb*param->sb);
876   param->d  = (param->slab_dip*param->cb-param->sb)/(param->slab_dip*param->slab_dip-param->sb*param->sb);
877 
878   /* length, velocity and time scale for non-dimensionalization */
879   param->L = PetscMin(param->width,param->depth);               /* km */
880   param->V = param->slab_velocity/100.0/SEC_PER_YR;             /* m/sec */
881 
882   /* other unit conversions and derived parameters */
883   param->scaled_width = param->width/param->L;                  /* dim'less */
884   param->scaled_depth = param->depth/param->L;                  /* dim'less */
885   param->lid_depth    = param->lid_depth/param->L;              /* dim'less */
886   param->fault_depth  = param->fault_depth/param->L;            /* dim'less */
887   grid->dx            = grid->dx/param->L;                      /* dim'less */
888   grid->dz            = grid->dz/param->L;                      /* dim'less */
889   grid->jlid          = (PetscInt)(param->lid_depth/grid->dz);       /* gridcells */
890   grid->jfault        = (PetscInt)(param->fault_depth/grid->dz);     /* gridcells */
891   param->lid_depth    = grid->jlid*grid->dz;                    /* dim'less */
892   param->fault_depth  = grid->jfault*grid->dz;                  /* dim'less */
893   grid->corner        = grid->jlid+1;                           /* gridcells */
894   param->peclet       = param->V                                /* m/sec */
895                         * param->L*1000.0                       /* m */
896                         / param->kappa;                         /* m^2/sec */
897   param->z_scale = param->L * alpha_g_on_cp_units_inverse_km;
898   param->skt     = PetscSqrtReal(param->kappa*param->slab_age*SEC_PER_YR);
899   ierr           = PetscOptionsGetReal(NULL,NULL,"-peclet",&(param->peclet),NULL);CHKERRQ(ierr);
900 
901   return ierr_out;
902 }
903 
904 /*---------------------------------------------------------------------*/
905 /*  prints a report of the problem parameters to stdout */
906 PetscErrorCode ReportParams(Parameter *param, GridInfo *grid)
907 /*---------------------------------------------------------------------*/
908 {
909   PetscErrorCode ierr, ierr_out=0;
910   char           date[30];
911 
912   ierr = PetscGetDate(date,30);CHKERRQ(ierr);
913 
914   if (!(param->quiet)) {
915     ierr = PetscPrintf(PETSC_COMM_WORLD,"---------------------BEGIN ex30 PARAM REPORT-------------------\n");CHKERRQ(ierr);
916     ierr = PetscPrintf(PETSC_COMM_WORLD,"Domain: \n");CHKERRQ(ierr);
917     ierr = PetscPrintf(PETSC_COMM_WORLD,"  Width = %g km,         Depth = %g km\n",(double)param->width,(double)param->depth);CHKERRQ(ierr);
918     ierr = PetscPrintf(PETSC_COMM_WORLD,"  Slab dip = %g degrees,  Slab velocity = %g cm/yr\n",(double)(param->slab_dip*180.0/PETSC_PI),(double)param->slab_velocity);CHKERRQ(ierr);
919     ierr = PetscPrintf(PETSC_COMM_WORLD,"  Lid depth = %5.2f km,   Fault depth = %5.2f km\n",(double)(param->lid_depth*param->L),(double)(param->fault_depth*param->L));CHKERRQ(ierr);
920 
921     ierr = PetscPrintf(PETSC_COMM_WORLD,"\nGrid: \n");CHKERRQ(ierr);
922     ierr = PetscPrintf(PETSC_COMM_WORLD,"  [ni,nj] = %D, %D       [dx,dz] = %g, %g km\n",grid->ni,grid->nj,(double)(grid->dx*param->L),(double)(grid->dz*param->L));CHKERRQ(ierr);
923     ierr = PetscPrintf(PETSC_COMM_WORLD,"  jlid = %3D              jfault = %3D \n",grid->jlid,grid->jfault);CHKERRQ(ierr);
924     ierr = PetscPrintf(PETSC_COMM_WORLD,"  Pe = %g\n",(double)param->peclet);CHKERRQ(ierr);
925 
926     ierr = PetscPrintf(PETSC_COMM_WORLD,"\nRheology:");CHKERRQ(ierr);
927     if (param->ivisc==VISC_CONST) {
928       ierr = PetscPrintf(PETSC_COMM_WORLD,"                 Isoviscous \n");CHKERRQ(ierr);
929       if (param->pv_analytic) {
930         ierr = PetscPrintf(PETSC_COMM_WORLD,"                          Pressure and Velocity prescribed! \n");CHKERRQ(ierr);
931       }
932     } else if (param->ivisc==VISC_DIFN) {
933       ierr = PetscPrintf(PETSC_COMM_WORLD,"                 Diffusion Creep (T-Dependent Newtonian) \n");CHKERRQ(ierr);
934       ierr = PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));CHKERRQ(ierr);
935     } else if (param->ivisc==VISC_DISL) {
936       ierr = PetscPrintf(PETSC_COMM_WORLD,"                 Dislocation Creep (T-Dependent Non-Newtonian) \n");CHKERRQ(ierr);
937       ierr = PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));CHKERRQ(ierr);
938     } else if (param->ivisc==VISC_FULL) {
939       ierr = PetscPrintf(PETSC_COMM_WORLD,"                 Full Rheology \n");CHKERRQ(ierr);
940       ierr = PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));CHKERRQ(ierr);
941     } else {
942       ierr     = PetscPrintf(PETSC_COMM_WORLD,"                 Invalid! \n");CHKERRQ(ierr);
943       ierr_out = 1;
944     }
945 
946     ierr = PetscPrintf(PETSC_COMM_WORLD,"Boundary condition:");CHKERRQ(ierr);
947     if (param->ibound==BC_ANALYTIC) {
948       ierr = PetscPrintf(PETSC_COMM_WORLD,"       Isoviscous Analytic Dirichlet \n");CHKERRQ(ierr);
949     } else if (param->ibound==BC_NOSTRESS) {
950       ierr = PetscPrintf(PETSC_COMM_WORLD,"       Stress-Free (normal & shear stress)\n");CHKERRQ(ierr);
951     } else if (param->ibound==BC_EXPERMNT) {
952       ierr = PetscPrintf(PETSC_COMM_WORLD,"       Experimental boundary condition \n");CHKERRQ(ierr);
953     } else {
954       ierr     = PetscPrintf(PETSC_COMM_WORLD,"       Invalid! \n");CHKERRQ(ierr);
955       ierr_out = 1;
956     }
957 
958     if (param->output_to_file)
959 #if defined(PETSC_HAVE_MATLAB_ENGINE)
960       ierr = PetscPrintf(PETSC_COMM_WORLD,"Output Destination:       Mat file \"%s\"\n",param->filename);CHKERRQ(ierr);
961 #else
962       ierr = PetscPrintf(PETSC_COMM_WORLD,"Output Destination:       PETSc binary file \"%s\"\n",param->filename);CHKERRQ(ierr);
963 #endif
964     if (param->output_ivisc != param->ivisc) {
965       ierr = PetscPrintf(PETSC_COMM_WORLD,"                          Output viscosity: -ivisc %D\n",param->output_ivisc);CHKERRQ(ierr);
966     }
967 
968     ierr = PetscPrintf(PETSC_COMM_WORLD,"---------------------END ex30 PARAM REPORT---------------------\n");CHKERRQ(ierr);
969   }
970   if (param->param_test) PetscEnd();
971   return ierr_out;
972 }
973 
974 /* ------------------------------------------------------------------- */
975 /*  generates an inital guess using the analytic solution for isoviscous
976     corner flow */
977 PetscErrorCode Initialize(DM da)
978 /* ------------------------------------------------------------------- */
979 {
980   AppCtx         *user;
981   Parameter      *param;
982   GridInfo       *grid;
983   PetscInt       i,j,is,js,im,jm;
984   PetscErrorCode ierr;
985   Field          **x;
986   Vec            Xguess;
987 
988   /* Get the fine grid */
989   ierr   = DMGetApplicationContext(da,&user);CHKERRQ(ierr);
990   Xguess = user->Xguess;
991   param  = user->param;
992   grid   = user->grid;
993   ierr   = DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);CHKERRQ(ierr);
994   ierr   = DMDAVecGetArray(da,Xguess,(void**)&x);CHKERRQ(ierr);
995 
996   /* Compute initial guess */
997   for (j=js; j<js+jm; j++) {
998     for (i=is; i<is+im; i++) {
999       if (i<j)                x[j][i].u = param->cb;
1000       else if (j<=grid->jlid) x[j][i].u = 0.0;
1001       else                    x[j][i].u = HorizVelocity(i,j,user);
1002 
1003       if (i<=j)               x[j][i].w = param->sb;
1004       else if (j<=grid->jlid) x[j][i].w = 0.0;
1005       else                    x[j][i].w = VertVelocity(i,j,user);
1006 
1007       if (i<j || j<=grid->jlid) x[j][i].p = 0.0;
1008       else                      x[j][i].p = Pressure(i,j,user);
1009 
1010       x[j][i].T = PetscMin(grid->dz*(j-0.5),1.0);
1011     }
1012   }
1013 
1014   /* Restore x to Xguess */
1015   ierr = DMDAVecRestoreArray(da,Xguess,(void**)&x);CHKERRQ(ierr);
1016 
1017   return 0;
1018 }
1019 
1020 /*---------------------------------------------------------------------*/
1021 /*  controls output to a file */
1022 PetscErrorCode DoOutput(SNES snes, PetscInt its)
1023 /*---------------------------------------------------------------------*/
1024 {
1025   AppCtx         *user;
1026   Parameter      *param;
1027   GridInfo       *grid;
1028   PetscInt       ivt;
1029   PetscErrorCode ierr;
1030   PetscMPIInt    rank;
1031   PetscViewer    viewer;
1032   Vec            res, pars;
1033   MPI_Comm       comm;
1034   DM             da;
1035 
1036   ierr  = SNESGetDM(snes,&da);CHKERRQ(ierr);
1037   ierr  = DMGetApplicationContext(da,&user);CHKERRQ(ierr);
1038   param = user->param;
1039   grid  = user->grid;
1040   ivt   = param->ivisc;
1041 
1042   param->ivisc = param->output_ivisc;
1043 
1044   /* compute final residual and final viscosity/strain rate fields */
1045   ierr = SNESGetFunction(snes, &res, NULL, NULL);CHKERRQ(ierr);
1046   ierr = ViscosityField(da, user->x, user->Xguess);CHKERRQ(ierr);
1047 
1048   /* get the communicator and the rank of the processor */
1049   ierr = PetscObjectGetComm((PetscObject)snes, &comm);CHKERRQ(ierr);
1050   ierr = MPI_Comm_rank(comm, &rank);CHKERRMPI(ierr);
1051 
1052   if (param->output_to_file) { /* send output to binary file */
1053     ierr = VecCreate(comm, &pars);CHKERRQ(ierr);
1054     if (!rank) { /* on processor 0 */
1055       ierr = VecSetSizes(pars, 20, PETSC_DETERMINE);CHKERRQ(ierr);
1056       ierr = VecSetFromOptions(pars);CHKERRQ(ierr);
1057       ierr = VecSetValue(pars,0, (PetscScalar)(grid->ni),INSERT_VALUES);CHKERRQ(ierr);
1058       ierr = VecSetValue(pars,1, (PetscScalar)(grid->nj),INSERT_VALUES);CHKERRQ(ierr);
1059       ierr = VecSetValue(pars,2, (PetscScalar)(grid->dx),INSERT_VALUES);CHKERRQ(ierr);
1060       ierr = VecSetValue(pars,3, (PetscScalar)(grid->dz),INSERT_VALUES);CHKERRQ(ierr);
1061       ierr = VecSetValue(pars,4, (PetscScalar)(param->L),INSERT_VALUES);CHKERRQ(ierr);
1062       ierr = VecSetValue(pars,5, (PetscScalar)(param->V),INSERT_VALUES);CHKERRQ(ierr);
1063       /* skipped 6 intentionally */
1064       ierr = VecSetValue(pars,7, (PetscScalar)(param->slab_dip),INSERT_VALUES);CHKERRQ(ierr);
1065       ierr = VecSetValue(pars,8, (PetscScalar)(grid->jlid),INSERT_VALUES);CHKERRQ(ierr);
1066       ierr = VecSetValue(pars,9, (PetscScalar)(param->lid_depth),INSERT_VALUES);CHKERRQ(ierr);
1067       ierr = VecSetValue(pars,10,(PetscScalar)(grid->jfault),INSERT_VALUES);CHKERRQ(ierr);
1068       ierr = VecSetValue(pars,11,(PetscScalar)(param->fault_depth),INSERT_VALUES);CHKERRQ(ierr);
1069       ierr = VecSetValue(pars,12,(PetscScalar)(param->potentialT),INSERT_VALUES);CHKERRQ(ierr);
1070       ierr = VecSetValue(pars,13,(PetscScalar)(param->ivisc),INSERT_VALUES);CHKERRQ(ierr);
1071       ierr = VecSetValue(pars,14,(PetscScalar)(param->visc_cutoff),INSERT_VALUES);CHKERRQ(ierr);
1072       ierr = VecSetValue(pars,15,(PetscScalar)(param->ibound),INSERT_VALUES);CHKERRQ(ierr);
1073       ierr = VecSetValue(pars,16,(PetscScalar)(its),INSERT_VALUES);CHKERRQ(ierr);
1074     } else { /* on some other processor */
1075       ierr = VecSetSizes(pars, 0, PETSC_DETERMINE);CHKERRQ(ierr);
1076       ierr = VecSetFromOptions(pars);CHKERRQ(ierr);
1077     }
1078     ierr = VecAssemblyBegin(pars);CHKERRQ(ierr); ierr = VecAssemblyEnd(pars);CHKERRQ(ierr);
1079 
1080     /* create viewer */
1081 #if defined(PETSC_HAVE_MATLAB_ENGINE)
1082     ierr = PetscViewerMatlabOpen(PETSC_COMM_WORLD,param->filename,FILE_MODE_WRITE,&viewer);CHKERRQ(ierr);
1083 #else
1084     ierr = PetscViewerBinaryOpen(PETSC_COMM_WORLD,param->filename,FILE_MODE_WRITE,&viewer);CHKERRQ(ierr);
1085 #endif
1086 
1087     /* send vectors to viewer */
1088     ierr = PetscObjectSetName((PetscObject)res,"res");CHKERRQ(ierr);
1089     ierr = VecView(res,viewer);CHKERRQ(ierr);
1090     ierr = PetscObjectSetName((PetscObject)user->x,"out");CHKERRQ(ierr);
1091     ierr = VecView(user->x, viewer);CHKERRQ(ierr);
1092     ierr = PetscObjectSetName((PetscObject)(user->Xguess),"aux");CHKERRQ(ierr);
1093     ierr = VecView(user->Xguess, viewer);CHKERRQ(ierr);
1094     ierr = StressField(da);CHKERRQ(ierr); /* compute stress fields */
1095     ierr = PetscObjectSetName((PetscObject)(user->Xguess),"str");CHKERRQ(ierr);
1096     ierr = VecView(user->Xguess, viewer);CHKERRQ(ierr);
1097     ierr = PetscObjectSetName((PetscObject)pars,"par");CHKERRQ(ierr);
1098     ierr = VecView(pars, viewer);CHKERRQ(ierr);
1099 
1100     /* destroy viewer and vector */
1101     ierr = PetscViewerDestroy(&viewer);CHKERRQ(ierr);
1102     ierr = VecDestroy(&pars);CHKERRQ(ierr);
1103   }
1104 
1105   param->ivisc = ivt;
1106   return 0;
1107 }
1108 
1109 /* ------------------------------------------------------------------- */
1110 /* Compute both the second invariant of the strain rate tensor and the viscosity, at both cell centers and cell corners */
1111 PetscErrorCode ViscosityField(DM da, Vec X, Vec V)
1112 /* ------------------------------------------------------------------- */
1113 {
1114   AppCtx         *user;
1115   Parameter      *param;
1116   GridInfo       *grid;
1117   Vec            localX;
1118   Field          **v, **x;
1119   PetscReal      eps, /* dx,*/ dz, T, epsC, TC;
1120   PetscInt       i,j,is,js,im,jm,ilim,jlim,ivt;
1121   PetscErrorCode ierr;
1122 
1123   PetscFunctionBeginUser;
1124   ierr         = DMGetApplicationContext(da,&user);CHKERRQ(ierr);
1125   param        = user->param;
1126   grid         = user->grid;
1127   ivt          = param->ivisc;
1128   param->ivisc = param->output_ivisc;
1129 
1130   ierr = DMGetLocalVector(da, &localX);CHKERRQ(ierr);
1131   ierr = DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX);CHKERRQ(ierr);
1132   ierr = DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX);CHKERRQ(ierr);
1133   ierr = DMDAVecGetArray(da,localX,(void**)&x);CHKERRQ(ierr);
1134   ierr = DMDAVecGetArray(da,V,(void**)&v);CHKERRQ(ierr);
1135 
1136   /* Parameters */
1137   /* dx = grid->dx; */ dz = grid->dz;
1138 
1139   ilim = grid->ni-1; jlim = grid->nj-1;
1140 
1141   /* Compute real temperature, strain rate and viscosity */
1142   ierr = DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);CHKERRQ(ierr);
1143   for (j=js; j<js+jm; j++) {
1144     for (i=is; i<is+im; i++) {
1145       T = PetscRealPart(param->potentialT * x[j][i].T * PetscExpScalar((j-0.5)*dz*param->z_scale));
1146       if (i<ilim && j<jlim) {
1147         TC = PetscRealPart(param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*param->z_scale));
1148       } else {
1149         TC = T;
1150       }
1151       eps  = PetscRealPart((CalcSecInv(x,i,j,CELL_CENTER,user)));
1152       epsC = PetscRealPart(CalcSecInv(x,i,j,CELL_CORNER,user));
1153 
1154       v[j][i].u = eps;
1155       v[j][i].w = epsC;
1156       v[j][i].p = Viscosity(T,eps,dz*(j-0.5),param);
1157       v[j][i].T = Viscosity(TC,epsC,dz*j,param);
1158     }
1159   }
1160   ierr = DMDAVecRestoreArray(da,V,(void**)&v);CHKERRQ(ierr);
1161   ierr = DMDAVecRestoreArray(da,localX,(void**)&x);CHKERRQ(ierr);
1162   ierr = DMRestoreLocalVector(da, &localX);CHKERRQ(ierr);
1163 
1164   param->ivisc = ivt;
1165   PetscFunctionReturn(0);
1166 }
1167 
1168 /* ------------------------------------------------------------------- */
1169 /* post-processing: compute stress everywhere */
1170 PetscErrorCode StressField(DM da)
1171 /* ------------------------------------------------------------------- */
1172 {
1173   AppCtx         *user;
1174   PetscInt       i,j,is,js,im,jm;
1175   PetscErrorCode ierr;
1176   Vec            locVec;
1177   Field          **x, **y;
1178 
1179   ierr = DMGetApplicationContext(da,&user);CHKERRQ(ierr);
1180 
1181   /* Get the fine grid of Xguess and X */
1182   ierr = DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);CHKERRQ(ierr);
1183   ierr = DMDAVecGetArray(da,user->Xguess,(void**)&x);CHKERRQ(ierr);
1184 
1185   ierr = DMGetLocalVector(da, &locVec);CHKERRQ(ierr);
1186   ierr = DMGlobalToLocalBegin(da, user->x, INSERT_VALUES, locVec);CHKERRQ(ierr);
1187   ierr = DMGlobalToLocalEnd(da, user->x, INSERT_VALUES, locVec);CHKERRQ(ierr);
1188   ierr = DMDAVecGetArray(da,locVec,(void**)&y);CHKERRQ(ierr);
1189 
1190   /* Compute stress on the corner points */
1191   for (j=js; j<js+jm; j++) {
1192     for (i=is; i<is+im; i++) {
1193       x[j][i].u = ShearStress(y,i,j,CELL_CENTER,user);
1194       x[j][i].w = ShearStress(y,i,j,CELL_CORNER,user);
1195       x[j][i].p = XNormalStress(y,i,j,CELL_CENTER,user);
1196       x[j][i].T = ZNormalStress(y,i,j,CELL_CENTER,user);
1197     }
1198   }
1199 
1200   /* Restore the fine grid of Xguess and X */
1201   ierr = DMDAVecRestoreArray(da,user->Xguess,(void**)&x);CHKERRQ(ierr);
1202   ierr = DMDAVecRestoreArray(da,locVec,(void**)&y);CHKERRQ(ierr);
1203   ierr = DMRestoreLocalVector(da, &locVec);CHKERRQ(ierr);
1204   return 0;
1205 }
1206 
1207 /*=====================================================================
1208   UTILITY FUNCTIONS
1209   =====================================================================*/
1210 
1211 /*---------------------------------------------------------------------*/
1212 /* returns the velocity of the subducting slab and handles fault nodes
1213    for BC */
1214 PETSC_STATIC_INLINE PetscScalar SlabVel(char c, PetscInt i, PetscInt j, AppCtx *user)
1215 /*---------------------------------------------------------------------*/
1216 {
1217   Parameter *param = user->param;
1218   GridInfo  *grid  = user->grid;
1219 
1220   if (c=='U' || c=='u') {
1221     if (i<j-1) return param->cb;
1222     else if (j<=grid->jfault) return 0.0;
1223     else return param->cb;
1224 
1225   } else {
1226     if (i<j) return param->sb;
1227     else if (j<=grid->jfault) return 0.0;
1228     else return param->sb;
1229   }
1230 }
1231 
1232 /*---------------------------------------------------------------------*/
1233 /*  solution to diffusive half-space cooling model for BC */
1234 PETSC_STATIC_INLINE PetscScalar PlateModel(PetscInt j, PetscInt plate, AppCtx *user)
1235 /*---------------------------------------------------------------------*/
1236 {
1237   Parameter     *param = user->param;
1238   PetscScalar   z;
1239   if (plate==PLATE_LID) z = (j-0.5)*user->grid->dz;
1240   else z = (j-0.5)*user->grid->dz*param->cb;  /* PLATE_SLAB */
1241 #if defined(PETSC_HAVE_ERF)
1242   return (PetscReal)(erf((double)PetscRealPart(z*param->L/2.0/param->skt)));
1243 #else
1244   (*PetscErrorPrintf)("erf() not available on this machine\n");
1245   MPI_Abort(PETSC_COMM_SELF,1);
1246 #endif
1247 }
1248 
1249 /*=====================================================================
1250   INTERACTIVE SIGNAL HANDLING
1251   =====================================================================*/
1252 
1253 /* ------------------------------------------------------------------- */
1254 PetscErrorCode SNESConverged_Interactive(SNES snes, PetscInt it,PetscReal xnorm, PetscReal snorm, PetscReal fnorm, SNESConvergedReason *reason, void *ctx)
1255 /* ------------------------------------------------------------------- */
1256 {
1257   AppCtx         *user  = (AppCtx*) ctx;
1258   Parameter      *param = user->param;
1259   KSP            ksp;
1260   PetscErrorCode ierr;
1261 
1262   PetscFunctionBeginUser;
1263   if (param->interrupted) {
1264     param->interrupted = PETSC_FALSE;
1265     ierr = PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: exiting SNES solve. \n");CHKERRQ(ierr);
1266     *reason = SNES_CONVERGED_FNORM_ABS;
1267     PetscFunctionReturn(0);
1268   } else if (param->toggle_kspmon) {
1269     param->toggle_kspmon = PETSC_FALSE;
1270 
1271     ierr = SNESGetKSP(snes, &ksp);CHKERRQ(ierr);
1272 
1273     if (param->kspmon) {
1274       ierr = KSPMonitorCancel(ksp);CHKERRQ(ierr);
1275 
1276       param->kspmon = PETSC_FALSE;
1277       ierr = PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: deactivating ksp singular value monitor. \n");CHKERRQ(ierr);
1278     } else {
1279       PetscViewerAndFormat *vf;
1280       ierr = PetscViewerAndFormatCreate(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_DEFAULT,&vf);CHKERRQ(ierr);
1281       ierr = KSPMonitorSet(ksp,(PetscErrorCode (*)(KSP,PetscInt,PetscReal,void*))KSPMonitorSingularValue,vf,(PetscErrorCode (*)(void**))PetscViewerAndFormatDestroy);CHKERRQ(ierr);
1282 
1283       param->kspmon = PETSC_TRUE;
1284       ierr = PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: activating ksp singular value monitor. \n");CHKERRQ(ierr);
1285     }
1286   }
1287   PetscFunctionReturn(SNESConvergedDefault(snes,it,xnorm,snorm,fnorm,reason,ctx));
1288 }
1289 
1290 /* ------------------------------------------------------------------- */
1291 #include <signal.h>
1292 PetscErrorCode InteractiveHandler(int signum, void *ctx)
1293 /* ------------------------------------------------------------------- */
1294 {
1295   AppCtx    *user  = (AppCtx*) ctx;
1296   Parameter *param = user->param;
1297 
1298   if (signum == SIGILL) {
1299     param->toggle_kspmon = PETSC_TRUE;
1300 #if !defined(PETSC_MISSING_SIGCONT)
1301   } else if (signum == SIGCONT) {
1302     param->interrupted = PETSC_TRUE;
1303 #endif
1304 #if !defined(PETSC_MISSING_SIGURG)
1305   } else if (signum == SIGURG) {
1306     param->stop_solve = PETSC_TRUE;
1307 #endif
1308   }
1309   return 0;
1310 }
1311 
1312 /*---------------------------------------------------------------------*/
1313 /*  main call-back function that computes the processor-local piece
1314     of the residual */
1315 PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,Field **x,Field **f,void *ptr)
1316 /*---------------------------------------------------------------------*/
1317 {
1318   AppCtx      *user  = (AppCtx*)ptr;
1319   Parameter   *param = user->param;
1320   GridInfo    *grid  = user->grid;
1321   PetscScalar mag_w, mag_u;
1322   PetscInt    i,j,mx,mz,ilim,jlim;
1323   PetscInt    is,ie,js,je,ibound;    /* ,ivisc */
1324 
1325   PetscFunctionBeginUser;
1326   /* Define global and local grid parameters */
1327   mx   = info->mx;     mz   = info->my;
1328   ilim = mx-1;         jlim = mz-1;
1329   is   = info->xs;     ie   = info->xs+info->xm;
1330   js   = info->ys;     je   = info->ys+info->ym;
1331 
1332   /* Define geometric and numeric parameters */
1333   /* ivisc = param->ivisc; */ ibound = param->ibound;
1334 
1335   for (j=js; j<je; j++) {
1336     for (i=is; i<ie; i++) {
1337 
1338       /************* X-MOMENTUM/VELOCITY *************/
1339       if (i<j) f[j][i].u = x[j][i].u - SlabVel('U',i,j,user);
1340       else if (j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
1341         /* in the lithospheric lid */
1342         f[j][i].u = x[j][i].u - 0.0;
1343       } else if (i==ilim) {
1344         /* on the right side boundary */
1345         if (ibound==BC_ANALYTIC) {
1346           f[j][i].u = x[j][i].u - HorizVelocity(i,j,user);
1347         } else {
1348           f[j][i].u = XNormalStress(x,i,j,CELL_CENTER,user) - EPS_ZERO;
1349         }
1350 
1351       } else if (j==jlim) {
1352         /* on the bottom boundary */
1353         if (ibound==BC_ANALYTIC) {
1354           f[j][i].u = x[j][i].u - HorizVelocity(i,j,user);
1355         } else if (ibound==BC_NOSTRESS) {
1356           f[j][i].u = XMomentumResidual(x,i,j,user);
1357         } else {
1358           /* experimental boundary condition */
1359         }
1360 
1361       } else {
1362         /* in the mantle wedge */
1363         f[j][i].u = XMomentumResidual(x,i,j,user);
1364       }
1365 
1366       /************* Z-MOMENTUM/VELOCITY *************/
1367       if (i<=j) {
1368         f[j][i].w = x[j][i].w - SlabVel('W',i,j,user);
1369 
1370       } else if (j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
1371         /* in the lithospheric lid */
1372         f[j][i].w = x[j][i].w - 0.0;
1373 
1374       } else if (j==jlim) {
1375         /* on the bottom boundary */
1376         if (ibound==BC_ANALYTIC) {
1377           f[j][i].w = x[j][i].w - VertVelocity(i,j,user);
1378         } else {
1379           f[j][i].w = ZNormalStress(x,i,j,CELL_CENTER,user) - EPS_ZERO;
1380         }
1381 
1382       } else if (i==ilim) {
1383         /* on the right side boundary */
1384         if (ibound==BC_ANALYTIC) {
1385           f[j][i].w = x[j][i].w - VertVelocity(i,j,user);
1386         } else if (ibound==BC_NOSTRESS) {
1387           f[j][i].w = ZMomentumResidual(x,i,j,user);
1388         } else {
1389           /* experimental boundary condition */
1390         }
1391 
1392       } else {
1393         /* in the mantle wedge */
1394         f[j][i].w =  ZMomentumResidual(x,i,j,user);
1395       }
1396 
1397       /************* CONTINUITY/PRESSURE *************/
1398       if (i<j || j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
1399         /* in the lid or slab */
1400         f[j][i].p = x[j][i].p;
1401 
1402       } else if ((i==ilim || j==jlim) && ibound==BC_ANALYTIC) {
1403         /* on an analytic boundary */
1404         f[j][i].p = x[j][i].p - Pressure(i,j,user);
1405 
1406       } else {
1407         /* in the mantle wedge */
1408         f[j][i].p = ContinuityResidual(x,i,j,user);
1409       }
1410 
1411       /************* TEMPERATURE *************/
1412       if (j==0) {
1413         /* on the surface */
1414         f[j][i].T = x[j][i].T + x[j+1][i].T + PetscMax(PetscRealPart(x[j][i].T),0.0);
1415 
1416       } else if (i==0) {
1417         /* slab inflow boundary */
1418         f[j][i].T = x[j][i].T - PlateModel(j,PLATE_SLAB,user);
1419 
1420       } else if (i==ilim) {
1421         /* right side boundary */
1422         mag_u = 1.0 - PetscPowRealInt((1.0-PetscMax(PetscMin(PetscRealPart(x[j][i-1].u)/param->cb,1.0),0.0)), 5);
1423         f[j][i].T = x[j][i].T - mag_u*x[j-1][i-1].T - (1.0-mag_u)*PlateModel(j,PLATE_LID,user);
1424 
1425       } else if (j==jlim) {
1426         /* bottom boundary */
1427         mag_w = 1.0 - PetscPowRealInt((1.0-PetscMax(PetscMin(PetscRealPart(x[j-1][i].w)/param->sb,1.0),0.0)), 5);
1428         f[j][i].T = x[j][i].T - mag_w*x[j-1][i-1].T - (1.0-mag_w);
1429 
1430       } else {
1431         /* in the mantle wedge */
1432         f[j][i].T = EnergyResidual(x,i,j,user);
1433       }
1434     }
1435   }
1436   PetscFunctionReturn(0);
1437 }
1438 
1439 /*TEST
1440 
1441    build:
1442       requires: !complex erf
1443 
1444    test:
1445       args: -ni 18
1446       filter: grep -v Destination
1447       requires: !single
1448 
1449 TEST*/
1450