Date: 2022-03-04T17:39:37+00:00

Actual source code: ex14.c

      1: static const char help[] = "Toy hydrostatic ice flow with multigrid in 3D.\n\
      2: \n\
      3: Solves the hydrostatic (aka Blatter/Pattyn/First Order) equations for ice sheet flow\n\
      4: using multigrid.  The ice uses a power-law rheology with \"Glen\" exponent 3 (corresponds\n\
      5: to p=4/3 in a p-Laplacian).  The focus is on ISMIP-HOM experiments which assume periodic\n\
      6: boundary conditions in the x- and y-directions.\n\
      7: \n\
      8: Equations are rescaled so that the domain size and solution are O(1), details of this scaling\n\
      9: can be controlled by the options -units_meter, -units_second, and -units_kilogram.\n\
     10: \n\
     11: A VTK StructuredGrid output file can be written using the option -o filename.vts\n\
     12: \n\n";

     14: /*
     15: The equations for horizontal velocity (u,v) are

     17:   - [eta (4 u_x + 2 v_y)]_x - [eta (u_y + v_x)]_y - [eta u_z]_z + rho g s_x = 0
     18:   - [eta (4 v_y + 2 u_x)]_y - [eta (u_y + v_x)]_x - [eta v_z]_z + rho g s_y = 0

     20: where

     22:   eta = B/2 (epsilon + gamma)^((p-2)/2)

     24: is the nonlinear effective viscosity with regularization epsilon and hardness parameter B,
     25: written in terms of the second invariant

     27:   gamma = u_x^2 + v_y^2 + u_x v_y + (1/4) (u_y + v_x)^2 + (1/4) u_z^2 + (1/4) v_z^2

     29: The surface boundary conditions are the natural conditions.  The basal boundary conditions
     30: are either no-slip, or Navier (linear) slip with spatially variant friction coefficient beta^2.

     32: In the code, the equations for (u,v) are multiplied through by 1/(rho g) so that residuals are O(1).

     34: The discretization is Q1 finite elements, managed by a DMDA.  The grid is never distorted in the
     35: map (x,y) plane, but the bed and surface may be bumpy.  This is handled as usual in FEM, through
     36: the Jacobian of the coordinate transformation from a reference element to the physical element.

     38: Since ice-flow is tightly coupled in the z-direction (within columns), the DMDA is managed
     39: specially so that columns are never distributed, and are always contiguous in memory.
     40: This amounts to reversing the meaning of X,Y,Z compared to the DMDA's internal interpretation,
     41: and then indexing as vec[i][j][k].  The exotic coarse spaces require 2D DMDAs which are made to
     42: use compatible domain decomposition relative to the 3D DMDAs.

     44: */

     46: #include <petscts.h>
     47: #include <petscdm.h>
     48: #include <petscdmda.h>
     49: #include <petscdmcomposite.h>
     50: #include <ctype.h>              /* toupper() */
     51: #include <petsc/private/petscimpl.h>

     53: #if defined __SSE2__
     54: #  include <emmintrin.h>
     55: #endif

     57: /* The SSE2 kernels are only for PetscScalar=double on architectures that support it */
     58: #define USE_SSE2_KERNELS (!defined NO_SSE2                              \
     59:                           && !defined PETSC_USE_COMPLEX                 \
     60:                           && !defined PETSC_USE_REAL_SINGLE           \
     61:                           && defined __SSE2__)

     63: #if !defined __STDC_VERSION__ || __STDC_VERSION__ < 199901L
     64: #  if defined __cplusplus       /* C++ restrict is nonstandard and compilers have inconsistent rules about where it can be used */
     65: #    define restrict
     66: #  else
     67: #    define restrict PETSC_RESTRICT
     68: #  endif
     69: #endif

     71: static PetscClassId THI_CLASSID;

     73: typedef enum {QUAD_GAUSS,QUAD_LOBATTO} QuadratureType;
     74: static const char *QuadratureTypes[] = {"gauss","lobatto","QuadratureType","QUAD_",0};
     75: static const PetscReal HexQWeights[8] = {1,1,1,1,1,1,1,1};
     76: static const PetscReal HexQNodes[]    = {-0.57735026918962573, 0.57735026918962573};
     77: #define G 0.57735026918962573
     78: #define H (0.5*(1.+G))
     79: #define L (0.5*(1.-G))
     80: #define M (-0.5)
     81: #define P (0.5)
     82: /* Special quadrature: Lobatto in horizontal, Gauss in vertical */
     83: static const PetscReal HexQInterp_Lobatto[8][8] = {{H,0,0,0,L,0,0,0},
     84:                                                    {0,H,0,0,0,L,0,0},
     85:                                                    {0,0,H,0,0,0,L,0},
     86:                                                    {0,0,0,H,0,0,0,L},
     87:                                                    {L,0,0,0,H,0,0,0},
     88:                                                    {0,L,0,0,0,H,0,0},
     89:                                                    {0,0,L,0,0,0,H,0},
     90:                                                    {0,0,0,L,0,0,0,H}};
     91: static const PetscReal HexQDeriv_Lobatto[8][8][3] = {
     92:   {{M*H,M*H,M},{P*H,0,0}  ,{0,0,0}    ,{0,P*H,0}  ,{M*L,M*L,P},{P*L,0,0}  ,{0,0,0}    ,{0,P*L,0}  },
     93:   {{M*H,0,0}  ,{P*H,M*H,M},{0,P*H,0}  ,{0,0,0}    ,{M*L,0,0}  ,{P*L,M*L,P},{0,P*L,0}  ,{0,0,0}    },
     94:   {{0,0,0}    ,{0,M*H,0}  ,{P*H,P*H,M},{M*H,0,0}  ,{0,0,0}    ,{0,M*L,0}  ,{P*L,P*L,P},{M*L,0,0}  },
     95:   {{0,M*H,0}  ,{0,0,0}    ,{P*H,0,0}  ,{M*H,P*H,M},{0,M*L,0}  ,{0,0,0}    ,{P*L,0,0}  ,{M*L,P*L,P}},
     96:   {{M*L,M*L,M},{P*L,0,0}  ,{0,0,0}    ,{0,P*L,0}  ,{M*H,M*H,P},{P*H,0,0}  ,{0,0,0}    ,{0,P*H,0}  },
     97:   {{M*L,0,0}  ,{P*L,M*L,M},{0,P*L,0}  ,{0,0,0}    ,{M*H,0,0}  ,{P*H,M*H,P},{0,P*H,0}  ,{0,0,0}    },
     98:   {{0,0,0}    ,{0,M*L,0}  ,{P*L,P*L,M},{M*L,0,0}  ,{0,0,0}    ,{0,M*H,0}  ,{P*H,P*H,P},{M*H,0,0}  },
     99:   {{0,M*L,0}  ,{0,0,0}    ,{P*L,0,0}  ,{M*L,P*L,M},{0,M*H,0}  ,{0,0,0}    ,{P*H,0,0}  ,{M*H,P*H,P}}};
    100: /* Stanndard Gauss */
    101: static const PetscReal HexQInterp_Gauss[8][8] = {{H*H*H,L*H*H,L*L*H,H*L*H, H*H*L,L*H*L,L*L*L,H*L*L},
    102:                                                  {L*H*H,H*H*H,H*L*H,L*L*H, L*H*L,H*H*L,H*L*L,L*L*L},
    103:                                                  {L*L*H,H*L*H,H*H*H,L*H*H, L*L*L,H*L*L,H*H*L,L*H*L},
    104:                                                  {H*L*H,L*L*H,L*H*H,H*H*H, H*L*L,L*L*L,L*H*L,H*H*L},
    105:                                                  {H*H*L,L*H*L,L*L*L,H*L*L, H*H*H,L*H*H,L*L*H,H*L*H},
    106:                                                  {L*H*L,H*H*L,H*L*L,L*L*L, L*H*H,H*H*H,H*L*H,L*L*H},
    107:                                                  {L*L*L,H*L*L,H*H*L,L*H*L, L*L*H,H*L*H,H*H*H,L*H*H},
    108:                                                  {H*L*L,L*L*L,L*H*L,H*H*L, H*L*H,L*L*H,L*H*H,H*H*H}};
    109: static const PetscReal HexQDeriv_Gauss[8][8][3] = {
    110:   {{M*H*H,H*M*H,H*H*M},{P*H*H,L*M*H,L*H*M},{P*L*H,L*P*H,L*L*M},{M*L*H,H*P*H,H*L*M}, {M*H*L,H*M*L,H*H*P},{P*H*L,L*M*L,L*H*P},{P*L*L,L*P*L,L*L*P},{M*L*L,H*P*L,H*L*P}},
    111:   {{M*H*H,L*M*H,L*H*M},{P*H*H,H*M*H,H*H*M},{P*L*H,H*P*H,H*L*M},{M*L*H,L*P*H,L*L*M}, {M*H*L,L*M*L,L*H*P},{P*H*L,H*M*L,H*H*P},{P*L*L,H*P*L,H*L*P},{M*L*L,L*P*L,L*L*P}},
    112:   {{M*L*H,L*M*H,L*L*M},{P*L*H,H*M*H,H*L*M},{P*H*H,H*P*H,H*H*M},{M*H*H,L*P*H,L*H*M}, {M*L*L,L*M*L,L*L*P},{P*L*L,H*M*L,H*L*P},{P*H*L,H*P*L,H*H*P},{M*H*L,L*P*L,L*H*P}},
    113:   {{M*L*H,H*M*H,H*L*M},{P*L*H,L*M*H,L*L*M},{P*H*H,L*P*H,L*H*M},{M*H*H,H*P*H,H*H*M}, {M*L*L,H*M*L,H*L*P},{P*L*L,L*M*L,L*L*P},{P*H*L,L*P*L,L*H*P},{M*H*L,H*P*L,H*H*P}},
    114:   {{M*H*L,H*M*L,H*H*M},{P*H*L,L*M*L,L*H*M},{P*L*L,L*P*L,L*L*M},{M*L*L,H*P*L,H*L*M}, {M*H*H,H*M*H,H*H*P},{P*H*H,L*M*H,L*H*P},{P*L*H,L*P*H,L*L*P},{M*L*H,H*P*H,H*L*P}},
    115:   {{M*H*L,L*M*L,L*H*M},{P*H*L,H*M*L,H*H*M},{P*L*L,H*P*L,H*L*M},{M*L*L,L*P*L,L*L*M}, {M*H*H,L*M*H,L*H*P},{P*H*H,H*M*H,H*H*P},{P*L*H,H*P*H,H*L*P},{M*L*H,L*P*H,L*L*P}},
    116:   {{M*L*L,L*M*L,L*L*M},{P*L*L,H*M*L,H*L*M},{P*H*L,H*P*L,H*H*M},{M*H*L,L*P*L,L*H*M}, {M*L*H,L*M*H,L*L*P},{P*L*H,H*M*H,H*L*P},{P*H*H,H*P*H,H*H*P},{M*H*H,L*P*H,L*H*P}},
    117:   {{M*L*L,H*M*L,H*L*M},{P*L*L,L*M*L,L*L*M},{P*H*L,L*P*L,L*H*M},{M*H*L,H*P*L,H*H*M}, {M*L*H,H*M*H,H*L*P},{P*L*H,L*M*H,L*L*P},{P*H*H,L*P*H,L*H*P},{M*H*H,H*P*H,H*H*P}}};
    118: static const PetscReal (*HexQInterp)[8],(*HexQDeriv)[8][3];
    119: /* Standard 2x2 Gauss quadrature for the bottom layer. */
    120: static const PetscReal QuadQInterp[4][4] = {{H*H,L*H,L*L,H*L},
    121:                                             {L*H,H*H,H*L,L*L},
    122:                                             {L*L,H*L,H*H,L*H},
    123:                                             {H*L,L*L,L*H,H*H}};
    124: static const PetscReal QuadQDeriv[4][4][2] = {
    125:   {{M*H,M*H},{P*H,M*L},{P*L,P*L},{M*L,P*H}},
    126:   {{M*H,M*L},{P*H,M*H},{P*L,P*H},{M*L,P*L}},
    127:   {{M*L,M*L},{P*L,M*H},{P*H,P*H},{M*H,P*L}},
    128:   {{M*L,M*H},{P*L,M*L},{P*H,P*L},{M*H,P*H}}};
    129: #undef G
    130: #undef H
    131: #undef L
    132: #undef M
    133: #undef P

    135: #define HexExtract(x,i,j,k,n) do {              \
    136:     (n)[0] = (x)[i][j][k];                      \
    137:     (n)[1] = (x)[i+1][j][k];                    \
    138:     (n)[2] = (x)[i+1][j+1][k];                  \
    139:     (n)[3] = (x)[i][j+1][k];                    \
    140:     (n)[4] = (x)[i][j][k+1];                    \
    141:     (n)[5] = (x)[i+1][j][k+1];                  \
    142:     (n)[6] = (x)[i+1][j+1][k+1];                \
    143:     (n)[7] = (x)[i][j+1][k+1];                  \
    144:   } while (0)

    146: #define HexExtractRef(x,i,j,k,n) do {           \
    147:     (n)[0] = &(x)[i][j][k];                     \
    148:     (n)[1] = &(x)[i+1][j][k];                   \
    149:     (n)[2] = &(x)[i+1][j+1][k];                 \
    150:     (n)[3] = &(x)[i][j+1][k];                   \
    151:     (n)[4] = &(x)[i][j][k+1];                   \
    152:     (n)[5] = &(x)[i+1][j][k+1];                 \
    153:     (n)[6] = &(x)[i+1][j+1][k+1];               \
    154:     (n)[7] = &(x)[i][j+1][k+1];                 \
    155:   } while (0)

    157: #define QuadExtract(x,i,j,n) do {               \
    158:     (n)[0] = (x)[i][j];                         \
    159:     (n)[1] = (x)[i+1][j];                       \
    160:     (n)[2] = (x)[i+1][j+1];                     \
    161:     (n)[3] = (x)[i][j+1];                       \
    162:   } while (0)

    164: static PetscScalar Sqr(PetscScalar a) {return a*a;}

    166: static void HexGrad(const PetscReal dphi[][3],const PetscReal zn[],PetscReal dz[])
    167: {
    168:   PetscInt i;
    169:   dz[0] = dz[1] = dz[2] = 0;
    170:   for (i=0; i<8; i++) {
    171:     dz[0] += dphi[i][0] * zn[i];
    172:     dz[1] += dphi[i][1] * zn[i];
    173:     dz[2] += dphi[i][2] * zn[i];
    174:   }
    175: }

    177: static void HexComputeGeometry(PetscInt q,PetscReal hx,PetscReal hy,const PetscReal dz[restrict],PetscReal phi[restrict],PetscReal dphi[restrict][3],PetscReal *restrict jw)
    178: {
    179:   const PetscReal
    180:     jac[3][3] = {{hx/2,0,0}, {0,hy/2,0}, {dz[0],dz[1],dz[2]}}
    181:   ,ijac[3][3] = {{1/jac[0][0],0,0}, {0,1/jac[1][1],0}, {-jac[2][0]/(jac[0][0]*jac[2][2]),-jac[2][1]/(jac[1][1]*jac[2][2]),1/jac[2][2]}}
    182:   ,jdet = jac[0][0]*jac[1][1]*jac[2][2];
    183:   PetscInt i;

    185:   for (i=0; i<8; i++) {
    186:     const PetscReal *dphir = HexQDeriv[q][i];
    187:     phi[i] = HexQInterp[q][i];
    188:     dphi[i][0] = dphir[0]*ijac[0][0] + dphir[1]*ijac[1][0] + dphir[2]*ijac[2][0];
    189:     dphi[i][1] = dphir[0]*ijac[0][1] + dphir[1]*ijac[1][1] + dphir[2]*ijac[2][1];
    190:     dphi[i][2] = dphir[0]*ijac[0][2] + dphir[1]*ijac[1][2] + dphir[2]*ijac[2][2];
    191:   }
    192:   *jw = 1.0 * jdet;
    193: }

    195: typedef struct _p_THI   *THI;
    196: typedef struct _n_Units *Units;

    198: typedef struct {
    199:   PetscScalar u,v;
    200: } Node;

    202: typedef struct {
    203:   PetscScalar b;                /* bed */
    204:   PetscScalar h;                /* thickness */
    205:   PetscScalar beta2;            /* friction */
    206: } PrmNode;

    208: #define FieldSize(ntype) ((PetscInt)(sizeof(ntype)/sizeof(PetscScalar)))
    209: #define FieldOffset(ntype,member) ((PetscInt)(offsetof(ntype,member)/sizeof(PetscScalar)))
    210: #define FieldIndex(ntype,i,member) ((PetscInt)((i)*FieldSize(ntype) + FieldOffset(ntype,member)))
    211: #define NODE_SIZE FieldSize(Node)
    212: #define PRMNODE_SIZE FieldSize(PrmNode)

    214: typedef struct {
    215:   PetscReal min,max,cmin,cmax;
    216: } PRange;

    218: struct _p_THI {
    219:   PETSCHEADER(int);
    220:   void      (*initialize)(THI,PetscReal x,PetscReal y,PrmNode *p);
    221:   PetscInt  nlevels;
    222:   PetscInt  zlevels;
    223:   PetscReal Lx,Ly,Lz;           /* Model domain */
    224:   PetscReal alpha;              /* Bed angle */
    225:   Units     units;
    226:   PetscReal dirichlet_scale;
    227:   PetscReal ssa_friction_scale;
    228:   PetscReal inertia;
    229:   PRange    eta;
    230:   PRange    beta2;
    231:   struct {
    232:     PetscReal Bd2,eps,exponent,glen_n;
    233:   } viscosity;
    234:   struct {
    235:     PetscReal irefgam,eps2,exponent;
    236:   } friction;
    237:   struct {
    238:     PetscReal rate,exponent,refvel;
    239:   } erosion;
    240:   PetscReal rhog;
    241:   PetscBool no_slip;
    242:   PetscBool verbose;
    243:   char      *mattype;
    244:   char      *monitor_basename;
    245:   PetscInt  monitor_interval;
    246: };

    248: struct _n_Units {
    249:   /* fundamental */
    250:   PetscReal meter;
    251:   PetscReal kilogram;
    252:   PetscReal second;
    253:   /* derived */
    254:   PetscReal Pascal;
    255:   PetscReal year;
    256: };

    258: static void PrmHexGetZ(const PrmNode pn[],PetscInt k,PetscInt zm,PetscReal zn[])
    259: {
    260:   const PetscScalar zm1 = zm-1,
    261:     znl[8] = {pn[0].b + pn[0].h*(PetscScalar)k/zm1,
    262:               pn[1].b + pn[1].h*(PetscScalar)k/zm1,
    263:               pn[2].b + pn[2].h*(PetscScalar)k/zm1,
    264:               pn[3].b + pn[3].h*(PetscScalar)k/zm1,
    265:               pn[0].b + pn[0].h*(PetscScalar)(k+1)/zm1,
    266:               pn[1].b + pn[1].h*(PetscScalar)(k+1)/zm1,
    267:               pn[2].b + pn[2].h*(PetscScalar)(k+1)/zm1,
    268:               pn[3].b + pn[3].h*(PetscScalar)(k+1)/zm1};
    269:   PetscInt i;
    270:   for (i=0; i<8; i++) zn[i] = PetscRealPart(znl[i]);
    271: }

    273: /* Compute a gradient of all the 2D fields at four quadrature points.  Output for [quadrature_point][direction].field_name */
    274: static PetscErrorCode QuadComputeGrad4(const PetscReal dphi[][4][2],PetscReal hx,PetscReal hy,const PrmNode pn[4],PrmNode dp[4][2])
    275: {
    277:   PetscInt       q,i,f;
    278:   const PetscScalar (*restrict pg)[PRMNODE_SIZE] = (const PetscScalar(*)[PRMNODE_SIZE])pn; /* Get generic array pointers to the node */
    279:   PetscScalar (*restrict dpg)[2][PRMNODE_SIZE]   = (PetscScalar(*)[2][PRMNODE_SIZE])dp;

    282:   PetscArrayzero(dpg,4);
    283:   for (q=0; q<4; q++) {
    284:     for (i=0; i<4; i++) {
    285:       for (f=0; f<PRMNODE_SIZE; f++) {
    286:         dpg[q][0][f] += dphi[q][i][0]/hx * pg[i][f];
    287:         dpg[q][1][f] += dphi[q][i][1]/hy * pg[i][f];
    288:       }
    289:     }
    290:   }
    291:   return(0);
    292: }

    294: static inline PetscReal StaggeredMidpoint2D(PetscScalar a,PetscScalar b,PetscScalar c,PetscScalar d)
    295: {return 0.5*PetscRealPart(0.75*a + 0.75*b + 0.25*c + 0.25*d);}
    296: static inline PetscReal UpwindFlux1D(PetscReal u,PetscReal hL,PetscReal hR)
    297: {return (u > 0) ? hL*u : hR*u;}

    299: #define UpwindFluxXW(x3,x2,h,i,j,k,dj) UpwindFlux1D(StaggeredMidpoint2D(x3[i][j][k].u,x3[i-1][j][k].u, x3[i-1][j+dj][k].u,x3[i][k+dj][k].u), \
    300:                                                     PetscRealPart(0.75*x2[i-1][j  ].h+0.25*x2[i-1][j+dj].h), PetscRealPart(0.75*x2[i  ][j  ].h+0.25*x2[i  ][j+dj].h))
    301: #define UpwindFluxXE(x3,x2,h,i,j,k,dj) UpwindFlux1D(StaggeredMidpoint2D(x3[i][j][k].u,x3[i+1][j][k].u, x3[i+1][j+dj][k].u,x3[i][k+dj][k].u), \
    302:                                                     PetscRealPart(0.75*x2[i  ][j  ].h+0.25*x2[i  ][j+dj].h), PetscRealPart(0.75*x2[i+1][j  ].h+0.25*x2[i+1][j+dj].h))
    303: #define UpwindFluxYS(x3,x2,h,i,j,k,di) UpwindFlux1D(StaggeredMidpoint2D(x3[i][j][k].v,x3[i][j-1][k].v, x3[i+di][j-1][k].v,x3[i+di][j][k].v), \
    304:                                                     PetscRealPart(0.75*x2[i  ][j-1].h+0.25*x2[i+di][j-1].h), PetscRealPart(0.75*x2[i  ][j  ].h+0.25*x2[i+di][j  ].h))
    305: #define UpwindFluxYN(x3,x2,h,i,j,k,di) UpwindFlux1D(StaggeredMidpoint2D(x3[i][j][k].v,x3[i][j+1][k].v, x3[i+di][j+1][k].v,x3[i+di][j][k].v), \
    306:                                                     PetscRealPart(0.75*x2[i  ][j  ].h+0.25*x2[i+di][j  ].h), PetscRealPart(0.75*x2[i  ][j+1].h+0.25*x2[i+di][j+1].h))

    308: static void PrmNodeGetFaceMeasure(const PrmNode **p,PetscInt i,PetscInt j,PetscScalar h[])
    309: {
    310:   /* West */
    311:   h[0] = StaggeredMidpoint2D(p[i][j].h,p[i-1][j].h,p[i-1][j-1].h,p[i][j-1].h);
    312:   h[1] = StaggeredMidpoint2D(p[i][j].h,p[i-1][j].h,p[i-1][j+1].h,p[i][j+1].h);
    313:   /* East */
    314:   h[2] = StaggeredMidpoint2D(p[i][j].h,p[i+1][j].h,p[i+1][j+1].h,p[i][j+1].h);
    315:   h[3] = StaggeredMidpoint2D(p[i][j].h,p[i+1][j].h,p[i+1][j-1].h,p[i][j-1].h);
    316:   /* South */
    317:   h[4] = StaggeredMidpoint2D(p[i][j].h,p[i][j-1].h,p[i+1][j-1].h,p[i+1][j].h);
    318:   h[5] = StaggeredMidpoint2D(p[i][j].h,p[i][j-1].h,p[i-1][j-1].h,p[i-1][j].h);
    319:   /* North */
    320:   h[6] = StaggeredMidpoint2D(p[i][j].h,p[i][j+1].h,p[i-1][j+1].h,p[i-1][j].h);
    321:   h[7] = StaggeredMidpoint2D(p[i][j].h,p[i][j+1].h,p[i+1][j+1].h,p[i+1][j].h);
    322: }

    324: /* Tests A and C are from the ISMIP-HOM paper (Pattyn et al. 2008) */
    325: static void THIInitialize_HOM_A(THI thi,PetscReal x,PetscReal y,PrmNode *p)
    326: {
    327:   Units units = thi->units;
    328:   PetscReal s = -x*PetscSinReal(thi->alpha);
    329:   p->b = s - 1000*units->meter + 500*units->meter * PetscSinReal(x*2*PETSC_PI/thi->Lx) * PetscSinReal(y*2*PETSC_PI/thi->Ly);
    330:   p->h = s - p->b;
    331:   p->beta2 = -1e-10;             /* This value is not used, but it should not be huge because that would change the finite difference step size  */
    332: }

    334: static void THIInitialize_HOM_C(THI thi,PetscReal x,PetscReal y,PrmNode *p)
    335: {
    336:   Units units = thi->units;
    337:   PetscReal s = -x*PetscSinReal(thi->alpha);
    338:   p->b = s - 1000*units->meter;
    339:   p->h = s - p->b;
    340:   /* tau_b = beta2 v   is a stress (Pa).
    341:    * This is a big number in our units (it needs to balance the driving force from the surface), so we scale it by 1/rhog, just like the residual. */
    342:   p->beta2 = 1000 * (1 + PetscSinReal(x*2*PETSC_PI/thi->Lx)*PetscSinReal(y*2*PETSC_PI/thi->Ly)) * units->Pascal * units->year / units->meter / thi->rhog;
    343: }

    345: /* These are just toys */

    347: /* From Fred Herman */
    348: static void THIInitialize_HOM_F(THI thi,PetscReal x,PetscReal y,PrmNode *p)
    349: {
    350:   Units units = thi->units;
    351:   PetscReal s = -x*PetscSinReal(thi->alpha);
    352:   p->b = s - 1000*units->meter + 100*units->meter * PetscSinReal(x*2*PETSC_PI/thi->Lx);/* * sin(y*2*PETSC_PI/thi->Ly); */
    353:   p->h = s - p->b;
    354:   p->h = (1-(atan((x-thi->Lx/2)/1.)+PETSC_PI/2.)/PETSC_PI)*500*units->meter+1*units->meter;
    355:   s = PetscRealPart(p->b + p->h);
    356:   p->beta2 = -1e-10;
    357:   /*  p->beta2 = 1000 * units->Pascal * units->year / units->meter; */
    358: }

    360: /* Same bed as test A, free slip everywhere except for a discontinuous jump to a circular sticky region in the middle. */
    361: static void THIInitialize_HOM_X(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
    362: {
    363:   Units units = thi->units;
    364:   PetscReal x = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
    365:   PetscReal r = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);
    366:   p->b = s - 1000*units->meter + 500*units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    367:   p->h = s - p->b;
    368:   p->beta2 = 1000 * (r < 1 ? 2 : 0) * units->Pascal * units->year / units->meter / thi->rhog;
    369: }

    371: /* Like Z, but with 200 meter cliffs */
    372: static void THIInitialize_HOM_Y(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
    373: {
    374:   Units units = thi->units;
    375:   PetscReal x = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
    376:   PetscReal r = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);
    377:   p->b = s - 1000*units->meter + 500*units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    378:   if (PetscRealPart(p->b) > -700*units->meter) p->b += 200*units->meter;
    379:   p->h = s - p->b;
    380:   p->beta2 = 1000 * (1. + PetscSinReal(PetscSqrtReal(16*r))/PetscSqrtReal(1e-2 + 16*r)*PetscCosReal(x*3/2)*PetscCosReal(y*3/2)) * units->Pascal * units->year / units->meter / thi->rhog;
    381: }

    383: /* Same bed as A, smoothly varying slipperiness, similar to MATLAB's "sombrero" (uncorrelated with bathymetry) */
    384: static void THIInitialize_HOM_Z(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
    385: {
    386:   Units units = thi->units;
    387:   PetscReal x = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
    388:   PetscReal r = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);
    389:   p->b = s - 1000*units->meter + 500*units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    390:   p->h = s - p->b;
    391:   p->beta2 = 1000 * (1. + PetscSinReal(PetscSqrtReal(16*r))/PetscSqrtReal(1e-2 + 16*r)*PetscCosReal(x*3/2)*PetscCosReal(y*3/2)) * units->Pascal * units->year / units->meter / thi->rhog;
    392: }

    394: static void THIFriction(THI thi,PetscReal rbeta2,PetscReal gam,PetscReal *beta2,PetscReal *dbeta2)
    395: {
    396:   if (thi->friction.irefgam == 0) {
    397:     Units units = thi->units;
    398:     thi->friction.irefgam = 1./(0.5*PetscSqr(100 * units->meter / units->year));
    399:     thi->friction.eps2 = 0.5*PetscSqr(1.e-4 / thi->friction.irefgam);
    400:   }
    401:   if (thi->friction.exponent == 0) {
    402:     *beta2  = rbeta2;
    403:     *dbeta2 = 0;
    404:   } else {
    405:     *beta2  = rbeta2 * PetscPowReal(thi->friction.eps2 + gam*thi->friction.irefgam,thi->friction.exponent);
    406:     *dbeta2 = thi->friction.exponent * *beta2 / (thi->friction.eps2 + gam*thi->friction.irefgam) * thi->friction.irefgam;
    407:   }
    408: }

    410: static void THIViscosity(THI thi,PetscReal gam,PetscReal *eta,PetscReal *deta)
    411: {
    412:   PetscReal Bd2,eps,exponent;
    413:   if (thi->viscosity.Bd2 == 0) {
    414:     Units units = thi->units;
    415:     const PetscReal
    416:       n = thi->viscosity.glen_n,                        /* Glen exponent */
    417:       p = 1. + 1./n,                                    /* for Stokes */
    418:       A = 1.e-16 * PetscPowReal(units->Pascal,-n) / units->year, /* softness parameter (Pa^{-n}/s) */
    419:       B = PetscPowReal(A,-1./n);                                 /* hardness parameter */
    420:     thi->viscosity.Bd2      = B/2;
    421:     thi->viscosity.exponent = (p-2)/2;
    422:     thi->viscosity.eps      = 0.5*PetscSqr(1e-5 / units->year);
    423:   }
    424:   Bd2      = thi->viscosity.Bd2;
    425:   exponent = thi->viscosity.exponent;
    426:   eps      = thi->viscosity.eps;
    427:   *eta     = Bd2 * PetscPowReal(eps + gam,exponent);
    428:   *deta    = exponent * (*eta) / (eps + gam);
    429: }

    431: static void THIErosion(THI thi,const Node *vel,PetscScalar *erate,Node *derate)
    432: {
    433:   const PetscScalar magref2 = 1.e-10 + (PetscSqr(vel->u) + PetscSqr(vel->v)) / PetscSqr(thi->erosion.refvel),
    434:                     rate    = -thi->erosion.rate*PetscPowScalar(magref2, 0.5*thi->erosion.exponent);
    435:   if (erate) *erate = rate;
    436:   if (derate) {
    437:     if (thi->erosion.exponent == 1) {
    438:       derate->u = 0;
    439:       derate->v = 0;
    440:     } else {
    441:       derate->u = 0.5*thi->erosion.exponent * rate / magref2 * 2. * vel->u / PetscSqr(thi->erosion.refvel);
    442:       derate->v = 0.5*thi->erosion.exponent * rate / magref2 * 2. * vel->v / PetscSqr(thi->erosion.refvel);
    443:     }
    444:   }
    445: }

    447: static void RangeUpdate(PetscReal *min,PetscReal *max,PetscReal x)
    448: {
    449:   if (x < *min) *min = x;
    450:   if (x > *max) *max = x;
    451: }

    453: static void PRangeClear(PRange *p)
    454: {
    455:   p->cmin = p->min = 1e100;
    456:   p->cmax = p->max = -1e100;
    457: }

    459: static PetscErrorCode PRangeMinMax(PRange *p,PetscReal min,PetscReal max)
    460: {

    463:   p->cmin = min;
    464:   p->cmax = max;
    465:   if (min < p->min) p->min = min;
    466:   if (max > p->max) p->max = max;
    467:   return(0);
    468: }

    470: static PetscErrorCode THIDestroy(THI *thi)
    471: {

    475:   if (--((PetscObject)(*thi))->refct > 0) return(0);
    476:   PetscFree((*thi)->units);
    477:   PetscFree((*thi)->mattype);
    478:   PetscFree((*thi)->monitor_basename);
    479:   PetscHeaderDestroy(thi);
    480:   return(0);
    481: }

    483: static PetscErrorCode THICreate(MPI_Comm comm,THI *inthi)
    484: {
    485:   static PetscBool registered = PETSC_FALSE;
    486:   THI              thi;
    487:   Units            units;
    488:   char             monitor_basename[PETSC_MAX_PATH_LEN] = "thi-";
    489:   PetscErrorCode   ierr;

    492:   *inthi = 0;
    493:   if (!registered) {
    494:     PetscClassIdRegister("Toy Hydrostatic Ice",&THI_CLASSID);
    495:     registered = PETSC_TRUE;
    496:   }
    497:   PetscHeaderCreate(thi,THI_CLASSID,"THI","Toy Hydrostatic Ice","THI",comm,THIDestroy,0);

    499:   PetscNew(&thi->units);

    501:   units           = thi->units;
    502:   units->meter    = 1e-2;
    503:   units->second   = 1e-7;
    504:   units->kilogram = 1e-12;

    506:   PetscOptionsBegin(comm,NULL,"Scaled units options","");
    507:   {
    508:     PetscOptionsReal("-units_meter","1 meter in scaled length units","",units->meter,&units->meter,NULL);
    509:     PetscOptionsReal("-units_second","1 second in scaled time units","",units->second,&units->second,NULL);
    510:     PetscOptionsReal("-units_kilogram","1 kilogram in scaled mass units","",units->kilogram,&units->kilogram,NULL);
    511:   }
    512:   PetscOptionsEnd();
    513:   units->Pascal = units->kilogram / (units->meter * PetscSqr(units->second));
    514:   units->year   = 31556926. * units->second, /* seconds per year */

    516:   thi->Lx              = 10.e3;
    517:   thi->Ly              = 10.e3;
    518:   thi->Lz              = 1000;
    519:   thi->nlevels         = 1;
    520:   thi->dirichlet_scale = 1;
    521:   thi->verbose         = PETSC_FALSE;

    523:   thi->viscosity.glen_n = 3.;
    524:   thi->erosion.rate     = 1e-3; /* m/a */
    525:   thi->erosion.exponent = 1.;
    526:   thi->erosion.refvel   = 1.;   /* m/a */

    528:   PetscOptionsBegin(comm,NULL,"Toy Hydrostatic Ice options","");
    529:   {
    530:     QuadratureType quad       = QUAD_GAUSS;
    531:     char           homexp[]   = "A";
    532:     char           mtype[256] = MATSBAIJ;
    533:     PetscReal      L,m = 1.0;
    534:     PetscBool      flg;
    535:     L    = thi->Lx;
    536:     PetscOptionsReal("-thi_L","Domain size (m)","",L,&L,&flg);
    537:     if (flg) thi->Lx = thi->Ly = L;
    538:     PetscOptionsReal("-thi_Lx","X Domain size (m)","",thi->Lx,&thi->Lx,NULL);
    539:     PetscOptionsReal("-thi_Ly","Y Domain size (m)","",thi->Ly,&thi->Ly,NULL);
    540:     PetscOptionsReal("-thi_Lz","Z Domain size (m)","",thi->Lz,&thi->Lz,NULL);
    541:     PetscOptionsString("-thi_hom","ISMIP-HOM experiment (A or C)","",homexp,homexp,sizeof(homexp),NULL);
    542:     switch (homexp[0] = toupper(homexp[0])) {
    543:     case 'A':
    544:       thi->initialize = THIInitialize_HOM_A;
    545:       thi->no_slip    = PETSC_TRUE;
    546:       thi->alpha      = 0.5;
    547:       break;
    548:     case 'C':
    549:       thi->initialize = THIInitialize_HOM_C;
    550:       thi->no_slip    = PETSC_FALSE;
    551:       thi->alpha      = 0.1;
    552:       break;
    553:     case 'F':
    554:       thi->initialize = THIInitialize_HOM_F;
    555:       thi->no_slip    = PETSC_FALSE;
    556:       thi->alpha      = 0.5;
    557:       break;
    558:     case 'X':
    559:       thi->initialize = THIInitialize_HOM_X;
    560:       thi->no_slip    = PETSC_FALSE;
    561:       thi->alpha      = 0.3;
    562:       break;
    563:     case 'Y':
    564:       thi->initialize = THIInitialize_HOM_Y;
    565:       thi->no_slip    = PETSC_FALSE;
    566:       thi->alpha      = 0.5;
    567:       break;
    568:     case 'Z':
    569:       thi->initialize = THIInitialize_HOM_Z;
    570:       thi->no_slip    = PETSC_FALSE;
    571:       thi->alpha      = 0.5;
    572:       break;
    573:     default:
    574:       SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"HOM experiment '%c' not implemented",homexp[0]);
    575:     }
    576:     PetscOptionsEnum("-thi_quadrature","Quadrature to use for 3D elements","",QuadratureTypes,(PetscEnum)quad,(PetscEnum*)&quad,NULL);
    577:     switch (quad) {
    578:     case QUAD_GAUSS:
    579:       HexQInterp = HexQInterp_Gauss;
    580:       HexQDeriv  = HexQDeriv_Gauss;
    581:       break;
    582:     case QUAD_LOBATTO:
    583:       HexQInterp = HexQInterp_Lobatto;
    584:       HexQDeriv  = HexQDeriv_Lobatto;
    585:       break;
    586:     }
    587:     PetscOptionsReal("-thi_alpha","Bed angle (degrees)","",thi->alpha,&thi->alpha,NULL);
    588:     PetscOptionsReal("-thi_viscosity_glen_n","Exponent in Glen flow law, 1=linear, infty=ideal plastic",NULL,thi->viscosity.glen_n,&thi->viscosity.glen_n,NULL);
    589:     PetscOptionsReal("-thi_friction_m","Friction exponent, 0=Coulomb, 1=Navier","",m,&m,NULL);
    590:     thi->friction.exponent = (m-1)/2;
    591:     PetscOptionsReal("-thi_erosion_rate","Rate of erosion relative to sliding velocity at reference velocity (m/a)",NULL,thi->erosion.rate,&thi->erosion.rate,NULL);
    592:     PetscOptionsReal("-thi_erosion_exponent","Power of sliding velocity appearing in erosion relation",NULL,thi->erosion.exponent,&thi->erosion.exponent,NULL);
    593:     PetscOptionsReal("-thi_erosion_refvel","Reference sliding velocity for erosion (m/a)",NULL,thi->erosion.refvel,&thi->erosion.refvel,NULL);
    594:     thi->erosion.rate   *= units->meter / units->year;
    595:     thi->erosion.refvel *= units->meter / units->year;
    596:     PetscOptionsReal("-thi_dirichlet_scale","Scale Dirichlet boundary conditions by this factor","",thi->dirichlet_scale,&thi->dirichlet_scale,NULL);
    597:     PetscOptionsReal("-thi_ssa_friction_scale","Scale slip boundary conditions by this factor in SSA (2D) assembly","",thi->ssa_friction_scale,&thi->ssa_friction_scale,NULL);
    598:     PetscOptionsReal("-thi_inertia","Coefficient of accelaration term in velocity system, physical is almost zero",NULL,thi->inertia,&thi->inertia,NULL);
    599:     PetscOptionsInt("-thi_nlevels","Number of levels of refinement","",thi->nlevels,&thi->nlevels,NULL);
    600:     PetscOptionsFList("-thi_mat_type","Matrix type","MatSetType",MatList,mtype,(char*)mtype,sizeof(mtype),NULL);
    601:     PetscStrallocpy(mtype,&thi->mattype);
    602:     PetscOptionsBool("-thi_verbose","Enable verbose output (like matrix sizes and statistics)","",thi->verbose,&thi->verbose,NULL);
    603:     PetscOptionsString("-thi_monitor","Basename to write state files to",NULL,monitor_basename,monitor_basename,sizeof(monitor_basename),&flg);
    604:     if (flg) {
    605:       PetscStrallocpy(monitor_basename,&thi->monitor_basename);
    606:       thi->monitor_interval = 1;
    607:       PetscOptionsInt("-thi_monitor_interval","Frequency at which to write state files",NULL,thi->monitor_interval,&thi->monitor_interval,NULL);
    608:     }
    609:   }
    610:   PetscOptionsEnd();

    612:   /* dimensionalize */
    613:   thi->Lx    *= units->meter;
    614:   thi->Ly    *= units->meter;
    615:   thi->Lz    *= units->meter;
    616:   thi->alpha *= PETSC_PI / 180;

    618:   PRangeClear(&thi->eta);
    619:   PRangeClear(&thi->beta2);

    621:   {
    622:     PetscReal u       = 1000*units->meter/(3e7*units->second),
    623:               gradu   = u / (100*units->meter),eta,deta,
    624:               rho     = 910 * units->kilogram/PetscPowRealInt(units->meter,3),
    625:               grav    = 9.81 * units->meter/PetscSqr(units->second),
    626:               driving = rho * grav * PetscSinReal(thi->alpha) * 1000*units->meter;
    627:     THIViscosity(thi,0.5*gradu*gradu,&eta,&deta);
    628:     thi->rhog = rho * grav;
    629:     if (thi->verbose) {
    630:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Units: meter %8.2g  second %8.2g  kg %8.2g  Pa %8.2g\n",units->meter,units->second,units->kilogram,units->Pascal);
    631:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Domain (%6.2g,%6.2g,%6.2g), pressure %8.2g, driving stress %8.2g\n",thi->Lx,thi->Ly,thi->Lz,rho*grav*1e3*units->meter,driving);
    632:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Large velocity 1km/a %8.2g, velocity gradient %8.2g, eta %8.2g, stress %8.2g, ratio %8.2g\n",u,gradu,eta,2*eta*gradu,2*eta*gradu/driving);
    633:       THIViscosity(thi,0.5*PetscSqr(1e-3*gradu),&eta,&deta);
    634:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Small velocity 1m/a  %8.2g, velocity gradient %8.2g, eta %8.2g, stress %8.2g, ratio %8.2g\n",1e-3*u,1e-3*gradu,eta,2*eta*1e-3*gradu,2*eta*1e-3*gradu/driving);
    635:     }
    636:   }

    638:   *inthi = thi;
    639:   return(0);
    640: }

    642: /* Our problem is periodic, but the domain has a mean slope of alpha so the bed does not line up between the upstream
    643:  * and downstream ends of the domain.  This function fixes the ghost values so that the domain appears truly periodic in
    644:  * the horizontal. */
    645: static PetscErrorCode THIFixGhosts(THI thi,DM da3,DM da2,Vec X3,Vec X2)
    646: {
    648:   DMDALocalInfo  info;
    649:   PrmNode        **x2;
    650:   PetscInt       i,j;

    653:   DMDAGetLocalInfo(da3,&info);
    654:   /* VecView(X2,PETSC_VIEWER_STDOUT_WORLD); */
    655:   DMDAVecGetArray(da2,X2,&x2);
    656:   for (i=info.gzs; i<info.gzs+info.gzm; i++) {
    657:     if (i > -1 && i < info.mz) continue;
    658:     for (j=info.gys; j<info.gys+info.gym; j++) {
    659:       x2[i][j].b += PetscSinReal(thi->alpha) * thi->Lx * (i<0 ? 1.0 : -1.0);
    660:     }
    661:   }
    662:   DMDAVecRestoreArray(da2,X2,&x2);
    663:   /* VecView(X2,PETSC_VIEWER_STDOUT_WORLD); */
    664:   return(0);
    665: }

    667: static PetscErrorCode THIInitializePrm(THI thi,DM da2prm,PrmNode **p)
    668: {
    669:   PetscInt       i,j,xs,xm,ys,ym,mx,my;

    673:   DMDAGetGhostCorners(da2prm,&ys,&xs,0,&ym,&xm,0);
    674:   DMDAGetInfo(da2prm,0, &my,&mx,0, 0,0,0, 0,0,0,0,0,0);
    675:   for (i=xs; i<xs+xm; i++) {
    676:     for (j=ys; j<ys+ym; j++) {
    677:       PetscReal xx = thi->Lx*i/mx,yy = thi->Ly*j/my;
    678:       thi->initialize(thi,xx,yy,&p[i][j]);
    679:     }
    680:   }
    681:   return(0);
    682: }

    684: static PetscErrorCode THIInitial(THI thi,DM pack,Vec X)
    685: {
    686:   DM             da3,da2;
    687:   PetscInt       i,j,k,xs,xm,ys,ym,zs,zm,mx,my;
    688:   PetscReal      hx,hy;
    689:   PrmNode        **prm;
    690:   Node           ***x;
    691:   Vec            X3g,X2g,X2;

    695:   DMCompositeGetEntries(pack,&da3,&da2);
    696:   DMCompositeGetAccess(pack,X,&X3g,&X2g);
    697:   DMGetLocalVector(da2,&X2);

    699:   DMDAGetInfo(da3,0, 0,&my,&mx, 0,0,0, 0,0,0,0,0,0);
    700:   DMDAGetCorners(da3,&zs,&ys,&xs,&zm,&ym,&xm);
    701:   DMDAVecGetArray(da3,X3g,&x);
    702:   DMDAVecGetArray(da2,X2,&prm);

    704:   THIInitializePrm(thi,da2,prm);

    706:   hx = thi->Lx / mx;
    707:   hy = thi->Ly / my;
    708:   for (i=xs; i<xs+xm; i++) {
    709:     for (j=ys; j<ys+ym; j++) {
    710:       for (k=zs; k<zs+zm; k++) {
    711:         const PetscScalar zm1      = zm-1,
    712:                           drivingx = thi->rhog * (prm[i+1][j].b+prm[i+1][j].h - prm[i-1][j].b-prm[i-1][j].h) / (2*hx),
    713:                           drivingy = thi->rhog * (prm[i][j+1].b+prm[i][j+1].h - prm[i][j-1].b-prm[i][j-1].h) / (2*hy);
    714:         x[i][j][k].u = 0. * drivingx * prm[i][j].h*(PetscScalar)k/zm1;
    715:         x[i][j][k].v = 0. * drivingy * prm[i][j].h*(PetscScalar)k/zm1;
    716:       }
    717:     }
    718:   }

    720:   DMDAVecRestoreArray(da3,X3g,&x);
    721:   DMDAVecRestoreArray(da2,X2,&prm);

    723:   DMLocalToGlobalBegin(da2,X2,INSERT_VALUES,X2g);
    724:   DMLocalToGlobalEnd  (da2,X2,INSERT_VALUES,X2g);
    725:   DMRestoreLocalVector(da2,&X2);

    727:   DMCompositeRestoreAccess(pack,X,&X3g,&X2g);
    728:   return(0);
    729: }

    731: static void PointwiseNonlinearity(THI thi,const Node n[restrict 8],const PetscReal phi[restrict 3],PetscReal dphi[restrict 8][3],PetscScalar *restrict u,PetscScalar *restrict v,PetscScalar du[restrict 3],PetscScalar dv[restrict 3],PetscReal *eta,PetscReal *deta)
    732: {
    733:   PetscInt    l,ll;
    734:   PetscScalar gam;

    736:   du[0] = du[1] = du[2] = 0;
    737:   dv[0] = dv[1] = dv[2] = 0;
    738:   *u    = 0;
    739:   *v    = 0;
    740:   for (l=0; l<8; l++) {
    741:     *u += phi[l] * n[l].u;
    742:     *v += phi[l] * n[l].v;
    743:     for (ll=0; ll<3; ll++) {
    744:       du[ll] += dphi[l][ll] * n[l].u;
    745:       dv[ll] += dphi[l][ll] * n[l].v;
    746:     }
    747:   }
    748:   gam = Sqr(du[0]) + Sqr(dv[1]) + du[0]*dv[1] + 0.25*Sqr(du[1]+dv[0]) + 0.25*Sqr(du[2]) + 0.25*Sqr(dv[2]);
    749:   THIViscosity(thi,PetscRealPart(gam),eta,deta);
    750: }

    752: static PetscErrorCode THIFunctionLocal_3D(DMDALocalInfo *info,const Node ***x,const PrmNode **prm,const Node ***xdot,Node ***f,THI thi)
    753: {
    754:   PetscInt       xs,ys,xm,ym,zm,i,j,k,q,l;
    755:   PetscReal      hx,hy,etamin,etamax,beta2min,beta2max;

    759:   xs = info->zs;
    760:   ys = info->ys;
    761:   xm = info->zm;
    762:   ym = info->ym;
    763:   zm = info->xm;
    764:   hx = thi->Lx / info->mz;
    765:   hy = thi->Ly / info->my;

    767:   etamin   = 1e100;
    768:   etamax   = 0;
    769:   beta2min = 1e100;
    770:   beta2max = 0;

    772:   for (i=xs; i<xs+xm; i++) {
    773:     for (j=ys; j<ys+ym; j++) {
    774:       PrmNode pn[4],dpn[4][2];
    775:       QuadExtract(prm,i,j,pn);
    776:       QuadComputeGrad4(QuadQDeriv,hx,hy,pn,dpn);
    777:       for (k=0; k<zm-1; k++) {
    778:         PetscInt  ls = 0;
    779:         Node      n[8],ndot[8],*fn[8];
    780:         PetscReal zn[8],etabase = 0;
    781:         PrmHexGetZ(pn,k,zm,zn);
    782:         HexExtract(x,i,j,k,n);
    783:         HexExtract(xdot,i,j,k,ndot);
    784:         HexExtractRef(f,i,j,k,fn);
    785:         if (thi->no_slip && k == 0) {
    786:           for (l=0; l<4; l++) n[l].u = n[l].v = 0;
    787:           /* The first 4 basis functions lie on the bottom layer, so their contribution is exactly 0, hence we can skip them */
    788:           ls = 4;
    789:         }
    790:         for (q=0; q<8; q++) {
    791:           PetscReal   dz[3],phi[8],dphi[8][3],jw,eta,deta;
    792:           PetscScalar du[3],dv[3],u,v,udot=0,vdot=0;
    793:           for (l=ls; l<8; l++) {
    794:             udot += HexQInterp[q][l]*ndot[l].u;
    795:             vdot += HexQInterp[q][l]*ndot[l].v;
    796:           }
    797:           HexGrad(HexQDeriv[q],zn,dz);
    798:           HexComputeGeometry(q,hx,hy,dz,phi,dphi,&jw);
    799:           PointwiseNonlinearity(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
    800:           jw /= thi->rhog;      /* scales residuals to be O(1) */
    801:           if (q == 0) etabase = eta;
    802:           RangeUpdate(&etamin,&etamax,eta);
    803:           for (l=ls; l<8; l++) { /* test functions */
    804:             const PetscScalar ds[2] = {dpn[q%4][0].h+dpn[q%4][0].b, dpn[q%4][1].h+dpn[q%4][1].b};
    805:             const PetscReal   pp    = phi[l],*dp = dphi[l];
    806:             fn[l]->u += dp[0]*jw*eta*(4.*du[0]+2.*dv[1]) + dp[1]*jw*eta*(du[1]+dv[0]) + dp[2]*jw*eta*du[2] + pp*jw*thi->rhog*ds[0];
    807:             fn[l]->v += dp[1]*jw*eta*(2.*du[0]+4.*dv[1]) + dp[0]*jw*eta*(du[1]+dv[0]) + dp[2]*jw*eta*dv[2] + pp*jw*thi->rhog*ds[1];
    808:             fn[l]->u += pp*jw*udot*thi->inertia*pp;
    809:             fn[l]->v += pp*jw*vdot*thi->inertia*pp;
    810:           }
    811:         }
    812:         if (k == 0) { /* we are on a bottom face */
    813:           if (thi->no_slip) {
    814:             /* Note: Non-Galerkin coarse grid operators are very sensitive to the scaling of Dirichlet boundary
    815:             * conditions.  After shenanigans above, etabase contains the effective viscosity at the closest quadrature
    816:             * point to the bed.  We want the diagonal entry in the Dirichlet condition to have similar magnitude to the
    817:             * diagonal entry corresponding to the adjacent node.  The fundamental scaling of the viscous part is in
    818:             * diagu, diagv below.  This scaling is easy to recognize by considering the finite difference operator after
    819:             * scaling by element size.  The no-slip Dirichlet condition is scaled by this factor, and also in the
    820:             * assembled matrix (see the similar block in THIJacobianLocal).
    821:             *
    822:             * Note that the residual at this Dirichlet node is linear in the state at this node, but also depends
    823:             * (nonlinearly in general) on the neighboring interior nodes through the local viscosity.  This will make
    824:             * a matrix-free Jacobian have extra entries in the corresponding row.  We assemble only the diagonal part,
    825:             * so the solution will exactly satisfy the boundary condition after the first linear iteration.
    826:             */
    827:             const PetscReal   hz    = PetscRealPart(pn[0].h)/(zm-1.);
    828:             const PetscScalar diagu = 2*etabase/thi->rhog*(hx*hy/hz + hx*hz/hy + 4*hy*hz/hx),diagv = 2*etabase/thi->rhog*(hx*hy/hz + 4*hx*hz/hy + hy*hz/hx);
    829:             fn[0]->u = thi->dirichlet_scale*diagu*x[i][j][k].u;
    830:             fn[0]->v = thi->dirichlet_scale*diagv*x[i][j][k].v;
    831:           } else {              /* Integrate over bottom face to apply boundary condition */
    832:             for (q=0; q<4; q++) { /* We remove the explicit scaling of the residual by 1/rhog because beta2 already has that scaling to be O(1) */
    833:               const PetscReal jw = 0.25*hx*hy,*phi = QuadQInterp[q];
    834:               PetscScalar     u  =0,v=0,rbeta2=0;
    835:               PetscReal       beta2,dbeta2;
    836:               for (l=0; l<4; l++) {
    837:                 u     += phi[l]*n[l].u;
    838:                 v     += phi[l]*n[l].v;
    839:                 rbeta2 += phi[l]*pn[l].beta2;
    840:               }
    841:               THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
    842:               RangeUpdate(&beta2min,&beta2max,beta2);
    843:               for (l=0; l<4; l++) {
    844:                 const PetscReal pp = phi[l];
    845:                 fn[ls+l]->u += pp*jw*beta2*u;
    846:                 fn[ls+l]->v += pp*jw*beta2*v;
    847:               }
    848:             }
    849:           }
    850:         }
    851:       }
    852:     }
    853:   }

    855:   PRangeMinMax(&thi->eta,etamin,etamax);
    856:   PRangeMinMax(&thi->beta2,beta2min,beta2max);
    857:   return(0);
    858: }

    860: static PetscErrorCode THIFunctionLocal_2D(DMDALocalInfo *info,const Node ***x,const PrmNode **prm,const PrmNode **prmdot,PrmNode **f,THI thi)
    861: {
    862:   PetscInt xs,ys,xm,ym,zm,i,j,k;

    865:   xs = info->zs;
    866:   ys = info->ys;
    867:   xm = info->zm;
    868:   ym = info->ym;
    869:   zm = info->xm;

    871:   for (i=xs; i<xs+xm; i++) {
    872:     for (j=ys; j<ys+ym; j++) {
    873:       PetscScalar div = 0,erate,h[8];
    874:       PrmNodeGetFaceMeasure(prm,i,j,h);
    875:       for (k=0; k<zm; k++) {
    876:         PetscScalar weight = (k==0 || k == zm-1) ? 0.5/(zm-1) : 1.0/(zm-1);
    877:         if (0) {                /* centered flux */
    878:           div += (- weight*h[0] * StaggeredMidpoint2D(x[i][j][k].u,x[i-1][j][k].u, x[i-1][j-1][k].u,x[i][j-1][k].u)
    879:                   - weight*h[1] * StaggeredMidpoint2D(x[i][j][k].u,x[i-1][j][k].u, x[i-1][j+1][k].u,x[i][j+1][k].u)
    880:                   + weight*h[2] * StaggeredMidpoint2D(x[i][j][k].u,x[i+1][j][k].u, x[i+1][j+1][k].u,x[i][j+1][k].u)
    881:                   + weight*h[3] * StaggeredMidpoint2D(x[i][j][k].u,x[i+1][j][k].u, x[i+1][j-1][k].u,x[i][j-1][k].u)
    882:                   - weight*h[4] * StaggeredMidpoint2D(x[i][j][k].v,x[i][j-1][k].v, x[i+1][j-1][k].v,x[i+1][j][k].v)
    883:                   - weight*h[5] * StaggeredMidpoint2D(x[i][j][k].v,x[i][j-1][k].v, x[i-1][j-1][k].v,x[i-1][j][k].v)
    884:                   + weight*h[6] * StaggeredMidpoint2D(x[i][j][k].v,x[i][j+1][k].v, x[i-1][j+1][k].v,x[i-1][j][k].v)
    885:                   + weight*h[7] * StaggeredMidpoint2D(x[i][j][k].v,x[i][j+1][k].v, x[i+1][j+1][k].v,x[i+1][j][k].v));
    886:         } else {                /* Upwind flux */
    887:           div += weight*(-UpwindFluxXW(x,prm,h,i,j,k, 1)
    888:                          -UpwindFluxXW(x,prm,h,i,j,k,-1)
    889:                          +UpwindFluxXE(x,prm,h,i,j,k, 1)
    890:                          +UpwindFluxXE(x,prm,h,i,j,k,-1)
    891:                          -UpwindFluxYS(x,prm,h,i,j,k, 1)
    892:                          -UpwindFluxYS(x,prm,h,i,j,k,-1)
    893:                          +UpwindFluxYN(x,prm,h,i,j,k, 1)
    894:                          +UpwindFluxYN(x,prm,h,i,j,k,-1));
    895:         }
    896:       }
    897:       /* printf("div[%d][%d] %g\n",i,j,div); */
    898:       THIErosion(thi,&x[i][j][0],&erate,NULL);
    899:       f[i][j].b     = prmdot[i][j].b - erate;
    900:       f[i][j].h     = prmdot[i][j].h + div;
    901:       f[i][j].beta2 = prmdot[i][j].beta2;
    902:     }
    903:   }
    904:   return(0);
    905: }

    907: static PetscErrorCode THIFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
    908: {
    910:   THI            thi = (THI)ctx;
    911:   DM             pack,da3,da2;
    912:   Vec            X3,X2,Xdot3,Xdot2,F3,F2,F3g,F2g;
    913:   const Node     ***x3,***xdot3;
    914:   const PrmNode  **x2,**xdot2;
    915:   Node           ***f3;
    916:   PrmNode        **f2;
    917:   DMDALocalInfo  info3;

    920:   TSGetDM(ts,&pack);
    921:   DMCompositeGetEntries(pack,&da3,&da2);
    922:   DMDAGetLocalInfo(da3,&info3);
    923:   DMCompositeGetLocalVectors(pack,&X3,&X2);
    924:   DMCompositeGetLocalVectors(pack,&Xdot3,&Xdot2);
    925:   DMCompositeScatter(pack,X,X3,X2);
    926:   THIFixGhosts(thi,da3,da2,X3,X2);
    927:   DMCompositeScatter(pack,Xdot,Xdot3,Xdot2);

    929:   DMGetLocalVector(da3,&F3);
    930:   DMGetLocalVector(da2,&F2);
    931:   VecZeroEntries(F3);

    933:   DMDAVecGetArray(da3,X3,&x3);
    934:   DMDAVecGetArray(da2,X2,&x2);
    935:   DMDAVecGetArray(da3,Xdot3,&xdot3);
    936:   DMDAVecGetArray(da2,Xdot2,&xdot2);
    937:   DMDAVecGetArray(da3,F3,&f3);
    938:   DMDAVecGetArray(da2,F2,&f2);

    940:   THIFunctionLocal_3D(&info3,x3,x2,xdot3,f3,thi);
    941:   THIFunctionLocal_2D(&info3,x3,x2,xdot2,f2,thi);

    943:   DMDAVecRestoreArray(da3,X3,&x3);
    944:   DMDAVecRestoreArray(da2,X2,&x2);
    945:   DMDAVecRestoreArray(da3,Xdot3,&xdot3);
    946:   DMDAVecRestoreArray(da2,Xdot2,&xdot2);
    947:   DMDAVecRestoreArray(da3,F3,&f3);
    948:   DMDAVecRestoreArray(da2,F2,&f2);

    950:   DMCompositeRestoreLocalVectors(pack,&X3,&X2);
    951:   DMCompositeRestoreLocalVectors(pack,&Xdot3,&Xdot2);

    953:   VecZeroEntries(F);
    954:   DMCompositeGetAccess(pack,F,&F3g,&F2g);
    955:   DMLocalToGlobalBegin(da3,F3,ADD_VALUES,F3g);
    956:   DMLocalToGlobalEnd  (da3,F3,ADD_VALUES,F3g);
    957:   DMLocalToGlobalBegin(da2,F2,INSERT_VALUES,F2g);
    958:   DMLocalToGlobalEnd  (da2,F2,INSERT_VALUES,F2g);

    960:   if (thi->verbose) {
    961:     PetscViewer viewer;
    962:     PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject)thi),&viewer);
    963:     PetscViewerASCIIPrintf(viewer,"3D_Velocity residual (bs=2):\n");
    964:     PetscViewerASCIIPushTab(viewer);
    965:     VecView(F3,viewer);
    966:     PetscViewerASCIIPopTab(viewer);
    967:     PetscViewerASCIIPrintf(viewer,"2D_Fields residual (bs=3):\n");
    968:     PetscViewerASCIIPushTab(viewer);
    969:     VecView(F2,viewer);
    970:     PetscViewerASCIIPopTab(viewer);
    971:   }

    973:   DMCompositeRestoreAccess(pack,F,&F3g,&F2g);

    975:   DMRestoreLocalVector(da3,&F3);
    976:   DMRestoreLocalVector(da2,&F2);
    977:   return(0);
    978: }

    980: static PetscErrorCode THIMatrixStatistics(THI thi,Mat B,PetscViewer viewer)
    981: {
    983:   PetscReal      nrm;
    984:   PetscInt       m;
    985:   PetscMPIInt    rank;

    988:   MatNorm(B,NORM_FROBENIUS,&nrm);
    989:   MatGetSize(B,&m,0);
    990:   MPI_Comm_rank(PetscObjectComm((PetscObject)B),&rank);
    991:   if (rank == 0) {
    992:     PetscScalar val0,val2;
    993:     MatGetValue(B,0,0,&val0);
    994:     MatGetValue(B,2,2,&val2);
    995:     PetscViewerASCIIPrintf(viewer,"Matrix dim %8d  norm %8.2e, (0,0) %8.2e  (2,2) %8.2e, eta [%8.2e,%8.2e] beta2 [%8.2e,%8.2e]\n",m,nrm,PetscRealPart(val0),PetscRealPart(val2),thi->eta.cmin,thi->eta.cmax,thi->beta2.cmin,thi->beta2.cmax);
    996:   }
    997:   return(0);
    998: }

    1000: static PetscErrorCode THISurfaceStatistics(DM pack,Vec X,PetscReal *min,PetscReal *max,PetscReal *mean)
    1001: {
    1003:   DM             da3,da2;
    1004:   Vec            X3,X2;
    1005:   Node           ***x;
    1006:   PetscInt       i,j,xs,ys,zs,xm,ym,zm,mx,my,mz;
    1007:   PetscReal      umin = 1e100,umax=-1e100;
    1008:   PetscScalar    usum =0.0,gusum;

    1011:   DMCompositeGetEntries(pack,&da3,&da2);
    1012:   DMCompositeGetAccess(pack,X,&X3,&X2);
    1013:   *min = *max = *mean = 0;
    1014:   DMDAGetInfo(da3,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
    1015:   DMDAGetCorners(da3,&zs,&ys,&xs,&zm,&ym,&xm);
    1016:   if (zs != 0 || zm != mz) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_PLIB,"Unexpected decomposition");
    1017:   DMDAVecGetArray(da3,X3,&x);
    1018:   for (i=xs; i<xs+xm; i++) {
    1019:     for (j=ys; j<ys+ym; j++) {
    1020:       PetscReal u = PetscRealPart(x[i][j][zm-1].u);
    1021:       RangeUpdate(&umin,&umax,u);
    1022:       usum += u;
    1023:     }
    1024:   }
    1025:   DMDAVecRestoreArray(da3,X3,&x);
    1026:   DMCompositeRestoreAccess(pack,X,&X3,&X2);

    1028:   MPI_Allreduce(&umin,min,1,MPIU_REAL,MPIU_MIN,PetscObjectComm((PetscObject)da3));
    1029:   MPI_Allreduce(&umax,max,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)da3));
    1030:   MPI_Allreduce(&usum,&gusum,1,MPIU_SCALAR,MPIU_SUM,PetscObjectComm((PetscObject)da3));
    1031:   *mean = PetscRealPart(gusum) / (mx*my);
    1032:   return(0);
    1033: }

    1035: static PetscErrorCode THISolveStatistics(THI thi,TS ts,PetscInt coarsened,const char name[])
    1036: {
    1037:   MPI_Comm       comm;
    1038:   DM             pack;
    1039:   Vec            X,X3,X2;

    1043:   PetscObjectGetComm((PetscObject)thi,&comm);
    1044:   TSGetDM(ts,&pack);
    1045:   TSGetSolution(ts,&X);
    1046:   DMCompositeGetAccess(pack,X,&X3,&X2);
    1047:   PetscPrintf(comm,"Solution statistics after solve: %s\n",name);
    1048:   {
    1049:     PetscInt            its,lits;
    1050:     SNESConvergedReason reason;
    1051:     SNES                snes;
    1052:     TSGetSNES(ts,&snes);
    1053:     SNESGetIterationNumber(snes,&its);
    1054:     SNESGetConvergedReason(snes,&reason);
    1055:     SNESGetLinearSolveIterations(snes,&lits);
    1056:     PetscPrintf(comm,"%s: Number of SNES iterations = %d, total linear iterations = %d\n",SNESConvergedReasons[reason],its,lits);
    1057:   }
    1058:   {
    1059:     PetscReal   nrm2,tmin[3]={1e100,1e100,1e100},tmax[3]={-1e100,-1e100,-1e100},min[3],max[3];
    1060:     PetscInt    i,j,m;
    1061:     PetscScalar *x;
    1062:     VecNorm(X3,NORM_2,&nrm2);
    1063:     VecGetLocalSize(X3,&m);
    1064:     VecGetArray(X3,&x);
    1065:     for (i=0; i<m; i+=2) {
    1066:       PetscReal u = PetscRealPart(x[i]),v = PetscRealPart(x[i+1]),c = PetscSqrtReal(u*u+v*v);
    1067:       tmin[0] = PetscMin(u,tmin[0]);
    1068:       tmin[1] = PetscMin(v,tmin[1]);
    1069:       tmin[2] = PetscMin(c,tmin[2]);
    1070:       tmax[0] = PetscMax(u,tmax[0]);
    1071:       tmax[1] = PetscMax(v,tmax[1]);
    1072:       tmax[2] = PetscMax(c,tmax[2]);
    1073:     }
    1074:     VecRestoreArray(X,&x);
    1075:     MPI_Allreduce(tmin,min,3,MPIU_REAL,MPIU_MIN,PetscObjectComm((PetscObject)thi));
    1076:     MPI_Allreduce(tmax,max,3,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)thi));
    1077:     /* Dimensionalize to meters/year */
    1078:     nrm2 *= thi->units->year / thi->units->meter;
    1079:     for (j=0; j<3; j++) {
    1080:       min[j] *= thi->units->year / thi->units->meter;
    1081:       max[j] *= thi->units->year / thi->units->meter;
    1082:     }
    1083:     PetscPrintf(comm,"|X|_2 %g   u in [%g, %g]   v in [%g, %g]   c in [%g, %g] \n",nrm2,min[0],max[0],min[1],max[1],min[2],max[2]);
    1084:     {
    1085:       PetscReal umin,umax,umean;
    1086:       THISurfaceStatistics(pack,X,&umin,&umax,&umean);
    1087:       umin  *= thi->units->year / thi->units->meter;
    1088:       umax  *= thi->units->year / thi->units->meter;
    1089:       umean *= thi->units->year / thi->units->meter;
    1090:       PetscPrintf(comm,"Surface statistics: u in [%12.6e, %12.6e] mean %12.6e\n",umin,umax,umean);
    1091:     }
    1092:     /* These values stay nondimensional */
    1093:     PetscPrintf(comm,"Global eta range   [%g, %g], converged range [%g, %g]\n",thi->eta.min,thi->eta.max,thi->eta.cmin,thi->eta.cmax);
    1094:     PetscPrintf(comm,"Global beta2 range [%g, %g], converged range [%g, %g]\n",thi->beta2.min,thi->beta2.max,thi->beta2.cmin,thi->beta2.cmax);
    1095:   }
    1096:   PetscPrintf(comm,"\n");
    1097:   DMCompositeRestoreAccess(pack,X,&X3,&X2);
    1098:   return(0);
    1099: }

    1101: static inline PetscInt DMDALocalIndex3D(DMDALocalInfo *info,PetscInt i,PetscInt j,PetscInt k)
    1102: {return ((i-info->gzs)*info->gym + (j-info->gys))*info->gxm + (k-info->gxs);}
    1103: static inline PetscInt DMDALocalIndex2D(DMDALocalInfo *info,PetscInt i,PetscInt j)
    1104: {return (i-info->gzs)*info->gym + (j-info->gys);}

    1106: static PetscErrorCode THIJacobianLocal_Momentum(DMDALocalInfo *info,const Node ***x,const PrmNode **prm,Mat B,Mat Bcpl,THI thi)
    1107: {
    1108:   PetscInt       xs,ys,xm,ym,zm,i,j,k,q,l,ll;
    1109:   PetscReal      hx,hy;

    1113:   xs = info->zs;
    1114:   ys = info->ys;
    1115:   xm = info->zm;
    1116:   ym = info->ym;
    1117:   zm = info->xm;
    1118:   hx = thi->Lx / info->mz;
    1119:   hy = thi->Ly / info->my;

    1121:   for (i=xs; i<xs+xm; i++) {
    1122:     for (j=ys; j<ys+ym; j++) {
    1123:       PrmNode pn[4],dpn[4][2];
    1124:       QuadExtract(prm,i,j,pn);
    1125:       QuadComputeGrad4(QuadQDeriv,hx,hy,pn,dpn);
    1126:       for (k=0; k<zm-1; k++) {
    1127:         Node        n[8];
    1128:         PetscReal   zn[8],etabase = 0;
    1129:         PetscScalar Ke[8*NODE_SIZE][8*NODE_SIZE],Kcpl[8*NODE_SIZE][4*PRMNODE_SIZE];
    1130:         PetscInt    ls = 0;

    1132:         PrmHexGetZ(pn,k,zm,zn);
    1133:         HexExtract(x,i,j,k,n);
    1134:         PetscMemzero(Ke,sizeof(Ke));
    1135:         PetscMemzero(Kcpl,sizeof(Kcpl));
    1136:         if (thi->no_slip && k == 0) {
    1137:           for (l=0; l<4; l++) n[l].u = n[l].v = 0;
    1138:           ls = 4;
    1139:         }
    1140:         for (q=0; q<8; q++) {
    1141:           PetscReal   dz[3],phi[8],dphi[8][3],jw,eta,deta;
    1142:           PetscScalar du[3],dv[3],u,v;
    1143:           HexGrad(HexQDeriv[q],zn,dz);
    1144:           HexComputeGeometry(q,hx,hy,dz,phi,dphi,&jw);
    1145:           PointwiseNonlinearity(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
    1146:           jw /= thi->rhog;      /* residuals are scaled by this factor */
    1147:           if (q == 0) etabase = eta;
    1148:           for (l=ls; l<8; l++) { /* test functions */
    1149:             const PetscReal pp=phi[l],*restrict dp = dphi[l];
    1150:             for (ll=ls; ll<8; ll++) { /* trial functions */
    1151:               const PetscReal *restrict dpl = dphi[ll];
    1152:               PetscScalar dgdu,dgdv;
    1153:               dgdu = 2.*du[0]*dpl[0] + dv[1]*dpl[0] + 0.5*(du[1]+dv[0])*dpl[1] + 0.5*du[2]*dpl[2];
    1154:               dgdv = 2.*dv[1]*dpl[1] + du[0]*dpl[1] + 0.5*(du[1]+dv[0])*dpl[0] + 0.5*dv[2]*dpl[2];
    1155:               /* Picard part */
    1156:               Ke[l*2+0][ll*2+0] += dp[0]*jw*eta*4.*dpl[0] + dp[1]*jw*eta*dpl[1] + dp[2]*jw*eta*dpl[2];
    1157:               Ke[l*2+0][ll*2+1] += dp[0]*jw*eta*2.*dpl[1] + dp[1]*jw*eta*dpl[0];
    1158:               Ke[l*2+1][ll*2+0] += dp[1]*jw*eta*2.*dpl[0] + dp[0]*jw*eta*dpl[1];
    1159:               Ke[l*2+1][ll*2+1] += dp[1]*jw*eta*4.*dpl[1] + dp[0]*jw*eta*dpl[0] + dp[2]*jw*eta*dpl[2];
    1160:               /* extra Newton terms */
    1161:               Ke[l*2+0][ll*2+0] += dp[0]*jw*deta*dgdu*(4.*du[0]+2.*dv[1]) + dp[1]*jw*deta*dgdu*(du[1]+dv[0]) + dp[2]*jw*deta*dgdu*du[2];
    1162:               Ke[l*2+0][ll*2+1] += dp[0]*jw*deta*dgdv*(4.*du[0]+2.*dv[1]) + dp[1]*jw*deta*dgdv*(du[1]+dv[0]) + dp[2]*jw*deta*dgdv*du[2];
    1163:               Ke[l*2+1][ll*2+0] += dp[1]*jw*deta*dgdu*(4.*dv[1]+2.*du[0]) + dp[0]*jw*deta*dgdu*(du[1]+dv[0]) + dp[2]*jw*deta*dgdu*dv[2];
    1164:               Ke[l*2+1][ll*2+1] += dp[1]*jw*deta*dgdv*(4.*dv[1]+2.*du[0]) + dp[0]*jw*deta*dgdv*(du[1]+dv[0]) + dp[2]*jw*deta*dgdv*dv[2];
    1165:               /* inertial part */
    1166:               Ke[l*2+0][ll*2+0] += pp*jw*thi->inertia*pp;
    1167:               Ke[l*2+1][ll*2+1] += pp*jw*thi->inertia*pp;
    1168:             }
    1169:             for (ll=0; ll<4; ll++) { /* Trial functions for surface/bed */
    1170:               const PetscReal dpl[] = {QuadQDeriv[q%4][ll][0]/hx, QuadQDeriv[q%4][ll][1]/hy}; /* surface = h + b */
    1171:               Kcpl[FieldIndex(Node,l,u)][FieldIndex(PrmNode,ll,h)] += pp*jw*thi->rhog*dpl[0];
    1172:               Kcpl[FieldIndex(Node,l,u)][FieldIndex(PrmNode,ll,b)] += pp*jw*thi->rhog*dpl[0];
    1173:               Kcpl[FieldIndex(Node,l,v)][FieldIndex(PrmNode,ll,h)] += pp*jw*thi->rhog*dpl[1];
    1174:               Kcpl[FieldIndex(Node,l,v)][FieldIndex(PrmNode,ll,b)] += pp*jw*thi->rhog*dpl[1];
    1175:             }
    1176:           }
    1177:         }
    1178:         if (k == 0) { /* on a bottom face */
    1179:           if (thi->no_slip) {
    1180:             const PetscReal   hz    = PetscRealPart(pn[0].h)/(zm-1);
    1181:             const PetscScalar diagu = 2*etabase/thi->rhog*(hx*hy/hz + hx*hz/hy + 4*hy*hz/hx),diagv = 2*etabase/thi->rhog*(hx*hy/hz + 4*hx*hz/hy + hy*hz/hx);
    1182:             Ke[0][0] = thi->dirichlet_scale*diagu;
    1183:             Ke[0][1] = 0;
    1184:             Ke[1][0] = 0;
    1185:             Ke[1][1] = thi->dirichlet_scale*diagv;
    1186:           } else {
    1187:             for (q=0; q<4; q++) { /* We remove the explicit scaling by 1/rhog because beta2 already has that scaling to be O(1) */
    1188:               const PetscReal jw = 0.25*hx*hy,*phi = QuadQInterp[q];
    1189:               PetscScalar     u  =0,v=0,rbeta2=0;
    1190:               PetscReal       beta2,dbeta2;
    1191:               for (l=0; l<4; l++) {
    1192:                 u      += phi[l]*n[l].u;
    1193:                 v      += phi[l]*n[l].v;
    1194:                 rbeta2 += phi[l]*pn[l].beta2;
    1195:               }
    1196:               THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
    1197:               for (l=0; l<4; l++) {
    1198:                 const PetscReal pp = phi[l];
    1199:                 for (ll=0; ll<4; ll++) {
    1200:                   const PetscReal ppl = phi[ll];
    1201:                   Ke[l*2+0][ll*2+0] += pp*jw*beta2*ppl + pp*jw*dbeta2*u*u*ppl;
    1202:                   Ke[l*2+0][ll*2+1] +=                   pp*jw*dbeta2*u*v*ppl;
    1203:                   Ke[l*2+1][ll*2+0] +=                   pp*jw*dbeta2*v*u*ppl;
    1204:                   Ke[l*2+1][ll*2+1] += pp*jw*beta2*ppl + pp*jw*dbeta2*v*v*ppl;
    1205:                 }
    1206:               }
    1207:             }
    1208:           }
    1209:         }
    1210:         {
    1211:           const PetscInt rc3blocked[8] = {
    1212:             DMDALocalIndex3D(info,i+0,j+0,k+0),
    1213:             DMDALocalIndex3D(info,i+1,j+0,k+0),
    1214:             DMDALocalIndex3D(info,i+1,j+1,k+0),
    1215:             DMDALocalIndex3D(info,i+0,j+1,k+0),
    1216:             DMDALocalIndex3D(info,i+0,j+0,k+1),
    1217:             DMDALocalIndex3D(info,i+1,j+0,k+1),
    1218:             DMDALocalIndex3D(info,i+1,j+1,k+1),
    1219:             DMDALocalIndex3D(info,i+0,j+1,k+1)
    1220:           },col2blocked[PRMNODE_SIZE*4] = {
    1221:             DMDALocalIndex2D(info,i+0,j+0),
    1222:             DMDALocalIndex2D(info,i+1,j+0),
    1223:             DMDALocalIndex2D(info,i+1,j+1),
    1224:             DMDALocalIndex2D(info,i+0,j+1)
    1225:           };
    1226: #if !defined COMPUTE_LOWER_TRIANGULAR /* fill in lower-triangular part, this is really cheap compared to computing the entries */
    1227:           for (l=0; l<8; l++) {
    1228:             for (ll=l+1; ll<8; ll++) {
    1229:               Ke[ll*2+0][l*2+0] = Ke[l*2+0][ll*2+0];
    1230:               Ke[ll*2+1][l*2+0] = Ke[l*2+0][ll*2+1];
    1231:               Ke[ll*2+0][l*2+1] = Ke[l*2+1][ll*2+0];
    1232:               Ke[ll*2+1][l*2+1] = Ke[l*2+1][ll*2+1];
    1233:             }
    1234:           }
    1235: #endif
    1236:           MatSetValuesBlockedLocal(B,8,rc3blocked,8,rc3blocked,&Ke[0][0],ADD_VALUES); /* velocity-velocity coupling can use blocked insertion */
    1237:           {                     /* The off-diagonal part cannot (yet) */
    1238:             PetscInt row3scalar[NODE_SIZE*8],col2scalar[PRMNODE_SIZE*4];
    1239:             for (l=0; l<8; l++) for (ll=0; ll<NODE_SIZE; ll++) row3scalar[l*NODE_SIZE+ll] = rc3blocked[l]*NODE_SIZE+ll;
    1240:             for (l=0; l<4; l++) for (ll=0; ll<PRMNODE_SIZE; ll++) col2scalar[l*PRMNODE_SIZE+ll] = col2blocked[l]*PRMNODE_SIZE+ll;
    1241:             MatSetValuesLocal(Bcpl,8*NODE_SIZE,row3scalar,4*PRMNODE_SIZE,col2scalar,&Kcpl[0][0],ADD_VALUES);
    1242:           }
    1243:         }
    1244:       }
    1245:     }
    1246:   }
    1247:   return(0);
    1248: }

    1250: static PetscErrorCode THIJacobianLocal_2D(DMDALocalInfo *info,const Node ***x3,const PrmNode **x2,const PrmNode **xdot2,PetscReal a,Mat B22,Mat B21,THI thi)
    1251: {
    1253:   PetscInt       xs,ys,xm,ym,zm,i,j,k;

    1256:   xs = info->zs;
    1257:   ys = info->ys;
    1258:   xm = info->zm;
    1259:   ym = info->ym;
    1260:   zm = info->xm;

    1262:   if (zm > 1024) SETERRQ(((PetscObject)info->da)->comm,PETSC_ERR_SUP,"Need to allocate more space");
    1263:   for (i=xs; i<xs+xm; i++) {
    1264:     for (j=ys; j<ys+ym; j++) {
    1265:       {                         /* Self-coupling */
    1266:         const PetscInt    row[]  = {DMDALocalIndex2D(info,i,j)};
    1267:         const PetscInt    col[]  = {DMDALocalIndex2D(info,i,j)};
    1268:         const PetscScalar vals[] = {
    1269:           a,0,0,
    1270:           0,a,0,
    1271:           0,0,a
    1272:         };
    1273:         MatSetValuesBlockedLocal(B22,1,row,1,col,vals,INSERT_VALUES);
    1274:       }
    1275:       for (k=0; k<zm; k++) {    /* Coupling to velocity problem */
    1276:         /* Use a cheaper quadrature than for residual evaluation, because it is much sparser */
    1277:         const PetscInt row[]  = {FieldIndex(PrmNode,DMDALocalIndex2D(info,i,j),h)};
    1278:         const PetscInt cols[] = {
    1279:           FieldIndex(Node,DMDALocalIndex3D(info,i-1,j,k),u),
    1280:           FieldIndex(Node,DMDALocalIndex3D(info,i  ,j,k),u),
    1281:           FieldIndex(Node,DMDALocalIndex3D(info,i+1,j,k),u),
    1282:           FieldIndex(Node,DMDALocalIndex3D(info,i,j-1,k),v),
    1283:           FieldIndex(Node,DMDALocalIndex3D(info,i,j  ,k),v),
    1284:           FieldIndex(Node,DMDALocalIndex3D(info,i,j+1,k),v)
    1285:         };
    1286:         const PetscScalar
    1287:           w  = (k && k<zm-1) ? 0.5 : 0.25,
    1288:           hW = w*(x2[i-1][j  ].h+x2[i  ][j  ].h)/(zm-1.),
    1289:           hE = w*(x2[i  ][j  ].h+x2[i+1][j  ].h)/(zm-1.),
    1290:           hS = w*(x2[i  ][j-1].h+x2[i  ][j  ].h)/(zm-1.),
    1291:           hN = w*(x2[i  ][j  ].h+x2[i  ][j+1].h)/(zm-1.);
    1292:         PetscScalar *vals,
    1293:                      vals_upwind[] = {((PetscRealPart(x3[i][j][k].u) > 0) ? -hW : 0),
    1294:                                       ((PetscRealPart(x3[i][j][k].u) > 0) ? +hE : -hW),
    1295:                                       ((PetscRealPart(x3[i][j][k].u) > 0) ?  0  : +hE),
    1296:                                       ((PetscRealPart(x3[i][j][k].v) > 0) ? -hS : 0),
    1297:                                       ((PetscRealPart(x3[i][j][k].v) > 0) ? +hN : -hS),
    1298:                                       ((PetscRealPart(x3[i][j][k].v) > 0) ?  0  : +hN)},
    1299:                      vals_centered[] = {-0.5*hW, 0.5*(-hW+hE), 0.5*hE,
    1300:                                         -0.5*hS, 0.5*(-hS+hN), 0.5*hN};
    1301:         vals = 1 ? vals_upwind : vals_centered;
    1302:         if (k == 0) {
    1303:           Node derate;
    1304:           THIErosion(thi,&x3[i][j][0],NULL,&derate);
    1305:           vals[1] -= derate.u;
    1306:           vals[4] -= derate.v;
    1307:         }
    1308:         MatSetValuesLocal(B21,1,row,6,cols,vals,INSERT_VALUES);
    1309:       }
    1310:     }
    1311:   }
    1312:   return(0);
    1313: }

    1315: static PetscErrorCode THIJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
    1316: {
    1318:   THI            thi = (THI)ctx;
    1319:   DM             pack,da3,da2;
    1320:   Vec            X3,X2,Xdot2;
    1321:   Mat            B11,B12,B21,B22;
    1322:   DMDALocalInfo  info3;
    1323:   IS             *isloc;
    1324:   const Node     ***x3;
    1325:   const PrmNode  **x2,**xdot2;

    1328:   TSGetDM(ts,&pack);
    1329:   DMCompositeGetEntries(pack,&da3,&da2);
    1330:   DMDAGetLocalInfo(da3,&info3);
    1331:   DMCompositeGetLocalVectors(pack,&X3,&X2);
    1332:   DMCompositeGetLocalVectors(pack,NULL,&Xdot2);
    1333:   DMCompositeScatter(pack,X,X3,X2);
    1334:   THIFixGhosts(thi,da3,da2,X3,X2);
    1335:   DMCompositeScatter(pack,Xdot,NULL,Xdot2);

    1337:   MatZeroEntries(B);

    1339:   DMCompositeGetLocalISs(pack,&isloc);
    1340:   MatGetLocalSubMatrix(B,isloc[0],isloc[0],&B11);
    1341:   MatGetLocalSubMatrix(B,isloc[0],isloc[1],&B12);
    1342:   MatGetLocalSubMatrix(B,isloc[1],isloc[0],&B21);
    1343:   MatGetLocalSubMatrix(B,isloc[1],isloc[1],&B22);

    1345:   DMDAVecGetArray(da3,X3,&x3);
    1346:   DMDAVecGetArray(da2,X2,&x2);
    1347:   DMDAVecGetArray(da2,Xdot2,&xdot2);

    1349:   THIJacobianLocal_Momentum(&info3,x3,x2,B11,B12,thi);

    1351:   /* Need to switch from ADD_VALUES to INSERT_VALUES */
    1352:   MatAssemblyBegin(B,MAT_FLUSH_ASSEMBLY);
    1353:   MatAssemblyEnd(B,MAT_FLUSH_ASSEMBLY);

    1355:   THIJacobianLocal_2D(&info3,x3,x2,xdot2,a,B22,B21,thi);

    1357:   DMDAVecRestoreArray(da3,X3,&x3);
    1358:   DMDAVecRestoreArray(da2,X2,&x2);
    1359:   DMDAVecRestoreArray(da2,Xdot2,&xdot2);

    1361:   MatRestoreLocalSubMatrix(B,isloc[0],isloc[0],&B11);
    1362:   MatRestoreLocalSubMatrix(B,isloc[0],isloc[1],&B12);
    1363:   MatRestoreLocalSubMatrix(B,isloc[1],isloc[0],&B21);
    1364:   MatRestoreLocalSubMatrix(B,isloc[1],isloc[1],&B22);
    1365:   ISDestroy(&isloc[0]);
    1366:   ISDestroy(&isloc[1]);
    1367:   PetscFree(isloc);

    1369:   DMCompositeRestoreLocalVectors(pack,&X3,&X2);
    1370:   DMCompositeRestoreLocalVectors(pack,0,&Xdot2);

    1372:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    1373:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    1374:   if (A != B) {
    1375:     MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    1376:     MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    1377:   }
    1378:   if (thi->verbose) {THIMatrixStatistics(thi,B,PETSC_VIEWER_STDOUT_WORLD);}
    1379:   return(0);
    1380: }

    1382: /* VTK's XML formats are so brain-dead that they can't handle multiple grids in the same file.  Since the communication
    1383:  * can be shared between the two grids, we write two files at once, one for velocity (living on a 3D grid defined by
    1384:  * h=thickness and b=bed) and another for all properties living on the 2D grid.
    1385:  */
    1386: static PetscErrorCode THIDAVecView_VTK_XML(THI thi,DM pack,Vec X,const char filename[],const char filename2[])
    1387: {
    1388:   const PetscInt dof   = NODE_SIZE,dof2 = PRMNODE_SIZE;
    1389:   Units          units = thi->units;
    1390:   MPI_Comm       comm;
    1392:   PetscViewer    viewer3,viewer2;
    1393:   PetscMPIInt    rank,size,tag,nn,nmax,nn2,nmax2;
    1394:   PetscInt       mx,my,mz,r,range[6];
    1395:   PetscScalar    *x,*x2;
    1396:   DM             da3,da2;
    1397:   Vec            X3,X2;

    1400:   PetscObjectGetComm((PetscObject)thi,&comm);
    1401:   DMCompositeGetEntries(pack,&da3,&da2);
    1402:   DMCompositeGetAccess(pack,X,&X3,&X2);
    1403:   DMDAGetInfo(da3,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
    1404:   MPI_Comm_size(comm,&size);
    1405:   MPI_Comm_rank(comm,&rank);
    1406:   PetscViewerASCIIOpen(comm,filename,&viewer3);
    1407:   PetscViewerASCIIOpen(comm,filename2,&viewer2);
    1408:   PetscViewerASCIIPrintf(viewer3,"<VTKFile type=\"StructuredGrid\" version=\"0.1\" byte_order=\"LittleEndian\">\n");
    1409:   PetscViewerASCIIPrintf(viewer2,"<VTKFile type=\"StructuredGrid\" version=\"0.1\" byte_order=\"LittleEndian\">\n");
    1410:   PetscViewerASCIIPrintf(viewer3,"  <StructuredGrid WholeExtent=\"%d %d %d %d %d %d\">\n",0,mz-1,0,my-1,0,mx-1);
    1411:   PetscViewerASCIIPrintf(viewer2,"  <StructuredGrid WholeExtent=\"%d %d %d %d %d %d\">\n",0,0,0,my-1,0,mx-1);

    1413:   DMDAGetCorners(da3,range,range+1,range+2,range+3,range+4,range+5);
    1414:   PetscMPIIntCast(range[3]*range[4]*range[5]*dof,&nn);
    1415:   MPI_Reduce(&nn,&nmax,1,MPI_INT,MPI_MAX,0,comm);
    1416:   PetscMPIIntCast(range[4]*range[5]*dof2,&nn2);
    1417:   MPI_Reduce(&nn2,&nmax2,1,MPI_INT,MPI_MAX,0,comm);
    1418:   tag  = ((PetscObject)viewer3)->tag;
    1419:   VecGetArrayRead(X3,(const PetscScalar**)&x);
    1420:   VecGetArrayRead(X2,(const PetscScalar**)&x2);
    1421:   if (rank == 0) {
    1422:     PetscScalar *array,*array2;
    1423:     PetscMalloc2(nmax,&array,nmax2,&array2);
    1424:     for (r=0; r<size; r++) {
    1425:       PetscInt    i,j,k,f,xs,xm,ys,ym,zs,zm;
    1426:       Node        *y3;
    1427:       PetscScalar (*y2)[PRMNODE_SIZE];
    1428:       MPI_Status status;

    1430:       if (r) {
    1431:         MPI_Recv(range,6,MPIU_INT,r,tag,comm,MPI_STATUS_IGNORE);
    1432:       }
    1433:       zs = range[0];ys = range[1];xs = range[2];zm = range[3];ym = range[4];xm = range[5];
    1434:       if (xm*ym*zm*dof > nmax) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"should not happen");
    1435:       if (r) {
    1436:         MPI_Recv(array,nmax,MPIU_SCALAR,r,tag,comm,&status);
    1437:         MPI_Get_count(&status,MPIU_SCALAR,&nn);
    1438:         if (nn != xm*ym*zm*dof) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"corrupt da3 send");
    1439:         y3   = (Node*)array;
    1440:         MPI_Recv(array2,nmax2,MPIU_SCALAR,r,tag,comm,&status);
    1441:         MPI_Get_count(&status,MPIU_SCALAR,&nn2);
    1442:         if (nn2 != xm*ym*dof2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"corrupt da2 send");
    1443:         y2 = (PetscScalar(*)[PRMNODE_SIZE])array2;
    1444:       } else {
    1445:         y3 = (Node*)x;
    1446:         y2 = (PetscScalar(*)[PRMNODE_SIZE])x2;
    1447:       }
    1448:       PetscViewerASCIIPrintf(viewer3,"    <Piece Extent=\"%D %D %D %D %D %D\">\n",zs,zs+zm-1,ys,ys+ym-1,xs,xs+xm-1);
    1449:       PetscViewerASCIIPrintf(viewer2,"    <Piece Extent=\"%d %d %D %D %D %D\">\n",0,0,ys,ys+ym-1,xs,xs+xm-1);

    1451:       PetscViewerASCIIPrintf(viewer3,"      <Points>\n");
    1452:       PetscViewerASCIIPrintf(viewer2,"      <Points>\n");
    1453:       PetscViewerASCIIPrintf(viewer3,"        <DataArray type=\"Float32\" NumberOfComponents=\"3\" format=\"ascii\">\n");
    1454:       PetscViewerASCIIPrintf(viewer2,"        <DataArray type=\"Float32\" NumberOfComponents=\"3\" format=\"ascii\">\n");
    1455:       for (i=xs; i<xs+xm; i++) {
    1456:         for (j=ys; j<ys+ym; j++) {
    1457:           PetscReal
    1458:             xx = thi->Lx*i/mx,
    1459:             yy = thi->Ly*j/my,
    1460:             b  = PetscRealPart(y2[i*ym+j][FieldOffset(PrmNode,b)]),
    1461:             h  = PetscRealPart(y2[i*ym+j][FieldOffset(PrmNode,h)]);
    1462:           for (k=zs; k<zs+zm; k++) {
    1463:             PetscReal zz = b + h*k/(mz-1.);
    1464:             PetscViewerASCIIPrintf(viewer3,"%f %f %f\n",xx,yy,zz);
    1465:           }
    1466:           PetscViewerASCIIPrintf(viewer2,"%f %f %f\n",xx,yy,(double)0.0);
    1467:         }
    1468:       }
    1469:       PetscViewerASCIIPrintf(viewer3,"        </DataArray>\n");
    1470:       PetscViewerASCIIPrintf(viewer2,"        </DataArray>\n");
    1471:       PetscViewerASCIIPrintf(viewer3,"      </Points>\n");
    1472:       PetscViewerASCIIPrintf(viewer2,"      </Points>\n");

    1474:       {                         /* Velocity and rank (3D) */
    1475:         PetscViewerASCIIPrintf(viewer3,"      <PointData>\n");
    1476:         PetscViewerASCIIPrintf(viewer3,"        <DataArray type=\"Float32\" Name=\"velocity\" NumberOfComponents=\"3\" format=\"ascii\">\n");
    1477:         for (i=0; i<nn/dof; i++) {
    1478:           PetscViewerASCIIPrintf(viewer3,"%f %f %f\n",PetscRealPart(y3[i].u)*units->year/units->meter,PetscRealPart(y3[i].v)*units->year/units->meter,0.0);
    1479:         }
    1480:         PetscViewerASCIIPrintf(viewer3,"        </DataArray>\n");

    1482:         PetscViewerASCIIPrintf(viewer3,"        <DataArray type=\"Int32\" Name=\"rank\" NumberOfComponents=\"1\" format=\"ascii\">\n");
    1483:         for (i=0; i<nn; i+=dof) {
    1484:           PetscViewerASCIIPrintf(viewer3,"%D\n",r);
    1485:         }
    1486:         PetscViewerASCIIPrintf(viewer3,"        </DataArray>\n");
    1487:         PetscViewerASCIIPrintf(viewer3,"      </PointData>\n");
    1488:       }

    1490:       {                         /* 2D */
    1491:         PetscViewerASCIIPrintf(viewer2,"      <PointData>\n");
    1492:         for (f=0; f<PRMNODE_SIZE; f++) {
    1493:           const char *fieldname;
    1494:           DMDAGetFieldName(da2,f,&fieldname);
    1495:           PetscViewerASCIIPrintf(viewer2,"        <DataArray type=\"Float32\" Name=\"%s\" format=\"ascii\">\n",fieldname);
    1496:           for (i=0; i<nn2/PRMNODE_SIZE; i++) {
    1497:             PetscViewerASCIIPrintf(viewer2,"%g\n",y2[i][f]);
    1498:           }
    1499:           PetscViewerASCIIPrintf(viewer2,"        </DataArray>\n");
    1500:         }
    1501:         PetscViewerASCIIPrintf(viewer2,"      </PointData>\n");
    1502:       }

    1504:       PetscViewerASCIIPrintf(viewer3,"    </Piece>\n");
    1505:       PetscViewerASCIIPrintf(viewer2,"    </Piece>\n");
    1506:     }
    1507:     PetscFree2(array,array2);
    1508:   } else {
    1509:     MPI_Send(range,6,MPIU_INT,0,tag,comm);
    1510:     MPI_Send(x,nn,MPIU_SCALAR,0,tag,comm);
    1511:     MPI_Send(x2,nn2,MPIU_SCALAR,0,tag,comm);
    1512:   }
    1513:   VecRestoreArrayRead(X3,(const PetscScalar**)&x);
    1514:   VecRestoreArrayRead(X2,(const PetscScalar**)&x2);
    1515:   PetscViewerASCIIPrintf(viewer3,"  </StructuredGrid>\n");
    1516:   PetscViewerASCIIPrintf(viewer2,"  </StructuredGrid>\n");

    1518:   DMCompositeRestoreAccess(pack,X,&X3,&X2);
    1519:   PetscViewerASCIIPrintf(viewer3,"</VTKFile>\n");
    1520:   PetscViewerASCIIPrintf(viewer2,"</VTKFile>\n");
    1521:   PetscViewerDestroy(&viewer3);
    1522:   PetscViewerDestroy(&viewer2);
    1523:   return(0);
    1524: }

    1526: static PetscErrorCode THITSMonitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
    1527: {
    1529:   THI            thi = (THI)ctx;
    1530:   DM             pack;
    1531:   char           filename3[PETSC_MAX_PATH_LEN],filename2[PETSC_MAX_PATH_LEN];

    1534:   if (step < 0) return(0); /* negative one is used to indicate an interpolated solution */
    1535:   PetscPrintf(PetscObjectComm((PetscObject)ts),"%3D: t=%g\n",step,(double)t);
    1536:   if (thi->monitor_interval && step % thi->monitor_interval) return(0);
    1537:   TSGetDM(ts,&pack);
    1538:   PetscSNPrintf(filename3,sizeof(filename3),"%s-3d-%03d.vts",thi->monitor_basename,step);
    1539:   PetscSNPrintf(filename2,sizeof(filename2),"%s-2d-%03d.vts",thi->monitor_basename,step);
    1540:   THIDAVecView_VTK_XML(thi,pack,X,filename3,filename2);
    1541:   return(0);
    1542: }

    1544: static PetscErrorCode THICreateDM3d(THI thi,DM *dm3d)
    1545: {
    1546:   MPI_Comm       comm;
    1547:   PetscInt       M    = 3,N = 3,P = 2;
    1548:   DM             da;

    1552:   PetscObjectGetComm((PetscObject)thi,&comm);
    1553:   PetscOptionsBegin(comm,NULL,"Grid resolution options","");
    1554:   {
    1555:     PetscOptionsInt("-M","Number of elements in x-direction on coarse level","",M,&M,NULL);
    1556:     N    = M;
    1557:     PetscOptionsInt("-N","Number of elements in y-direction on coarse level (if different from M)","",N,&N,NULL);
    1558:     PetscOptionsInt("-P","Number of elements in z-direction on coarse level","",P,&P,NULL);
    1559:   }
    1560:   PetscOptionsEnd();
    1561:   DMDACreate3d(comm,DM_BOUNDARY_NONE,DM_BOUNDARY_PERIODIC,DM_BOUNDARY_PERIODIC,DMDA_STENCIL_BOX,P,N,M,1,PETSC_DETERMINE,PETSC_DETERMINE,sizeof(Node)/sizeof(PetscScalar),1,0,0,0,&da);
    1562:   DMSetFromOptions(da);
    1563:   DMSetUp(da);
    1564:   DMDASetFieldName(da,0,"x-velocity");
    1565:   DMDASetFieldName(da,1,"y-velocity");
    1566:   *dm3d = da;
    1567:   return(0);
    1568: }

    1570: int main(int argc,char *argv[])
    1571: {
    1572:   MPI_Comm       comm;
    1573:   DM             pack,da3,da2;
    1574:   TS             ts;
    1575:   THI            thi;
    1576:   Vec            X;
    1577:   Mat            B;
    1578:   PetscInt       i,steps;
    1579:   PetscReal      ftime;

    1582:   PetscInitialize(&argc,&argv,0,help);if (ierr) return ierr;
    1583:   comm = PETSC_COMM_WORLD;

    1585:   THICreate(comm,&thi);
    1586:   THICreateDM3d(thi,&da3);
    1587:   {
    1588:     PetscInt        Mx,My,mx,my,s;
    1589:     DMDAStencilType st;
    1590:     DMDAGetInfo(da3,0, 0,&My,&Mx, 0,&my,&mx, 0,&s,0,0,0,&st);
    1591:     DMDACreate2d(PetscObjectComm((PetscObject)thi),DM_BOUNDARY_PERIODIC,DM_BOUNDARY_PERIODIC,st,My,Mx,my,mx,sizeof(PrmNode)/sizeof(PetscScalar),s,0,0,&da2);
    1592:     DMSetUp(da2);
    1593:   }

    1595:   PetscObjectSetName((PetscObject)da3,"3D_Velocity");
    1596:   DMSetOptionsPrefix(da3,"f3d_");
    1597:   DMDASetFieldName(da3,0,"u");
    1598:   DMDASetFieldName(da3,1,"v");
    1599:   PetscObjectSetName((PetscObject)da2,"2D_Fields");
    1600:   DMSetOptionsPrefix(da2,"f2d_");
    1601:   DMDASetFieldName(da2,0,"b");
    1602:   DMDASetFieldName(da2,1,"h");
    1603:   DMDASetFieldName(da2,2,"beta2");
    1604:   DMCompositeCreate(comm,&pack);
    1605:   DMCompositeAddDM(pack,da3);
    1606:   DMCompositeAddDM(pack,da2);
    1607:   DMDestroy(&da3);
    1608:   DMDestroy(&da2);
    1609:   DMSetUp(pack);
    1610:   DMCreateMatrix(pack,&B);
    1611:   MatSetOption(B,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_FALSE);
    1612:   MatSetOptionsPrefix(B,"thi_");

    1614:   for (i=0; i<thi->nlevels; i++) {
    1615:     PetscReal Lx = thi->Lx / thi->units->meter,Ly = thi->Ly / thi->units->meter,Lz = thi->Lz / thi->units->meter;
    1616:     PetscInt  Mx,My,Mz;
    1617:     DMCompositeGetEntries(pack,&da3,&da2);
    1618:     DMDAGetInfo(da3,0, &Mz,&My,&Mx, 0,0,0, 0,0,0,0,0,0);
    1619:     PetscPrintf(PetscObjectComm((PetscObject)thi),"Level %D domain size (m) %8.2g x %8.2g x %8.2g, num elements %3d x %3d x %3d (%8d), size (m) %g x %g x %g\n",i,Lx,Ly,Lz,Mx,My,Mz,Mx*My*Mz,Lx/Mx,Ly/My,1000./(Mz-1));
    1620:   }

    1622:   DMCreateGlobalVector(pack,&X);
    1623:   THIInitial(thi,pack,X);

    1625:   TSCreate(comm,&ts);
    1626:   TSSetDM(ts,pack);
    1627:   TSSetProblemType(ts,TS_NONLINEAR);
    1628:   TSMonitorSet(ts,THITSMonitor,thi,NULL);
    1629:   TSSetType(ts,TSTHETA);
    1630:   TSSetIFunction(ts,NULL,THIFunction,thi);
    1631:   TSSetIJacobian(ts,B,B,THIJacobian,thi);
    1632:   TSSetMaxTime(ts,10.0);
    1633:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
    1634:   TSSetSolution(ts,X);
    1635:   TSSetTimeStep(ts,1e-3);
    1636:   TSSetFromOptions(ts);

    1638:   TSSolve(ts,X);
    1639:   TSGetSolveTime(ts,&ftime);
    1640:   TSGetStepNumber(ts,&steps);
    1641:   PetscPrintf(PETSC_COMM_WORLD,"Steps %D  final time %g\n",steps,(double)ftime);

    1643:   if (0) {THISolveStatistics(thi,ts,0,"Full");}

    1645:   {
    1646:     PetscBool flg;
    1647:     char      filename[PETSC_MAX_PATH_LEN] = "";
    1648:     PetscOptionsGetString(NULL,NULL,"-o",filename,sizeof(filename),&flg);
    1649:     if (flg) {
    1650:       THIDAVecView_VTK_XML(thi,pack,X,filename,NULL);
    1651:     }
    1652:   }

    1654:   VecDestroy(&X);
    1655:   MatDestroy(&B);
    1656:   DMDestroy(&pack);
    1657:   TSDestroy(&ts);
    1658:   THIDestroy(&thi);
    1659:   PetscFinalize();
    1660:   return ierr;
    1661: }