Date: 2021-03-03T22:27:45+00:00

Actual source code: ex30.c
petsc-3.14.5 2021-03-03
Report Typos and Errors

      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";

This PETSc 2.2.0 example by Richard F. Katz
http://www.ldeo.columbia.edu/~katz/
The problem is modeled by the partial differential equation system
\begin{eqnarray}
-\nabla P + \nabla \cdot [\eta (\nabla v + \nabla v^T)] & = & 0 \\
\nabla \cdot v & = & 0 \\
dT/dt + \nabla \cdot (vT) - 1/Pe \triangle^2(T) & = & 0 \\
\end{eqnarray}
\begin{eqnarray}
\eta(T,Eps\_dot) & = & \hbox{constant } \hbox{if ivisc} ==0 \\
& = & \hbox{diffusion creep (T,P-dependent) } \hbox{if ivisc} ==1 \\
& = & \hbox{dislocation creep (T,P,v-dependent)} \hbox{if ivisc} ==2 \\
& = & \hbox{mantle viscosity (difn and disl) } \hbox{if ivisc} ==3
\end{eqnarray}
which is uniformly discretized on a staggered mesh:
-------$w_{ij}$------
$u_{i-1j}$ $P,T_{ij}$ $u_{ij}$
------$w_{ij-1}$-----

     55: #include <petscsnes.h>
     56: #include <petscdm.h>
     57: #include <petscdmda.h>

     59: #define VISC_CONST   0
     60: #define VISC_DIFN    1
     61: #define VISC_DISL    2
     62: #define VISC_FULL    3
     63: #define CELL_CENTER  0
     64: #define CELL_CORNER  1
     65: #define BC_ANALYTIC  0
     66: #define BC_NOSTRESS  1
     67: #define BC_EXPERMNT  2
     68: #define ADVECT_FV    0
     69: #define ADVECT_FROMM 1
     70: #define PLATE_SLAB   0
     71: #define PLATE_LID    1
     72: #define EPS_ZERO     0.00000001

     74: typedef struct { /* holds the variables to be solved for */
     75:   PetscScalar u,w,p,T;
     76: } Field;

     78: typedef struct { /* parameters needed to compute viscosity */
     79:   PetscReal A,n,Estar,Vstar;
     80: } ViscParam;

     82: typedef struct { /* physical and miscelaneous parameters */
     83:   PetscReal width, depth, scaled_width, scaled_depth, peclet, potentialT;
     84:   PetscReal slab_dip, slab_age, slab_velocity, kappa, z_scale;
     85:   PetscReal c, d, sb, cb, skt, visc_cutoff, lid_age, eta0, continuation;
     86:   PetscReal L, V, lid_depth, fault_depth;
     87:   ViscParam diffusion, dislocation;
     88:   PetscInt  ivisc, adv_scheme, ibound, output_ivisc;
     89:   PetscBool quiet, param_test, output_to_file, pv_analytic;
     90:   PetscBool interrupted, stop_solve, toggle_kspmon, kspmon;
     91:   char      filename[PETSC_MAX_PATH_LEN];
     92: } Parameter;

     94: typedef struct { /* grid parameters */
     95:   DMBoundaryType   bx,by;
     96:   DMDAStencilType  stencil;
     97:   PetscInt         corner,ni,nj,jlid,jfault,inose;
     98:   PetscInt         dof,stencil_width,mglevels;
     99:   PetscReal        dx,dz;
    100: } GridInfo;

    102: typedef struct { /* application context */
    103:   Vec       x,Xguess;
    104:   Parameter *param;
    105:   GridInfo  *grid;
    106: } AppCtx;

    108: /* Callback functions (static interface) */
    109: extern PetscErrorCode FormFunctionLocal(DMDALocalInfo*,Field**,Field**,void*);

    111: /* Main routines */
    112: extern PetscErrorCode SetParams(Parameter*, GridInfo*);
    113: extern PetscErrorCode ReportParams(Parameter*, GridInfo*);
    114: extern PetscErrorCode Initialize(DM);
    115: extern PetscErrorCode UpdateSolution(SNES,AppCtx*, PetscInt*);
    116: extern PetscErrorCode DoOutput(SNES,PetscInt);

    118: /* Post-processing & misc */
    119: extern PetscErrorCode ViscosityField(DM,Vec,Vec);
    120: extern PetscErrorCode StressField(DM);
    121: extern PetscErrorCode SNESConverged_Interactive(SNES, PetscInt, PetscReal, PetscReal, PetscReal, SNESConvergedReason*, void*);
    122: extern PetscErrorCode InteractiveHandler(int, void*);

    124: /*-----------------------------------------------------------------------*/
    125: int main(int argc,char **argv)
    126: /*-----------------------------------------------------------------------*/
    127: {
    128:   SNES           snes;
    129:   AppCtx         *user;               /* user-defined work context */
    130:   Parameter      param;
    131:   GridInfo       grid;
    132:   PetscInt       nits;
    134:   MPI_Comm       comm;
    135:   DM             da;

    137:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
    138:   PetscOptionsSetValue(NULL,"-file","ex30_output");
    139:   PetscOptionsSetValue(NULL,"-snes_monitor_short",NULL);
    140:   PetscOptionsSetValue(NULL,"-snes_max_it","20");
    141:   PetscOptionsSetValue(NULL,"-ksp_max_it","1500");
    142:   PetscOptionsSetValue(NULL,"-ksp_gmres_restart","300");
    143:   PetscOptionsInsert(NULL,&argc,&argv,NULL);

    145:   comm = PETSC_COMM_WORLD;

    147:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    148:      Set up the problem parameters.
    149:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    150:   SetParams(&param,&grid);
    151:   ReportParams(&param,&grid);

    153:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    154:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    155:   SNESCreate(comm,&snes);
    156:   DMDACreate2d(comm,grid.bx,grid.by,grid.stencil,grid.ni,grid.nj,PETSC_DECIDE,PETSC_DECIDE,grid.dof,grid.stencil_width,0,0,&da);
    157:   DMSetFromOptions(da);
    158:   DMSetUp(da);
    159:   SNESSetDM(snes,da);
    160:   DMDASetFieldName(da,0,"x-velocity");
    161:   DMDASetFieldName(da,1,"y-velocity");
    162:   DMDASetFieldName(da,2,"pressure");
    163:   DMDASetFieldName(da,3,"temperature");

    166:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    167:      Create user context, set problem data, create vector data structures.
    168:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    169:   PetscNew(&user);
    170:   user->param = &param;
    171:   user->grid  = &grid;
    172:   DMSetApplicationContext(da,user);
    173:   DMCreateGlobalVector(da,&(user->Xguess));

    176:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    177:      Set up the SNES solver with callback functions.
    178:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    179:   DMDASNESSetFunctionLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionLocal,(void*)user);
    180:   SNESSetFromOptions(snes);

    183:   SNESSetConvergenceTest(snes,SNESConverged_Interactive,(void*)user,NULL);
    184:   PetscPushSignalHandler(InteractiveHandler,(void*)user);

    186:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    187:      Initialize and solve the nonlinear system
    188:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    189:   Initialize(da);
    190:   UpdateSolution(snes,user,&nits);

    192:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    193:      Output variables.
    194:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    195:   DoOutput(snes,nits);

    197:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    198:      Free work space.
    199:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    200:   VecDestroy(&user->Xguess);
    201:   VecDestroy(&user->x);
    202:   PetscFree(user);
    203:   SNESDestroy(&snes);
    204:   DMDestroy(&da);
    205:   PetscPopSignalHandler();
    206:   PetscFinalize();
    207:   return ierr;
    208: }

    210: /*=====================================================================
    211:   PETSc INTERACTION FUNCTIONS (initialize & call SNESSolve)
    212:   =====================================================================*/

    214: /*---------------------------------------------------------------------*/
    215: /*  manages solve: adaptive continuation method  */
    216: PetscErrorCode UpdateSolution(SNES snes, AppCtx *user, PetscInt *nits)
    217: {
    218:   KSP                 ksp;
    219:   PC                  pc;
    220:   SNESConvergedReason reason = SNES_CONVERGED_ITERATING;
    221:   Parameter           *param   = user->param;
    222:   PetscReal           cont_incr=0.3;
    223:   PetscInt            its;
    224:   PetscErrorCode      ierr;
    225:   PetscBool           q = PETSC_FALSE;
    226:   DM                  dm;

    229:   SNESGetDM(snes,&dm);
    230:   DMCreateGlobalVector(dm,&user->x);
    231:   SNESGetKSP(snes,&ksp);
    232:   KSPGetPC(ksp, &pc);
    233:   KSPSetComputeSingularValues(ksp, PETSC_TRUE);

    235:   *nits=0;

    237:   /* Isoviscous solve */
    238:   if (param->ivisc == VISC_CONST && !param->stop_solve) {
    239:     param->ivisc = VISC_CONST;

    241:     SNESSolve(snes,0,user->x);
    242:     SNESGetConvergedReason(snes,&reason);
    243:     SNESGetIterationNumber(snes,&its);
    244:     *nits += its;
    245:     VecCopy(user->x,user->Xguess);
    246:     if (param->stop_solve) goto done;
    247:   }

    249:   /* Olivine diffusion creep */
    250:   if (param->ivisc >= VISC_DIFN && !param->stop_solve) {
    251:     if (!q) {PetscPrintf(PETSC_COMM_WORLD,"Computing Variable Viscosity Solution\n");}

    253:     /* continuation method on viscosity cutoff */
    254:     for (param->continuation=0.0;; param->continuation+=cont_incr) {
    255:       if (!q) {PetscPrintf(PETSC_COMM_WORLD," Continuation parameter = %g\n", (double)param->continuation);}

    257:       /* solve the non-linear system */
    258:       VecCopy(user->Xguess,user->x);
    259:       SNESSolve(snes,0,user->x);
    260:       SNESGetConvergedReason(snes,&reason);
    261:       SNESGetIterationNumber(snes,&its);
    262:       *nits += its;
    263:       if (!q) {PetscPrintf(PETSC_COMM_WORLD," SNES iterations: %D, Cumulative: %D\n", its, *nits);}
    264:       if (param->stop_solve) goto done;

    266:       if (reason<0) {
    267:         /* NOT converged */
    268:         cont_incr = -PetscAbsReal(cont_incr)/2.0;
    269:         if (PetscAbsReal(cont_incr)<0.01) goto done;

    271:       } else {
    272:         /* converged */
    273:         VecCopy(user->x,user->Xguess);
    274:         if (param->continuation >= 1.0) goto done;
    275:         if (its<=3)      cont_incr = 0.30001;
    276:         else if (its<=8) cont_incr = 0.15001;
    277:         else             cont_incr = 0.10001;

    279:         if (param->continuation+cont_incr > 1.0) cont_incr = 1.0 - param->continuation;
    280:       } /* endif reason<0 */
    281:     }
    282:   }
    283: done:
    284:   if (param->stop_solve && !q) {PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: stopping solve.\n");}
    285:   if (reason<0 && !q) {PetscPrintf(PETSC_COMM_WORLD,"FAILED TO CONVERGE: stopping solve.\n");}
    286:   return(0);
    287: }

    290: /*=====================================================================
    291:   PHYSICS FUNCTIONS (compute the discrete residual)
    292:   =====================================================================*/

    295: /*---------------------------------------------------------------------*/
    296: PETSC_STATIC_INLINE PetscScalar UInterp(Field **x, PetscInt i, PetscInt j)
    297: /*---------------------------------------------------------------------*/
    298: {
    299:   return 0.25*(x[j][i].u+x[j+1][i].u+x[j][i+1].u+x[j+1][i+1].u);
    300: }

    302: /*---------------------------------------------------------------------*/
    303: PETSC_STATIC_INLINE PetscScalar WInterp(Field **x, PetscInt i, PetscInt j)
    304: /*---------------------------------------------------------------------*/
    305: {
    306:   return 0.25*(x[j][i].w+x[j+1][i].w+x[j][i+1].w+x[j+1][i+1].w);
    307: }

    309: /*---------------------------------------------------------------------*/
    310: PETSC_STATIC_INLINE PetscScalar PInterp(Field **x, PetscInt i, PetscInt j)
    311: /*---------------------------------------------------------------------*/
    312: {
    313:   return 0.25*(x[j][i].p+x[j+1][i].p+x[j][i+1].p+x[j+1][i+1].p);
    314: }

    316: /*---------------------------------------------------------------------*/
    317: PETSC_STATIC_INLINE PetscScalar TInterp(Field **x, PetscInt i, PetscInt j)
    318: /*---------------------------------------------------------------------*/
    319: {
    320:   return 0.25*(x[j][i].T+x[j+1][i].T+x[j][i+1].T+x[j+1][i+1].T);
    321: }

    323: /*---------------------------------------------------------------------*/
    324: /*  isoviscous analytic solution for IC */
    325: PETSC_STATIC_INLINE PetscScalar HorizVelocity(PetscInt i, PetscInt j, AppCtx *user)
    326: /*---------------------------------------------------------------------*/
    327: {
    328:   Parameter   *param = user->param;
    329:   GridInfo    *grid  = user->grid;
    330:   PetscScalar st, ct, th, c=param->c, d=param->d;
    331:   PetscReal   x, z,r;

    333:   x  = (i - grid->jlid)*grid->dx;  z = (j - grid->jlid - 0.5)*grid->dz;
    334:   r  = PetscSqrtReal(x*x+z*z);
    335:   st = z/r;
    336:   ct = x/r;
    337:   th = PetscAtanReal(z/x);
    338:   return ct*(c*th*st+d*(st+th*ct)) + st*(c*(st-th*ct)+d*th*st);
    339: }

    341: /*---------------------------------------------------------------------*/
    342: /*  isoviscous analytic solution for IC */
    343: PETSC_STATIC_INLINE PetscScalar VertVelocity(PetscInt i, PetscInt j, AppCtx *user)
    344: /*---------------------------------------------------------------------*/
    345: {
    346:   Parameter   *param = user->param;
    347:   GridInfo    *grid  = user->grid;
    348:   PetscScalar st, ct, th, c=param->c, d=param->d;
    349:   PetscReal   x, z, r;

    351:   x = (i - grid->jlid - 0.5)*grid->dx;  z = (j - grid->jlid)*grid->dz;
    352:   r = PetscSqrtReal(x*x+z*z); st = z/r;  ct = x/r;  th = PetscAtanReal(z/x);
    353:   return st*(c*th*st+d*(st+th*ct)) - ct*(c*(st-th*ct)+d*th*st);
    354: }

    356: /*---------------------------------------------------------------------*/
    357: /*  isoviscous analytic solution for IC */
    358: PETSC_STATIC_INLINE PetscScalar Pressure(PetscInt i, PetscInt j, AppCtx *user)
    359: /*---------------------------------------------------------------------*/
    360: {
    361:   Parameter   *param = user->param;
    362:   GridInfo    *grid  = user->grid;
    363:   PetscScalar x, z, r, st, ct, c=param->c, d=param->d;

    365:   x = (i - grid->jlid - 0.5)*grid->dx;  z = (j - grid->jlid - 0.5)*grid->dz;
    366:   r = PetscSqrtReal(x*x+z*z);  st = z/r;  ct = x/r;
    367:   return (-2.0*(c*ct-d*st)/r);
    368: }

    370: /*  computes the second invariant of the strain rate tensor */
    371: PETSC_STATIC_INLINE PetscScalar CalcSecInv(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    372: /*---------------------------------------------------------------------*/
    373: {
    374:   Parameter   *param = user->param;
    375:   GridInfo    *grid  = user->grid;
    376:   PetscInt    ilim   =grid->ni-1, jlim=grid->nj-1;
    377:   PetscScalar uN,uS,uE,uW,wN,wS,wE,wW;
    378:   PetscScalar eps11, eps12, eps22;

    380:   if (i<j) return EPS_ZERO;
    381:   if (i==ilim) i--;
    382:   if (j==jlim) j--;

    384:   if (ipos==CELL_CENTER) { /* on cell center */
    385:     if (j<=grid->jlid) return EPS_ZERO;

    387:     uE = x[j][i].u; uW = x[j][i-1].u;
    388:     wN = x[j][i].w; wS = x[j-1][i].w;
    389:     wE = WInterp(x,i,j-1);
    390:     if (i==j) {
    391:       uN = param->cb; wW = param->sb;
    392:     } else {
    393:       uN = UInterp(x,i-1,j); wW = WInterp(x,i-1,j-1);
    394:     }

    396:     if (j==grid->jlid+1) uS = 0.0;
    397:     else                 uS = UInterp(x,i-1,j-1);

    399:   } else {       /* on CELL_CORNER */
    400:     if (j<grid->jlid) return EPS_ZERO;

    402:     uN = x[j+1][i].u;  uS = x[j][i].u;
    403:     wE = x[j][i+1].w;  wW = x[j][i].w;
    404:     if (i==j) {
    405:       wN = param->sb;
    406:       uW = param->cb;
    407:     } else {
    408:       wN = WInterp(x,i,j);
    409:       uW = UInterp(x,i-1,j);
    410:     }

    412:     if (j==grid->jlid) {
    413:       uE = 0.0;  uW = 0.0;
    414:       uS = -uN;
    415:       wS = -wN;
    416:     } else {
    417:       uE = UInterp(x,i,j);
    418:       wS = WInterp(x,i,j-1);
    419:     }
    420:   }

    422:   eps11 = (uE-uW)/grid->dx;  eps22 = (wN-wS)/grid->dz;
    423:   eps12 = 0.5*((uN-uS)/grid->dz + (wE-wW)/grid->dx);

    425:   return PetscSqrtReal(0.5*(eps11*eps11 + 2.0*eps12*eps12 + eps22*eps22));
    426: }

    428: /*---------------------------------------------------------------------*/
    429: /*  computes the shear viscosity */
    430: PETSC_STATIC_INLINE PetscScalar Viscosity(PetscScalar T, PetscScalar eps, PetscScalar z, Parameter *param)
    431: /*---------------------------------------------------------------------*/
    432: {
    433:   PetscReal   result   =0.0;
    434:   ViscParam   difn     =param->diffusion, disl=param->dislocation;
    435:   PetscInt    iVisc    =param->ivisc;
    436:   PetscScalar eps_scale=param->V/(param->L*1000.0);
    437:   PetscScalar strain_power, v1, v2, P;
    438:   PetscScalar rho_g = 32340.0, R=8.3144;

    440:   P = rho_g*(z*param->L*1000.0); /* Pa */

    442:   if (iVisc==VISC_CONST) {
    443:     /* constant viscosity */
    444:     return 1.0;
    445:   } else if (iVisc==VISC_DIFN) {
    446:     /* diffusion creep rheology */
    447:     result = PetscRealPart((difn.A*PetscExpScalar((difn.Estar + P*difn.Vstar)/R/(T+273.0))/param->eta0));
    448:   } else if (iVisc==VISC_DISL) {
    449:     /* dislocation creep rheology */
    450:     strain_power = PetscPowScalar(eps*eps_scale, (1.0-disl.n)/disl.n);

    452:     result = PetscRealPart(disl.A*PetscExpScalar((disl.Estar + P*disl.Vstar)/disl.n/R/(T+273.0))*strain_power/param->eta0);
    453:   } else if (iVisc==VISC_FULL) {
    454:     /* dislocation/diffusion creep rheology */
    455:     strain_power = PetscPowScalar(eps*eps_scale, (1.0-disl.n)/disl.n);

    457:     v1 = difn.A*PetscExpScalar((difn.Estar + P*difn.Vstar)/R/(T+273.0))/param->eta0;
    458:     v2 = disl.A*PetscExpScalar((disl.Estar + P*disl.Vstar)/disl.n/R/(T+273.0))*strain_power/param->eta0;

    460:     result = PetscRealPart(1.0/(1.0/v1 + 1.0/v2));
    461:   }

    463:   /* max viscosity is param->eta0 */
    464:   result = PetscMin(result, 1.0);
    465:   /* min viscosity is param->visc_cutoff */
    466:   result = PetscMax(result, param->visc_cutoff);
    467:   /* continuation method */
    468:   result = PetscPowReal(result,param->continuation);
    469:   return result;
    470: }

    472: /*---------------------------------------------------------------------*/
    473: /*  computes the residual of the x-component of eqn (1) above */
    474: PETSC_STATIC_INLINE PetscScalar XMomentumResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    475: /*---------------------------------------------------------------------*/
    476: {
    477:   Parameter   *param=user->param;
    478:   GridInfo    *grid =user->grid;
    479:   PetscScalar dx    = grid->dx, dz=grid->dz;
    480:   PetscScalar etaN,etaS,etaE,etaW,epsN=0.0,epsS=0.0,epsE=0.0,epsW=0.0;
    481:   PetscScalar TE=0.0,TN=0.0,TS=0.0,TW=0.0, dPdx, residual, z_scale;
    482:   PetscScalar dudxW,dudxE,dudzN,dudzS,dwdxN,dwdxS;
    483:   PetscInt    jlim = grid->nj-1;

    485:   z_scale = param->z_scale;

    487:   if (param->ivisc==VISC_DIFN || param->ivisc>=VISC_DISL) { /* viscosity is T-dependent */
    488:     TS = param->potentialT * TInterp(x,i,j-1) * PetscExpScalar((j-1.0)*dz*z_scale);
    489:     if (j==jlim) TN = TS;
    490:     else         TN = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
    491:     TW = param->potentialT * x[j][i].T        * PetscExpScalar((j-0.5)*dz*z_scale);
    492:     TE = param->potentialT * x[j][i+1].T      * PetscExpScalar((j-0.5)*dz*z_scale);
    493:     if (param->ivisc>=VISC_DISL) { /* olivine dislocation creep */
    494:       epsN = CalcSecInv(x,i,j,  CELL_CORNER,user);
    495:       epsS = CalcSecInv(x,i,j-1,CELL_CORNER,user);
    496:       epsE = CalcSecInv(x,i+1,j,CELL_CENTER,user);
    497:       epsW = CalcSecInv(x,i,j,  CELL_CENTER,user);
    498:     }
    499:   }
    500:   etaN = Viscosity(TN,epsN,dz*(j+0.5),param);
    501:   etaS = Viscosity(TS,epsS,dz*(j-0.5),param);
    502:   etaW = Viscosity(TW,epsW,dz*j,param);
    503:   etaE = Viscosity(TE,epsE,dz*j,param);

    505:   dPdx = (x[j][i+1].p - x[j][i].p)/dx;
    506:   if (j==jlim) dudzN = etaN * (x[j][i].w   - x[j][i+1].w)/dx;
    507:   else         dudzN = etaN * (x[j+1][i].u - x[j][i].u)  /dz;
    508:   dudzS = etaS * (x[j][i].u    - x[j-1][i].u)/dz;
    509:   dudxE = etaE * (x[j][i+1].u  - x[j][i].u)  /dx;
    510:   dudxW = etaW * (x[j][i].u    - x[j][i-1].u)/dx;

    512:   residual = -dPdx                          /* X-MOMENTUM EQUATION*/
    513:              +(dudxE - dudxW)/dx
    514:              +(dudzN - dudzS)/dz;

    516:   if (param->ivisc!=VISC_CONST) {
    517:     dwdxN = etaN * (x[j][i+1].w   - x[j][i].w)  /dx;
    518:     dwdxS = etaS * (x[j-1][i+1].w - x[j-1][i].w)/dx;

    520:     residual += (dudxE - dudxW)/dx + (dwdxN - dwdxS)/dz;
    521:   }

    523:   return residual;
    524: }

    526: /*---------------------------------------------------------------------*/
    527: /*  computes the residual of the z-component of eqn (1) above */
    528: PETSC_STATIC_INLINE PetscScalar ZMomentumResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    529: /*---------------------------------------------------------------------*/
    530: {
    531:   Parameter   *param=user->param;
    532:   GridInfo    *grid =user->grid;
    533:   PetscScalar dx    = grid->dx, dz=grid->dz;
    534:   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;
    535:   PetscScalar TE    =0.0,TN=0.0,TS=0.0,TW=0.0, dPdz, residual,z_scale;
    536:   PetscScalar dudzE,dudzW,dwdxW,dwdxE,dwdzN,dwdzS;
    537:   PetscInt    ilim = grid->ni-1;

    539:   /* geometric and other parameters */
    540:   z_scale = param->z_scale;

    542:   /* viscosity */
    543:   if (param->ivisc==VISC_DIFN || param->ivisc>=VISC_DISL) { /* viscosity is T-dependent */
    544:     TN = param->potentialT * x[j+1][i].T      * PetscExpScalar((j+0.5)*dz*z_scale);
    545:     TS = param->potentialT * x[j][i].T        * PetscExpScalar((j-0.5)*dz*z_scale);
    546:     TW = param->potentialT * TInterp(x,i-1,j) * PetscExpScalar(j*dz*z_scale);
    547:     if (i==ilim) TE = TW;
    548:     else         TE = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
    549:     if (param->ivisc>=VISC_DISL) { /* olivine dislocation creep */
    550:       epsN = CalcSecInv(x,i,j+1,CELL_CENTER,user);
    551:       epsS = CalcSecInv(x,i,j,  CELL_CENTER,user);
    552:       epsE = CalcSecInv(x,i,j,  CELL_CORNER,user);
    553:       epsW = CalcSecInv(x,i-1,j,CELL_CORNER,user);
    554:     }
    555:   }
    556:   etaN = Viscosity(TN,epsN,dz*(j+1.0),param);
    557:   etaS = Viscosity(TS,epsS,dz*(j+0.0),param);
    558:   etaW = Viscosity(TW,epsW,dz*(j+0.5),param);
    559:   etaE = Viscosity(TE,epsE,dz*(j+0.5),param);

    561:   dPdz  = (x[j+1][i].p - x[j][i].p)/dz;
    562:   dwdzN = etaN * (x[j+1][i].w - x[j][i].w)/dz;
    563:   dwdzS = etaS * (x[j][i].w - x[j-1][i].w)/dz;
    564:   if (i==ilim) dwdxE = etaE * (x[j][i].u   - x[j+1][i].u)/dz;
    565:   else         dwdxE = etaE * (x[j][i+1].w - x[j][i].w)  /dx;
    566:   dwdxW = 2.0*etaW * (x[j][i].w - x[j][i-1].w)/dx;

    568:   /* Z-MOMENTUM */
    569:   residual = -dPdz                 /* constant viscosity terms */
    570:              +(dwdzN - dwdzS)/dz
    571:              +(dwdxE - dwdxW)/dx;

    573:   if (param->ivisc!=VISC_CONST) {
    574:     dudzE = etaE * (x[j+1][i].u - x[j][i].u)/dz;
    575:     dudzW = etaW * (x[j+1][i-1].u - x[j][i-1].u)/dz;

    577:     residual += (dwdzN - dwdzS)/dz + (dudzE - dudzW)/dx;
    578:   }

    580:   return residual;
    581: }

    583: /*---------------------------------------------------------------------*/
    584: /*  computes the residual of eqn (2) above */
    585: PETSC_STATIC_INLINE PetscScalar ContinuityResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    586: /*---------------------------------------------------------------------*/
    587: {
    588:   GridInfo    *grid =user->grid;
    589:   PetscScalar uE,uW,wN,wS,dudx,dwdz;

    591:   uW = x[j][i-1].u; uE = x[j][i].u; dudx = (uE - uW)/grid->dx;
    592:   wS = x[j-1][i].w; wN = x[j][i].w; dwdz = (wN - wS)/grid->dz;

    594:   return dudx + dwdz;
    595: }

    597: /*---------------------------------------------------------------------*/
    598: /*  computes the residual of eqn (3) above */
    599: PETSC_STATIC_INLINE PetscScalar EnergyResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    600: /*---------------------------------------------------------------------*/
    601: {
    602:   Parameter   *param=user->param;
    603:   GridInfo    *grid =user->grid;
    604:   PetscScalar dx    = grid->dx, dz=grid->dz;
    605:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, jlid=grid->jlid;
    606:   PetscScalar TE, TN, TS, TW, residual;
    607:   PetscScalar uE,uW,wN,wS;
    608:   PetscScalar fN,fS,fE,fW,dTdxW,dTdxE,dTdzN,dTdzS;

    610:   dTdzN = (x[j+1][i].T - x[j][i].T)  /dz;
    611:   dTdzS = (x[j][i].T   - x[j-1][i].T)/dz;
    612:   dTdxE = (x[j][i+1].T - x[j][i].T)  /dx;
    613:   dTdxW = (x[j][i].T   - x[j][i-1].T)/dx;

    615:   residual = ((dTdzN - dTdzS)/dz + /* diffusion term */
    616:               (dTdxE - dTdxW)/dx)*dx*dz/param->peclet;

    618:   if (j<=jlid && i>=j) {
    619:     /* don't advect in the lid */
    620:     return residual;
    621:   } else if (i<j) {
    622:     /* beneath the slab sfc */
    623:     uW = uE = param->cb;
    624:     wS = wN = param->sb;
    625:   } else {
    626:     /* advect in the slab and wedge */
    627:     uW = x[j][i-1].u; uE = x[j][i].u;
    628:     wS = x[j-1][i].w; wN = x[j][i].w;
    629:   }

    631:   if (param->adv_scheme==ADVECT_FV || i==ilim-1 || j==jlim-1 || i==1 || j==1) {
    632:     /* finite volume advection */
    633:     TS = (x[j][i].T + x[j-1][i].T)/2.0;
    634:     TN = (x[j][i].T + x[j+1][i].T)/2.0;
    635:     TE = (x[j][i].T + x[j][i+1].T)/2.0;
    636:     TW = (x[j][i].T + x[j][i-1].T)/2.0;
    637:     fN = wN*TN*dx; fS = wS*TS*dx;
    638:     fE = uE*TE*dz; fW = uW*TW*dz;

    640:   } else {
    641:     /* Fromm advection scheme */
    642:     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
    643:               - 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;
    644:     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
    645:               - 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;
    646:     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
    647:               - 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;
    648:     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
    649:               - 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;
    650:   }

    652:   residual -= (fE - fW + fN - fS);

    654:   return residual;
    655: }

    657: /*---------------------------------------------------------------------*/
    658: /*  computes the shear stress---used on the boundaries */
    659: PETSC_STATIC_INLINE PetscScalar ShearStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    660: /*---------------------------------------------------------------------*/
    661: {
    662:   Parameter   *param=user->param;
    663:   GridInfo    *grid =user->grid;
    664:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1;
    665:   PetscScalar uN, uS, wE, wW;

    667:   if (j<=grid->jlid || i<j || i==ilim || j==jlim) return EPS_ZERO;

    669:   if (ipos==CELL_CENTER) { /* on cell center */

    671:     wE = WInterp(x,i,j-1);
    672:     if (i==j) {
    673:       wW = param->sb;
    674:       uN = param->cb;
    675:     } else {
    676:       wW = WInterp(x,i-1,j-1);
    677:       uN = UInterp(x,i-1,j);
    678:     }
    679:     if (j==grid->jlid+1) uS = 0.0;
    680:     else                 uS = UInterp(x,i-1,j-1);

    682:   } else { /* on cell corner */

    684:     uN = x[j+1][i].u;         uS = x[j][i].u;
    685:     wW = x[j][i].w;           wE = x[j][i+1].w;

    687:   }

    689:   return (uN-uS)/grid->dz + (wE-wW)/grid->dx;
    690: }

    692: /*---------------------------------------------------------------------*/
    693: /*  computes the normal stress---used on the boundaries */
    694: PETSC_STATIC_INLINE PetscScalar XNormalStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    695: /*---------------------------------------------------------------------*/
    696: {
    697:   Parameter   *param=user->param;
    698:   GridInfo    *grid =user->grid;
    699:   PetscScalar dx    = grid->dx, dz=grid->dz;
    700:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, ivisc;
    701:   PetscScalar epsC  =0.0, etaC, TC, uE, uW, pC, z_scale;
    702:   if (i<j || j<=grid->jlid) return EPS_ZERO;

    704:   ivisc=param->ivisc;  z_scale = param->z_scale;

    706:   if (ipos==CELL_CENTER) { /* on cell center */

    708:     TC = param->potentialT * x[j][i].T * PetscExpScalar((j-0.5)*dz*z_scale);
    709:     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CENTER,user);
    710:     etaC = Viscosity(TC,epsC,dz*j,param);

    712:     uW = x[j][i-1].u;   uE = x[j][i].u;
    713:     pC = x[j][i].p;

    715:   } else { /* on cell corner */
    716:     if (i==ilim || j==jlim) return EPS_ZERO;

    718:     TC = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
    719:     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CORNER,user);
    720:     etaC = Viscosity(TC,epsC,dz*(j+0.5),param);

    722:     if (i==j) uW = param->sb;
    723:     else      uW = UInterp(x,i-1,j);
    724:     uE = UInterp(x,i,j); pC = PInterp(x,i,j);
    725:   }

    727:   return 2.0*etaC*(uE-uW)/dx - pC;
    728: }

    730: /*---------------------------------------------------------------------*/
    731: /*  computes the normal stress---used on the boundaries */
    732: PETSC_STATIC_INLINE PetscScalar ZNormalStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    733: /*---------------------------------------------------------------------*/
    734: {
    735:   Parameter   *param=user->param;
    736:   GridInfo    *grid =user->grid;
    737:   PetscScalar dz    =grid->dz;
    738:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, ivisc;
    739:   PetscScalar epsC  =0.0, etaC, TC;
    740:   PetscScalar pC, wN, wS, z_scale;
    741:   if (i<j || j<=grid->jlid) return EPS_ZERO;

    743:   ivisc=param->ivisc;  z_scale = param->z_scale;

    745:   if (ipos==CELL_CENTER) { /* on cell center */

    747:     TC = param->potentialT * x[j][i].T * PetscExpScalar((j-0.5)*dz*z_scale);
    748:     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CENTER,user);
    749:     etaC = Viscosity(TC,epsC,dz*j,param);
    750:     wN   = x[j][i].w; wS = x[j-1][i].w; pC = x[j][i].p;

    752:   } else { /* on cell corner */
    753:     if ((i==ilim) || (j==jlim)) return EPS_ZERO;

    755:     TC = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
    756:     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CORNER,user);
    757:     etaC = Viscosity(TC,epsC,dz*(j+0.5),param);
    758:     if (i==j) wN = param->sb;
    759:     else      wN = WInterp(x,i,j);
    760:     wS = WInterp(x,i,j-1); pC = PInterp(x,i,j);
    761:   }

    763:   return 2.0*etaC*(wN-wS)/dz - pC;
    764: }

    766: /*---------------------------------------------------------------------*/

    768: /*=====================================================================
    769:   INITIALIZATION, POST-PROCESSING AND OUTPUT FUNCTIONS
    770:   =====================================================================*/

    772: /*---------------------------------------------------------------------*/
    773: /* initializes the problem parameters and checks for
    774:    command line changes */
    775: PetscErrorCode SetParams(Parameter *param, GridInfo *grid)
    776: /*---------------------------------------------------------------------*/
    777: {
    778:   PetscErrorCode ierr, ierr_out=0;
    779:   PetscReal      SEC_PER_YR                    = 3600.00*24.00*365.2500;
    780:   PetscReal      alpha_g_on_cp_units_inverse_km=4.0e-5*9.8;

    782:   /* domain geometry */
    783:   param->slab_dip    = 45.0;
    784:   param->width       = 320.0;                                              /* km */
    785:   param->depth       = 300.0;                                              /* km */
    786:   param->lid_depth   = 35.0;                                               /* km */
    787:   param->fault_depth = 35.0;                                               /* km */

    789:   PetscOptionsGetReal(NULL,NULL,"-slab_dip",&(param->slab_dip),NULL);
    790:   PetscOptionsGetReal(NULL,NULL,"-width",&(param->width),NULL);
    791:   PetscOptionsGetReal(NULL,NULL,"-depth",&(param->depth),NULL);
    792:   PetscOptionsGetReal(NULL,NULL,"-lid_depth",&(param->lid_depth),NULL);
    793:   PetscOptionsGetReal(NULL,NULL,"-fault_depth",&(param->fault_depth),NULL);

    795:   param->slab_dip = param->slab_dip*PETSC_PI/180.0;                    /* radians */

    797:   /* grid information */
    798:   PetscOptionsGetInt(NULL,NULL, "-jfault",&(grid->jfault),NULL);
    799:   grid->ni = 82;
    800:   PetscOptionsGetInt(NULL,NULL, "-ni",&(grid->ni),NULL);

    802:   grid->dx            = param->width/((PetscReal)(grid->ni-2));               /* km */
    803:   grid->dz            = grid->dx*PetscTanReal(param->slab_dip);               /* km */
    804:   grid->nj            = (PetscInt)(param->depth/grid->dz + 3.0);         /* gridpoints*/
    805:   param->depth        = grid->dz*(grid->nj-2);                             /* km */
    806:   grid->inose         = 0;                                          /* gridpoints*/
    807:   PetscOptionsGetInt(NULL,NULL,"-inose",&(grid->inose),NULL);
    808:   grid->bx            = DM_BOUNDARY_NONE;
    809:   grid->by            = DM_BOUNDARY_NONE;
    810:   grid->stencil       = DMDA_STENCIL_BOX;
    811:   grid->dof           = 4;
    812:   grid->stencil_width = 2;
    813:   grid->mglevels      = 1;

    815:   /* boundary conditions */
    816:   param->pv_analytic = PETSC_FALSE;
    817:   param->ibound      = BC_NOSTRESS;
    818:   PetscOptionsGetInt(NULL,NULL,"-ibound",&(param->ibound),NULL);

    820:   /* physical constants */
    821:   param->slab_velocity = 5.0;               /* cm/yr */
    822:   param->slab_age      = 50.0;              /* Ma */
    823:   param->lid_age       = 50.0;              /* Ma */
    824:   param->kappa         = 0.7272e-6;         /* m^2/sec */
    825:   param->potentialT    = 1300.0;            /* degrees C */

    827:   PetscOptionsGetReal(NULL,NULL,"-slab_velocity",&(param->slab_velocity),NULL);
    828:   PetscOptionsGetReal(NULL,NULL,"-slab_age",&(param->slab_age),NULL);
    829:   PetscOptionsGetReal(NULL,NULL,"-lid_age",&(param->lid_age),NULL);
    830:   PetscOptionsGetReal(NULL,NULL,"-kappa",&(param->kappa),NULL);
    831:   PetscOptionsGetReal(NULL,NULL,"-potentialT",&(param->potentialT),NULL);

    833:   /* viscosity */
    834:   param->ivisc        = 3;                  /* 0=isovisc, 1=difn creep, 2=disl creep, 3=full */
    835:   param->eta0         = 1e24;               /* Pa-s */
    836:   param->visc_cutoff  = 0.0;                /* factor of eta_0 */
    837:   param->continuation = 1.0;

    839:   /* constants for diffusion creep */
    840:   param->diffusion.A     = 1.8e7;             /* Pa-s */
    841:   param->diffusion.n     = 1.0;               /* dim'less */
    842:   param->diffusion.Estar = 375e3;             /* J/mol */
    843:   param->diffusion.Vstar = 5e-6;              /* m^3/mol */

    845:   /* constants for param->dislocationocation creep */
    846:   param->dislocation.A     = 2.8969e4;        /* Pa-s */
    847:   param->dislocation.n     = 3.5;             /* dim'less */
    848:   param->dislocation.Estar = 530e3;           /* J/mol */
    849:   param->dislocation.Vstar = 14e-6;           /* m^3/mol */

    851:   PetscOptionsGetInt(NULL,NULL, "-ivisc",&(param->ivisc),NULL);
    852:   PetscOptionsGetReal(NULL,NULL,"-visc_cutoff",&(param->visc_cutoff),NULL);

    854:   param->output_ivisc = param->ivisc;

    856:   PetscOptionsGetInt(NULL,NULL,"-output_ivisc",&(param->output_ivisc),NULL);
    857:   PetscOptionsGetReal(NULL,NULL,"-vstar",&(param->dislocation.Vstar),NULL);

    859:   /* output options */
    860:   param->quiet      = PETSC_FALSE;
    861:   param->param_test = PETSC_FALSE;

    863:   PetscOptionsHasName(NULL,NULL,"-quiet",&(param->quiet));
    864:   PetscOptionsHasName(NULL,NULL,"-test",&(param->param_test));
    865:   PetscOptionsGetString(NULL,NULL,"-file",param->filename,sizeof(param->filename),&(param->output_to_file));

    867:   /* advection */
    868:   param->adv_scheme = ADVECT_FROMM;       /* advection scheme: 0=finite vol, 1=Fromm */

    870:   PetscOptionsGetInt(NULL,NULL,"-adv_scheme",&(param->adv_scheme),NULL);

    872:   /* misc. flags */
    873:   param->stop_solve    = PETSC_FALSE;
    874:   param->interrupted   = PETSC_FALSE;
    875:   param->kspmon        = PETSC_FALSE;
    876:   param->toggle_kspmon = PETSC_FALSE;

    878:   /* derived parameters for slab angle */
    879:   param->sb = PetscSinReal(param->slab_dip);
    880:   param->cb = PetscCosReal(param->slab_dip);
    881:   param->c  =  param->slab_dip*param->sb/(param->slab_dip*param->slab_dip-param->sb*param->sb);
    882:   param->d  = (param->slab_dip*param->cb-param->sb)/(param->slab_dip*param->slab_dip-param->sb*param->sb);

    884:   /* length, velocity and time scale for non-dimensionalization */
    885:   param->L = PetscMin(param->width,param->depth);               /* km */
    886:   param->V = param->slab_velocity/100.0/SEC_PER_YR;             /* m/sec */

    888:   /* other unit conversions and derived parameters */
    889:   param->scaled_width = param->width/param->L;                  /* dim'less */
    890:   param->scaled_depth = param->depth/param->L;                  /* dim'less */
    891:   param->lid_depth    = param->lid_depth/param->L;              /* dim'less */
    892:   param->fault_depth  = param->fault_depth/param->L;            /* dim'less */
    893:   grid->dx            = grid->dx/param->L;                      /* dim'less */
    894:   grid->dz            = grid->dz/param->L;                      /* dim'less */
    895:   grid->jlid          = (PetscInt)(param->lid_depth/grid->dz);       /* gridcells */
    896:   grid->jfault        = (PetscInt)(param->fault_depth/grid->dz);     /* gridcells */
    897:   param->lid_depth    = grid->jlid*grid->dz;                    /* dim'less */
    898:   param->fault_depth  = grid->jfault*grid->dz;                  /* dim'less */
    899:   grid->corner        = grid->jlid+1;                           /* gridcells */
    900:   param->peclet       = param->V                                /* m/sec */
    901:                         * param->L*1000.0                       /* m */
    902:                         / param->kappa;                         /* m^2/sec */
    903:   param->z_scale = param->L * alpha_g_on_cp_units_inverse_km;
    904:   param->skt     = PetscSqrtReal(param->kappa*param->slab_age*SEC_PER_YR);
    905:   PetscOptionsGetReal(NULL,NULL,"-peclet",&(param->peclet),NULL);

    907:   return ierr_out;
    908: }

    910: /*---------------------------------------------------------------------*/
    911: /*  prints a report of the problem parameters to stdout */
    912: PetscErrorCode ReportParams(Parameter *param, GridInfo *grid)
    913: /*---------------------------------------------------------------------*/
    914: {
    915:   PetscErrorCode ierr, ierr_out=0;
    916:   char           date[30];

    918:   PetscGetDate(date,30);

    920:   if (!(param->quiet)) {
    921:     PetscPrintf(PETSC_COMM_WORLD,"---------------------BEGIN ex30 PARAM REPORT-------------------\n");
    922:     PetscPrintf(PETSC_COMM_WORLD,"Domain: \n");
    923:     PetscPrintf(PETSC_COMM_WORLD,"  Width = %g km,         Depth = %g km\n",(double)param->width,(double)param->depth);
    924:     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);
    925:     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));

    927:     PetscPrintf(PETSC_COMM_WORLD,"\nGrid: \n");
    928:     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));
    929:     PetscPrintf(PETSC_COMM_WORLD,"  jlid = %3D              jfault = %3D \n",grid->jlid,grid->jfault);
    930:     PetscPrintf(PETSC_COMM_WORLD,"  Pe = %g\n",(double)param->peclet);

    932:     PetscPrintf(PETSC_COMM_WORLD,"\nRheology:");
    933:     if (param->ivisc==VISC_CONST) {
    934:       PetscPrintf(PETSC_COMM_WORLD,"                 Isoviscous \n");
    935:       if (param->pv_analytic) {
    936:         PetscPrintf(PETSC_COMM_WORLD,"                          Pressure and Velocity prescribed! \n");
    937:       }
    938:     } else if (param->ivisc==VISC_DIFN) {
    939:       PetscPrintf(PETSC_COMM_WORLD,"                 Diffusion Creep (T-Dependent Newtonian) \n");
    940:       PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));
    941:     } else if (param->ivisc==VISC_DISL) {
    942:       PetscPrintf(PETSC_COMM_WORLD,"                 Dislocation Creep (T-Dependent Non-Newtonian) \n");
    943:       PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));
    944:     } else if (param->ivisc==VISC_FULL) {
    945:       PetscPrintf(PETSC_COMM_WORLD,"                 Full Rheology \n");
    946:       PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));
    947:     } else {
    948:       PetscPrintf(PETSC_COMM_WORLD,"                 Invalid! \n");
    949:       ierr_out = 1;
    950:     }

    952:     PetscPrintf(PETSC_COMM_WORLD,"Boundary condition:");
    953:     if (param->ibound==BC_ANALYTIC) {
    954:       PetscPrintf(PETSC_COMM_WORLD,"       Isoviscous Analytic Dirichlet \n");
    955:     } else if (param->ibound==BC_NOSTRESS) {
    956:       PetscPrintf(PETSC_COMM_WORLD,"       Stress-Free (normal & shear stress)\n");
    957:     } else if (param->ibound==BC_EXPERMNT) {
    958:       PetscPrintf(PETSC_COMM_WORLD,"       Experimental boundary condition \n");
    959:     } else {
    960:       PetscPrintf(PETSC_COMM_WORLD,"       Invalid! \n");
    961:       ierr_out = 1;
    962:     }

    964:     if (param->output_to_file)
    965: #if defined(PETSC_HAVE_MATLAB_ENGINE)
    966:       PetscPrintf(PETSC_COMM_WORLD,"Output Destination:       Mat file \"%s\"\n",param->filename);
    967: #else
    968:       PetscPrintf(PETSC_COMM_WORLD,"Output Destination:       PETSc binary file \"%s\"\n",param->filename);
    969: #endif
    970:     if (param->output_ivisc != param->ivisc) {
    971:       PetscPrintf(PETSC_COMM_WORLD,"                          Output viscosity: -ivisc %D\n",param->output_ivisc);
    972:     }

    974:     PetscPrintf(PETSC_COMM_WORLD,"---------------------END ex30 PARAM REPORT---------------------\n");
    975:   }
    976:   if (param->param_test) PetscEnd();
    977:   return ierr_out;
    978: }

    980: /* ------------------------------------------------------------------- */
    981: /*  generates an inital guess using the analytic solution for isoviscous
    982:     corner flow */
    983: PetscErrorCode Initialize(DM da)
    984: /* ------------------------------------------------------------------- */
    985: {
    986:   AppCtx         *user;
    987:   Parameter      *param;
    988:   GridInfo       *grid;
    989:   PetscInt       i,j,is,js,im,jm;
    991:   Field          **x;
    992:   Vec            Xguess;

    994:   /* Get the fine grid */
    995:   DMGetApplicationContext(da,&user);
    996:   Xguess = user->Xguess;
    997:   param  = user->param;
    998:   grid   = user->grid;
    999:   DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);
    1000:   DMDAVecGetArray(da,Xguess,(void**)&x);

    1002:   /* Compute initial guess */
    1003:   for (j=js; j<js+jm; j++) {
    1004:     for (i=is; i<is+im; i++) {
    1005:       if (i<j)                x[j][i].u = param->cb;
    1006:       else if (j<=grid->jlid) x[j][i].u = 0.0;
    1007:       else                    x[j][i].u = HorizVelocity(i,j,user);

    1009:       if (i<=j)               x[j][i].w = param->sb;
    1010:       else if (j<=grid->jlid) x[j][i].w = 0.0;
    1011:       else                    x[j][i].w = VertVelocity(i,j,user);

    1013:       if (i<j || j<=grid->jlid) x[j][i].p = 0.0;
    1014:       else                      x[j][i].p = Pressure(i,j,user);

    1016:       x[j][i].T = PetscMin(grid->dz*(j-0.5),1.0);
    1017:     }
    1018:   }

    1020:   /* Restore x to Xguess */
    1021:   DMDAVecRestoreArray(da,Xguess,(void**)&x);

    1023:   return 0;
    1024: }

    1026: /*---------------------------------------------------------------------*/
    1027: /*  controls output to a file */
    1028: PetscErrorCode DoOutput(SNES snes, PetscInt its)
    1029: /*---------------------------------------------------------------------*/
    1030: {
    1031:   AppCtx         *user;
    1032:   Parameter      *param;
    1033:   GridInfo       *grid;
    1034:   PetscInt       ivt;
    1036:   PetscMPIInt    rank;
    1037:   PetscViewer    viewer;
    1038:   Vec            res, pars;
    1039:   MPI_Comm       comm;
    1040:   DM             da;

    1042:   SNESGetDM(snes,&da);
    1043:   DMGetApplicationContext(da,&user);
    1044:   param = user->param;
    1045:   grid  = user->grid;
    1046:   ivt   = param->ivisc;

    1048:   param->ivisc = param->output_ivisc;

    1050:   /* compute final residual and final viscosity/strain rate fields */
    1051:   SNESGetFunction(snes, &res, NULL, NULL);
    1052:   ViscosityField(da, user->x, user->Xguess);

    1054:   /* get the communicator and the rank of the processor */
    1055:   PetscObjectGetComm((PetscObject)snes, &comm);
    1056:   MPI_Comm_rank(comm, &rank);

    1058:   if (param->output_to_file) { /* send output to binary file */
    1059:     VecCreate(comm, &pars);
    1060:     if (!rank) { /* on processor 0 */
    1061:       VecSetSizes(pars, 20, PETSC_DETERMINE);
    1062:       VecSetFromOptions(pars);
    1063:       VecSetValue(pars,0, (PetscScalar)(grid->ni),INSERT_VALUES);
    1064:       VecSetValue(pars,1, (PetscScalar)(grid->nj),INSERT_VALUES);
    1065:       VecSetValue(pars,2, (PetscScalar)(grid->dx),INSERT_VALUES);
    1066:       VecSetValue(pars,3, (PetscScalar)(grid->dz),INSERT_VALUES);
    1067:       VecSetValue(pars,4, (PetscScalar)(param->L),INSERT_VALUES);
    1068:       VecSetValue(pars,5, (PetscScalar)(param->V),INSERT_VALUES);
    1069:       /* skipped 6 intentionally */
    1070:       VecSetValue(pars,7, (PetscScalar)(param->slab_dip),INSERT_VALUES);
    1071:       VecSetValue(pars,8, (PetscScalar)(grid->jlid),INSERT_VALUES);
    1072:       VecSetValue(pars,9, (PetscScalar)(param->lid_depth),INSERT_VALUES);
    1073:       VecSetValue(pars,10,(PetscScalar)(grid->jfault),INSERT_VALUES);
    1074:       VecSetValue(pars,11,(PetscScalar)(param->fault_depth),INSERT_VALUES);
    1075:       VecSetValue(pars,12,(PetscScalar)(param->potentialT),INSERT_VALUES);
    1076:       VecSetValue(pars,13,(PetscScalar)(param->ivisc),INSERT_VALUES);
    1077:       VecSetValue(pars,14,(PetscScalar)(param->visc_cutoff),INSERT_VALUES);
    1078:       VecSetValue(pars,15,(PetscScalar)(param->ibound),INSERT_VALUES);
    1079:       VecSetValue(pars,16,(PetscScalar)(its),INSERT_VALUES);
    1080:     } else { /* on some other processor */
    1081:       VecSetSizes(pars, 0, PETSC_DETERMINE);
    1082:       VecSetFromOptions(pars);
    1083:     }
    1084:     VecAssemblyBegin(pars); VecAssemblyEnd(pars);

    1086:     /* create viewer */
    1087: #if defined(PETSC_HAVE_MATLAB_ENGINE)
    1088:     PetscViewerMatlabOpen(PETSC_COMM_WORLD,param->filename,FILE_MODE_WRITE,&viewer);
    1089: #else
    1090:     PetscViewerBinaryOpen(PETSC_COMM_WORLD,param->filename,FILE_MODE_WRITE,&viewer);
    1091: #endif

    1093:     /* send vectors to viewer */
    1094:     PetscObjectSetName((PetscObject)res,"res");
    1095:     VecView(res,viewer);
    1096:     PetscObjectSetName((PetscObject)user->x,"out");
    1097:     VecView(user->x, viewer);
    1098:     PetscObjectSetName((PetscObject)(user->Xguess),"aux");
    1099:     VecView(user->Xguess, viewer);
    1100:     StressField(da); /* compute stress fields */
    1101:     PetscObjectSetName((PetscObject)(user->Xguess),"str");
    1102:     VecView(user->Xguess, viewer);
    1103:     PetscObjectSetName((PetscObject)pars,"par");
    1104:     VecView(pars, viewer);

    1106:     /* destroy viewer and vector */
    1107:     PetscViewerDestroy(&viewer);
    1108:     VecDestroy(&pars);
    1109:   }

    1111:   param->ivisc = ivt;
    1112:   return 0;
    1113: }

    1115: /* ------------------------------------------------------------------- */
    1116: /* Compute both the second invariant of the strain rate tensor and the viscosity, at both cell centers and cell corners */
    1117: PetscErrorCode ViscosityField(DM da, Vec X, Vec V)
    1118: /* ------------------------------------------------------------------- */
    1119: {
    1120:   AppCtx         *user;
    1121:   Parameter      *param;
    1122:   GridInfo       *grid;
    1123:   Vec            localX;
    1124:   Field          **v, **x;
    1125:   PetscReal      eps, /* dx,*/ dz, T, epsC, TC;
    1126:   PetscInt       i,j,is,js,im,jm,ilim,jlim,ivt;

    1130:   DMGetApplicationContext(da,&user);
    1131:   param        = user->param;
    1132:   grid         = user->grid;
    1133:   ivt          = param->ivisc;
    1134:   param->ivisc = param->output_ivisc;

    1136:   DMGetLocalVector(da, &localX);
    1137:   DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX);
    1138:   DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX);
    1139:   DMDAVecGetArray(da,localX,(void**)&x);
    1140:   DMDAVecGetArray(da,V,(void**)&v);

    1142:   /* Parameters */
    1143:   /* dx = grid->dx; */ dz = grid->dz;

    1145:   ilim = grid->ni-1; jlim = grid->nj-1;

    1147:   /* Compute real temperature, strain rate and viscosity */
    1148:   DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);
    1149:   for (j=js; j<js+jm; j++) {
    1150:     for (i=is; i<is+im; i++) {
    1151:       T = PetscRealPart(param->potentialT * x[j][i].T * PetscExpScalar((j-0.5)*dz*param->z_scale));
    1152:       if (i<ilim && j<jlim) {
    1153:         TC = PetscRealPart(param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*param->z_scale));
    1154:       } else {
    1155:         TC = T;
    1156:       }
    1157:       eps  = PetscRealPart((CalcSecInv(x,i,j,CELL_CENTER,user)));
    1158:       epsC = PetscRealPart(CalcSecInv(x,i,j,CELL_CORNER,user));

    1160:       v[j][i].u = eps;
    1161:       v[j][i].w = epsC;
    1162:       v[j][i].p = Viscosity(T,eps,dz*(j-0.5),param);
    1163:       v[j][i].T = Viscosity(TC,epsC,dz*j,param);
    1164:     }
    1165:   }
    1166:   DMDAVecRestoreArray(da,V,(void**)&v);
    1167:   DMDAVecRestoreArray(da,localX,(void**)&x);
    1168:   DMRestoreLocalVector(da, &localX);

    1170:   param->ivisc = ivt;
    1171:   return(0);
    1172: }

    1174: /* ------------------------------------------------------------------- */
    1175: /* post-processing: compute stress everywhere */
    1176: PetscErrorCode StressField(DM da)
    1177: /* ------------------------------------------------------------------- */
    1178: {
    1179:   AppCtx         *user;
    1180:   PetscInt       i,j,is,js,im,jm;
    1182:   Vec            locVec;
    1183:   Field          **x, **y;

    1185:   DMGetApplicationContext(da,&user);

    1187:   /* Get the fine grid of Xguess and X */
    1188:   DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);
    1189:   DMDAVecGetArray(da,user->Xguess,(void**)&x);

    1191:   DMGetLocalVector(da, &locVec);
    1192:   DMGlobalToLocalBegin(da, user->x, INSERT_VALUES, locVec);
    1193:   DMGlobalToLocalEnd(da, user->x, INSERT_VALUES, locVec);
    1194:   DMDAVecGetArray(da,locVec,(void**)&y);

    1196:   /* Compute stress on the corner points */
    1197:   for (j=js; j<js+jm; j++) {
    1198:     for (i=is; i<is+im; i++) {
    1199:       x[j][i].u = ShearStress(y,i,j,CELL_CENTER,user);
    1200:       x[j][i].w = ShearStress(y,i,j,CELL_CORNER,user);
    1201:       x[j][i].p = XNormalStress(y,i,j,CELL_CENTER,user);
    1202:       x[j][i].T = ZNormalStress(y,i,j,CELL_CENTER,user);
    1203:     }
    1204:   }

    1206:   /* Restore the fine grid of Xguess and X */
    1207:   DMDAVecRestoreArray(da,user->Xguess,(void**)&x);
    1208:   DMDAVecRestoreArray(da,locVec,(void**)&y);
    1209:   DMRestoreLocalVector(da, &locVec);
    1210:   return 0;
    1211: }

    1213: /*=====================================================================
    1214:   UTILITY FUNCTIONS
    1215:   =====================================================================*/

    1217: /*---------------------------------------------------------------------*/
    1218: /* returns the velocity of the subducting slab and handles fault nodes
    1219:    for BC */
    1220: PETSC_STATIC_INLINE PetscScalar SlabVel(char c, PetscInt i, PetscInt j, AppCtx *user)
    1221: /*---------------------------------------------------------------------*/
    1222: {
    1223:   Parameter *param = user->param;
    1224:   GridInfo  *grid  = user->grid;

    1226:   if (c=='U' || c=='u') {
    1227:     if (i<j-1) return param->cb;
    1228:     else if (j<=grid->jfault) return 0.0;
    1229:     else return param->cb;

    1231:   } else {
    1232:     if (i<j) return param->sb;
    1233:     else if (j<=grid->jfault) return 0.0;
    1234:     else return param->sb;
    1235:   }
    1236: }

    1238: /*---------------------------------------------------------------------*/
    1239: /*  solution to diffusive half-space cooling model for BC */
    1240: PETSC_STATIC_INLINE PetscScalar PlateModel(PetscInt j, PetscInt plate, AppCtx *user)
    1241: /*---------------------------------------------------------------------*/
    1242: {
    1243:   Parameter     *param = user->param;
    1244:   PetscScalar   z;
    1245:   if (plate==PLATE_LID) z = (j-0.5)*user->grid->dz;
    1246:   else z = (j-0.5)*user->grid->dz*param->cb;  /* PLATE_SLAB */
    1247: #if defined(PETSC_HAVE_ERF)
    1248:   return (PetscReal)(erf((double)PetscRealPart(z*param->L/2.0/param->skt)));
    1249: #else
    1250:   (*PetscErrorPrintf)("erf() not available on this machine\n");
    1251:   MPI_Abort(PETSC_COMM_SELF,1);
    1252: #endif
    1253: }

    1255: /*=====================================================================
    1256:   INTERACTIVE SIGNAL HANDLING
    1257:   =====================================================================*/

    1259: /* ------------------------------------------------------------------- */
    1260: PetscErrorCode SNESConverged_Interactive(SNES snes, PetscInt it,PetscReal xnorm, PetscReal snorm, PetscReal fnorm, SNESConvergedReason *reason, void *ctx)
    1261: /* ------------------------------------------------------------------- */
    1262: {
    1263:   AppCtx         *user  = (AppCtx*) ctx;
    1264:   Parameter      *param = user->param;
    1265:   KSP            ksp;

    1269:   if (param->interrupted) {
    1270:     param->interrupted = PETSC_FALSE;
    1271:     PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: exiting SNES solve. \n");
    1272:     *reason = SNES_CONVERGED_FNORM_ABS;
    1273:     return(0);
    1274:   } else if (param->toggle_kspmon) {
    1275:     param->toggle_kspmon = PETSC_FALSE;

    1277:     SNESGetKSP(snes, &ksp);

    1279:     if (param->kspmon) {
    1280:       KSPMonitorCancel(ksp);

    1282:       param->kspmon = PETSC_FALSE;
    1283:       PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: deactivating ksp singular value monitor. \n");
    1284:     } else {
    1285:       PetscViewerAndFormat *vf;
    1286:       PetscViewerAndFormatCreate(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_DEFAULT,&vf);
    1287:       KSPMonitorSet(ksp,(PetscErrorCode (*)(KSP,PetscInt,PetscReal,void*))KSPMonitorSingularValue,vf,(PetscErrorCode (*)(void**))PetscViewerAndFormatDestroy);

    1289:       param->kspmon = PETSC_TRUE;
    1290:       PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: activating ksp singular value monitor. \n");
    1291:     }
    1292:   }
    1293:   PetscFunctionReturn(SNESConvergedDefault(snes,it,xnorm,snorm,fnorm,reason,ctx));
    1294: }

    1296: /* ------------------------------------------------------------------- */
    1297: #include <signal.h>
    1298: PetscErrorCode InteractiveHandler(int signum, void *ctx)
    1299: /* ------------------------------------------------------------------- */
    1300: {
    1301:   AppCtx    *user  = (AppCtx*) ctx;
    1302:   Parameter *param = user->param;

    1304:   if (signum == SIGILL) {
    1305:     param->toggle_kspmon = PETSC_TRUE;
    1306: #if !defined(PETSC_MISSING_SIGCONT)
    1307:   } else if (signum == SIGCONT) {
    1308:     param->interrupted = PETSC_TRUE;
    1309: #endif
    1310: #if !defined(PETSC_MISSING_SIGURG)
    1311:   } else if (signum == SIGURG) {
    1312:     param->stop_solve = PETSC_TRUE;
    1313: #endif
    1314:   }
    1315:   return 0;
    1316: }

    1318: /*---------------------------------------------------------------------*/
    1319: /*  main call-back function that computes the processor-local piece
    1320:     of the residual */
    1321: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,Field **x,Field **f,void *ptr)
    1322: /*---------------------------------------------------------------------*/
    1323: {
    1324:   AppCtx      *user  = (AppCtx*)ptr;
    1325:   Parameter   *param = user->param;
    1326:   GridInfo    *grid  = user->grid;
    1327:   PetscScalar mag_w, mag_u;
    1328:   PetscInt    i,j,mx,mz,ilim,jlim;
    1329:   PetscInt    is,ie,js,je,ibound;    /* ,ivisc */

    1332:   /* Define global and local grid parameters */
    1333:   mx   = info->mx;     mz   = info->my;
    1334:   ilim = mx-1;         jlim = mz-1;
    1335:   is   = info->xs;     ie   = info->xs+info->xm;
    1336:   js   = info->ys;     je   = info->ys+info->ym;

    1338:   /* Define geometric and numeric parameters */
    1339:   /* ivisc = param->ivisc; */ ibound = param->ibound;

    1341:   for (j=js; j<je; j++) {
    1342:     for (i=is; i<ie; i++) {

    1344:       /************* X-MOMENTUM/VELOCITY *************/
    1345:       if (i<j) f[j][i].u = x[j][i].u - SlabVel('U',i,j,user);
    1346:       else if (j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
    1347:         /* in the lithospheric lid */
    1348:         f[j][i].u = x[j][i].u - 0.0;
    1349:       } else if (i==ilim) {
    1350:         /* on the right side boundary */
    1351:         if (ibound==BC_ANALYTIC) {
    1352:           f[j][i].u = x[j][i].u - HorizVelocity(i,j,user);
    1353:         } else {
    1354:           f[j][i].u = XNormalStress(x,i,j,CELL_CENTER,user) - EPS_ZERO;
    1355:         }

    1357:       } else if (j==jlim) {
    1358:         /* on the bottom boundary */
    1359:         if (ibound==BC_ANALYTIC) {
    1360:           f[j][i].u = x[j][i].u - HorizVelocity(i,j,user);
    1361:         } else if (ibound==BC_NOSTRESS) {
    1362:           f[j][i].u = XMomentumResidual(x,i,j,user);
    1363:         } else {
    1364:           /* experimental boundary condition */
    1365:         }

    1367:       } else {
    1368:         /* in the mantle wedge */
    1369:         f[j][i].u = XMomentumResidual(x,i,j,user);
    1370:       }

    1372:       /************* Z-MOMENTUM/VELOCITY *************/
    1373:       if (i<=j) {
    1374:         f[j][i].w = x[j][i].w - SlabVel('W',i,j,user);

    1376:       } else if (j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
    1377:         /* in the lithospheric lid */
    1378:         f[j][i].w = x[j][i].w - 0.0;

    1380:       } else if (j==jlim) {
    1381:         /* on the bottom boundary */
    1382:         if (ibound==BC_ANALYTIC) {
    1383:           f[j][i].w = x[j][i].w - VertVelocity(i,j,user);
    1384:         } else {
    1385:           f[j][i].w = ZNormalStress(x,i,j,CELL_CENTER,user) - EPS_ZERO;
    1386:         }

    1388:       } else if (i==ilim) {
    1389:         /* on the right side boundary */
    1390:         if (ibound==BC_ANALYTIC) {
    1391:           f[j][i].w = x[j][i].w - VertVelocity(i,j,user);
    1392:         } else if (ibound==BC_NOSTRESS) {
    1393:           f[j][i].w = ZMomentumResidual(x,i,j,user);
    1394:         } else {
    1395:           /* experimental boundary condition */
    1396:         }

    1398:       } else {
    1399:         /* in the mantle wedge */
    1400:         f[j][i].w =  ZMomentumResidual(x,i,j,user);
    1401:       }

    1403:       /************* CONTINUITY/PRESSURE *************/
    1404:       if (i<j || j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
    1405:         /* in the lid or slab */
    1406:         f[j][i].p = x[j][i].p;

    1408:       } else if ((i==ilim || j==jlim) && ibound==BC_ANALYTIC) {
    1409:         /* on an analytic boundary */
    1410:         f[j][i].p = x[j][i].p - Pressure(i,j,user);

    1412:       } else {
    1413:         /* in the mantle wedge */
    1414:         f[j][i].p = ContinuityResidual(x,i,j,user);
    1415:       }

    1417:       /************* TEMPERATURE *************/
    1418:       if (j==0) {
    1419:         /* on the surface */
    1420:         f[j][i].T = x[j][i].T + x[j+1][i].T + PetscMax(PetscRealPart(x[j][i].T),0.0);

    1422:       } else if (i==0) {
    1423:         /* slab inflow boundary */
    1424:         f[j][i].T = x[j][i].T - PlateModel(j,PLATE_SLAB,user);

    1426:       } else if (i==ilim) {
    1427:         /* right side boundary */
    1428:         mag_u = 1.0 - PetscPowRealInt((1.0-PetscMax(PetscMin(PetscRealPart(x[j][i-1].u)/param->cb,1.0),0.0)), 5);
    1429:         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);

    1431:       } else if (j==jlim) {
    1432:         /* bottom boundary */
    1433:         mag_w = 1.0 - PetscPowRealInt((1.0-PetscMax(PetscMin(PetscRealPart(x[j-1][i].w)/param->sb,1.0),0.0)), 5);
    1434:         f[j][i].T = x[j][i].T - mag_w*x[j-1][i-1].T - (1.0-mag_w);

    1436:       } else {
    1437:         /* in the mantle wedge */
    1438:         f[j][i].T = EnergyResidual(x,i,j,user);
    1439:       }
    1440:     }
    1441:   }
    1442:   return(0);
    1443: }

    1446: /*TEST

    1448:    build:
    1449:       requires: !complex erf

    1451:    test:
    1452:       args: -ni 18
    1453:       filter: grep -v Destination
    1454:       requires: !single

    1456: TEST*/