Actual source code: ex30.c

      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:
-------$w_{ij}$------
$u_{i-1j}$ $P,T_{ij}$ $u_{ij}$
------$w_{ij-1}$-----

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

     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

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

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

     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;

     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;

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

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

    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);

    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*);

    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:   MPI_Comm       comm;
    133:   DM             da;

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

    143:   comm = PETSC_COMM_WORLD;

    145:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    146:      Set up the problem parameters.
    147:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    148:   SetParams(&param,&grid);
    149:   ReportParams(&param,&grid);

    151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    152:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    153:   SNESCreate(comm,&snes);
    154:   DMDACreate2d(comm,grid.bx,grid.by,grid.stencil,grid.ni,grid.nj,PETSC_DECIDE,PETSC_DECIDE,grid.dof,grid.stencil_width,0,0,&da);
    155:   DMSetFromOptions(da);
    156:   DMSetUp(da);
    157:   SNESSetDM(snes,da);
    158:   DMDASetFieldName(da,0,"x-velocity");
    159:   DMDASetFieldName(da,1,"y-velocity");
    160:   DMDASetFieldName(da,2,"pressure");
    161:   DMDASetFieldName(da,3,"temperature");

    163:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    164:      Create user context, set problem data, create vector data structures.
    165:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    166:   PetscNew(&user);
    167:   user->param = &param;
    168:   user->grid  = &grid;
    169:   DMSetApplicationContext(da,user);
    170:   DMCreateGlobalVector(da,&(user->Xguess));

    172:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    173:      Set up the SNES solver with callback functions.
    174:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    175:   DMDASNESSetFunctionLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionLocal,(void*)user);
    176:   SNESSetFromOptions(snes);

    178:   SNESSetConvergenceTest(snes,SNESConverged_Interactive,(void*)user,NULL);
    179:   PetscPushSignalHandler(InteractiveHandler,(void*)user);

    181:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    182:      Initialize and solve the nonlinear system
    183:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    184:   Initialize(da);
    185:   UpdateSolution(snes,user,&nits);

    187:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    188:      Output variables.
    189:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    190:   DoOutput(snes,nits);

    192:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    193:      Free work space.
    194:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    195:   VecDestroy(&user->Xguess);
    196:   VecDestroy(&user->x);
    197:   PetscFree(user);
    198:   SNESDestroy(&snes);
    199:   DMDestroy(&da);
    200:   PetscPopSignalHandler();
    201:   PetscFinalize();
    202:   return 0;
    203: }

    205: /*=====================================================================
    206:   PETSc INTERACTION FUNCTIONS (initialize & call SNESSolve)
    207:   =====================================================================*/

    209: /*---------------------------------------------------------------------*/
    210: /*  manages solve: adaptive continuation method  */
    211: PetscErrorCode UpdateSolution(SNES snes, AppCtx *user, PetscInt *nits)
    212: {
    213:   KSP                 ksp;
    214:   PC                  pc;
    215:   SNESConvergedReason reason = SNES_CONVERGED_ITERATING;
    216:   Parameter           *param   = user->param;
    217:   PetscReal           cont_incr=0.3;
    218:   PetscInt            its;
    219:   PetscBool           q = PETSC_FALSE;
    220:   DM                  dm;

    223:   SNESGetDM(snes,&dm);
    224:   DMCreateGlobalVector(dm,&user->x);
    225:   SNESGetKSP(snes,&ksp);
    226:   KSPGetPC(ksp, &pc);
    227:   KSPSetComputeSingularValues(ksp, PETSC_TRUE);

    229:   *nits=0;

    231:   /* Isoviscous solve */
    232:   if (param->ivisc == VISC_CONST && !param->stop_solve) {
    233:     param->ivisc = VISC_CONST;

    235:     SNESSolve(snes,0,user->x);
    236:     SNESGetConvergedReason(snes,&reason);
    237:     SNESGetIterationNumber(snes,&its);
    238:     *nits += its;
    239:     VecCopy(user->x,user->Xguess);
    240:     if (param->stop_solve) goto done;
    241:   }

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

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

    251:       /* solve the non-linear system */
    252:       VecCopy(user->Xguess,user->x);
    253:       SNESSolve(snes,0,user->x);
    254:       SNESGetConvergedReason(snes,&reason);
    255:       SNESGetIterationNumber(snes,&its);
    256:       *nits += its;
    257:       if (!q) PetscPrintf(PETSC_COMM_WORLD," SNES iterations: %D, Cumulative: %D\n", its, *nits);
    258:       if (param->stop_solve) goto done;

    260:       if (reason<0) {
    261:         /* NOT converged */
    262:         cont_incr = -PetscAbsReal(cont_incr)/2.0;
    263:         if (PetscAbsReal(cont_incr)<0.01) goto done;

    265:       } else {
    266:         /* converged */
    267:         VecCopy(user->x,user->Xguess);
    268:         if (param->continuation >= 1.0) goto done;
    269:         if (its<=3)      cont_incr = 0.30001;
    270:         else if (its<=8) cont_incr = 0.15001;
    271:         else             cont_incr = 0.10001;

    273:         if (param->continuation+cont_incr > 1.0) cont_incr = 1.0 - param->continuation;
    274:       } /* endif reason<0 */
    275:     }
    276:   }
    277: done:
    278:   if (param->stop_solve && !q) PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: stopping solve.\n");
    279:   if (reason<0 && !q) PetscPrintf(PETSC_COMM_WORLD,"FAILED TO CONVERGE: stopping solve.\n");
    280:   return 0;
    281: }

    283: /*=====================================================================
    284:   PHYSICS FUNCTIONS (compute the discrete residual)
    285:   =====================================================================*/

    287: /*---------------------------------------------------------------------*/
    288: static inline PetscScalar UInterp(Field **x, PetscInt i, PetscInt j)
    289: /*---------------------------------------------------------------------*/
    290: {
    291:   return 0.25*(x[j][i].u+x[j+1][i].u+x[j][i+1].u+x[j+1][i+1].u);
    292: }

    294: /*---------------------------------------------------------------------*/
    295: static inline PetscScalar WInterp(Field **x, PetscInt i, PetscInt j)
    296: /*---------------------------------------------------------------------*/
    297: {
    298:   return 0.25*(x[j][i].w+x[j+1][i].w+x[j][i+1].w+x[j+1][i+1].w);
    299: }

    301: /*---------------------------------------------------------------------*/
    302: static inline PetscScalar PInterp(Field **x, PetscInt i, PetscInt j)
    303: /*---------------------------------------------------------------------*/
    304: {
    305:   return 0.25*(x[j][i].p+x[j+1][i].p+x[j][i+1].p+x[j+1][i+1].p);
    306: }

    308: /*---------------------------------------------------------------------*/
    309: static inline PetscScalar TInterp(Field **x, PetscInt i, PetscInt j)
    310: /*---------------------------------------------------------------------*/
    311: {
    312:   return 0.25*(x[j][i].T+x[j+1][i].T+x[j][i+1].T+x[j+1][i+1].T);
    313: }

    315: /*---------------------------------------------------------------------*/
    316: /*  isoviscous analytic solution for IC */
    317: static inline PetscScalar HorizVelocity(PetscInt i, PetscInt j, AppCtx *user)
    318: /*---------------------------------------------------------------------*/
    319: {
    320:   Parameter   *param = user->param;
    321:   GridInfo    *grid  = user->grid;
    322:   PetscScalar st, ct, th, c=param->c, d=param->d;
    323:   PetscReal   x, z,r;

    325:   x  = (i - grid->jlid)*grid->dx;  z = (j - grid->jlid - 0.5)*grid->dz;
    326:   r  = PetscSqrtReal(x*x+z*z);
    327:   st = z/r;
    328:   ct = x/r;
    329:   th = PetscAtanReal(z/x);
    330:   return ct*(c*th*st+d*(st+th*ct)) + st*(c*(st-th*ct)+d*th*st);
    331: }

    333: /*---------------------------------------------------------------------*/
    334: /*  isoviscous analytic solution for IC */
    335: static inline PetscScalar VertVelocity(PetscInt i, PetscInt j, AppCtx *user)
    336: /*---------------------------------------------------------------------*/
    337: {
    338:   Parameter   *param = user->param;
    339:   GridInfo    *grid  = user->grid;
    340:   PetscScalar st, ct, th, c=param->c, d=param->d;
    341:   PetscReal   x, z, r;

    343:   x = (i - grid->jlid - 0.5)*grid->dx;  z = (j - grid->jlid)*grid->dz;
    344:   r = PetscSqrtReal(x*x+z*z); st = z/r;  ct = x/r;  th = PetscAtanReal(z/x);
    345:   return st*(c*th*st+d*(st+th*ct)) - ct*(c*(st-th*ct)+d*th*st);
    346: }

    348: /*---------------------------------------------------------------------*/
    349: /*  isoviscous analytic solution for IC */
    350: static inline PetscScalar Pressure(PetscInt i, PetscInt j, AppCtx *user)
    351: /*---------------------------------------------------------------------*/
    352: {
    353:   Parameter   *param = user->param;
    354:   GridInfo    *grid  = user->grid;
    355:   PetscScalar x, z, r, st, ct, c=param->c, d=param->d;

    357:   x = (i - grid->jlid - 0.5)*grid->dx;  z = (j - grid->jlid - 0.5)*grid->dz;
    358:   r = PetscSqrtReal(x*x+z*z);  st = z/r;  ct = x/r;
    359:   return (-2.0*(c*ct-d*st)/r);
    360: }

    362: /*  computes the second invariant of the strain rate tensor */
    363: static inline PetscScalar CalcSecInv(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    364: /*---------------------------------------------------------------------*/
    365: {
    366:   Parameter   *param = user->param;
    367:   GridInfo    *grid  = user->grid;
    368:   PetscInt    ilim   =grid->ni-1, jlim=grid->nj-1;
    369:   PetscScalar uN,uS,uE,uW,wN,wS,wE,wW;
    370:   PetscScalar eps11, eps12, eps22;

    372:   if (i<j) return EPS_ZERO;
    373:   if (i==ilim) i--;
    374:   if (j==jlim) j--;

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

    379:     uE = x[j][i].u; uW = x[j][i-1].u;
    380:     wN = x[j][i].w; wS = x[j-1][i].w;
    381:     wE = WInterp(x,i,j-1);
    382:     if (i==j) {
    383:       uN = param->cb; wW = param->sb;
    384:     } else {
    385:       uN = UInterp(x,i-1,j); wW = WInterp(x,i-1,j-1);
    386:     }

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

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

    394:     uN = x[j+1][i].u;  uS = x[j][i].u;
    395:     wE = x[j][i+1].w;  wW = x[j][i].w;
    396:     if (i==j) {
    397:       wN = param->sb;
    398:       uW = param->cb;
    399:     } else {
    400:       wN = WInterp(x,i,j);
    401:       uW = UInterp(x,i-1,j);
    402:     }

    404:     if (j==grid->jlid) {
    405:       uE = 0.0;  uW = 0.0;
    406:       uS = -uN;
    407:       wS = -wN;
    408:     } else {
    409:       uE = UInterp(x,i,j);
    410:       wS = WInterp(x,i,j-1);
    411:     }
    412:   }

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

    417:   return PetscSqrtReal(0.5*(eps11*eps11 + 2.0*eps12*eps12 + eps22*eps22));
    418: }

    420: /*---------------------------------------------------------------------*/
    421: /*  computes the shear viscosity */
    422: static inline PetscScalar Viscosity(PetscScalar T, PetscScalar eps, PetscScalar z, Parameter *param)
    423: /*---------------------------------------------------------------------*/
    424: {
    425:   PetscReal   result   =0.0;
    426:   ViscParam   difn     =param->diffusion, disl=param->dislocation;
    427:   PetscInt    iVisc    =param->ivisc;
    428:   PetscScalar eps_scale=param->V/(param->L*1000.0);
    429:   PetscScalar strain_power, v1, v2, P;
    430:   PetscScalar rho_g = 32340.0, R=8.3144;

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

    434:   if (iVisc==VISC_CONST) {
    435:     /* constant viscosity */
    436:     return 1.0;
    437:   } else if (iVisc==VISC_DIFN) {
    438:     /* diffusion creep rheology */
    439:     result = PetscRealPart((difn.A*PetscExpScalar((difn.Estar + P*difn.Vstar)/R/(T+273.0))/param->eta0));
    440:   } else if (iVisc==VISC_DISL) {
    441:     /* dislocation creep rheology */
    442:     strain_power = PetscPowScalar(eps*eps_scale, (1.0-disl.n)/disl.n);

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

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

    452:     result = PetscRealPart(1.0/(1.0/v1 + 1.0/v2));
    453:   }

    455:   /* max viscosity is param->eta0 */
    456:   result = PetscMin(result, 1.0);
    457:   /* min viscosity is param->visc_cutoff */
    458:   result = PetscMax(result, param->visc_cutoff);
    459:   /* continuation method */
    460:   result = PetscPowReal(result,param->continuation);
    461:   return result;
    462: }

    464: /*---------------------------------------------------------------------*/
    465: /*  computes the residual of the x-component of eqn (1) above */
    466: static inline PetscScalar XMomentumResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    467: /*---------------------------------------------------------------------*/
    468: {
    469:   Parameter   *param=user->param;
    470:   GridInfo    *grid =user->grid;
    471:   PetscScalar dx    = grid->dx, dz=grid->dz;
    472:   PetscScalar etaN,etaS,etaE,etaW,epsN=0.0,epsS=0.0,epsE=0.0,epsW=0.0;
    473:   PetscScalar TE=0.0,TN=0.0,TS=0.0,TW=0.0, dPdx, residual, z_scale;
    474:   PetscScalar dudxW,dudxE,dudzN,dudzS,dwdxN,dwdxS;
    475:   PetscInt    jlim = grid->nj-1;

    477:   z_scale = param->z_scale;

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

    497:   dPdx = (x[j][i+1].p - x[j][i].p)/dx;
    498:   if (j==jlim) dudzN = etaN * (x[j][i].w   - x[j][i+1].w)/dx;
    499:   else         dudzN = etaN * (x[j+1][i].u - x[j][i].u)  /dz;
    500:   dudzS = etaS * (x[j][i].u    - x[j-1][i].u)/dz;
    501:   dudxE = etaE * (x[j][i+1].u  - x[j][i].u)  /dx;
    502:   dudxW = etaW * (x[j][i].u    - x[j][i-1].u)/dx;

    504:   residual = -dPdx                          /* X-MOMENTUM EQUATION*/
    505:              +(dudxE - dudxW)/dx
    506:              +(dudzN - dudzS)/dz;

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

    512:     residual += (dudxE - dudxW)/dx + (dwdxN - dwdxS)/dz;
    513:   }

    515:   return residual;
    516: }

    518: /*---------------------------------------------------------------------*/
    519: /*  computes the residual of the z-component of eqn (1) above */
    520: static inline PetscScalar ZMomentumResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    521: /*---------------------------------------------------------------------*/
    522: {
    523:   Parameter   *param=user->param;
    524:   GridInfo    *grid =user->grid;
    525:   PetscScalar dx    = grid->dx, dz=grid->dz;
    526:   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;
    527:   PetscScalar TE    =0.0,TN=0.0,TS=0.0,TW=0.0, dPdz, residual,z_scale;
    528:   PetscScalar dudzE,dudzW,dwdxW,dwdxE,dwdzN,dwdzS;
    529:   PetscInt    ilim = grid->ni-1;

    531:   /* geometric and other parameters */
    532:   z_scale = param->z_scale;

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

    553:   dPdz  = (x[j+1][i].p - x[j][i].p)/dz;
    554:   dwdzN = etaN * (x[j+1][i].w - x[j][i].w)/dz;
    555:   dwdzS = etaS * (x[j][i].w - x[j-1][i].w)/dz;
    556:   if (i==ilim) dwdxE = etaE * (x[j][i].u   - x[j+1][i].u)/dz;
    557:   else         dwdxE = etaE * (x[j][i+1].w - x[j][i].w)  /dx;
    558:   dwdxW = 2.0*etaW * (x[j][i].w - x[j][i-1].w)/dx;

    560:   /* Z-MOMENTUM */
    561:   residual = -dPdz                 /* constant viscosity terms */
    562:              +(dwdzN - dwdzS)/dz
    563:              +(dwdxE - dwdxW)/dx;

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

    569:     residual += (dwdzN - dwdzS)/dz + (dudzE - dudzW)/dx;
    570:   }

    572:   return residual;
    573: }

    575: /*---------------------------------------------------------------------*/
    576: /*  computes the residual of eqn (2) above */
    577: static inline PetscScalar ContinuityResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    578: /*---------------------------------------------------------------------*/
    579: {
    580:   GridInfo    *grid =user->grid;
    581:   PetscScalar uE,uW,wN,wS,dudx,dwdz;

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

    586:   return dudx + dwdz;
    587: }

    589: /*---------------------------------------------------------------------*/
    590: /*  computes the residual of eqn (3) above */
    591: static inline PetscScalar EnergyResidual(Field **x, PetscInt i, PetscInt j, AppCtx *user)
    592: /*---------------------------------------------------------------------*/
    593: {
    594:   Parameter   *param=user->param;
    595:   GridInfo    *grid =user->grid;
    596:   PetscScalar dx    = grid->dx, dz=grid->dz;
    597:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, jlid=grid->jlid;
    598:   PetscScalar TE, TN, TS, TW, residual;
    599:   PetscScalar uE,uW,wN,wS;
    600:   PetscScalar fN,fS,fE,fW,dTdxW,dTdxE,dTdzN,dTdzS;

    602:   dTdzN = (x[j+1][i].T - x[j][i].T)  /dz;
    603:   dTdzS = (x[j][i].T   - x[j-1][i].T)/dz;
    604:   dTdxE = (x[j][i+1].T - x[j][i].T)  /dx;
    605:   dTdxW = (x[j][i].T   - x[j][i-1].T)/dx;

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

    610:   if (j<=jlid && i>=j) {
    611:     /* don't advect in the lid */
    612:     return residual;
    613:   } else if (i<j) {
    614:     /* beneath the slab sfc */
    615:     uW = uE = param->cb;
    616:     wS = wN = param->sb;
    617:   } else {
    618:     /* advect in the slab and wedge */
    619:     uW = x[j][i-1].u; uE = x[j][i].u;
    620:     wS = x[j-1][i].w; wN = x[j][i].w;
    621:   }

    623:   if (param->adv_scheme==ADVECT_FV || i==ilim-1 || j==jlim-1 || i==1 || j==1) {
    624:     /* finite volume advection */
    625:     TS = (x[j][i].T + x[j-1][i].T)/2.0;
    626:     TN = (x[j][i].T + x[j+1][i].T)/2.0;
    627:     TE = (x[j][i].T + x[j][i+1].T)/2.0;
    628:     TW = (x[j][i].T + x[j][i-1].T)/2.0;
    629:     fN = wN*TN*dx; fS = wS*TS*dx;
    630:     fE = uE*TE*dz; fW = uW*TW*dz;

    632:   } else {
    633:     /* Fromm advection scheme */
    634:     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
    635:               - 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;
    636:     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
    637:               - 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;
    638:     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
    639:               - 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;
    640:     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
    641:               - 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;
    642:   }

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

    646:   return residual;
    647: }

    649: /*---------------------------------------------------------------------*/
    650: /*  computes the shear stress---used on the boundaries */
    651: static inline PetscScalar ShearStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    652: /*---------------------------------------------------------------------*/
    653: {
    654:   Parameter   *param=user->param;
    655:   GridInfo    *grid =user->grid;
    656:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1;
    657:   PetscScalar uN, uS, wE, wW;

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

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

    663:     wE = WInterp(x,i,j-1);
    664:     if (i==j) {
    665:       wW = param->sb;
    666:       uN = param->cb;
    667:     } else {
    668:       wW = WInterp(x,i-1,j-1);
    669:       uN = UInterp(x,i-1,j);
    670:     }
    671:     if (j==grid->jlid+1) uS = 0.0;
    672:     else                 uS = UInterp(x,i-1,j-1);

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

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

    679:   }

    681:   return (uN-uS)/grid->dz + (wE-wW)/grid->dx;
    682: }

    684: /*---------------------------------------------------------------------*/
    685: /*  computes the normal stress---used on the boundaries */
    686: static inline PetscScalar XNormalStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    687: /*---------------------------------------------------------------------*/
    688: {
    689:   Parameter   *param=user->param;
    690:   GridInfo    *grid =user->grid;
    691:   PetscScalar dx    = grid->dx, dz=grid->dz;
    692:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, ivisc;
    693:   PetscScalar epsC  =0.0, etaC, TC, uE, uW, pC, z_scale;
    694:   if (i<j || j<=grid->jlid) return EPS_ZERO;

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

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

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

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

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

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

    714:     if (i==j) uW = param->sb;
    715:     else      uW = UInterp(x,i-1,j);
    716:     uE = UInterp(x,i,j); pC = PInterp(x,i,j);
    717:   }

    719:   return 2.0*etaC*(uE-uW)/dx - pC;
    720: }

    722: /*---------------------------------------------------------------------*/
    723: /*  computes the normal stress---used on the boundaries */
    724: static inline PetscScalar ZNormalStress(Field **x, PetscInt i, PetscInt j, PetscInt ipos, AppCtx *user)
    725: /*---------------------------------------------------------------------*/
    726: {
    727:   Parameter   *param=user->param;
    728:   GridInfo    *grid =user->grid;
    729:   PetscScalar dz    =grid->dz;
    730:   PetscInt    ilim  =grid->ni-1, jlim=grid->nj-1, ivisc;
    731:   PetscScalar epsC  =0.0, etaC, TC;
    732:   PetscScalar pC, wN, wS, z_scale;
    733:   if (i<j || j<=grid->jlid) return EPS_ZERO;

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

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

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

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

    747:     TC = param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*z_scale);
    748:     if (ivisc>=VISC_DISL) epsC = CalcSecInv(x,i,j,CELL_CORNER,user);
    749:     etaC = Viscosity(TC,epsC,dz*(j+0.5),param);
    750:     if (i==j) wN = param->sb;
    751:     else      wN = WInterp(x,i,j);
    752:     wS = WInterp(x,i,j-1); pC = PInterp(x,i,j);
    753:   }

    755:   return 2.0*etaC*(wN-wS)/dz - pC;
    756: }

    758: /*---------------------------------------------------------------------*/

    760: /*=====================================================================
    761:   INITIALIZATION, POST-PROCESSING AND OUTPUT FUNCTIONS
    762:   =====================================================================*/

    764: /*---------------------------------------------------------------------*/
    765: /* initializes the problem parameters and checks for
    766:    command line changes */
    767: PetscErrorCode SetParams(Parameter *param, GridInfo *grid)
    768: /*---------------------------------------------------------------------*/
    769: {
    770:   PetscReal SEC_PER_YR                     = 3600.00*24.00*365.2500;
    771:   PetscReal alpha_g_on_cp_units_inverse_km = 4.0e-5*9.8;

    773:   /* domain geometry */
    774:   param->slab_dip    = 45.0;
    775:   param->width       = 320.0;                                              /* km */
    776:   param->depth       = 300.0;                                              /* km */
    777:   param->lid_depth   = 35.0;                                               /* km */
    778:   param->fault_depth = 35.0;                                               /* km */

    780:   PetscOptionsGetReal(NULL,NULL,"-slab_dip",&(param->slab_dip),NULL);
    781:   PetscOptionsGetReal(NULL,NULL,"-width",&(param->width),NULL);
    782:   PetscOptionsGetReal(NULL,NULL,"-depth",&(param->depth),NULL);
    783:   PetscOptionsGetReal(NULL,NULL,"-lid_depth",&(param->lid_depth),NULL);
    784:   PetscOptionsGetReal(NULL,NULL,"-fault_depth",&(param->fault_depth),NULL);

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

    788:   /* grid information */
    789:   PetscOptionsGetInt(NULL,NULL, "-jfault",&(grid->jfault),NULL);
    790:   grid->ni = 82;
    791:   PetscOptionsGetInt(NULL,NULL, "-ni",&(grid->ni),NULL);

    793:   grid->dx            = param->width/((PetscReal)(grid->ni-2));               /* km */
    794:   grid->dz            = grid->dx*PetscTanReal(param->slab_dip);               /* km */
    795:   grid->nj            = (PetscInt)(param->depth/grid->dz + 3.0);         /* gridpoints*/
    796:   param->depth        = grid->dz*(grid->nj-2);                             /* km */
    797:   grid->inose         = 0;                                          /* gridpoints*/
    798:   PetscOptionsGetInt(NULL,NULL,"-inose",&(grid->inose),NULL);
    799:   grid->bx            = DM_BOUNDARY_NONE;
    800:   grid->by            = DM_BOUNDARY_NONE;
    801:   grid->stencil       = DMDA_STENCIL_BOX;
    802:   grid->dof           = 4;
    803:   grid->stencil_width = 2;
    804:   grid->mglevels      = 1;

    806:   /* boundary conditions */
    807:   param->pv_analytic = PETSC_FALSE;
    808:   param->ibound      = BC_NOSTRESS;
    809:   PetscOptionsGetInt(NULL,NULL,"-ibound",&(param->ibound),NULL);

    811:   /* physical constants */
    812:   param->slab_velocity = 5.0;               /* cm/yr */
    813:   param->slab_age      = 50.0;              /* Ma */
    814:   param->lid_age       = 50.0;              /* Ma */
    815:   param->kappa         = 0.7272e-6;         /* m^2/sec */
    816:   param->potentialT    = 1300.0;            /* degrees C */

    818:   PetscOptionsGetReal(NULL,NULL,"-slab_velocity",&(param->slab_velocity),NULL);
    819:   PetscOptionsGetReal(NULL,NULL,"-slab_age",&(param->slab_age),NULL);
    820:   PetscOptionsGetReal(NULL,NULL,"-lid_age",&(param->lid_age),NULL);
    821:   PetscOptionsGetReal(NULL,NULL,"-kappa",&(param->kappa),NULL);
    822:   PetscOptionsGetReal(NULL,NULL,"-potentialT",&(param->potentialT),NULL);

    824:   /* viscosity */
    825:   param->ivisc        = 3;                  /* 0=isovisc, 1=difn creep, 2=disl creep, 3=full */
    826:   param->eta0         = 1e24;               /* Pa-s */
    827:   param->visc_cutoff  = 0.0;                /* factor of eta_0 */
    828:   param->continuation = 1.0;

    830:   /* constants for diffusion creep */
    831:   param->diffusion.A     = 1.8e7;             /* Pa-s */
    832:   param->diffusion.n     = 1.0;               /* dim'less */
    833:   param->diffusion.Estar = 375e3;             /* J/mol */
    834:   param->diffusion.Vstar = 5e-6;              /* m^3/mol */

    836:   /* constants for param->dislocationocation creep */
    837:   param->dislocation.A     = 2.8969e4;        /* Pa-s */
    838:   param->dislocation.n     = 3.5;             /* dim'less */
    839:   param->dislocation.Estar = 530e3;           /* J/mol */
    840:   param->dislocation.Vstar = 14e-6;           /* m^3/mol */

    842:   PetscOptionsGetInt(NULL,NULL, "-ivisc",&(param->ivisc),NULL);
    843:   PetscOptionsGetReal(NULL,NULL,"-visc_cutoff",&(param->visc_cutoff),NULL);

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

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

    850:   /* output options */
    851:   param->quiet      = PETSC_FALSE;
    852:   param->param_test = PETSC_FALSE;

    854:   PetscOptionsHasName(NULL,NULL,"-quiet",&(param->quiet));
    855:   PetscOptionsHasName(NULL,NULL,"-test",&(param->param_test));
    856:   PetscOptionsGetString(NULL,NULL,"-file",param->filename,sizeof(param->filename),&(param->output_to_file));

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

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

    863:   /* misc. flags */
    864:   param->stop_solve    = PETSC_FALSE;
    865:   param->interrupted   = PETSC_FALSE;
    866:   param->kspmon        = PETSC_FALSE;
    867:   param->toggle_kspmon = PETSC_FALSE;

    869:   /* derived parameters for slab angle */
    870:   param->sb = PetscSinReal(param->slab_dip);
    871:   param->cb = PetscCosReal(param->slab_dip);
    872:   param->c  =  param->slab_dip*param->sb/(param->slab_dip*param->slab_dip-param->sb*param->sb);
    873:   param->d  = (param->slab_dip*param->cb-param->sb)/(param->slab_dip*param->slab_dip-param->sb*param->sb);

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

    879:   /* other unit conversions and derived parameters */
    880:   param->scaled_width = param->width/param->L;                  /* dim'less */
    881:   param->scaled_depth = param->depth/param->L;                  /* dim'less */
    882:   param->lid_depth    = param->lid_depth/param->L;              /* dim'less */
    883:   param->fault_depth  = param->fault_depth/param->L;            /* dim'less */
    884:   grid->dx            = grid->dx/param->L;                      /* dim'less */
    885:   grid->dz            = grid->dz/param->L;                      /* dim'less */
    886:   grid->jlid          = (PetscInt)(param->lid_depth/grid->dz);       /* gridcells */
    887:   grid->jfault        = (PetscInt)(param->fault_depth/grid->dz);     /* gridcells */
    888:   param->lid_depth    = grid->jlid*grid->dz;                    /* dim'less */
    889:   param->fault_depth  = grid->jfault*grid->dz;                  /* dim'less */
    890:   grid->corner        = grid->jlid+1;                           /* gridcells */
    891:   param->peclet       = param->V                                /* m/sec */
    892:                         * param->L*1000.0                       /* m */
    893:                         / param->kappa;                         /* m^2/sec */
    894:   param->z_scale = param->L * alpha_g_on_cp_units_inverse_km;
    895:   param->skt     = PetscSqrtReal(param->kappa*param->slab_age*SEC_PER_YR);
    896:   PetscOptionsGetReal(NULL,NULL,"-peclet",&(param->peclet),NULL);

    898:   return 0;
    899: }

    901: /*---------------------------------------------------------------------*/
    902: /*  prints a report of the problem parameters to stdout */
    903: PetscErrorCode ReportParams(Parameter *param, GridInfo *grid)
    904: /*---------------------------------------------------------------------*/
    905: {
    906:   char           date[30];

    908:   PetscGetDate(date,30);

    910:   if (!(param->quiet)) {
    911:     PetscPrintf(PETSC_COMM_WORLD,"---------------------BEGIN ex30 PARAM REPORT-------------------\n");
    912:     PetscPrintf(PETSC_COMM_WORLD,"Domain: \n");
    913:     PetscPrintf(PETSC_COMM_WORLD,"  Width = %g km,         Depth = %g km\n",(double)param->width,(double)param->depth);
    914:     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);
    915:     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));

    917:     PetscPrintf(PETSC_COMM_WORLD,"\nGrid: \n");
    918:     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));
    919:     PetscPrintf(PETSC_COMM_WORLD,"  jlid = %3D              jfault = %3D \n",grid->jlid,grid->jfault);
    920:     PetscPrintf(PETSC_COMM_WORLD,"  Pe = %g\n",(double)param->peclet);

    922:     PetscPrintf(PETSC_COMM_WORLD,"\nRheology:");
    923:     if (param->ivisc==VISC_CONST) {
    924:       PetscPrintf(PETSC_COMM_WORLD,"                 Isoviscous \n");
    925:       if (param->pv_analytic) {
    926:         PetscPrintf(PETSC_COMM_WORLD,"                          Pressure and Velocity prescribed! \n");
    927:       }
    928:     } else if (param->ivisc==VISC_DIFN) {
    929:       PetscPrintf(PETSC_COMM_WORLD,"                 Diffusion Creep (T-Dependent Newtonian) \n");
    930:       PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));
    931:     } else if (param->ivisc==VISC_DISL) {
    932:       PetscPrintf(PETSC_COMM_WORLD,"                 Dislocation Creep (T-Dependent Non-Newtonian) \n");
    933:       PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));
    934:     } else if (param->ivisc==VISC_FULL) {
    935:       PetscPrintf(PETSC_COMM_WORLD,"                 Full Rheology \n");
    936:       PetscPrintf(PETSC_COMM_WORLD,"                          Viscosity range: %g--%g Pa-sec \n",(double)param->eta0,(double)(param->visc_cutoff*param->eta0));
    937:     } else {
    938:       PetscPrintf(PETSC_COMM_WORLD,"                 Invalid! \n");
    939:       return 1;
    940:     }

    942:     PetscPrintf(PETSC_COMM_WORLD,"Boundary condition:");
    943:     if (param->ibound==BC_ANALYTIC) {
    944:       PetscPrintf(PETSC_COMM_WORLD,"       Isoviscous Analytic Dirichlet \n");
    945:     } else if (param->ibound==BC_NOSTRESS) {
    946:       PetscPrintf(PETSC_COMM_WORLD,"       Stress-Free (normal & shear stress)\n");
    947:     } else if (param->ibound==BC_EXPERMNT) {
    948:       PetscPrintf(PETSC_COMM_WORLD,"       Experimental boundary condition \n");
    949:     } else {
    950:       PetscPrintf(PETSC_COMM_WORLD,"       Invalid! \n");
    951:       return 1;
    952:     }

    954:     if (param->output_to_file) {
    955: #if defined(PETSC_HAVE_MATLAB_ENGINE)
    956:       PetscPrintf(PETSC_COMM_WORLD,"Output Destination:       Mat file \"%s\"\n",param->filename);
    957: #else
    958:       PetscPrintf(PETSC_COMM_WORLD,"Output Destination:       PETSc binary file \"%s\"\n",param->filename);
    959: #endif
    960:     }
    961:     if (param->output_ivisc != param->ivisc) {
    962:       PetscPrintf(PETSC_COMM_WORLD,"                          Output viscosity: -ivisc %D\n",param->output_ivisc);
    963:     }

    965:     PetscPrintf(PETSC_COMM_WORLD,"---------------------END ex30 PARAM REPORT---------------------\n");
    966:   }
    967:   if (param->param_test) PetscEnd();
    968:   return 0;
    969: }

    971: /* ------------------------------------------------------------------- */
    972: /*  generates an initial guess using the analytic solution for isoviscous
    973:     corner flow */
    974: PetscErrorCode Initialize(DM da)
    975: /* ------------------------------------------------------------------- */
    976: {
    977:   AppCtx         *user;
    978:   Parameter      *param;
    979:   GridInfo       *grid;
    980:   PetscInt       i,j,is,js,im,jm;
    981:   Field          **x;
    982:   Vec            Xguess;

    984:   /* Get the fine grid */
    985:   DMGetApplicationContext(da,&user);
    986:   Xguess = user->Xguess;
    987:   param  = user->param;
    988:   grid   = user->grid;
    989:   DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);
    990:   DMDAVecGetArray(da,Xguess,(void**)&x);

    992:   /* Compute initial guess */
    993:   for (j=js; j<js+jm; j++) {
    994:     for (i=is; i<is+im; i++) {
    995:       if (i<j)                x[j][i].u = param->cb;
    996:       else if (j<=grid->jlid) x[j][i].u = 0.0;
    997:       else                    x[j][i].u = HorizVelocity(i,j,user);

    999:       if (i<=j)               x[j][i].w = param->sb;
    1000:       else if (j<=grid->jlid) x[j][i].w = 0.0;
    1001:       else                    x[j][i].w = VertVelocity(i,j,user);

    1003:       if (i<j || j<=grid->jlid) x[j][i].p = 0.0;
    1004:       else                      x[j][i].p = Pressure(i,j,user);

    1006:       x[j][i].T = PetscMin(grid->dz*(j-0.5),1.0);
    1007:     }
    1008:   }

    1010:   /* Restore x to Xguess */
    1011:   DMDAVecRestoreArray(da,Xguess,(void**)&x);

    1013:   return 0;
    1014: }

    1016: /*---------------------------------------------------------------------*/
    1017: /*  controls output to a file */
    1018: PetscErrorCode DoOutput(SNES snes, PetscInt its)
    1019: /*---------------------------------------------------------------------*/
    1020: {
    1021:   AppCtx         *user;
    1022:   Parameter      *param;
    1023:   GridInfo       *grid;
    1024:   PetscInt       ivt;
    1025:   PetscMPIInt    rank;
    1026:   PetscViewer    viewer;
    1027:   Vec            res, pars;
    1028:   MPI_Comm       comm;
    1029:   DM             da;

    1031:   SNESGetDM(snes,&da);
    1032:   DMGetApplicationContext(da,&user);
    1033:   param = user->param;
    1034:   grid  = user->grid;
    1035:   ivt   = param->ivisc;

    1037:   param->ivisc = param->output_ivisc;

    1039:   /* compute final residual and final viscosity/strain rate fields */
    1040:   SNESGetFunction(snes, &res, NULL, NULL);
    1041:   ViscosityField(da, user->x, user->Xguess);

    1043:   /* get the communicator and the rank of the processor */
    1044:   PetscObjectGetComm((PetscObject)snes, &comm);
    1045:   MPI_Comm_rank(comm, &rank);

    1047:   if (param->output_to_file) { /* send output to binary file */
    1048:     VecCreate(comm, &pars);
    1049:     if (rank == 0) { /* on processor 0 */
    1050:       VecSetSizes(pars, 20, PETSC_DETERMINE);
    1051:       VecSetFromOptions(pars);
    1052:       VecSetValue(pars,0, (PetscScalar)(grid->ni),INSERT_VALUES);
    1053:       VecSetValue(pars,1, (PetscScalar)(grid->nj),INSERT_VALUES);
    1054:       VecSetValue(pars,2, (PetscScalar)(grid->dx),INSERT_VALUES);
    1055:       VecSetValue(pars,3, (PetscScalar)(grid->dz),INSERT_VALUES);
    1056:       VecSetValue(pars,4, (PetscScalar)(param->L),INSERT_VALUES);
    1057:       VecSetValue(pars,5, (PetscScalar)(param->V),INSERT_VALUES);
    1058:       /* skipped 6 intentionally */
    1059:       VecSetValue(pars,7, (PetscScalar)(param->slab_dip),INSERT_VALUES);
    1060:       VecSetValue(pars,8, (PetscScalar)(grid->jlid),INSERT_VALUES);
    1061:       VecSetValue(pars,9, (PetscScalar)(param->lid_depth),INSERT_VALUES);
    1062:       VecSetValue(pars,10,(PetscScalar)(grid->jfault),INSERT_VALUES);
    1063:       VecSetValue(pars,11,(PetscScalar)(param->fault_depth),INSERT_VALUES);
    1064:       VecSetValue(pars,12,(PetscScalar)(param->potentialT),INSERT_VALUES);
    1065:       VecSetValue(pars,13,(PetscScalar)(param->ivisc),INSERT_VALUES);
    1066:       VecSetValue(pars,14,(PetscScalar)(param->visc_cutoff),INSERT_VALUES);
    1067:       VecSetValue(pars,15,(PetscScalar)(param->ibound),INSERT_VALUES);
    1068:       VecSetValue(pars,16,(PetscScalar)(its),INSERT_VALUES);
    1069:     } else { /* on some other processor */
    1070:       VecSetSizes(pars, 0, PETSC_DETERMINE);
    1071:       VecSetFromOptions(pars);
    1072:     }
    1073:     VecAssemblyBegin(pars)); PetscCall(VecAssemblyEnd(pars);

    1075:     /* create viewer */
    1076: #if defined(PETSC_HAVE_MATLAB_ENGINE)
    1077:     PetscViewerMatlabOpen(PETSC_COMM_WORLD,param->filename,FILE_MODE_WRITE,&viewer);
    1078: #else
    1079:     PetscViewerBinaryOpen(PETSC_COMM_WORLD,param->filename,FILE_MODE_WRITE,&viewer);
    1080: #endif

    1082:     /* send vectors to viewer */
    1083:     PetscObjectSetName((PetscObject)res,"res");
    1084:     VecView(res,viewer);
    1085:     PetscObjectSetName((PetscObject)user->x,"out");
    1086:     VecView(user->x, viewer);
    1087:     PetscObjectSetName((PetscObject)(user->Xguess),"aux");
    1088:     VecView(user->Xguess, viewer);
    1089:     StressField(da); /* compute stress fields */
    1090:     PetscObjectSetName((PetscObject)(user->Xguess),"str");
    1091:     VecView(user->Xguess, viewer);
    1092:     PetscObjectSetName((PetscObject)pars,"par");
    1093:     VecView(pars, viewer);

    1095:     /* destroy viewer and vector */
    1096:     PetscViewerDestroy(&viewer);
    1097:     VecDestroy(&pars);
    1098:   }

    1100:   param->ivisc = ivt;
    1101:   return 0;
    1102: }

    1104: /* ------------------------------------------------------------------- */
    1105: /* Compute both the second invariant of the strain rate tensor and the viscosity, at both cell centers and cell corners */
    1106: PetscErrorCode ViscosityField(DM da, Vec X, Vec V)
    1107: /* ------------------------------------------------------------------- */
    1108: {
    1109:   AppCtx         *user;
    1110:   Parameter      *param;
    1111:   GridInfo       *grid;
    1112:   Vec            localX;
    1113:   Field          **v, **x;
    1114:   PetscReal      eps, /* dx,*/ dz, T, epsC, TC;
    1115:   PetscInt       i,j,is,js,im,jm,ilim,jlim,ivt;

    1118:   DMGetApplicationContext(da,&user);
    1119:   param        = user->param;
    1120:   grid         = user->grid;
    1121:   ivt          = param->ivisc;
    1122:   param->ivisc = param->output_ivisc;

    1124:   DMGetLocalVector(da, &localX);
    1125:   DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX);
    1126:   DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX);
    1127:   DMDAVecGetArray(da,localX,(void**)&x);
    1128:   DMDAVecGetArray(da,V,(void**)&v);

    1130:   /* Parameters */
    1131:   /* dx = grid->dx; */ dz = grid->dz;

    1133:   ilim = grid->ni-1; jlim = grid->nj-1;

    1135:   /* Compute real temperature, strain rate and viscosity */
    1136:   DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);
    1137:   for (j=js; j<js+jm; j++) {
    1138:     for (i=is; i<is+im; i++) {
    1139:       T = PetscRealPart(param->potentialT * x[j][i].T * PetscExpScalar((j-0.5)*dz*param->z_scale));
    1140:       if (i<ilim && j<jlim) {
    1141:         TC = PetscRealPart(param->potentialT * TInterp(x,i,j) * PetscExpScalar(j*dz*param->z_scale));
    1142:       } else {
    1143:         TC = T;
    1144:       }
    1145:       eps  = PetscRealPart((CalcSecInv(x,i,j,CELL_CENTER,user)));
    1146:       epsC = PetscRealPart(CalcSecInv(x,i,j,CELL_CORNER,user));

    1148:       v[j][i].u = eps;
    1149:       v[j][i].w = epsC;
    1150:       v[j][i].p = Viscosity(T,eps,dz*(j-0.5),param);
    1151:       v[j][i].T = Viscosity(TC,epsC,dz*j,param);
    1152:     }
    1153:   }
    1154:   DMDAVecRestoreArray(da,V,(void**)&v);
    1155:   DMDAVecRestoreArray(da,localX,(void**)&x);
    1156:   DMRestoreLocalVector(da, &localX);

    1158:   param->ivisc = ivt;
    1159:   return 0;
    1160: }

    1162: /* ------------------------------------------------------------------- */
    1163: /* post-processing: compute stress everywhere */
    1164: PetscErrorCode StressField(DM da)
    1165: /* ------------------------------------------------------------------- */
    1166: {
    1167:   AppCtx         *user;
    1168:   PetscInt       i,j,is,js,im,jm;
    1169:   Vec            locVec;
    1170:   Field          **x, **y;

    1172:   DMGetApplicationContext(da,&user);

    1174:   /* Get the fine grid of Xguess and X */
    1175:   DMDAGetCorners(da,&is,&js,NULL,&im,&jm,NULL);
    1176:   DMDAVecGetArray(da,user->Xguess,(void**)&x);

    1178:   DMGetLocalVector(da, &locVec);
    1179:   DMGlobalToLocalBegin(da, user->x, INSERT_VALUES, locVec);
    1180:   DMGlobalToLocalEnd(da, user->x, INSERT_VALUES, locVec);
    1181:   DMDAVecGetArray(da,locVec,(void**)&y);

    1183:   /* Compute stress on the corner points */
    1184:   for (j=js; j<js+jm; j++) {
    1185:     for (i=is; i<is+im; i++) {
    1186:       x[j][i].u = ShearStress(y,i,j,CELL_CENTER,user);
    1187:       x[j][i].w = ShearStress(y,i,j,CELL_CORNER,user);
    1188:       x[j][i].p = XNormalStress(y,i,j,CELL_CENTER,user);
    1189:       x[j][i].T = ZNormalStress(y,i,j,CELL_CENTER,user);
    1190:     }
    1191:   }

    1193:   /* Restore the fine grid of Xguess and X */
    1194:   DMDAVecRestoreArray(da,user->Xguess,(void**)&x);
    1195:   DMDAVecRestoreArray(da,locVec,(void**)&y);
    1196:   DMRestoreLocalVector(da, &locVec);
    1197:   return 0;
    1198: }

    1200: /*=====================================================================
    1201:   UTILITY FUNCTIONS
    1202:   =====================================================================*/

    1204: /*---------------------------------------------------------------------*/
    1205: /* returns the velocity of the subducting slab and handles fault nodes
    1206:    for BC */
    1207: static inline PetscScalar SlabVel(char c, PetscInt i, PetscInt j, AppCtx *user)
    1208: /*---------------------------------------------------------------------*/
    1209: {
    1210:   Parameter *param = user->param;
    1211:   GridInfo  *grid  = user->grid;

    1213:   if (c=='U' || c=='u') {
    1214:     if (i<j-1) return param->cb;
    1215:     else if (j<=grid->jfault) return 0.0;
    1216:     else return param->cb;

    1218:   } else {
    1219:     if (i<j) return param->sb;
    1220:     else if (j<=grid->jfault) return 0.0;
    1221:     else return param->sb;
    1222:   }
    1223: }

    1225: /*---------------------------------------------------------------------*/
    1226: /*  solution to diffusive half-space cooling model for BC */
    1227: static inline PetscScalar PlateModel(PetscInt j, PetscInt plate, AppCtx *user)
    1228: /*---------------------------------------------------------------------*/
    1229: {
    1230:   Parameter     *param = user->param;
    1231:   PetscScalar   z;
    1232:   if (plate==PLATE_LID) z = (j-0.5)*user->grid->dz;
    1233:   else z = (j-0.5)*user->grid->dz*param->cb;  /* PLATE_SLAB */
    1234: #if defined(PETSC_HAVE_ERF)
    1235:   return (PetscReal)(erf((double)PetscRealPart(z*param->L/2.0/param->skt)));
    1236: #else
    1237:   (*PetscErrorPrintf)("erf() not available on this machine\n");
    1238:   MPI_Abort(PETSC_COMM_SELF,1);
    1239: #endif
    1240: }

    1242: /*=====================================================================
    1243:   INTERACTIVE SIGNAL HANDLING
    1244:   =====================================================================*/

    1246: /* ------------------------------------------------------------------- */
    1247: PetscErrorCode SNESConverged_Interactive(SNES snes, PetscInt it,PetscReal xnorm, PetscReal snorm, PetscReal fnorm, SNESConvergedReason *reason, void *ctx)
    1248: /* ------------------------------------------------------------------- */
    1249: {
    1250:   AppCtx         *user  = (AppCtx*) ctx;
    1251:   Parameter      *param = user->param;
    1252:   KSP            ksp;

    1255:   if (param->interrupted) {
    1256:     param->interrupted = PETSC_FALSE;
    1257:     PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: exiting SNES solve. \n");
    1258:     *reason = SNES_CONVERGED_FNORM_ABS;
    1259:     return 0;
    1260:   } else if (param->toggle_kspmon) {
    1261:     param->toggle_kspmon = PETSC_FALSE;

    1263:     SNESGetKSP(snes, &ksp);

    1265:     if (param->kspmon) {
    1266:       KSPMonitorCancel(ksp);

    1268:       param->kspmon = PETSC_FALSE;
    1269:       PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: deactivating ksp singular value monitor. \n");
    1270:     } else {
    1271:       PetscViewerAndFormat *vf;
    1272:       PetscViewerAndFormatCreate(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_DEFAULT,&vf);
    1273:       KSPMonitorSet(ksp,(PetscErrorCode (*)(KSP,PetscInt,PetscReal,void*))KSPMonitorSingularValue,vf,(PetscErrorCode (*)(void**))PetscViewerAndFormatDestroy);

    1275:       param->kspmon = PETSC_TRUE;
    1276:       PetscPrintf(PETSC_COMM_WORLD,"USER SIGNAL: activating ksp singular value monitor. \n");
    1277:     }
    1278:   }
    1279:   PetscFunctionReturn(SNESConvergedDefault(snes,it,xnorm,snorm,fnorm,reason,ctx));
    1280: }

    1282: /* ------------------------------------------------------------------- */
    1283: #include <signal.h>
    1284: PetscErrorCode InteractiveHandler(int signum, void *ctx)
    1285: /* ------------------------------------------------------------------- */
    1286: {
    1287:   AppCtx    *user  = (AppCtx*) ctx;
    1288:   Parameter *param = user->param;

    1290:   if (signum == SIGILL) {
    1291:     param->toggle_kspmon = PETSC_TRUE;
    1292: #if !defined(PETSC_MISSING_SIGCONT)
    1293:   } else if (signum == SIGCONT) {
    1294:     param->interrupted = PETSC_TRUE;
    1295: #endif
    1296: #if !defined(PETSC_MISSING_SIGURG)
    1297:   } else if (signum == SIGURG) {
    1298:     param->stop_solve = PETSC_TRUE;
    1299: #endif
    1300:   }
    1301:   return 0;
    1302: }

    1304: /*---------------------------------------------------------------------*/
    1305: /*  main call-back function that computes the processor-local piece
    1306:     of the residual */
    1307: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,Field **x,Field **f,void *ptr)
    1308: /*---------------------------------------------------------------------*/
    1309: {
    1310:   AppCtx      *user  = (AppCtx*)ptr;
    1311:   Parameter   *param = user->param;
    1312:   GridInfo    *grid  = user->grid;
    1313:   PetscScalar mag_w, mag_u;
    1314:   PetscInt    i,j,mx,mz,ilim,jlim;
    1315:   PetscInt    is,ie,js,je,ibound;    /* ,ivisc */

    1318:   /* Define global and local grid parameters */
    1319:   mx   = info->mx;     mz   = info->my;
    1320:   ilim = mx-1;         jlim = mz-1;
    1321:   is   = info->xs;     ie   = info->xs+info->xm;
    1322:   js   = info->ys;     je   = info->ys+info->ym;

    1324:   /* Define geometric and numeric parameters */
    1325:   /* ivisc = param->ivisc; */ ibound = param->ibound;

    1327:   for (j=js; j<je; j++) {
    1328:     for (i=is; i<ie; i++) {

    1330:       /************* X-MOMENTUM/VELOCITY *************/
    1331:       if (i<j) f[j][i].u = x[j][i].u - SlabVel('U',i,j,user);
    1332:       else if (j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
    1333:         /* in the lithospheric lid */
    1334:         f[j][i].u = x[j][i].u - 0.0;
    1335:       } else if (i==ilim) {
    1336:         /* on the right side boundary */
    1337:         if (ibound==BC_ANALYTIC) {
    1338:           f[j][i].u = x[j][i].u - HorizVelocity(i,j,user);
    1339:         } else {
    1340:           f[j][i].u = XNormalStress(x,i,j,CELL_CENTER,user) - EPS_ZERO;
    1341:         }

    1343:       } else if (j==jlim) {
    1344:         /* on the bottom boundary */
    1345:         if (ibound==BC_ANALYTIC) {
    1346:           f[j][i].u = x[j][i].u - HorizVelocity(i,j,user);
    1347:         } else if (ibound==BC_NOSTRESS) {
    1348:           f[j][i].u = XMomentumResidual(x,i,j,user);
    1349:         } else {
    1350:           /* experimental boundary condition */
    1351:         }

    1353:       } else {
    1354:         /* in the mantle wedge */
    1355:         f[j][i].u = XMomentumResidual(x,i,j,user);
    1356:       }

    1358:       /************* Z-MOMENTUM/VELOCITY *************/
    1359:       if (i<=j) {
    1360:         f[j][i].w = x[j][i].w - SlabVel('W',i,j,user);

    1362:       } else if (j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
    1363:         /* in the lithospheric lid */
    1364:         f[j][i].w = x[j][i].w - 0.0;

    1366:       } else if (j==jlim) {
    1367:         /* on the bottom boundary */
    1368:         if (ibound==BC_ANALYTIC) {
    1369:           f[j][i].w = x[j][i].w - VertVelocity(i,j,user);
    1370:         } else {
    1371:           f[j][i].w = ZNormalStress(x,i,j,CELL_CENTER,user) - EPS_ZERO;
    1372:         }

    1374:       } else if (i==ilim) {
    1375:         /* on the right side boundary */
    1376:         if (ibound==BC_ANALYTIC) {
    1377:           f[j][i].w = x[j][i].w - VertVelocity(i,j,user);
    1378:         } else if (ibound==BC_NOSTRESS) {
    1379:           f[j][i].w = ZMomentumResidual(x,i,j,user);
    1380:         } else {
    1381:           /* experimental boundary condition */
    1382:         }

    1384:       } else {
    1385:         /* in the mantle wedge */
    1386:         f[j][i].w =  ZMomentumResidual(x,i,j,user);
    1387:       }

    1389:       /************* CONTINUITY/PRESSURE *************/
    1390:       if (i<j || j<=grid->jlid || (j<grid->corner+grid->inose && i<grid->corner+grid->inose)) {
    1391:         /* in the lid or slab */
    1392:         f[j][i].p = x[j][i].p;

    1394:       } else if ((i==ilim || j==jlim) && ibound==BC_ANALYTIC) {
    1395:         /* on an analytic boundary */
    1396:         f[j][i].p = x[j][i].p - Pressure(i,j,user);

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

    1403:       /************* TEMPERATURE *************/
    1404:       if (j==0) {
    1405:         /* on the surface */
    1406:         f[j][i].T = x[j][i].T + x[j+1][i].T + PetscMax(PetscRealPart(x[j][i].T),0.0);

    1408:       } else if (i==0) {
    1409:         /* slab inflow boundary */
    1410:         f[j][i].T = x[j][i].T - PlateModel(j,PLATE_SLAB,user);

    1412:       } else if (i==ilim) {
    1413:         /* right side boundary */
    1414:         mag_u = 1.0 - PetscPowRealInt((1.0-PetscMax(PetscMin(PetscRealPart(x[j][i-1].u)/param->cb,1.0),0.0)), 5);
    1415:         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);

    1417:       } else if (j==jlim) {
    1418:         /* bottom boundary */
    1419:         mag_w = 1.0 - PetscPowRealInt((1.0-PetscMax(PetscMin(PetscRealPart(x[j-1][i].w)/param->sb,1.0),0.0)), 5);
    1420:         f[j][i].T = x[j][i].T - mag_w*x[j-1][i-1].T - (1.0-mag_w);

    1422:       } else {
    1423:         /* in the mantle wedge */
    1424:         f[j][i].T = EnergyResidual(x,i,j,user);
    1425:       }
    1426:     }
    1427:   }
    1428:   return 0;
    1429: }

    1431: /*TEST

    1433:    build:
    1434:       requires: !complex erf

    1436:    test:
    1437:       args: -ni 18
    1438:       filter: grep -v Destination
    1439:       requires: !single

    1441: TEST*/