Actual source code: ex48.c
petsc-3.12.4 2020-02-04
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      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: There are two compile-time options:

     46:   NO_SSE2:
     47:     If the host supports SSE2, we use integration code that has been vectorized with SSE2
     48:     intrinsics, unless this macro is defined.  The intrinsics speed up integration by about
     49:     30% on my architecture (P8700, gcc-4.5 snapshot).

     51:   COMPUTE_LOWER_TRIANGULAR:
     52:     The element matrices we assemble are lower-triangular so it is not necessary to compute
     53:     all entries explicitly.  If this macro is defined, the lower-triangular entries are
     54:     computed explicitly.

     56: */

     58: #if defined(PETSC_APPLE_FRAMEWORK)
     59: #import <PETSc/petscsnes.h>
     60: #import <PETSc/petsc/private/dmdaimpl.h>     /* There is not yet a public interface to manipulate dm->ops */
     61: #else

     63:  #include <petscsnes.h>
     64: #include <petsc/private/dmdaimpl.h>     /* There is not yet a public interface to manipulate dm->ops */
     65: #endif
     66: #include <ctype.h>              /* toupper() */

     68: #if defined(__cplusplus) || defined (PETSC_HAVE_WINDOWS_COMPILERS) || defined (__PGI)
     69: /*  c++ cannot handle  [_restrict_] notation like C does */
     70: #undef PETSC_RESTRICT
     71: #define PETSC_RESTRICT
     72: #endif

     74: #if defined __SSE2__
     75: #  include <emmintrin.h>
     76: #endif

     78: /* The SSE2 kernels are only for PetscScalar=double on architectures that support it */
     79: #if !defined NO_SSE2                           \
     80:      && !defined PETSC_USE_COMPLEX             \
     81:      && !defined PETSC_USE_REAL_SINGLE         \
     82:      && !defined PETSC_USE_REAL___FLOAT128     \
     83:      && !defined PETSC_USE_REAL___FP16         \
     84:      && defined __SSE2__
     85: #define USE_SSE2_KERNELS 1
     86: #else
     87: #define USE_SSE2_KERNELS 0
     88: #endif

     90: static PetscClassId THI_CLASSID;

     92: typedef enum {QUAD_GAUSS,QUAD_LOBATTO} QuadratureType;
     93: static const char      *QuadratureTypes[] = {"gauss","lobatto","QuadratureType","QUAD_",0};
     94: PETSC_UNUSED static const PetscReal HexQWeights[8]     = {1,1,1,1,1,1,1,1};
     95: PETSC_UNUSED static const PetscReal HexQNodes[]        = {-0.57735026918962573, 0.57735026918962573};
     96: #define G 0.57735026918962573
     97: #define H (0.5*(1.+G))
     98: #define L (0.5*(1.-G))
     99: #define M (-0.5)
    100: #define P (0.5)
    101: /* Special quadrature: Lobatto in horizontal, Gauss in vertical */
    102: static const PetscReal HexQInterp_Lobatto[8][8] = {{H,0,0,0,L,0,0,0},
    103:                                                    {0,H,0,0,0,L,0,0},
    104:                                                    {0,0,H,0,0,0,L,0},
    105:                                                    {0,0,0,H,0,0,0,L},
    106:                                                    {L,0,0,0,H,0,0,0},
    107:                                                    {0,L,0,0,0,H,0,0},
    108:                                                    {0,0,L,0,0,0,H,0},
    109:                                                    {0,0,0,L,0,0,0,H}};
    110: static const PetscReal HexQDeriv_Lobatto[8][8][3] = {
    111:   {{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}  },
    112:   {{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}    },
    113:   {{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}  },
    114:   {{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}},
    115:   {{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}  },
    116:   {{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}    },
    117:   {{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}  },
    118:   {{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}}};
    119: /* Stanndard Gauss */
    120: 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},
    121:                                                  {L*H*H,H*H*H,H*L*H,L*L*H, L*H*L,H*H*L,H*L*L,L*L*L},
    122:                                                  {L*L*H,H*L*H,H*H*H,L*H*H, L*L*L,H*L*L,H*H*L,L*H*L},
    123:                                                  {H*L*H,L*L*H,L*H*H,H*H*H, H*L*L,L*L*L,L*H*L,H*H*L},
    124:                                                  {H*H*L,L*H*L,L*L*L,H*L*L, H*H*H,L*H*H,L*L*H,H*L*H},
    125:                                                  {L*H*L,H*H*L,H*L*L,L*L*L, L*H*H,H*H*H,H*L*H,L*L*H},
    126:                                                  {L*L*L,H*L*L,H*H*L,L*H*L, L*L*H,H*L*H,H*H*H,L*H*H},
    127:                                                  {H*L*L,L*L*L,L*H*L,H*H*L, H*L*H,L*L*H,L*H*H,H*H*H}};
    128: static const PetscReal HexQDeriv_Gauss[8][8][3] = {
    129:   {{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}},
    130:   {{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}},
    131:   {{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}},
    132:   {{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}},
    133:   {{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}},
    134:   {{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}},
    135:   {{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}},
    136:   {{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}}};
    137: static const PetscReal (*HexQInterp)[8],(*HexQDeriv)[8][3];
    138: /* Standard 2x2 Gauss quadrature for the bottom layer. */
    139: static const PetscReal QuadQInterp[4][4] = {{H*H,L*H,L*L,H*L},
    140:                                             {L*H,H*H,H*L,L*L},
    141:                                             {L*L,H*L,H*H,L*H},
    142:                                             {H*L,L*L,L*H,H*H}};
    143: static const PetscReal QuadQDeriv[4][4][2] = {
    144:   {{M*H,M*H},{P*H,M*L},{P*L,P*L},{M*L,P*H}},
    145:   {{M*H,M*L},{P*H,M*H},{P*L,P*H},{M*L,P*L}},
    146:   {{M*L,M*L},{P*L,M*H},{P*H,P*H},{M*H,P*L}},
    147:   {{M*L,M*H},{P*L,M*L},{P*H,P*L},{M*H,P*H}}};
    148: #undef G
    149: #undef H
    150: #undef L
    151: #undef M
    152: #undef P

    154: #define HexExtract(x,i,j,k,n) do {              \
    155:     (n)[0] = (x)[i][j][k];                      \
    156:     (n)[1] = (x)[i+1][j][k];                    \
    157:     (n)[2] = (x)[i+1][j+1][k];                  \
    158:     (n)[3] = (x)[i][j+1][k];                    \
    159:     (n)[4] = (x)[i][j][k+1];                    \
    160:     (n)[5] = (x)[i+1][j][k+1];                  \
    161:     (n)[6] = (x)[i+1][j+1][k+1];                \
    162:     (n)[7] = (x)[i][j+1][k+1];                  \
    163:   } while (0)

    165: #define HexExtractRef(x,i,j,k,n) do {           \
    166:     (n)[0] = &(x)[i][j][k];                     \
    167:     (n)[1] = &(x)[i+1][j][k];                   \
    168:     (n)[2] = &(x)[i+1][j+1][k];                 \
    169:     (n)[3] = &(x)[i][j+1][k];                   \
    170:     (n)[4] = &(x)[i][j][k+1];                   \
    171:     (n)[5] = &(x)[i+1][j][k+1];                 \
    172:     (n)[6] = &(x)[i+1][j+1][k+1];               \
    173:     (n)[7] = &(x)[i][j+1][k+1];                 \
    174:   } while (0)

    176: #define QuadExtract(x,i,j,n) do {               \
    177:     (n)[0] = (x)[i][j];                         \
    178:     (n)[1] = (x)[i+1][j];                       \
    179:     (n)[2] = (x)[i+1][j+1];                     \
    180:     (n)[3] = (x)[i][j+1];                       \
    181:   } while (0)

    183: static void HexGrad(const PetscReal dphi[][3],const PetscReal zn[],PetscReal dz[])
    184: {
    185:   PetscInt i;
    186:   dz[0] = dz[1] = dz[2] = 0;
    187:   for (i=0; i<8; i++) {
    188:     dz[0] += dphi[i][0] * zn[i];
    189:     dz[1] += dphi[i][1] * zn[i];
    190:     dz[2] += dphi[i][2] * zn[i];
    191:   }
    192: }

    194: static void HexComputeGeometry(PetscInt q,PetscReal hx,PetscReal hy,const PetscReal dz[PETSC_RESTRICT],PetscReal phi[PETSC_RESTRICT],PetscReal dphi[PETSC_RESTRICT][3],PetscReal *PETSC_RESTRICT jw)
    195: {
    196:   const PetscReal jac[3][3]  = {{hx/2,0,0}, {0,hy/2,0}, {dz[0],dz[1],dz[2]}};
    197:   const PetscReal 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]}};
    198:   const PetscReal jdet       = jac[0][0]*jac[1][1]*jac[2][2];
    199:   PetscInt        i;

    201:   for (i=0; i<8; i++) {
    202:     const PetscReal *dphir = HexQDeriv[q][i];
    203:     phi[i]     = HexQInterp[q][i];
    204:     dphi[i][0] = dphir[0]*ijac[0][0] + dphir[1]*ijac[1][0] + dphir[2]*ijac[2][0];
    205:     dphi[i][1] = dphir[0]*ijac[0][1] + dphir[1]*ijac[1][1] + dphir[2]*ijac[2][1];
    206:     dphi[i][2] = dphir[0]*ijac[0][2] + dphir[1]*ijac[1][2] + dphir[2]*ijac[2][2];
    207:   }
    208:   *jw = 1.0 * jdet;
    209: }

    211: typedef struct _p_THI   *THI;
    212: typedef struct _n_Units *Units;

    214: typedef struct {
    215:   PetscScalar u,v;
    216: } Node;

    218: typedef struct {
    219:   PetscScalar b;                /* bed */
    220:   PetscScalar h;                /* thickness */
    221:   PetscScalar beta2;            /* friction */
    222: } PrmNode;

    224: typedef struct {
    225:   PetscReal min,max,cmin,cmax;
    226: } PRange;

    228: typedef enum {THIASSEMBLY_TRIDIAGONAL,THIASSEMBLY_FULL} THIAssemblyMode;

    230: struct _p_THI {
    231:   PETSCHEADER(int);
    232:   void      (*initialize)(THI,PetscReal x,PetscReal y,PrmNode *p);
    233:   PetscInt  zlevels;
    234:   PetscReal Lx,Ly,Lz;           /* Model domain */
    235:   PetscReal alpha;              /* Bed angle */
    236:   Units     units;
    237:   PetscReal dirichlet_scale;
    238:   PetscReal ssa_friction_scale;
    239:   PRange    eta;
    240:   PRange    beta2;
    241:   struct {
    242:     PetscReal Bd2,eps,exponent;
    243:   } viscosity;
    244:   struct {
    245:     PetscReal irefgam,eps2,exponent,refvel,epsvel;
    246:   } friction;
    247:   PetscReal rhog;
    248:   PetscBool no_slip;
    249:   PetscBool tridiagonal;
    250:   PetscBool coarse2d;
    251:   PetscBool verbose;
    252:   MatType   mattype;
    253: };

    255: struct _n_Units {
    256:   /* fundamental */
    257:   PetscReal meter;
    258:   PetscReal kilogram;
    259:   PetscReal second;
    260:   /* derived */
    261:   PetscReal Pascal;
    262:   PetscReal year;
    263: };

    265: static PetscErrorCode THIJacobianLocal_3D_Full(DMDALocalInfo*,Node***,Mat,Mat,THI);
    266: static PetscErrorCode THIJacobianLocal_3D_Tridiagonal(DMDALocalInfo*,Node***,Mat,Mat,THI);
    267: static PetscErrorCode THIJacobianLocal_2D(DMDALocalInfo*,Node**,Mat,Mat,THI);

    269: static void PrmHexGetZ(const PrmNode pn[],PetscInt k,PetscInt zm,PetscReal zn[])
    270: {
    271:   const PetscScalar zm1    = zm-1,
    272:                     znl[8] = {pn[0].b + pn[0].h*(PetscScalar)k/zm1,
    273:                               pn[1].b + pn[1].h*(PetscScalar)k/zm1,
    274:                               pn[2].b + pn[2].h*(PetscScalar)k/zm1,
    275:                               pn[3].b + pn[3].h*(PetscScalar)k/zm1,
    276:                               pn[0].b + pn[0].h*(PetscScalar)(k+1)/zm1,
    277:                               pn[1].b + pn[1].h*(PetscScalar)(k+1)/zm1,
    278:                               pn[2].b + pn[2].h*(PetscScalar)(k+1)/zm1,
    279:                               pn[3].b + pn[3].h*(PetscScalar)(k+1)/zm1};
    280:   PetscInt i;
    281:   for (i=0; i<8; i++) zn[i] = PetscRealPart(znl[i]);
    282: }

    284: /* Tests A and C are from the ISMIP-HOM paper (Pattyn et al. 2008) */
    285: static void THIInitialize_HOM_A(THI thi,PetscReal x,PetscReal y,PrmNode *p)
    286: {
    287:   Units     units = thi->units;
    288:   PetscReal s     = -x*PetscSinReal(thi->alpha);

    290:   p->b     = s - 1000*units->meter + 500*units->meter * PetscSinReal(x*2*PETSC_PI/thi->Lx) * PetscSinReal(y*2*PETSC_PI/thi->Ly);
    291:   p->h     = s - p->b;
    292:   p->beta2 = 1e30;
    293: }

    295: static void THIInitialize_HOM_C(THI thi,PetscReal x,PetscReal y,PrmNode *p)
    296: {
    297:   Units     units = thi->units;
    298:   PetscReal s     = -x*PetscSinReal(thi->alpha);

    300:   p->b = s - 1000*units->meter;
    301:   p->h = s - p->b;
    302:   /* tau_b = beta2 v   is a stress (Pa) */
    303:   p->beta2 = 1000 * (1 + PetscSinReal(x*2*PETSC_PI/thi->Lx)*PetscSinReal(y*2*PETSC_PI/thi->Ly)) * units->Pascal * units->year / units->meter;
    304: }

    306: /* These are just toys */

    308: /* Same bed as test A, free slip everywhere except for a discontinuous jump to a circular sticky region in the middle. */
    309: static void THIInitialize_HOM_X(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
    310: {
    311:   Units     units = thi->units;
    312:   PetscReal x     = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
    313:   PetscReal r     = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);
    314:   p->b     = s - 1000*units->meter + 500*units->meter*PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    315:   p->h     = s - p->b;
    316:   p->beta2 = 1000 * (r < 1 ? 2 : 0) * units->Pascal * units->year / units->meter;
    317: }

    319: /* Like Z, but with 200 meter cliffs */
    320: static void THIInitialize_HOM_Y(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
    321: {
    322:   Units     units = thi->units;
    323:   PetscReal x     = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
    324:   PetscReal r     = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);

    326:   p->b = s - 1000*units->meter + 500*units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    327:   if (PetscRealPart(p->b) > -700*units->meter) p->b += 200*units->meter;
    328:   p->h     = s - p->b;
    329:   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;
    330: }

    332: /* Same bed as A, smoothly varying slipperiness, similar to MATLAB's "sombrero" (uncorrelated with bathymetry) */
    333: static void THIInitialize_HOM_Z(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
    334: {
    335:   Units     units = thi->units;
    336:   PetscReal x     = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
    337:   PetscReal r     = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);

    339:   p->b     = s - 1000*units->meter + 500*units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    340:   p->h     = s - p->b;
    341:   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;
    342: }

    344: static void THIFriction(THI thi,PetscReal rbeta2,PetscReal gam,PetscReal *beta2,PetscReal *dbeta2)
    345: {
    346:   if (thi->friction.irefgam == 0) {
    347:     Units units = thi->units;
    348:     thi->friction.irefgam = 1./(0.5*PetscSqr(thi->friction.refvel * units->meter / units->year));
    349:     thi->friction.eps2    = 0.5*PetscSqr(thi->friction.epsvel * units->meter / units->year) * thi->friction.irefgam;
    350:   }
    351:   if (thi->friction.exponent == 0) {
    352:     *beta2  = rbeta2;
    353:     *dbeta2 = 0;
    354:   } else {
    355:     *beta2  = rbeta2 * PetscPowReal(thi->friction.eps2 + gam*thi->friction.irefgam,thi->friction.exponent);
    356:     *dbeta2 = thi->friction.exponent * *beta2 / (thi->friction.eps2 + gam*thi->friction.irefgam) * thi->friction.irefgam;
    357:   }
    358: }

    360: static void THIViscosity(THI thi,PetscReal gam,PetscReal *eta,PetscReal *deta)
    361: {
    362:   PetscReal Bd2,eps,exponent;
    363:   if (thi->viscosity.Bd2 == 0) {
    364:     Units units = thi->units;
    365:     const PetscReal
    366:       n = 3.,                                           /* Glen exponent */
    367:       p = 1. + 1./n,                                    /* for Stokes */
    368:       A = 1.e-16 * PetscPowReal(units->Pascal,-n) / units->year, /* softness parameter (Pa^{-n}/s) */
    369:       B = PetscPowReal(A,-1./n);                                 /* hardness parameter */
    370:     thi->viscosity.Bd2      = B/2;
    371:     thi->viscosity.exponent = (p-2)/2;
    372:     thi->viscosity.eps      = 0.5*PetscSqr(1e-5 / units->year);
    373:   }
    374:   Bd2      = thi->viscosity.Bd2;
    375:   exponent = thi->viscosity.exponent;
    376:   eps      = thi->viscosity.eps;
    377:   *eta     = Bd2 * PetscPowReal(eps + gam,exponent);
    378:   *deta    = exponent * (*eta) / (eps + gam);
    379: }

    381: static void RangeUpdate(PetscReal *min,PetscReal *max,PetscReal x)
    382: {
    383:   if (x < *min) *min = x;
    384:   if (x > *max) *max = x;
    385: }

    387: static void PRangeClear(PRange *p)
    388: {
    389:   p->cmin = p->min = 1e100;
    390:   p->cmax = p->max = -1e100;
    391: }

    393: static PetscErrorCode PRangeMinMax(PRange *p,PetscReal min,PetscReal max)
    394: {

    397:   p->cmin = min;
    398:   p->cmax = max;
    399:   if (min < p->min) p->min = min;
    400:   if (max > p->max) p->max = max;
    401:   return(0);
    402: }

    404: static PetscErrorCode THIDestroy(THI *thi)
    405: {

    409:   if (!*thi) return(0);
    410:   if (--((PetscObject)(*thi))->refct > 0) {*thi = 0; return(0);}
    411:   PetscFree((*thi)->units);
    412:   PetscFree((*thi)->mattype);
    413:   PetscHeaderDestroy(thi);
    414:   return(0);
    415: }

    417: static PetscErrorCode THICreate(MPI_Comm comm,THI *inthi)
    418: {
    419:   static PetscBool registered = PETSC_FALSE;
    420:   THI              thi;
    421:   Units            units;
    422:   PetscErrorCode   ierr;

    425:   *inthi = 0;
    426:   if (!registered) {
    427:     PetscClassIdRegister("Toy Hydrostatic Ice",&THI_CLASSID);
    428:     registered = PETSC_TRUE;
    429:   }
    430:   PetscHeaderCreate(thi,THI_CLASSID,"THI","Toy Hydrostatic Ice","",comm,THIDestroy,0);

    432:   PetscNew(&thi->units);
    433:   units           = thi->units;
    434:   units->meter    = 1e-2;
    435:   units->second   = 1e-7;
    436:   units->kilogram = 1e-12;

    438:   PetscOptionsBegin(comm,NULL,"Scaled units options","");
    439:   {
    440:     PetscOptionsReal("-units_meter","1 meter in scaled length units","",units->meter,&units->meter,NULL);
    441:     PetscOptionsReal("-units_second","1 second in scaled time units","",units->second,&units->second,NULL);
    442:     PetscOptionsReal("-units_kilogram","1 kilogram in scaled mass units","",units->kilogram,&units->kilogram,NULL);
    443:   }
    444:   PetscOptionsEnd();
    445:   units->Pascal = units->kilogram / (units->meter * PetscSqr(units->second));
    446:   units->year   = 31556926. * units->second; /* seconds per year */

    448:   thi->Lx              = 10.e3;
    449:   thi->Ly              = 10.e3;
    450:   thi->Lz              = 1000;
    451:   thi->dirichlet_scale = 1;
    452:   thi->verbose         = PETSC_FALSE;

    454:   PetscOptionsBegin(comm,NULL,"Toy Hydrostatic Ice options","");
    455:   {
    456:     QuadratureType quad       = QUAD_GAUSS;
    457:     char           homexp[]   = "A";
    458:     char           mtype[256] = MATSBAIJ;
    459:     PetscReal      L,m = 1.0;
    460:     PetscBool      flg;
    461:     L    = thi->Lx;
    462:     PetscOptionsReal("-thi_L","Domain size (m)","",L,&L,&flg);
    463:     if (flg) thi->Lx = thi->Ly = L;
    464:     PetscOptionsReal("-thi_Lx","X Domain size (m)","",thi->Lx,&thi->Lx,NULL);
    465:     PetscOptionsReal("-thi_Ly","Y Domain size (m)","",thi->Ly,&thi->Ly,NULL);
    466:     PetscOptionsReal("-thi_Lz","Z Domain size (m)","",thi->Lz,&thi->Lz,NULL);
    467:     PetscOptionsString("-thi_hom","ISMIP-HOM experiment (A or C)","",homexp,homexp,sizeof(homexp),NULL);
    468:     switch (homexp[0] = toupper(homexp[0])) {
    469:     case 'A':
    470:       thi->initialize = THIInitialize_HOM_A;
    471:       thi->no_slip    = PETSC_TRUE;
    472:       thi->alpha      = 0.5;
    473:       break;
    474:     case 'C':
    475:       thi->initialize = THIInitialize_HOM_C;
    476:       thi->no_slip    = PETSC_FALSE;
    477:       thi->alpha      = 0.1;
    478:       break;
    479:     case 'X':
    480:       thi->initialize = THIInitialize_HOM_X;
    481:       thi->no_slip    = PETSC_FALSE;
    482:       thi->alpha      = 0.3;
    483:       break;
    484:     case 'Y':
    485:       thi->initialize = THIInitialize_HOM_Y;
    486:       thi->no_slip    = PETSC_FALSE;
    487:       thi->alpha      = 0.5;
    488:       break;
    489:     case 'Z':
    490:       thi->initialize = THIInitialize_HOM_Z;
    491:       thi->no_slip    = PETSC_FALSE;
    492:       thi->alpha      = 0.5;
    493:       break;
    494:     default:
    495:       SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"HOM experiment '%c' not implemented",homexp[0]);
    496:     }
    497:     PetscOptionsEnum("-thi_quadrature","Quadrature to use for 3D elements","",QuadratureTypes,(PetscEnum)quad,(PetscEnum*)&quad,NULL);
    498:     switch (quad) {
    499:     case QUAD_GAUSS:
    500:       HexQInterp = HexQInterp_Gauss;
    501:       HexQDeriv  = HexQDeriv_Gauss;
    502:       break;
    503:     case QUAD_LOBATTO:
    504:       HexQInterp = HexQInterp_Lobatto;
    505:       HexQDeriv  = HexQDeriv_Lobatto;
    506:       break;
    507:     }
    508:     PetscOptionsReal("-thi_alpha","Bed angle (degrees)","",thi->alpha,&thi->alpha,NULL);

    510:     thi->friction.refvel = 100.;
    511:     thi->friction.epsvel = 1.;

    513:     PetscOptionsReal("-thi_friction_refvel","Reference velocity for sliding","",thi->friction.refvel,&thi->friction.refvel,NULL);
    514:     PetscOptionsReal("-thi_friction_epsvel","Regularization velocity for sliding","",thi->friction.epsvel,&thi->friction.epsvel,NULL);
    515:     PetscOptionsReal("-thi_friction_m","Friction exponent, 0=Coulomb, 1=Navier","",m,&m,NULL);

    517:     thi->friction.exponent = (m-1)/2;

    519:     PetscOptionsReal("-thi_dirichlet_scale","Scale Dirichlet boundary conditions by this factor","",thi->dirichlet_scale,&thi->dirichlet_scale,NULL);
    520:     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);
    521:     PetscOptionsBool("-thi_coarse2d","Use a 2D coarse space corresponding to SSA","",thi->coarse2d,&thi->coarse2d,NULL);
    522:     PetscOptionsBool("-thi_tridiagonal","Assemble a tridiagonal system (column coupling only) on the finest level","",thi->tridiagonal,&thi->tridiagonal,NULL);
    523:     PetscOptionsFList("-thi_mat_type","Matrix type","MatSetType",MatList,mtype,(char*)mtype,sizeof(mtype),NULL);
    524:     PetscStrallocpy(mtype,(char**)&thi->mattype);
    525:     PetscOptionsBool("-thi_verbose","Enable verbose output (like matrix sizes and statistics)","",thi->verbose,&thi->verbose,NULL);
    526:   }
    527:   PetscOptionsEnd();

    529:   /* dimensionalize */
    530:   thi->Lx    *= units->meter;
    531:   thi->Ly    *= units->meter;
    532:   thi->Lz    *= units->meter;
    533:   thi->alpha *= PETSC_PI / 180;

    535:   PRangeClear(&thi->eta);
    536:   PRangeClear(&thi->beta2);

    538:   {
    539:     PetscReal u       = 1000*units->meter/(3e7*units->second),
    540:               gradu   = u / (100*units->meter),eta,deta,
    541:               rho     = 910 * units->kilogram/PetscPowReal(units->meter,3),
    542:               grav    = 9.81 * units->meter/PetscSqr(units->second),
    543:               driving = rho * grav * PetscSinReal(thi->alpha) * 1000*units->meter;
    544:     THIViscosity(thi,0.5*gradu*gradu,&eta,&deta);
    545:     thi->rhog = rho * grav;
    546:     if (thi->verbose) {
    547:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Units: meter %8.2g  second %8.2g  kg %8.2g  Pa %8.2g\n",(double)units->meter,(double)units->second,(double)units->kilogram,(double)units->Pascal);
    548:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Domain (%6.2g,%6.2g,%6.2g), pressure %8.2g, driving stress %8.2g\n",(double)thi->Lx,(double)thi->Ly,(double)thi->Lz,(double)(rho*grav*1e3*units->meter),(double)driving);
    549:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Large velocity 1km/a %8.2g, velocity gradient %8.2g, eta %8.2g, stress %8.2g, ratio %8.2g\n",(double)u,(double)gradu,(double)eta,(double)(2*eta*gradu),(double)(2*eta*gradu/driving));
    550:       THIViscosity(thi,0.5*PetscSqr(1e-3*gradu),&eta,&deta);
    551:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Small velocity 1m/a  %8.2g, velocity gradient %8.2g, eta %8.2g, stress %8.2g, ratio %8.2g\n",(double)(1e-3*u),(double)(1e-3*gradu),(double)eta,(double)(2*eta*1e-3*gradu),(double)(2*eta*1e-3*gradu/driving));
    552:     }
    553:   }

    555:   *inthi = thi;
    556:   return(0);
    557: }

    559: static PetscErrorCode THIInitializePrm(THI thi,DM da2prm,Vec prm)
    560: {
    561:   PrmNode        **p;
    562:   PetscInt       i,j,xs,xm,ys,ym,mx,my;

    566:   DMDAGetGhostCorners(da2prm,&ys,&xs,0,&ym,&xm,0);
    567:   DMDAGetInfo(da2prm,0, &my,&mx,0, 0,0,0, 0,0,0,0,0,0);
    568:   DMDAVecGetArray(da2prm,prm,&p);
    569:   for (i=xs; i<xs+xm; i++) {
    570:     for (j=ys; j<ys+ym; j++) {
    571:       PetscReal xx = thi->Lx*i/mx,yy = thi->Ly*j/my;
    572:       thi->initialize(thi,xx,yy,&p[i][j]);
    573:     }
    574:   }
    575:   DMDAVecRestoreArray(da2prm,prm,&p);
    576:   return(0);
    577: }

    579: static PetscErrorCode THISetUpDM(THI thi,DM dm)
    580: {
    581:   PetscErrorCode  ierr;
    582:   PetscInt        refinelevel,coarsenlevel,level,dim,Mx,My,Mz,mx,my,s;
    583:   DMDAStencilType st;
    584:   DM              da2prm;
    585:   Vec             X;

    588:   DMDAGetInfo(dm,&dim, &Mz,&My,&Mx, 0,&my,&mx, 0,&s,0,0,0,&st);
    589:   if (dim == 2) {
    590:     DMDAGetInfo(dm,&dim, &My,&Mx,0, &my,&mx,0, 0,&s,0,0,0,&st);
    591:   }
    592:   DMGetRefineLevel(dm,&refinelevel);
    593:   DMGetCoarsenLevel(dm,&coarsenlevel);
    594:   level = refinelevel - coarsenlevel;
    595:   DMDACreate2d(PetscObjectComm((PetscObject)thi),DM_BOUNDARY_PERIODIC,DM_BOUNDARY_PERIODIC,st,My,Mx,my,mx,sizeof(PrmNode)/sizeof(PetscScalar),s,0,0,&da2prm);
    596:   DMSetUp(da2prm);
    597:   DMCreateLocalVector(da2prm,&X);
    598:   {
    599:     PetscReal Lx = thi->Lx / thi->units->meter,Ly = thi->Ly / thi->units->meter,Lz = thi->Lz / thi->units->meter;
    600:     if (dim == 2) {
    601:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Level %D domain size (m) %8.2g x %8.2g, num elements %D x %D (%D), size (m) %g x %g\n",level,(double)Lx,(double)Ly,Mx,My,Mx*My,(double)(Lx/Mx),(double)(Ly/My));
    602:     } else {
    603:       PetscPrintf(PetscObjectComm((PetscObject)thi),"Level %D domain size (m) %8.2g x %8.2g x %8.2g, num elements %D x %D x %D (%D), size (m) %g x %g x %g\n",level,(double)Lx,(double)Ly,(double)Lz,Mx,My,Mz,Mx*My*Mz,(double)(Lx/Mx),(double)(Ly/My),(double)(1000./(Mz-1)));
    604:     }
    605:   }
    606:   THIInitializePrm(thi,da2prm,X);
    607:   if (thi->tridiagonal) {       /* Reset coarse Jacobian evaluation */
    608:     DMDASNESSetJacobianLocal(dm,(DMDASNESJacobian)THIJacobianLocal_3D_Full,thi);
    609:   }
    610:   if (thi->coarse2d) {
    611:     DMDASNESSetJacobianLocal(dm,(DMDASNESJacobian)THIJacobianLocal_2D,thi);
    612:   }
    613:   PetscObjectCompose((PetscObject)dm,"DMDA2Prm",(PetscObject)da2prm);
    614:   PetscObjectCompose((PetscObject)dm,"DMDA2Prm_Vec",(PetscObject)X);
    615:   DMDestroy(&da2prm);
    616:   VecDestroy(&X);
    617:   return(0);
    618: }

    620: static PetscErrorCode DMCoarsenHook_THI(DM dmf,DM dmc,void *ctx)
    621: {
    622:   THI            thi = (THI)ctx;
    624:   PetscInt       rlevel,clevel;

    627:   THISetUpDM(thi,dmc);
    628:   DMGetRefineLevel(dmc,&rlevel);
    629:   DMGetCoarsenLevel(dmc,&clevel);
    630:   if (rlevel-clevel == 0) {DMSetMatType(dmc,MATAIJ);}
    631:   DMCoarsenHookAdd(dmc,DMCoarsenHook_THI,NULL,thi);
    632:   return(0);
    633: }

    635: static PetscErrorCode DMRefineHook_THI(DM dmc,DM dmf,void *ctx)
    636: {
    637:   THI            thi = (THI)ctx;

    641:   THISetUpDM(thi,dmf);
    642:   DMSetMatType(dmf,thi->mattype);
    643:   DMRefineHookAdd(dmf,DMRefineHook_THI,NULL,thi);
    644:   /* With grid sequencing, a formerly-refined DM will later be coarsened by PCSetUp_MG */
    645:   DMCoarsenHookAdd(dmf,DMCoarsenHook_THI,NULL,thi);
    646:   return(0);
    647: }

    649: static PetscErrorCode THIDAGetPrm(DM da,PrmNode ***prm)
    650: {
    652:   DM             da2prm;
    653:   Vec            X;

    656:   PetscObjectQuery((PetscObject)da,"DMDA2Prm",(PetscObject*)&da2prm);
    657:   if (!da2prm) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm composed with given DMDA");
    658:   PetscObjectQuery((PetscObject)da,"DMDA2Prm_Vec",(PetscObject*)&X);
    659:   if (!X) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm_Vec composed with given DMDA");
    660:   DMDAVecGetArray(da2prm,X,prm);
    661:   return(0);
    662: }

    664: static PetscErrorCode THIDARestorePrm(DM da,PrmNode ***prm)
    665: {
    667:   DM             da2prm;
    668:   Vec            X;

    671:   PetscObjectQuery((PetscObject)da,"DMDA2Prm",(PetscObject*)&da2prm);
    672:   if (!da2prm) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm composed with given DMDA");
    673:   PetscObjectQuery((PetscObject)da,"DMDA2Prm_Vec",(PetscObject*)&X);
    674:   if (!X) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm_Vec composed with given DMDA");
    675:   DMDAVecRestoreArray(da2prm,X,prm);
    676:   return(0);
    677: }

    679: static PetscErrorCode THIInitial(SNES snes,Vec X,void *ctx)
    680: {
    681:   THI            thi;
    682:   PetscInt       i,j,k,xs,xm,ys,ym,zs,zm,mx,my;
    683:   PetscReal      hx,hy;
    684:   PrmNode        **prm;
    685:   Node           ***x;
    687:   DM             da;

    690:   SNESGetDM(snes,&da);
    691:   DMGetApplicationContext(da,&thi);
    692:   DMDAGetInfo(da,0, 0,&my,&mx, 0,0,0, 0,0,0,0,0,0);
    693:   DMDAGetCorners(da,&zs,&ys,&xs,&zm,&ym,&xm);
    694:   DMDAVecGetArray(da,X,&x);
    695:   THIDAGetPrm(da,&prm);
    696:   hx   = thi->Lx / mx;
    697:   hy   = thi->Ly / my;
    698:   for (i=xs; i<xs+xm; i++) {
    699:     for (j=ys; j<ys+ym; j++) {
    700:       for (k=zs; k<zs+zm; k++) {
    701:         const PetscScalar zm1      = zm-1,
    702:                           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),
    703:                           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);
    704:         x[i][j][k].u = 0. * drivingx * prm[i][j].h*(PetscScalar)k/zm1;
    705:         x[i][j][k].v = 0. * drivingy * prm[i][j].h*(PetscScalar)k/zm1;
    706:       }
    707:     }
    708:   }
    709:   DMDAVecRestoreArray(da,X,&x);
    710:   THIDARestorePrm(da,&prm);
    711:   return(0);
    712: }

    714: static void PointwiseNonlinearity(THI thi,const Node n[PETSC_RESTRICT],const PetscReal phi[PETSC_RESTRICT],PetscReal dphi[PETSC_RESTRICT][3],PetscScalar *PETSC_RESTRICT u,PetscScalar *PETSC_RESTRICT v,PetscScalar du[PETSC_RESTRICT],PetscScalar dv[PETSC_RESTRICT],PetscReal *eta,PetscReal *deta)
    715: {
    716:   PetscInt    l,ll;
    717:   PetscScalar gam;

    719:   du[0] = du[1] = du[2] = 0;
    720:   dv[0] = dv[1] = dv[2] = 0;
    721:   *u    = 0;
    722:   *v    = 0;
    723:   for (l=0; l<8; l++) {
    724:     *u += phi[l] * n[l].u;
    725:     *v += phi[l] * n[l].v;
    726:     for (ll=0; ll<3; ll++) {
    727:       du[ll] += dphi[l][ll] * n[l].u;
    728:       dv[ll] += dphi[l][ll] * n[l].v;
    729:     }
    730:   }
    731:   gam = PetscSqr(du[0]) + PetscSqr(dv[1]) + du[0]*dv[1] + 0.25*PetscSqr(du[1]+dv[0]) + 0.25*PetscSqr(du[2]) + 0.25*PetscSqr(dv[2]);
    732:   THIViscosity(thi,PetscRealPart(gam),eta,deta);
    733: }

    735: static void PointwiseNonlinearity2D(THI thi,Node n[],PetscReal phi[],PetscReal dphi[4][2],PetscScalar *u,PetscScalar *v,PetscScalar du[],PetscScalar dv[],PetscReal *eta,PetscReal *deta)
    736: {
    737:   PetscInt    l,ll;
    738:   PetscScalar gam;

    740:   du[0] = du[1] = 0;
    741:   dv[0] = dv[1] = 0;
    742:   *u    = 0;
    743:   *v    = 0;
    744:   for (l=0; l<4; l++) {
    745:     *u += phi[l] * n[l].u;
    746:     *v += phi[l] * n[l].v;
    747:     for (ll=0; ll<2; ll++) {
    748:       du[ll] += dphi[l][ll] * n[l].u;
    749:       dv[ll] += dphi[l][ll] * n[l].v;
    750:     }
    751:   }
    752:   gam = PetscSqr(du[0]) + PetscSqr(dv[1]) + du[0]*dv[1] + 0.25*PetscSqr(du[1]+dv[0]);
    753:   THIViscosity(thi,PetscRealPart(gam),eta,deta);
    754: }

    756: static PetscErrorCode THIFunctionLocal(DMDALocalInfo *info,Node ***x,Node ***f,THI thi)
    757: {
    758:   PetscInt       xs,ys,xm,ym,zm,i,j,k,q,l;
    759:   PetscReal      hx,hy,etamin,etamax,beta2min,beta2max;
    760:   PrmNode        **prm;

    764:   xs = info->zs;
    765:   ys = info->ys;
    766:   xm = info->zm;
    767:   ym = info->ym;
    768:   zm = info->xm;
    769:   hx = thi->Lx / info->mz;
    770:   hy = thi->Ly / info->my;

    772:   etamin   = 1e100;
    773:   etamax   = 0;
    774:   beta2min = 1e100;
    775:   beta2max = 0;

    777:   THIDAGetPrm(info->da,&prm);

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

    854:   THIDARestorePrm(info->da,&prm);

    856:   PRangeMinMax(&thi->eta,etamin,etamax);
    857:   PRangeMinMax(&thi->beta2,beta2min,beta2max);
    858:   return(0);
    859: }

    861: static PetscErrorCode THIMatrixStatistics(THI thi,Mat B,PetscViewer viewer)
    862: {
    864:   PetscReal      nrm;
    865:   PetscInt       m;
    866:   PetscMPIInt    rank;

    869:   MatNorm(B,NORM_FROBENIUS,&nrm);
    870:   MatGetSize(B,&m,0);
    871:   MPI_Comm_rank(PetscObjectComm((PetscObject)B),&rank);
    872:   if (!rank) {
    873:     PetscScalar val0,val2;
    874:     MatGetValue(B,0,0,&val0);
    875:     MatGetValue(B,2,2,&val2);
    876:     PetscViewerASCIIPrintf(viewer,"Matrix dim %D norm %8.2e (0,0) %8.2e  (2,2) %8.2e %8.2e <= eta <= %8.2e %8.2e <= beta2 <= %8.2e\n",m,(double)nrm,(double)PetscRealPart(val0),(double)PetscRealPart(val2),(double)thi->eta.cmin,(double)thi->eta.cmax,(double)thi->beta2.cmin,(double)thi->beta2.cmax);
    877:   }
    878:   return(0);
    879: }

    881: static PetscErrorCode THISurfaceStatistics(DM da,Vec X,PetscReal *min,PetscReal *max,PetscReal *mean)
    882: {
    884:   Node           ***x;
    885:   PetscInt       i,j,xs,ys,zs,xm,ym,zm,mx,my,mz;
    886:   PetscReal      umin = 1e100,umax=-1e100;
    887:   PetscScalar    usum = 0.0,gusum;

    890:   *min = *max = *mean = 0;
    891:   DMDAGetInfo(da,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
    892:   DMDAGetCorners(da,&zs,&ys,&xs,&zm,&ym,&xm);
    893:   if (zs != 0 || zm != mz) SETERRQ(PETSC_COMM_SELF,1,"Unexpected decomposition");
    894:   DMDAVecGetArray(da,X,&x);
    895:   for (i=xs; i<xs+xm; i++) {
    896:     for (j=ys; j<ys+ym; j++) {
    897:       PetscReal u = PetscRealPart(x[i][j][zm-1].u);
    898:       RangeUpdate(&umin,&umax,u);
    899:       usum += u;
    900:     }
    901:   }
    902:   DMDAVecRestoreArray(da,X,&x);
    903:   MPI_Allreduce(&umin,min,1,MPIU_REAL,MPIU_MIN,PetscObjectComm((PetscObject)da));
    904:   MPI_Allreduce(&umax,max,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)da));
    905:   MPI_Allreduce(&usum,&gusum,1,MPIU_SCALAR,MPIU_SUM,PetscObjectComm((PetscObject)da));
    906:   *mean = PetscRealPart(gusum) / (mx*my);
    907:   return(0);
    908: }

    910: static PetscErrorCode THISolveStatistics(THI thi,SNES snes,PetscInt coarsened,const char name[])
    911: {
    912:   MPI_Comm       comm;
    913:   Vec            X;
    914:   DM             dm;

    918:   PetscObjectGetComm((PetscObject)thi,&comm);
    919:   SNESGetSolution(snes,&X);
    920:   SNESGetDM(snes,&dm);
    921:   PetscPrintf(comm,"Solution statistics after solve: %s\n",name);
    922:   {
    923:     PetscInt            its,lits;
    924:     SNESConvergedReason reason;
    925:     SNESGetIterationNumber(snes,&its);
    926:     SNESGetConvergedReason(snes,&reason);
    927:     SNESGetLinearSolveIterations(snes,&lits);
    928:     PetscPrintf(comm,"%s: Number of SNES iterations = %D, total linear iterations = %D\n",SNESConvergedReasons[reason],its,lits);
    929:   }
    930:   {
    931:     PetscReal         nrm2,tmin[3]={1e100,1e100,1e100},tmax[3]={-1e100,-1e100,-1e100},min[3],max[3];
    932:     PetscInt          i,j,m;
    933:     const PetscScalar *x;
    934:     VecNorm(X,NORM_2,&nrm2);
    935:     VecGetLocalSize(X,&m);
    936:     VecGetArrayRead(X,&x);
    937:     for (i=0; i<m; i+=2) {
    938:       PetscReal u = PetscRealPart(x[i]),v = PetscRealPart(x[i+1]),c = PetscSqrtReal(u*u+v*v);
    939:       tmin[0] = PetscMin(u,tmin[0]);
    940:       tmin[1] = PetscMin(v,tmin[1]);
    941:       tmin[2] = PetscMin(c,tmin[2]);
    942:       tmax[0] = PetscMax(u,tmax[0]);
    943:       tmax[1] = PetscMax(v,tmax[1]);
    944:       tmax[2] = PetscMax(c,tmax[2]);
    945:     }
    946:     VecRestoreArrayRead(X,&x);
    947:     MPI_Allreduce(tmin,min,3,MPIU_REAL,MPIU_MIN,PetscObjectComm((PetscObject)thi));
    948:     MPI_Allreduce(tmax,max,3,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)thi));
    949:     /* Dimensionalize to meters/year */
    950:     nrm2 *= thi->units->year / thi->units->meter;
    951:     for (j=0; j<3; j++) {
    952:       min[j] *= thi->units->year / thi->units->meter;
    953:       max[j] *= thi->units->year / thi->units->meter;
    954:     }
    955:     if (min[0] == 0.0) min[0] = 0.0;
    956:     PetscPrintf(comm,"|X|_2 %g   %g <= u <=  %g   %g <= v <=  %g   %g <= c <=  %g \n",(double)nrm2,(double)min[0],(double)max[0],(double)min[1],(double)max[1],(double)min[2],(double)max[2]);
    957:     {
    958:       PetscReal umin,umax,umean;
    959:       THISurfaceStatistics(dm,X,&umin,&umax,&umean);
    960:       umin  *= thi->units->year / thi->units->meter;
    961:       umax  *= thi->units->year / thi->units->meter;
    962:       umean *= thi->units->year / thi->units->meter;
    963:       PetscPrintf(comm,"Surface statistics: u in [%12.6e, %12.6e] mean %12.6e\n",(double)umin,(double)umax,(double)umean);
    964:     }
    965:     /* These values stay nondimensional */
    966:     PetscPrintf(comm,"Global eta range   %g to %g converged range %g to %g\n",(double)thi->eta.min,(double)thi->eta.max,(double)thi->eta.cmin,(double)thi->eta.cmax);
    967:     PetscPrintf(comm,"Global beta2 range %g to %g converged range %g to %g\n",(double)thi->beta2.min,(double)thi->beta2.max,(double)thi->beta2.cmin,(double)thi->beta2.cmax);
    968:   }
    969:   return(0);
    970: }

    972: static PetscErrorCode THIJacobianLocal_2D(DMDALocalInfo *info,Node **x,Mat J,Mat B,THI thi)
    973: {
    974:   PetscInt       xs,ys,xm,ym,i,j,q,l,ll;
    975:   PetscReal      hx,hy;
    976:   PrmNode        **prm;

    980:   xs = info->ys;
    981:   ys = info->xs;
    982:   xm = info->ym;
    983:   ym = info->xm;
    984:   hx = thi->Lx / info->my;
    985:   hy = thi->Ly / info->mx;

    987:   MatZeroEntries(B);
    988:   THIDAGetPrm(info->da,&prm);

    990:   for (i=xs; i<xs+xm; i++) {
    991:     for (j=ys; j<ys+ym; j++) {
    992:       Node        n[4];
    993:       PrmNode     pn[4];
    994:       PetscScalar Ke[4*2][4*2];
    995:       QuadExtract(prm,i,j,pn);
    996:       QuadExtract(x,i,j,n);
    997:       PetscMemzero(Ke,sizeof(Ke));
    998:       for (q=0; q<4; q++) {
    999:         PetscReal   phi[4],dphi[4][2],jw,eta,deta,beta2,dbeta2;
    1000:         PetscScalar u,v,du[2],dv[2],h = 0,rbeta2 = 0;
    1001:         for (l=0; l<4; l++) {
    1002:           phi[l]     = QuadQInterp[q][l];
    1003:           dphi[l][0] = QuadQDeriv[q][l][0]*2./hx;
    1004:           dphi[l][1] = QuadQDeriv[q][l][1]*2./hy;
    1005:           h         += phi[l] * pn[l].h;
    1006:           rbeta2    += phi[l] * pn[l].beta2;
    1007:         }
    1008:         jw = 0.25*hx*hy / thi->rhog; /* rhog is only scaling */
    1009:         PointwiseNonlinearity2D(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
    1010:         THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
    1011:         for (l=0; l<4; l++) {
    1012:           const PetscReal pp = phi[l],*dp = dphi[l];
    1013:           for (ll=0; ll<4; ll++) {
    1014:             const PetscReal ppl = phi[ll],*dpl = dphi[ll];
    1015:             PetscScalar     dgdu,dgdv;
    1016:             dgdu = 2.*du[0]*dpl[0] + dv[1]*dpl[0] + 0.5*(du[1]+dv[0])*dpl[1];
    1017:             dgdv = 2.*dv[1]*dpl[1] + du[0]*dpl[1] + 0.5*(du[1]+dv[0])*dpl[0];
    1018:             /* Picard part */
    1019:             Ke[l*2+0][ll*2+0] += dp[0]*jw*eta*4.*dpl[0] + dp[1]*jw*eta*dpl[1] + pp*jw*(beta2/h)*ppl*thi->ssa_friction_scale;
    1020:             Ke[l*2+0][ll*2+1] += dp[0]*jw*eta*2.*dpl[1] + dp[1]*jw*eta*dpl[0];
    1021:             Ke[l*2+1][ll*2+0] += dp[1]*jw*eta*2.*dpl[0] + dp[0]*jw*eta*dpl[1];
    1022:             Ke[l*2+1][ll*2+1] += dp[1]*jw*eta*4.*dpl[1] + dp[0]*jw*eta*dpl[0] + pp*jw*(beta2/h)*ppl*thi->ssa_friction_scale;
    1023:             /* extra Newton terms */
    1024:             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]) + pp*jw*(dbeta2/h)*u*u*ppl*thi->ssa_friction_scale;
    1025:             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]) + pp*jw*(dbeta2/h)*u*v*ppl*thi->ssa_friction_scale;
    1026:             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]) + pp*jw*(dbeta2/h)*v*u*ppl*thi->ssa_friction_scale;
    1027:             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]) + pp*jw*(dbeta2/h)*v*v*ppl*thi->ssa_friction_scale;
    1028:           }
    1029:         }
    1030:       }
    1031:       {
    1032:         const MatStencil rc[4] = {{0,i,j,0},{0,i+1,j,0},{0,i+1,j+1,0},{0,i,j+1,0}};
    1033:         MatSetValuesBlockedStencil(B,4,rc,4,rc,&Ke[0][0],ADD_VALUES);
    1034:       }
    1035:     }
    1036:   }
    1037:   THIDARestorePrm(info->da,&prm);

    1039:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    1040:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    1041:   MatSetOption(B,MAT_SYMMETRIC,PETSC_TRUE);
    1042:   if (thi->verbose) {THIMatrixStatistics(thi,B,PETSC_VIEWER_STDOUT_WORLD);}
    1043:   return(0);
    1044: }

    1046: static PetscErrorCode THIJacobianLocal_3D(DMDALocalInfo *info,Node ***x,Mat B,THI thi,THIAssemblyMode amode)
    1047: {
    1048:   PetscInt       xs,ys,xm,ym,zm,i,j,k,q,l,ll;
    1049:   PetscReal      hx,hy;
    1050:   PrmNode        **prm;

    1054:   xs = info->zs;
    1055:   ys = info->ys;
    1056:   xm = info->zm;
    1057:   ym = info->ym;
    1058:   zm = info->xm;
    1059:   hx = thi->Lx / info->mz;
    1060:   hy = thi->Ly / info->my;

    1062:   MatZeroEntries(B);
    1063:   MatSetOption(B,MAT_SUBSET_OFF_PROC_ENTRIES,PETSC_TRUE);
    1064:   THIDAGetPrm(info->da,&prm);

    1066:   for (i=xs; i<xs+xm; i++) {
    1067:     for (j=ys; j<ys+ym; j++) {
    1068:       PrmNode pn[4];
    1069:       QuadExtract(prm,i,j,pn);
    1070:       for (k=0; k<zm-1; k++) {
    1071:         Node        n[8];
    1072:         PetscReal   zn[8],etabase = 0;
    1073:         PetscScalar Ke[8*2][8*2];
    1074:         PetscInt    ls = 0;

    1076:         PrmHexGetZ(pn,k,zm,zn);
    1077:         HexExtract(x,i,j,k,n);
    1078:         PetscMemzero(Ke,sizeof(Ke));
    1079:         if (thi->no_slip && k == 0) {
    1080:           for (l=0; l<4; l++) n[l].u = n[l].v = 0;
    1081:           ls = 4;
    1082:         }
    1083:         for (q=0; q<8; q++) {
    1084:           PetscReal   dz[3],phi[8],dphi[8][3],jw,eta,deta;
    1085:           PetscScalar du[3],dv[3],u,v;
    1086:           HexGrad(HexQDeriv[q],zn,dz);
    1087:           HexComputeGeometry(q,hx,hy,dz,phi,dphi,&jw);
    1088:           PointwiseNonlinearity(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
    1089:           jw /= thi->rhog;      /* residuals are scaled by this factor */
    1090:           if (q == 0) etabase = eta;
    1091:           for (l=ls; l<8; l++) { /* test functions */
    1092:             const PetscReal *PETSC_RESTRICT dp = dphi[l];
    1093: #if USE_SSE2_KERNELS
    1094:             /* gcc (up to my 4.5 snapshot) is really bad at hoisting intrinsics so we do it manually */
    1095:             __m128d
    1096:               p4         = _mm_set1_pd(4),p2 = _mm_set1_pd(2),p05 = _mm_set1_pd(0.5),
    1097:               p42        = _mm_setr_pd(4,2),p24 = _mm_shuffle_pd(p42,p42,_MM_SHUFFLE2(0,1)),
    1098:               du0        = _mm_set1_pd(du[0]),du1 = _mm_set1_pd(du[1]),du2 = _mm_set1_pd(du[2]),
    1099:               dv0        = _mm_set1_pd(dv[0]),dv1 = _mm_set1_pd(dv[1]),dv2 = _mm_set1_pd(dv[2]),
    1100:               jweta      = _mm_set1_pd(jw*eta),jwdeta = _mm_set1_pd(jw*deta),
    1101:               dp0        = _mm_set1_pd(dp[0]),dp1 = _mm_set1_pd(dp[1]),dp2 = _mm_set1_pd(dp[2]),
    1102:               dp0jweta   = _mm_mul_pd(dp0,jweta),dp1jweta = _mm_mul_pd(dp1,jweta),dp2jweta = _mm_mul_pd(dp2,jweta),
    1103:               p4du0p2dv1 = _mm_add_pd(_mm_mul_pd(p4,du0),_mm_mul_pd(p2,dv1)), /* 4 du0 + 2 dv1 */
    1104:               p4dv1p2du0 = _mm_add_pd(_mm_mul_pd(p4,dv1),_mm_mul_pd(p2,du0)), /* 4 dv1 + 2 du0 */
    1105:               pdu2dv2    = _mm_unpacklo_pd(du2,dv2),                          /* [du2, dv2] */
    1106:               du1pdv0    = _mm_add_pd(du1,dv0),                               /* du1 + dv0 */
    1107:               t1         = _mm_mul_pd(dp0,p4du0p2dv1),                        /* dp0 (4 du0 + 2 dv1) */
    1108:               t2         = _mm_mul_pd(dp1,p4dv1p2du0);                        /* dp1 (4 dv1 + 2 du0) */

    1110: #endif
    1111: #if defined COMPUTE_LOWER_TRIANGULAR  /* The element matrices are always symmetric so computing the lower-triangular part is not necessary */
    1112:             for (ll=ls; ll<8; ll++) { /* trial functions */
    1113: #else
    1114:             for (ll=l; ll<8; ll++) {
    1115: #endif
    1116:               const PetscReal *PETSC_RESTRICT dpl = dphi[ll];
    1117:               if (amode == THIASSEMBLY_TRIDIAGONAL && (l-ll)%4) continue; /* these entries would not be inserted */
    1118: #if !USE_SSE2_KERNELS
    1119:               /* The analytic Jacobian in nice, easy-to-read form */
    1120:               {
    1121:                 PetscScalar dgdu,dgdv;
    1122:                 dgdu = 2.*du[0]*dpl[0] + dv[1]*dpl[0] + 0.5*(du[1]+dv[0])*dpl[1] + 0.5*du[2]*dpl[2];
    1123:                 dgdv = 2.*dv[1]*dpl[1] + du[0]*dpl[1] + 0.5*(du[1]+dv[0])*dpl[0] + 0.5*dv[2]*dpl[2];
    1124:                 /* Picard part */
    1125:                 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];
    1126:                 Ke[l*2+0][ll*2+1] += dp[0]*jw*eta*2.*dpl[1] + dp[1]*jw*eta*dpl[0];
    1127:                 Ke[l*2+1][ll*2+0] += dp[1]*jw*eta*2.*dpl[0] + dp[0]*jw*eta*dpl[1];
    1128:                 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];
    1129:                 /* extra Newton terms */
    1130:                 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];
    1131:                 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];
    1132:                 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];
    1133:                 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];
    1134:               }
    1135: #else
    1136:               /* This SSE2 code is an exact replica of above, but uses explicit packed instructions for some speed
    1137:               * benefit.  On my hardware, these intrinsics are almost twice as fast as above, reducing total assembly cost
    1138:               * by 25 to 30 percent. */
    1139:               {
    1140:                 __m128d
    1141:                   keu   = _mm_loadu_pd(&Ke[l*2+0][ll*2+0]),
    1142:                   kev   = _mm_loadu_pd(&Ke[l*2+1][ll*2+0]),
    1143:                   dpl01 = _mm_loadu_pd(&dpl[0]),dpl10 = _mm_shuffle_pd(dpl01,dpl01,_MM_SHUFFLE2(0,1)),dpl2 = _mm_set_sd(dpl[2]),
    1144:                   t0,t3,pdgduv;
    1145:                 keu = _mm_add_pd(keu,_mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp0jweta,p42),dpl01),
    1146:                                                 _mm_add_pd(_mm_mul_pd(dp1jweta,dpl10),
    1147:                                                            _mm_mul_pd(dp2jweta,dpl2))));
    1148:                 kev = _mm_add_pd(kev,_mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp1jweta,p24),dpl01),
    1149:                                                 _mm_add_pd(_mm_mul_pd(dp0jweta,dpl10),
    1150:                                                            _mm_mul_pd(dp2jweta,_mm_shuffle_pd(dpl2,dpl2,_MM_SHUFFLE2(0,1))))));
    1151:                 pdgduv = _mm_mul_pd(p05,_mm_add_pd(_mm_add_pd(_mm_mul_pd(p42,_mm_mul_pd(du0,dpl01)),
    1152:                                                               _mm_mul_pd(p24,_mm_mul_pd(dv1,dpl01))),
    1153:                                                    _mm_add_pd(_mm_mul_pd(du1pdv0,dpl10),
    1154:                                                               _mm_mul_pd(pdu2dv2,_mm_set1_pd(dpl[2]))))); /* [dgdu, dgdv] */
    1155:                 t0 = _mm_mul_pd(jwdeta,pdgduv);  /* jw deta [dgdu, dgdv] */
    1156:                 t3 = _mm_mul_pd(t0,du1pdv0);     /* t0 (du1 + dv0) */
    1157:                 _mm_storeu_pd(&Ke[l*2+0][ll*2+0],_mm_add_pd(keu,_mm_add_pd(_mm_mul_pd(t1,t0),
    1158:                                                                            _mm_add_pd(_mm_mul_pd(dp1,t3),
    1159:                                                                                       _mm_mul_pd(t0,_mm_mul_pd(dp2,du2))))));
    1160:                 _mm_storeu_pd(&Ke[l*2+1][ll*2+0],_mm_add_pd(kev,_mm_add_pd(_mm_mul_pd(t2,t0),
    1161:                                                                            _mm_add_pd(_mm_mul_pd(dp0,t3),
    1162:                                                                                       _mm_mul_pd(t0,_mm_mul_pd(dp2,dv2))))));
    1163:               }
    1164: #endif
    1165:             }
    1166:           }
    1167:         }
    1168:         if (k == 0) { /* on a bottom face */
    1169:           if (thi->no_slip) {
    1170:             const PetscReal   hz    = PetscRealPart(pn[0].h)/(zm-1);
    1171:             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);
    1172:             Ke[0][0] = thi->dirichlet_scale*diagu;
    1173:             Ke[1][1] = thi->dirichlet_scale*diagv;
    1174:           } else {
    1175:             for (q=0; q<4; q++) {
    1176:               const PetscReal jw = 0.25*hx*hy/thi->rhog,*phi = QuadQInterp[q];
    1177:               PetscScalar     u  =0,v=0,rbeta2=0;
    1178:               PetscReal       beta2,dbeta2;
    1179:               for (l=0; l<4; l++) {
    1180:                 u      += phi[l]*n[l].u;
    1181:                 v      += phi[l]*n[l].v;
    1182:                 rbeta2 += phi[l]*pn[l].beta2;
    1183:               }
    1184:               THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
    1185:               for (l=0; l<4; l++) {
    1186:                 const PetscReal pp = phi[l];
    1187:                 for (ll=0; ll<4; ll++) {
    1188:                   const PetscReal ppl = phi[ll];
    1189:                   Ke[l*2+0][ll*2+0] += pp*jw*beta2*ppl + pp*jw*dbeta2*u*u*ppl;
    1190:                   Ke[l*2+0][ll*2+1] +=                   pp*jw*dbeta2*u*v*ppl;
    1191:                   Ke[l*2+1][ll*2+0] +=                   pp*jw*dbeta2*v*u*ppl;
    1192:                   Ke[l*2+1][ll*2+1] += pp*jw*beta2*ppl + pp*jw*dbeta2*v*v*ppl;
    1193:                 }
    1194:               }
    1195:             }
    1196:           }
    1197:         }
    1198:         {
    1199:           const MatStencil rc[8] = {{i,j,k,0},{i+1,j,k,0},{i+1,j+1,k,0},{i,j+1,k,0},{i,j,k+1,0},{i+1,j,k+1,0},{i+1,j+1,k+1,0},{i,j+1,k+1,0}};
    1200:           if (amode == THIASSEMBLY_TRIDIAGONAL) {
    1201:             for (l=0; l<4; l++) { /* Copy out each of the blocks, discarding horizontal coupling */
    1202:               const PetscInt   l4     = l+4;
    1203:               const MatStencil rcl[2] = {{rc[l].k,rc[l].j,rc[l].i,0},{rc[l4].k,rc[l4].j,rc[l4].i,0}};
    1204: #if defined COMPUTE_LOWER_TRIANGULAR
    1205:               const PetscScalar Kel[4][4] = {{Ke[2*l+0][2*l+0] ,Ke[2*l+0][2*l+1] ,Ke[2*l+0][2*l4+0] ,Ke[2*l+0][2*l4+1]},
    1206:                                              {Ke[2*l+1][2*l+0] ,Ke[2*l+1][2*l+1] ,Ke[2*l+1][2*l4+0] ,Ke[2*l+1][2*l4+1]},
    1207:                                              {Ke[2*l4+0][2*l+0],Ke[2*l4+0][2*l+1],Ke[2*l4+0][2*l4+0],Ke[2*l4+0][2*l4+1]},
    1208:                                              {Ke[2*l4+1][2*l+0],Ke[2*l4+1][2*l+1],Ke[2*l4+1][2*l4+0],Ke[2*l4+1][2*l4+1]}};
    1209: #else
    1210:               /* Same as above except for the lower-left block */
    1211:               const PetscScalar Kel[4][4] = {{Ke[2*l+0][2*l+0] ,Ke[2*l+0][2*l+1] ,Ke[2*l+0][2*l4+0] ,Ke[2*l+0][2*l4+1]},
    1212:                                              {Ke[2*l+1][2*l+0] ,Ke[2*l+1][2*l+1] ,Ke[2*l+1][2*l4+0] ,Ke[2*l+1][2*l4+1]},
    1213:                                              {Ke[2*l+0][2*l4+0],Ke[2*l+1][2*l4+0],Ke[2*l4+0][2*l4+0],Ke[2*l4+0][2*l4+1]},
    1214:                                              {Ke[2*l+0][2*l4+1],Ke[2*l+1][2*l4+1],Ke[2*l4+1][2*l4+0],Ke[2*l4+1][2*l4+1]}};
    1215: #endif
    1216:               MatSetValuesBlockedStencil(B,2,rcl,2,rcl,&Kel[0][0],ADD_VALUES);
    1217:             }
    1218:           } else {
    1219: #if !defined COMPUTE_LOWER_TRIANGULAR /* fill in lower-triangular part, this is really cheap compared to computing the entries */
    1220:             for (l=0; l<8; l++) {
    1221:               for (ll=l+1; ll<8; ll++) {
    1222:                 Ke[ll*2+0][l*2+0] = Ke[l*2+0][ll*2+0];
    1223:                 Ke[ll*2+1][l*2+0] = Ke[l*2+0][ll*2+1];
    1224:                 Ke[ll*2+0][l*2+1] = Ke[l*2+1][ll*2+0];
    1225:                 Ke[ll*2+1][l*2+1] = Ke[l*2+1][ll*2+1];
    1226:               }
    1227:             }
    1228: #endif
    1229:             MatSetValuesBlockedStencil(B,8,rc,8,rc,&Ke[0][0],ADD_VALUES);
    1230:           }
    1231:         }
    1232:       }
    1233:     }
    1234:   }
    1235:   THIDARestorePrm(info->da,&prm);

    1237:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    1238:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    1239:   MatSetOption(B,MAT_SYMMETRIC,PETSC_TRUE);
    1240:   if (thi->verbose) {THIMatrixStatistics(thi,B,PETSC_VIEWER_STDOUT_WORLD);}
    1241:   return(0);
    1242: }

    1244: static PetscErrorCode THIJacobianLocal_3D_Full(DMDALocalInfo *info,Node ***x,Mat A,Mat B,THI thi)
    1245: {

    1249:   THIJacobianLocal_3D(info,x,B,thi,THIASSEMBLY_FULL);
    1250:   return(0);
    1251: }

    1253: static PetscErrorCode THIJacobianLocal_3D_Tridiagonal(DMDALocalInfo *info,Node ***x,Mat A,Mat B,THI thi)
    1254: {

    1258:   THIJacobianLocal_3D(info,x,B,thi,THIASSEMBLY_TRIDIAGONAL);
    1259:   return(0);
    1260: }

    1262: static PetscErrorCode DMRefineHierarchy_THI(DM dac0,PetscInt nlevels,DM hierarchy[])
    1263: {
    1264:   PetscErrorCode  ierr;
    1265:   THI             thi;
    1266:   PetscInt        dim,M,N,m,n,s,dof;
    1267:   DM              dac,daf;
    1268:   DMDAStencilType st;
    1269:   DM_DA           *ddf,*ddc;

    1272:   PetscObjectQuery((PetscObject)dac0,"THI",(PetscObject*)&thi);
    1273:   if (!thi) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Cannot refine this DMDA, missing composed THI instance");
    1274:   if (nlevels > 1) {
    1275:     DMRefineHierarchy(dac0,nlevels-1,hierarchy);
    1276:     dac  = hierarchy[nlevels-2];
    1277:   } else {
    1278:     dac = dac0;
    1279:   }
    1280:   DMDAGetInfo(dac,&dim, &N,&M,0, &n,&m,0, &dof,&s,0,0,0,&st);
    1281:   if (dim != 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"This function can only refine 2D DMDAs");

    1283:   /* Creates a 3D DMDA with the same map-plane layout as the 2D one, with contiguous columns */
    1284:   DMDACreate3d(PetscObjectComm((PetscObject)dac),DM_BOUNDARY_NONE,DM_BOUNDARY_PERIODIC,DM_BOUNDARY_PERIODIC,st,thi->zlevels,N,M,1,n,m,dof,s,NULL,NULL,NULL,&daf);
    1285:   DMSetUp(daf);

    1287:   daf->ops->creatematrix        = dac->ops->creatematrix;
    1288:   daf->ops->createinterpolation = dac->ops->createinterpolation;
    1289:   daf->ops->getcoloring         = dac->ops->getcoloring;
    1290:   ddf                           = (DM_DA*)daf->data;
    1291:   ddc                           = (DM_DA*)dac->data;
    1292:   ddf->interptype               = ddc->interptype;

    1294:   DMDASetFieldName(daf,0,"x-velocity");
    1295:   DMDASetFieldName(daf,1,"y-velocity");

    1297:   hierarchy[nlevels-1] = daf;
    1298:   return(0);
    1299: }

    1301: static PetscErrorCode DMCreateInterpolation_DA_THI(DM dac,DM daf,Mat *A,Vec *scale)
    1302: {
    1304:   PetscInt       dim;

    1311:   DMDAGetInfo(daf,&dim,0,0,0,0,0,0,0,0,0,0,0,0);
    1312:   if (dim  == 2) {
    1313:     /* We are in the 2D problem and use normal DMDA interpolation */
    1314:     DMCreateInterpolation(dac,daf,A,scale);
    1315:   } else {
    1316:     PetscInt i,j,k,xs,ys,zs,xm,ym,zm,mx,my,mz,rstart,cstart;
    1317:     Mat      B;

    1319:     DMDAGetInfo(daf,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
    1320:     DMDAGetCorners(daf,&zs,&ys,&xs,&zm,&ym,&xm);
    1321:     if (zs != 0) SETERRQ(PETSC_COMM_SELF,1,"unexpected");
    1322:     MatCreate(PetscObjectComm((PetscObject)daf),&B);
    1323:     MatSetSizes(B,xm*ym*zm,xm*ym,mx*my*mz,mx*my);

    1325:     MatSetType(B,MATAIJ);
    1326:     MatSeqAIJSetPreallocation(B,1,NULL);
    1327:     MatMPIAIJSetPreallocation(B,1,NULL,0,NULL);
    1328:     MatGetOwnershipRange(B,&rstart,NULL);
    1329:     MatGetOwnershipRangeColumn(B,&cstart,NULL);
    1330:     for (i=xs; i<xs+xm; i++) {
    1331:       for (j=ys; j<ys+ym; j++) {
    1332:         for (k=zs; k<zs+zm; k++) {
    1333:           PetscInt    i2  = i*ym+j,i3 = i2*zm+k;
    1334:           PetscScalar val = ((k == 0 || k == mz-1) ? 0.5 : 1.) / (mz-1.); /* Integration using trapezoid rule */
    1335:           MatSetValue(B,cstart+i3,rstart+i2,val,INSERT_VALUES);
    1336:         }
    1337:       }
    1338:     }
    1339:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    1340:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    1341:     MatCreateMAIJ(B,sizeof(Node)/sizeof(PetscScalar),A);
    1342:     MatDestroy(&B);
    1343:   }
    1344:   return(0);
    1345: }

    1347: static PetscErrorCode DMCreateMatrix_THI_Tridiagonal(DM da,Mat *J)
    1348: {
    1349:   PetscErrorCode         ierr;
    1350:   Mat                    A;
    1351:   PetscInt               xm,ym,zm,dim,dof = 2,starts[3],dims[3];
    1352:   ISLocalToGlobalMapping ltog;

    1355:   DMDAGetInfo(da,&dim, 0,0,0, 0,0,0, 0,0,0,0,0,0);
    1356:   if (dim != 3) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Expected DMDA to be 3D");
    1357:   DMDAGetCorners(da,0,0,0,&zm,&ym,&xm);
    1358:   DMGetLocalToGlobalMapping(da,&ltog);
    1359:   MatCreate(PetscObjectComm((PetscObject)da),&A);
    1360:   MatSetSizes(A,dof*xm*ym*zm,dof*xm*ym*zm,PETSC_DETERMINE,PETSC_DETERMINE);
    1361:   MatSetType(A,da->mattype);
    1362:   MatSetFromOptions(A);
    1363:   MatSeqAIJSetPreallocation(A,3*2,NULL);
    1364:   MatMPIAIJSetPreallocation(A,3*2,NULL,0,NULL);
    1365:   MatSeqBAIJSetPreallocation(A,2,3,NULL);
    1366:   MatMPIBAIJSetPreallocation(A,2,3,NULL,0,NULL);
    1367:   MatSeqSBAIJSetPreallocation(A,2,2,NULL);
    1368:   MatMPISBAIJSetPreallocation(A,2,2,NULL,0,NULL);
    1369:   MatSetLocalToGlobalMapping(A,ltog,ltog);
    1370:   DMDAGetGhostCorners(da,&starts[0],&starts[1],&starts[2],&dims[0],&dims[1],&dims[2]);
    1371:   MatSetStencil(A,dim,dims,starts,dof);
    1372:   *J   = A;
    1373:   return(0);
    1374: }

    1376: static PetscErrorCode THIDAVecView_VTK_XML(THI thi,DM da,Vec X,const char filename[])
    1377: {
    1378:   const PetscInt    dof   = 2;
    1379:   Units             units = thi->units;
    1380:   MPI_Comm          comm;
    1381:   PetscErrorCode    ierr;
    1382:   PetscViewer       viewer;
    1383:   PetscMPIInt       rank,size,tag,nn,nmax;
    1384:   PetscInt          mx,my,mz,r,range[6];
    1385:   const PetscScalar *x;

    1388:   PetscObjectGetComm((PetscObject)thi,&comm);
    1389:   DMDAGetInfo(da,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
    1390:   MPI_Comm_size(comm,&size);
    1391:   MPI_Comm_rank(comm,&rank);
    1392:   PetscViewerASCIIOpen(comm,filename,&viewer);
    1393:   PetscViewerASCIIPrintf(viewer,"<VTKFile type=\"StructuredGrid\" version=\"0.1\" byte_order=\"LittleEndian\">\n");
    1394:   PetscViewerASCIIPrintf(viewer,"  <StructuredGrid WholeExtent=\"%d %D %d %D %d %D\">\n",0,mz-1,0,my-1,0,mx-1);

    1396:   DMDAGetCorners(da,range,range+1,range+2,range+3,range+4,range+5);
    1397:   PetscMPIIntCast(range[3]*range[4]*range[5]*dof,&nn);
    1398:   MPI_Reduce(&nn,&nmax,1,MPI_INT,MPI_MAX,0,comm);
    1399:   tag  = ((PetscObject) viewer)->tag;
    1400:   VecGetArrayRead(X,&x);
    1401:   if (!rank) {
    1402:     PetscScalar *array;
    1403:     PetscMalloc1(nmax,&array);
    1404:     for (r=0; r<size; r++) {
    1405:       PetscInt          i,j,k,xs,xm,ys,ym,zs,zm;
    1406:       const PetscScalar *ptr;
    1407:       MPI_Status        status;
    1408:       if (r) {
    1409:         MPI_Recv(range,6,MPIU_INT,r,tag,comm,MPI_STATUS_IGNORE);
    1410:       }
    1411:       zs = range[0];ys = range[1];xs = range[2];zm = range[3];ym = range[4];xm = range[5];
    1412:       if (xm*ym*zm*dof > nmax) SETERRQ(PETSC_COMM_SELF,1,"should not happen");
    1413:       if (r) {
    1414:         MPI_Recv(array,nmax,MPIU_SCALAR,r,tag,comm,&status);
    1415:         MPI_Get_count(&status,MPIU_SCALAR,&nn);
    1416:         if (nn != xm*ym*zm*dof) SETERRQ(PETSC_COMM_SELF,1,"should not happen");
    1417:         ptr = array;
    1418:       } else ptr = x;
    1419:       PetscViewerASCIIPrintf(viewer,"    <Piece Extent=\"%D %D %D %D %D %D\">\n",zs,zs+zm-1,ys,ys+ym-1,xs,xs+xm-1);

    1421:       PetscViewerASCIIPrintf(viewer,"      <Points>\n");
    1422:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Float32\" NumberOfComponents=\"3\" format=\"ascii\">\n");
    1423:       for (i=xs; i<xs+xm; i++) {
    1424:         for (j=ys; j<ys+ym; j++) {
    1425:           for (k=zs; k<zs+zm; k++) {
    1426:             PrmNode   p;
    1427:             PetscReal xx = thi->Lx*i/mx,yy = thi->Ly*j/my,zz;
    1428:             thi->initialize(thi,xx,yy,&p);
    1429:             zz   = PetscRealPart(p.b) + PetscRealPart(p.h)*k/(mz-1);
    1430:             PetscViewerASCIIPrintf(viewer,"%f %f %f\n",(double)xx,(double)yy,(double)zz);
    1431:           }
    1432:         }
    1433:       }
    1434:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");
    1435:       PetscViewerASCIIPrintf(viewer,"      </Points>\n");

    1437:       PetscViewerASCIIPrintf(viewer,"      <PointData>\n");
    1438:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Float32\" Name=\"velocity\" NumberOfComponents=\"3\" format=\"ascii\">\n");
    1439:       for (i=0; i<nn; i+=dof) {
    1440:         PetscViewerASCIIPrintf(viewer,"%f %f %f\n",(double)(PetscRealPart(ptr[i])*units->year/units->meter),(double)(PetscRealPart(ptr[i+1])*units->year/units->meter),0.0);
    1441:       }
    1442:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");

    1444:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Int32\" Name=\"rank\" NumberOfComponents=\"1\" format=\"ascii\">\n");
    1445:       for (i=0; i<nn; i+=dof) {
    1446:         PetscViewerASCIIPrintf(viewer,"%D\n",r);
    1447:       }
    1448:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");
    1449:       PetscViewerASCIIPrintf(viewer,"      </PointData>\n");

    1451:       PetscViewerASCIIPrintf(viewer,"    </Piece>\n");
    1452:     }
    1453:     PetscFree(array);
    1454:   } else {
    1455:     MPI_Send(range,6,MPIU_INT,0,tag,comm);
    1456:     MPI_Send((PetscScalar*)x,nn,MPIU_SCALAR,0,tag,comm);
    1457:   }
    1458:   VecRestoreArrayRead(X,&x);
    1459:   PetscViewerASCIIPrintf(viewer,"  </StructuredGrid>\n");
    1460:   PetscViewerASCIIPrintf(viewer,"</VTKFile>\n");
    1461:   PetscViewerDestroy(&viewer);
    1462:   return(0);
    1463: }

    1465: int main(int argc,char *argv[])
    1466: {
    1467:   MPI_Comm       comm;
    1468:   THI            thi;
    1470:   DM             da;
    1471:   SNES           snes;

    1473:   PetscInitialize(&argc,&argv,0,help);if (ierr) return ierr;
    1474:   comm = PETSC_COMM_WORLD;

    1476:   THICreate(comm,&thi);
    1477:   {
    1478:     PetscInt M = 3,N = 3,P = 2;
    1479:     PetscOptionsBegin(comm,NULL,"Grid resolution options","");
    1480:     {
    1481:       PetscOptionsInt("-M","Number of elements in x-direction on coarse level","",M,&M,NULL);
    1482:       N    = M;
    1483:       PetscOptionsInt("-N","Number of elements in y-direction on coarse level (if different from M)","",N,&N,NULL);
    1484:       if (thi->coarse2d) {
    1485:         PetscOptionsInt("-zlevels","Number of elements in z-direction on fine level","",thi->zlevels,&thi->zlevels,NULL);
    1486:       } else {
    1487:         PetscOptionsInt("-P","Number of elements in z-direction on coarse level","",P,&P,NULL);
    1488:       }
    1489:     }
    1490:     PetscOptionsEnd();
    1491:     if (thi->coarse2d) {
    1492:       DMDACreate2d(comm,DM_BOUNDARY_PERIODIC,DM_BOUNDARY_PERIODIC,DMDA_STENCIL_BOX,N,M,PETSC_DETERMINE,PETSC_DETERMINE,sizeof(Node)/sizeof(PetscScalar),1,0,0,&da);
    1493:       DMSetFromOptions(da);
    1494:       DMSetUp(da);
    1495:       da->ops->refinehierarchy     = DMRefineHierarchy_THI;
    1496:       da->ops->createinterpolation = DMCreateInterpolation_DA_THI;

    1498:       PetscObjectCompose((PetscObject)da,"THI",(PetscObject)thi);
    1499:     } else {
    1500:       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);
    1501:       DMSetFromOptions(da);
    1502:       DMSetUp(da);
    1503:     }
    1504:     DMDASetFieldName(da,0,"x-velocity");
    1505:     DMDASetFieldName(da,1,"y-velocity");
    1506:   }
    1507:   THISetUpDM(thi,da);
    1508:   if (thi->tridiagonal) da->ops->creatematrix = DMCreateMatrix_THI_Tridiagonal;

    1510:   {                             /* Set the fine level matrix type if -da_refine */
    1511:     PetscInt rlevel,clevel;
    1512:     DMGetRefineLevel(da,&rlevel);
    1513:     DMGetCoarsenLevel(da,&clevel);
    1514:     if (rlevel - clevel > 0) {DMSetMatType(da,thi->mattype);}
    1515:   }

    1517:   DMDASNESSetFunctionLocal(da,ADD_VALUES,(DMDASNESFunction)THIFunctionLocal,thi);
    1518:   if (thi->tridiagonal) {
    1519:     DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)THIJacobianLocal_3D_Tridiagonal,thi);
    1520:   } else {
    1521:     DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)THIJacobianLocal_3D_Full,thi);
    1522:   }
    1523:   DMCoarsenHookAdd(da,DMCoarsenHook_THI,NULL,thi);
    1524:   DMRefineHookAdd(da,DMRefineHook_THI,NULL,thi);

    1526:   DMSetApplicationContext(da,thi);

    1528:   SNESCreate(comm,&snes);
    1529:   SNESSetDM(snes,da);
    1530:   DMDestroy(&da);
    1531:   SNESSetComputeInitialGuess(snes,THIInitial,NULL);
    1532:   SNESSetFromOptions(snes);

    1534:   SNESSolve(snes,NULL,NULL);

    1536:   THISolveStatistics(thi,snes,0,"Full");

    1538:   {
    1539:     PetscBool flg;
    1540:     char      filename[PETSC_MAX_PATH_LEN] = "";
    1541:     PetscOptionsGetString(NULL,NULL,"-o",filename,sizeof(filename),&flg);
    1542:     if (flg) {
    1543:       Vec X;
    1544:       DM  dm;
    1545:       SNESGetSolution(snes,&X);
    1546:       SNESGetDM(snes,&dm);
    1547:       THIDAVecView_VTK_XML(thi,dm,X,filename);
    1548:     }
    1549:   }

    1551:   DMDestroy(&da);
    1552:   SNESDestroy(&snes);
    1553:   THIDestroy(&thi);
    1554:   PetscFinalize();
    1555:   return ierr;
    1556: }

    1559: /*TEST

    1561:    build:
    1562:       requires: c99

    1564:    test:
    1565:       args: -M 6 -P 4 -da_refine 1 -snes_monitor_short -snes_converged_reason -ksp_monitor_short -ksp_converged_reason -thi_mat_type sbaij -ksp_type fgmres -pc_type mg -pc_mg_type full -mg_levels_ksp_type gmres -mg_levels_ksp_max_it 1 -mg_levels_pc_type icc
    1566:       requires: !single

    1568:    test:
    1569:       suffix: 2
    1570:       nsize: 2
    1571:       args: -M 6 -P 4 -thi_hom z -snes_monitor_short -snes_converged_reason -ksp_monitor_short -ksp_converged_reason -thi_mat_type sbaij -ksp_type fgmres -pc_type mg -pc_mg_type full -mg_levels_ksp_type gmres -mg_levels_ksp_max_it 1 -mg_levels_pc_type asm -mg_levels_pc_asm_blocks 6 -mg_levels_0_pc_type redundant -snes_grid_sequence 1 -mat_partitioning_type current -ksp_atol -1
    1572:       requires: !single

    1574:    test:
    1575:       suffix: 3
    1576:       nsize: 3
    1577:       args: -M 7 -P 4 -thi_hom z -da_refine 1 -snes_monitor_short -snes_converged_reason -ksp_monitor_short -ksp_converged_reason -thi_mat_type baij -ksp_type fgmres -pc_type mg -pc_mg_type full -mg_levels_pc_asm_type restrict -mg_levels_pc_type asm -mg_levels_pc_asm_blocks 9 -mg_levels_ksp_type gmres -mg_levels_ksp_max_it 1 -mat_partitioning_type current
    1578:       requires: !single

    1580:    test:
    1581:       suffix: 4
    1582:       nsize: 6
    1583:       args: -M 4 -P 2 -da_refine_hierarchy_x 1,1,3 -da_refine_hierarchy_y 2,2,1 -da_refine_hierarchy_z 2,2,1 -snes_grid_sequence 3 -ksp_converged_reason -ksp_type fgmres -ksp_rtol 1e-2 -pc_type mg -mg_levels_ksp_type gmres -mg_levels_ksp_max_it 1 -mg_levels_pc_type bjacobi -mg_levels_1_sub_pc_type cholesky -pc_mg_type multiplicative -snes_converged_reason -snes_stol 1e-12 -thi_L 80e3 -thi_alpha 0.05 -thi_friction_m 1 -thi_hom x -snes_view -mg_levels_0_pc_type redundant -mg_levels_0_ksp_type preonly -ksp_atol -1
    1584:       requires: !single

    1586:    test:
    1587:       suffix: 5
    1588:       nsize: 6
    1589:       args: -M 12 -P 5 -snes_monitor_short -ksp_converged_reason -pc_type asm -pc_asm_type restrict -dm_mat_type {{aij baij sbaij}}
    1590:       requires: !single

    1592: TEST*/