Actual source code: ex48.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: 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>
     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 && !defined PETSC_USE_COMPLEX && !defined PETSC_USE_REAL_SINGLE && !defined PETSC_USE_REAL___FLOAT128 && !defined PETSC_USE_REAL___FP16 && defined __SSE2__
     80:   #define USE_SSE2_KERNELS 1
     81: #else
     82:   #define USE_SSE2_KERNELS 0
     83: #endif

     85: static PetscClassId THI_CLASSID;

     87: typedef enum {
     88:   QUAD_GAUSS,
     89:   QUAD_LOBATTO
     90: } QuadratureType;
     91: static const char                  *QuadratureTypes[] = {"gauss", "lobatto", "QuadratureType", "QUAD_", 0};
     92: PETSC_UNUSED static const PetscReal HexQWeights[8]    = {1, 1, 1, 1, 1, 1, 1, 1};
     93: PETSC_UNUSED static const PetscReal HexQNodes[]       = {-0.57735026918962573, 0.57735026918962573};
     94: #define G 0.57735026918962573
     95: #define H (0.5 * (1. + G))
     96: #define L (0.5 * (1. - G))
     97: #define M (-0.5)
     98: #define P (0.5)
     99: /* Special quadrature: Lobatto in horizontal, Gauss in vertical */
    100: static const PetscReal HexQInterp_Lobatto[8][8] = {
    101:   {H, 0, 0, 0, L, 0, 0, 0},
    102:   {0, H, 0, 0, 0, L, 0, 0},
    103:   {0, 0, H, 0, 0, 0, L, 0},
    104:   {0, 0, 0, H, 0, 0, 0, L},
    105:   {L, 0, 0, 0, H, 0, 0, 0},
    106:   {0, L, 0, 0, 0, H, 0, 0},
    107:   {0, 0, L, 0, 0, 0, H, 0},
    108:   {0, 0, 0, L, 0, 0, 0, H}
    109: };
    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: };
    120: /* Stanndard Gauss */
    121: static const PetscReal HexQInterp_Gauss[8][8] = {
    122:   {H * H * H, L *H *H, L *L *H, H *L *H, H *H *L, L *H *L, L *L *L, H *L *L},
    123:   {L * H * H, H *H *H, H *L *H, L *L *H, L *H *L, H *H *L, H *L *L, L *L *L},
    124:   {L * L * H, H *L *H, H *H *H, L *H *H, L *L *L, H *L *L, H *H *L, L *H *L},
    125:   {H * L * H, L *L *H, L *H *H, H *H *H, H *L *L, L *L *L, L *H *L, H *H *L},
    126:   {H * H * L, L *H *L, L *L *L, H *L *L, H *H *H, L *H *H, L *L *H, H *L *H},
    127:   {L * H * L, H *H *L, H *L *L, L *L *L, L *H *H, H *H *H, H *L *H, L *L *H},
    128:   {L * L * L, H *L *L, H *H *L, L *H *L, L *L *H, H *L *H, H *H *H, L *H *H},
    129:   {H * L * L, L *L *L, L *H *L, H *H *L, H *L *H, L *L *H, L *H *H, H *H *H}
    130: };
    131: static const PetscReal HexQDeriv_Gauss[8][8][3] = {
    132:   {{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}},
    133:   {{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}},
    134:   {{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}},
    135:   {{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}},
    136:   {{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}},
    137:   {{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}},
    138:   {{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}},
    139:   {{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}}
    140: };
    141: static const PetscReal (*HexQInterp)[8], (*HexQDeriv)[8][3];
    142: /* Standard 2x2 Gauss quadrature for the bottom layer. */
    143: static const PetscReal QuadQInterp[4][4] = {
    144:   {H * H, L *H, L *L, H *L},
    145:   {L * H, H *H, H *L, L *L},
    146:   {L * L, H *L, H *H, L *H},
    147:   {H * L, L *L, L *H, H *H}
    148: };
    149: static const PetscReal QuadQDeriv[4][4][2] = {
    150:   {{M * H, M *H}, {P * H, M *L}, {P * L, P *L}, {M * L, P *H}},
    151:   {{M * H, M *L}, {P * H, M *H}, {P * L, P *H}, {M * L, P *L}},
    152:   {{M * L, M *L}, {P * L, M *H}, {P * H, P *H}, {M * H, P *L}},
    153:   {{M * L, M *H}, {P * L, M *L}, {P * H, P *L}, {M * H, P *H}}
    154: };
    155: #undef G
    156: #undef H
    157: #undef L
    158: #undef M
    159: #undef P

    161: #define HexExtract(x, i, j, k, n) \
    162:   do { \
    163:     (n)[0] = (x)[i][j][k]; \
    164:     (n)[1] = (x)[i + 1][j][k]; \
    165:     (n)[2] = (x)[i + 1][j + 1][k]; \
    166:     (n)[3] = (x)[i][j + 1][k]; \
    167:     (n)[4] = (x)[i][j][k + 1]; \
    168:     (n)[5] = (x)[i + 1][j][k + 1]; \
    169:     (n)[6] = (x)[i + 1][j + 1][k + 1]; \
    170:     (n)[7] = (x)[i][j + 1][k + 1]; \
    171:   } while (0)

    173: #define HexExtractRef(x, i, j, k, n) \
    174:   do { \
    175:     (n)[0] = &(x)[i][j][k]; \
    176:     (n)[1] = &(x)[i + 1][j][k]; \
    177:     (n)[2] = &(x)[i + 1][j + 1][k]; \
    178:     (n)[3] = &(x)[i][j + 1][k]; \
    179:     (n)[4] = &(x)[i][j][k + 1]; \
    180:     (n)[5] = &(x)[i + 1][j][k + 1]; \
    181:     (n)[6] = &(x)[i + 1][j + 1][k + 1]; \
    182:     (n)[7] = &(x)[i][j + 1][k + 1]; \
    183:   } while (0)

    185: #define QuadExtract(x, i, j, n) \
    186:   do { \
    187:     (n)[0] = (x)[i][j]; \
    188:     (n)[1] = (x)[i + 1][j]; \
    189:     (n)[2] = (x)[i + 1][j + 1]; \
    190:     (n)[3] = (x)[i][j + 1]; \
    191:   } while (0)

    193: static void HexGrad(const PetscReal dphi[][3], const PetscReal zn[], PetscReal dz[])
    194: {
    195:   PetscInt i;
    196:   dz[0] = dz[1] = dz[2] = 0;
    197:   for (i = 0; i < 8; i++) {
    198:     dz[0] += dphi[i][0] * zn[i];
    199:     dz[1] += dphi[i][1] * zn[i];
    200:     dz[2] += dphi[i][2] * zn[i];
    201:   }
    202: }

    204: 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)
    205: {
    206:   const PetscReal jac[3][3] = {
    207:     {hx / 2, 0,      0    },
    208:     {0,      hy / 2, 0    },
    209:     {dz[0],  dz[1],  dz[2]}
    210:   };
    211:   const PetscReal ijac[3][3] = {
    212:     {1 / jac[0][0],                        0,                                    0            },
    213:     {0,                                    1 / jac[1][1],                        0            },
    214:     {-jac[2][0] / (jac[0][0] * jac[2][2]), -jac[2][1] / (jac[1][1] * jac[2][2]), 1 / jac[2][2]}
    215:   };
    216:   const PetscReal jdet = jac[0][0] * jac[1][1] * jac[2][2];
    217:   PetscInt        i;

    219:   for (i = 0; i < 8; i++) {
    220:     const PetscReal *dphir = HexQDeriv[q][i];
    221:     phi[i]                 = HexQInterp[q][i];
    222:     dphi[i][0]             = dphir[0] * ijac[0][0] + dphir[1] * ijac[1][0] + dphir[2] * ijac[2][0];
    223:     dphi[i][1]             = dphir[0] * ijac[0][1] + dphir[1] * ijac[1][1] + dphir[2] * ijac[2][1];
    224:     dphi[i][2]             = dphir[0] * ijac[0][2] + dphir[1] * ijac[1][2] + dphir[2] * ijac[2][2];
    225:   }
    226:   *jw = 1.0 * jdet;
    227: }

    229: typedef struct _p_THI   *THI;
    230: typedef struct _n_Units *Units;

    232: typedef struct {
    233:   PetscScalar u, v;
    234: } Node;

    236: typedef struct {
    237:   PetscScalar b;     /* bed */
    238:   PetscScalar h;     /* thickness */
    239:   PetscScalar beta2; /* friction */
    240: } PrmNode;

    242: typedef struct {
    243:   PetscReal min, max, cmin, cmax;
    244: } PRange;

    246: typedef enum {
    247:   THIASSEMBLY_TRIDIAGONAL,
    248:   THIASSEMBLY_FULL
    249: } THIAssemblyMode;

    251: struct _p_THI {
    252:   PETSCHEADER(int);
    253:   void (*initialize)(THI, PetscReal x, PetscReal y, PrmNode *p);
    254:   PetscInt  zlevels;
    255:   PetscReal Lx, Ly, Lz; /* Model domain */
    256:   PetscReal alpha;      /* Bed angle */
    257:   Units     units;
    258:   PetscReal dirichlet_scale;
    259:   PetscReal ssa_friction_scale;
    260:   PRange    eta;
    261:   PRange    beta2;
    262:   struct {
    263:     PetscReal Bd2, eps, exponent;
    264:   } viscosity;
    265:   struct {
    266:     PetscReal irefgam, eps2, exponent, refvel, epsvel;
    267:   } friction;
    268:   PetscReal rhog;
    269:   PetscBool no_slip;
    270:   PetscBool tridiagonal;
    271:   PetscBool coarse2d;
    272:   PetscBool verbose;
    273:   MatType   mattype;
    274: };

    276: struct _n_Units {
    277:   /* fundamental */
    278:   PetscReal meter;
    279:   PetscReal kilogram;
    280:   PetscReal second;
    281:   /* derived */
    282:   PetscReal Pascal;
    283:   PetscReal year;
    284: };

    286: static PetscErrorCode THIJacobianLocal_3D_Full(DMDALocalInfo *, Node ***, Mat, Mat, THI);
    287: static PetscErrorCode THIJacobianLocal_3D_Tridiagonal(DMDALocalInfo *, Node ***, Mat, Mat, THI);
    288: static PetscErrorCode THIJacobianLocal_2D(DMDALocalInfo *, Node **, Mat, Mat, THI);

    290: static void PrmHexGetZ(const PrmNode pn[], PetscInt k, PetscInt zm, PetscReal zn[])
    291: {
    292:   const PetscScalar zm1 = zm - 1, znl[8] = {pn[0].b + pn[0].h * (PetscScalar)k / zm1,       pn[1].b + pn[1].h * (PetscScalar)k / zm1,       pn[2].b + pn[2].h * (PetscScalar)k / zm1,       pn[3].b + pn[3].h * (PetscScalar)k / zm1,
    293:                                             pn[0].b + pn[0].h * (PetscScalar)(k + 1) / zm1, pn[1].b + pn[1].h * (PetscScalar)(k + 1) / zm1, pn[2].b + pn[2].h * (PetscScalar)(k + 1) / zm1, pn[3].b + pn[3].h * (PetscScalar)(k + 1) / zm1};
    294:   PetscInt          i;
    295:   for (i = 0; i < 8; i++) zn[i] = PetscRealPart(znl[i]);
    296: }

    298: /* Tests A and C are from the ISMIP-HOM paper (Pattyn et al. 2008) */
    299: static void THIInitialize_HOM_A(THI thi, PetscReal x, PetscReal y, PrmNode *p)
    300: {
    301:   Units     units = thi->units;
    302:   PetscReal s     = -x * PetscSinReal(thi->alpha);

    304:   p->b     = s - 1000 * units->meter + 500 * units->meter * PetscSinReal(x * 2 * PETSC_PI / thi->Lx) * PetscSinReal(y * 2 * PETSC_PI / thi->Ly);
    305:   p->h     = s - p->b;
    306:   p->beta2 = 1e30;
    307: }

    309: static void THIInitialize_HOM_C(THI thi, PetscReal x, PetscReal y, PrmNode *p)
    310: {
    311:   Units     units = thi->units;
    312:   PetscReal s     = -x * PetscSinReal(thi->alpha);

    314:   p->b = s - 1000 * units->meter;
    315:   p->h = s - p->b;
    316:   /* tau_b = beta2 v   is a stress (Pa) */
    317:   p->beta2 = 1000 * (1 + PetscSinReal(x * 2 * PETSC_PI / thi->Lx) * PetscSinReal(y * 2 * PETSC_PI / thi->Ly)) * units->Pascal * units->year / units->meter;
    318: }

    320: /* These are just toys */

    322: /* Same bed as test A, free slip everywhere except for a discontinuous jump to a circular sticky region in the middle. */
    323: static void THIInitialize_HOM_X(THI thi, PetscReal xx, PetscReal yy, PrmNode *p)
    324: {
    325:   Units     units = thi->units;
    326:   PetscReal x = xx * 2 * PETSC_PI / thi->Lx - PETSC_PI, y = yy * 2 * PETSC_PI / thi->Ly - PETSC_PI; /* [-pi,pi] */
    327:   PetscReal r = PetscSqrtReal(x * x + y * y), s = -x * PetscSinReal(thi->alpha);
    328:   p->b     = s - 1000 * units->meter + 500 * units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    329:   p->h     = s - p->b;
    330:   p->beta2 = 1000 * (r < 1 ? 2 : 0) * units->Pascal * units->year / units->meter;
    331: }

    333: /* Like Z, but with 200 meter cliffs */
    334: static void THIInitialize_HOM_Y(THI thi, PetscReal xx, PetscReal yy, PrmNode *p)
    335: {
    336:   Units     units = thi->units;
    337:   PetscReal x = xx * 2 * PETSC_PI / thi->Lx - PETSC_PI, y = yy * 2 * PETSC_PI / thi->Ly - PETSC_PI; /* [-pi,pi] */
    338:   PetscReal r = PetscSqrtReal(x * x + y * y), s = -x * PetscSinReal(thi->alpha);

    340:   p->b = s - 1000 * units->meter + 500 * units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    341:   if (PetscRealPart(p->b) > -700 * units->meter) p->b += 200 * units->meter;
    342:   p->h     = s - p->b;
    343:   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;
    344: }

    346: /* Same bed as A, smoothly varying slipperiness, similar to MATLAB's "sombrero" (uncorrelated with bathymetry) */
    347: static void THIInitialize_HOM_Z(THI thi, PetscReal xx, PetscReal yy, PrmNode *p)
    348: {
    349:   Units     units = thi->units;
    350:   PetscReal x = xx * 2 * PETSC_PI / thi->Lx - PETSC_PI, y = yy * 2 * PETSC_PI / thi->Ly - PETSC_PI; /* [-pi,pi] */
    351:   PetscReal r = PetscSqrtReal(x * x + y * y), s = -x * PetscSinReal(thi->alpha);

    353:   p->b     = s - 1000 * units->meter + 500 * units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
    354:   p->h     = s - p->b;
    355:   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;
    356: }

    358: static void THIFriction(THI thi, PetscReal rbeta2, PetscReal gam, PetscReal *beta2, PetscReal *dbeta2)
    359: {
    360:   if (thi->friction.irefgam == 0) {
    361:     Units units           = thi->units;
    362:     thi->friction.irefgam = 1. / (0.5 * PetscSqr(thi->friction.refvel * units->meter / units->year));
    363:     thi->friction.eps2    = 0.5 * PetscSqr(thi->friction.epsvel * units->meter / units->year) * thi->friction.irefgam;
    364:   }
    365:   if (thi->friction.exponent == 0) {
    366:     *beta2  = rbeta2;
    367:     *dbeta2 = 0;
    368:   } else {
    369:     *beta2  = rbeta2 * PetscPowReal(thi->friction.eps2 + gam * thi->friction.irefgam, thi->friction.exponent);
    370:     *dbeta2 = thi->friction.exponent * *beta2 / (thi->friction.eps2 + gam * thi->friction.irefgam) * thi->friction.irefgam;
    371:   }
    372: }

    374: static void THIViscosity(THI thi, PetscReal gam, PetscReal *eta, PetscReal *deta)
    375: {
    376:   PetscReal Bd2, eps, exponent;
    377:   if (thi->viscosity.Bd2 == 0) {
    378:     Units           units   = thi->units;
    379:     const PetscReal n       = 3.,                                                     /* Glen exponent */
    380:       p                     = 1. + 1. / n,                                            /* for Stokes */
    381:       A                     = 1.e-16 * PetscPowReal(units->Pascal, -n) / units->year, /* softness parameter (Pa^{-n}/s) */
    382:       B                     = PetscPowReal(A, -1. / n);                               /* hardness parameter */
    383:     thi->viscosity.Bd2      = B / 2;
    384:     thi->viscosity.exponent = (p - 2) / 2;
    385:     thi->viscosity.eps      = 0.5 * PetscSqr(1e-5 / units->year);
    386:   }
    387:   Bd2      = thi->viscosity.Bd2;
    388:   exponent = thi->viscosity.exponent;
    389:   eps      = thi->viscosity.eps;
    390:   *eta     = Bd2 * PetscPowReal(eps + gam, exponent);
    391:   *deta    = exponent * (*eta) / (eps + gam);
    392: }

    394: static void RangeUpdate(PetscReal *min, PetscReal *max, PetscReal x)
    395: {
    396:   if (x < *min) *min = x;
    397:   if (x > *max) *max = x;
    398: }

    400: static void PRangeClear(PRange *p)
    401: {
    402:   p->cmin = p->min = 1e100;
    403:   p->cmax = p->max = -1e100;
    404: }

    406: static PetscErrorCode PRangeMinMax(PRange *p, PetscReal min, PetscReal max)
    407: {
    408:   PetscFunctionBeginUser;
    409:   p->cmin = min;
    410:   p->cmax = max;
    411:   if (min < p->min) p->min = min;
    412:   if (max > p->max) p->max = max;
    413:   PetscFunctionReturn(PETSC_SUCCESS);
    414: }

    416: static PetscErrorCode THIDestroy(THI *thi)
    417: {
    418:   PetscFunctionBeginUser;
    419:   if (!*thi) PetscFunctionReturn(PETSC_SUCCESS);
    420:   if (--((PetscObject)*thi)->refct > 0) {
    421:     *thi = 0;
    422:     PetscFunctionReturn(PETSC_SUCCESS);
    423:   }
    424:   PetscCall(PetscFree((*thi)->units));
    425:   PetscCall(PetscFree((*thi)->mattype));
    426:   PetscCall(PetscHeaderDestroy(thi));
    427:   PetscFunctionReturn(PETSC_SUCCESS);
    428: }

    430: static PetscErrorCode THICreate(MPI_Comm comm, THI *inthi)
    431: {
    432:   static PetscBool registered = PETSC_FALSE;
    433:   THI              thi;
    434:   Units            units;

    436:   PetscFunctionBeginUser;
    437:   *inthi = 0;
    438:   if (!registered) {
    439:     PetscCall(PetscClassIdRegister("Toy Hydrostatic Ice", &THI_CLASSID));
    440:     registered = PETSC_TRUE;
    441:   }
    442:   PetscCall(PetscHeaderCreate(thi, THI_CLASSID, "THI", "Toy Hydrostatic Ice", "", comm, THIDestroy, 0));

    444:   PetscCall(PetscNew(&thi->units));
    445:   units           = thi->units;
    446:   units->meter    = 1e-2;
    447:   units->second   = 1e-7;
    448:   units->kilogram = 1e-12;

    450:   PetscOptionsBegin(comm, NULL, "Scaled units options", "");
    451:   {
    452:     PetscCall(PetscOptionsReal("-units_meter", "1 meter in scaled length units", "", units->meter, &units->meter, NULL));
    453:     PetscCall(PetscOptionsReal("-units_second", "1 second in scaled time units", "", units->second, &units->second, NULL));
    454:     PetscCall(PetscOptionsReal("-units_kilogram", "1 kilogram in scaled mass units", "", units->kilogram, &units->kilogram, NULL));
    455:   }
    456:   PetscOptionsEnd();
    457:   units->Pascal = units->kilogram / (units->meter * PetscSqr(units->second));
    458:   units->year   = 31556926. * units->second; /* seconds per year */

    460:   thi->Lx              = 10.e3;
    461:   thi->Ly              = 10.e3;
    462:   thi->Lz              = 1000;
    463:   thi->dirichlet_scale = 1;
    464:   thi->verbose         = PETSC_FALSE;

    466:   PetscOptionsBegin(comm, NULL, "Toy Hydrostatic Ice options", "");
    467:   {
    468:     QuadratureType quad       = QUAD_GAUSS;
    469:     char           homexp[]   = "A";
    470:     char           mtype[256] = MATSBAIJ;
    471:     PetscReal      L, m = 1.0;
    472:     PetscBool      flg;
    473:     L = thi->Lx;
    474:     PetscCall(PetscOptionsReal("-thi_L", "Domain size (m)", "", L, &L, &flg));
    475:     if (flg) thi->Lx = thi->Ly = L;
    476:     PetscCall(PetscOptionsReal("-thi_Lx", "X Domain size (m)", "", thi->Lx, &thi->Lx, NULL));
    477:     PetscCall(PetscOptionsReal("-thi_Ly", "Y Domain size (m)", "", thi->Ly, &thi->Ly, NULL));
    478:     PetscCall(PetscOptionsReal("-thi_Lz", "Z Domain size (m)", "", thi->Lz, &thi->Lz, NULL));
    479:     PetscCall(PetscOptionsString("-thi_hom", "ISMIP-HOM experiment (A or C)", "", homexp, homexp, sizeof(homexp), NULL));
    480:     switch (homexp[0] = (char)toupper(homexp[0])) {
    481:     case 'A':
    482:       thi->initialize = THIInitialize_HOM_A;
    483:       thi->no_slip    = PETSC_TRUE;
    484:       thi->alpha      = 0.5;
    485:       break;
    486:     case 'C':
    487:       thi->initialize = THIInitialize_HOM_C;
    488:       thi->no_slip    = PETSC_FALSE;
    489:       thi->alpha      = 0.1;
    490:       break;
    491:     case 'X':
    492:       thi->initialize = THIInitialize_HOM_X;
    493:       thi->no_slip    = PETSC_FALSE;
    494:       thi->alpha      = 0.3;
    495:       break;
    496:     case 'Y':
    497:       thi->initialize = THIInitialize_HOM_Y;
    498:       thi->no_slip    = PETSC_FALSE;
    499:       thi->alpha      = 0.5;
    500:       break;
    501:     case 'Z':
    502:       thi->initialize = THIInitialize_HOM_Z;
    503:       thi->no_slip    = PETSC_FALSE;
    504:       thi->alpha      = 0.5;
    505:       break;
    506:     default:
    507:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "HOM experiment '%c' not implemented", homexp[0]);
    508:     }
    509:     PetscCall(PetscOptionsEnum("-thi_quadrature", "Quadrature to use for 3D elements", "", QuadratureTypes, (PetscEnum)quad, (PetscEnum *)&quad, NULL));
    510:     switch (quad) {
    511:     case QUAD_GAUSS:
    512:       HexQInterp = HexQInterp_Gauss;
    513:       HexQDeriv  = HexQDeriv_Gauss;
    514:       break;
    515:     case QUAD_LOBATTO:
    516:       HexQInterp = HexQInterp_Lobatto;
    517:       HexQDeriv  = HexQDeriv_Lobatto;
    518:       break;
    519:     }
    520:     PetscCall(PetscOptionsReal("-thi_alpha", "Bed angle (degrees)", "", thi->alpha, &thi->alpha, NULL));

    522:     thi->friction.refvel = 100.;
    523:     thi->friction.epsvel = 1.;

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

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

    531:     PetscCall(PetscOptionsReal("-thi_dirichlet_scale", "Scale Dirichlet boundary conditions by this factor", "", thi->dirichlet_scale, &thi->dirichlet_scale, NULL));
    532:     PetscCall(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));
    533:     PetscCall(PetscOptionsBool("-thi_coarse2d", "Use a 2D coarse space corresponding to SSA", "", thi->coarse2d, &thi->coarse2d, NULL));
    534:     PetscCall(PetscOptionsBool("-thi_tridiagonal", "Assemble a tridiagonal system (column coupling only) on the finest level", "", thi->tridiagonal, &thi->tridiagonal, NULL));
    535:     PetscCall(PetscOptionsFList("-thi_mat_type", "Matrix type", "MatSetType", MatList, mtype, (char *)mtype, sizeof(mtype), NULL));
    536:     PetscCall(PetscStrallocpy(mtype, (char **)&thi->mattype));
    537:     PetscCall(PetscOptionsBool("-thi_verbose", "Enable verbose output (like matrix sizes and statistics)", "", thi->verbose, &thi->verbose, NULL));
    538:   }
    539:   PetscOptionsEnd();

    541:   /* dimensionalize */
    542:   thi->Lx *= units->meter;
    543:   thi->Ly *= units->meter;
    544:   thi->Lz *= units->meter;
    545:   thi->alpha *= PETSC_PI / 180;

    547:   PRangeClear(&thi->eta);
    548:   PRangeClear(&thi->beta2);

    550:   {
    551:     PetscReal u = 1000 * units->meter / (3e7 * units->second), gradu = u / (100 * units->meter), eta, deta, rho = 910 * units->kilogram / PetscPowReal(units->meter, 3), grav = 9.81 * units->meter / PetscSqr(units->second),
    552:               driving = rho * grav * PetscSinReal(thi->alpha) * 1000 * units->meter;
    553:     THIViscosity(thi, 0.5 * gradu * gradu, &eta, &deta);
    554:     thi->rhog = rho * grav;
    555:     if (thi->verbose) {
    556:       PetscCall(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));
    557:       PetscCall(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));
    558:       PetscCall(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)));
    559:       THIViscosity(thi, 0.5 * PetscSqr(1e-3 * gradu), &eta, &deta);
    560:       PetscCall(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)));
    561:     }
    562:   }

    564:   *inthi = thi;
    565:   PetscFunctionReturn(PETSC_SUCCESS);
    566: }

    568: static PetscErrorCode THIInitializePrm(THI thi, DM da2prm, Vec prm)
    569: {
    570:   PrmNode **p;
    571:   PetscInt  i, j, xs, xm, ys, ym, mx, my;

    573:   PetscFunctionBeginUser;
    574:   PetscCall(DMDAGetGhostCorners(da2prm, &ys, &xs, 0, &ym, &xm, 0));
    575:   PetscCall(DMDAGetInfo(da2prm, 0, &my, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    576:   PetscCall(DMDAVecGetArray(da2prm, prm, &p));
    577:   for (i = xs; i < xs + xm; i++) {
    578:     for (j = ys; j < ys + ym; j++) {
    579:       PetscReal xx = thi->Lx * i / mx, yy = thi->Ly * j / my;
    580:       thi->initialize(thi, xx, yy, &p[i][j]);
    581:     }
    582:   }
    583:   PetscCall(DMDAVecRestoreArray(da2prm, prm, &p));
    584:   PetscFunctionReturn(PETSC_SUCCESS);
    585: }

    587: static PetscErrorCode THISetUpDM(THI thi, DM dm)
    588: {
    589:   PetscInt        refinelevel, coarsenlevel, level, dim, Mx, My, Mz, mx, my, s;
    590:   DMDAStencilType st;
    591:   DM              da2prm;
    592:   Vec             X;

    594:   PetscFunctionBeginUser;
    595:   PetscCall(DMDAGetInfo(dm, &dim, &Mz, &My, &Mx, 0, &my, &mx, 0, &s, 0, 0, 0, &st));
    596:   if (dim == 2) PetscCall(DMDAGetInfo(dm, &dim, &My, &Mx, 0, &my, &mx, 0, 0, &s, 0, 0, 0, &st));
    597:   PetscCall(DMGetRefineLevel(dm, &refinelevel));
    598:   PetscCall(DMGetCoarsenLevel(dm, &coarsenlevel));
    599:   level = refinelevel - coarsenlevel;
    600:   PetscCall(DMDACreate2d(PetscObjectComm((PetscObject)thi), DM_BOUNDARY_PERIODIC, DM_BOUNDARY_PERIODIC, st, My, Mx, my, mx, sizeof(PrmNode) / sizeof(PetscScalar), s, 0, 0, &da2prm));
    601:   PetscCall(DMSetUp(da2prm));
    602:   PetscCall(DMCreateLocalVector(da2prm, &X));
    603:   {
    604:     PetscReal Lx = thi->Lx / thi->units->meter, Ly = thi->Ly / thi->units->meter, Lz = thi->Lz / thi->units->meter;
    605:     if (dim == 2) {
    606:       PetscCall(PetscPrintf(PetscObjectComm((PetscObject)thi), "Level %" PetscInt_FMT " domain size (m) %8.2g x %8.2g, num elements %" PetscInt_FMT " x %" PetscInt_FMT " (%" PetscInt_FMT "), size (m) %g x %g\n", level, (double)Lx, (double)Ly, Mx, My, Mx * My, (double)(Lx / Mx), (double)(Ly / My)));
    607:     } else {
    608:       PetscCall(PetscPrintf(PetscObjectComm((PetscObject)thi), "Level %" PetscInt_FMT " domain size (m) %8.2g x %8.2g x %8.2g, num elements %" PetscInt_FMT " x %" PetscInt_FMT " x %" PetscInt_FMT " (%" PetscInt_FMT "), 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))));
    609:     }
    610:   }
    611:   PetscCall(THIInitializePrm(thi, da2prm, X));
    612:   if (thi->tridiagonal) { /* Reset coarse Jacobian evaluation */
    613:     PetscCall(DMDASNESSetJacobianLocal(dm, (DMDASNESJacobianFn *)THIJacobianLocal_3D_Full, thi));
    614:   }
    615:   if (thi->coarse2d) PetscCall(DMDASNESSetJacobianLocal(dm, (DMDASNESJacobianFn *)THIJacobianLocal_2D, thi));
    616:   PetscCall(PetscObjectCompose((PetscObject)dm, "DMDA2Prm", (PetscObject)da2prm));
    617:   PetscCall(PetscObjectCompose((PetscObject)dm, "DMDA2Prm_Vec", (PetscObject)X));
    618:   PetscCall(DMDestroy(&da2prm));
    619:   PetscCall(VecDestroy(&X));
    620:   PetscFunctionReturn(PETSC_SUCCESS);
    621: }

    623: static PetscErrorCode DMCoarsenHook_THI(DM dmf, DM dmc, void *ctx)
    624: {
    625:   THI      thi = (THI)ctx;
    626:   PetscInt rlevel, clevel;

    628:   PetscFunctionBeginUser;
    629:   PetscCall(THISetUpDM(thi, dmc));
    630:   PetscCall(DMGetRefineLevel(dmc, &rlevel));
    631:   PetscCall(DMGetCoarsenLevel(dmc, &clevel));
    632:   if (rlevel - clevel == 0) PetscCall(DMSetMatType(dmc, MATAIJ));
    633:   PetscCall(DMCoarsenHookAdd(dmc, DMCoarsenHook_THI, NULL, thi));
    634:   PetscFunctionReturn(PETSC_SUCCESS);
    635: }

    637: static PetscErrorCode DMRefineHook_THI(DM dmc, DM dmf, void *ctx)
    638: {
    639:   THI thi = (THI)ctx;

    641:   PetscFunctionBeginUser;
    642:   PetscCall(THISetUpDM(thi, dmf));
    643:   PetscCall(DMSetMatType(dmf, thi->mattype));
    644:   PetscCall(DMRefineHookAdd(dmf, DMRefineHook_THI, NULL, thi));
    645:   /* With grid sequencing, a formerly-refined DM will later be coarsened by PCSetUp_MG */
    646:   PetscCall(DMCoarsenHookAdd(dmf, DMCoarsenHook_THI, NULL, thi));
    647:   PetscFunctionReturn(PETSC_SUCCESS);
    648: }

    650: static PetscErrorCode THIDAGetPrm(DM da, PrmNode ***prm)
    651: {
    652:   DM  da2prm;
    653:   Vec X;

    655:   PetscFunctionBeginUser;
    656:   PetscCall(PetscObjectQuery((PetscObject)da, "DMDA2Prm", (PetscObject *)&da2prm));
    657:   PetscCheck(da2prm, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No DMDA2Prm composed with given DMDA");
    658:   PetscCall(PetscObjectQuery((PetscObject)da, "DMDA2Prm_Vec", (PetscObject *)&X));
    659:   PetscCheck(X, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No DMDA2Prm_Vec composed with given DMDA");
    660:   PetscCall(DMDAVecGetArray(da2prm, X, prm));
    661:   PetscFunctionReturn(PETSC_SUCCESS);
    662: }

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

    669:   PetscFunctionBeginUser;
    670:   PetscCall(PetscObjectQuery((PetscObject)da, "DMDA2Prm", (PetscObject *)&da2prm));
    671:   PetscCheck(da2prm, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No DMDA2Prm composed with given DMDA");
    672:   PetscCall(PetscObjectQuery((PetscObject)da, "DMDA2Prm_Vec", (PetscObject *)&X));
    673:   PetscCheck(X, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No DMDA2Prm_Vec composed with given DMDA");
    674:   PetscCall(DMDAVecRestoreArray(da2prm, X, prm));
    675:   PetscFunctionReturn(PETSC_SUCCESS);
    676: }

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

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

    710: 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)
    711: {
    712:   PetscInt    l, ll;
    713:   PetscScalar gam;

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

    731: 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)
    732: {
    733:   PetscInt    l, ll;
    734:   PetscScalar gam;

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

    752: static PetscErrorCode THIFunctionLocal(DMDALocalInfo *info, Node ***x, 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;
    756:   PrmNode **prm;

    758:   PetscFunctionBeginUser;
    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:   PetscCall(THIDAGetPrm(info->da, &prm));

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

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

    851:   PetscCall(PRangeMinMax(&thi->eta, etamin, etamax));
    852:   PetscCall(PRangeMinMax(&thi->beta2, beta2min, beta2max));
    853:   PetscFunctionReturn(PETSC_SUCCESS);
    854: }

    856: static PetscErrorCode THIMatrixStatistics(THI thi, Mat B, PetscViewer viewer)
    857: {
    858:   PetscReal   nrm;
    859:   PetscInt    m;
    860:   PetscMPIInt rank;

    862:   PetscFunctionBeginUser;
    863:   PetscCall(MatNorm(B, NORM_FROBENIUS, &nrm));
    864:   PetscCall(MatGetSize(B, &m, 0));
    865:   PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)B), &rank));
    866:   if (rank == 0) {
    867:     PetscScalar val0, val2;
    868:     PetscCall(MatGetValue(B, 0, 0, &val0));
    869:     PetscCall(MatGetValue(B, 2, 2, &val2));
    870:     PetscCall(PetscViewerASCIIPrintf(viewer, "Matrix dim %" PetscInt_FMT " 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),
    871:                                      (double)thi->eta.cmin, (double)thi->eta.cmax, (double)thi->beta2.cmin, (double)thi->beta2.cmax));
    872:   }
    873:   PetscFunctionReturn(PETSC_SUCCESS);
    874: }

    876: static PetscErrorCode THISurfaceStatistics(DM da, Vec X, PetscReal *min, PetscReal *max, PetscReal *mean)
    877: {
    878:   Node     ***x;
    879:   PetscInt    i, j, xs, ys, zs, xm, ym, zm, mx, my, mz;
    880:   PetscReal   umin = 1e100, umax = -1e100;
    881:   PetscScalar usum = 0.0, gusum;

    883:   PetscFunctionBeginUser;
    884:   *min = *max = *mean = 0;
    885:   PetscCall(DMDAGetInfo(da, 0, &mz, &my, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    886:   PetscCall(DMDAGetCorners(da, &zs, &ys, &xs, &zm, &ym, &xm));
    887:   PetscCheck(zs == 0 && zm == mz, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Unexpected decomposition");
    888:   PetscCall(DMDAVecGetArray(da, X, &x));
    889:   for (i = xs; i < xs + xm; i++) {
    890:     for (j = ys; j < ys + ym; j++) {
    891:       PetscReal u = PetscRealPart(x[i][j][zm - 1].u);
    892:       RangeUpdate(&umin, &umax, u);
    893:       usum += u;
    894:     }
    895:   }
    896:   PetscCall(DMDAVecRestoreArray(da, X, &x));
    897:   PetscCallMPI(MPIU_Allreduce(&umin, min, 1, MPIU_REAL, MPIU_MIN, PetscObjectComm((PetscObject)da)));
    898:   PetscCallMPI(MPIU_Allreduce(&umax, max, 1, MPIU_REAL, MPIU_MAX, PetscObjectComm((PetscObject)da)));
    899:   PetscCallMPI(MPIU_Allreduce(&usum, &gusum, 1, MPIU_SCALAR, MPIU_SUM, PetscObjectComm((PetscObject)da)));
    900:   *mean = PetscRealPart(gusum) / (mx * my);
    901:   PetscFunctionReturn(PETSC_SUCCESS);
    902: }

    904: static PetscErrorCode THISolveStatistics(THI thi, SNES snes, PetscInt coarsened, const char name[])
    905: {
    906:   MPI_Comm comm;
    907:   Vec      X;
    908:   DM       dm;

    910:   PetscFunctionBeginUser;
    911:   PetscCall(PetscObjectGetComm((PetscObject)thi, &comm));
    912:   PetscCall(SNESGetSolution(snes, &X));
    913:   PetscCall(SNESGetDM(snes, &dm));
    914:   PetscCall(PetscPrintf(comm, "Solution statistics after solve: %s\n", name));
    915:   {
    916:     PetscInt            its, lits;
    917:     SNESConvergedReason reason;
    918:     PetscCall(SNESGetIterationNumber(snes, &its));
    919:     PetscCall(SNESGetConvergedReason(snes, &reason));
    920:     PetscCall(SNESGetLinearSolveIterations(snes, &lits));
    921:     PetscCall(PetscPrintf(comm, "%s: Number of SNES iterations = %" PetscInt_FMT ", total linear iterations = %" PetscInt_FMT "\n", SNESConvergedReasons[reason], its, lits));
    922:   }
    923:   {
    924:     PetscReal          nrm2, tmin[3] = {1e100, 1e100, 1e100}, tmax[3] = {-1e100, -1e100, -1e100}, min[3], max[3];
    925:     PetscInt           i, j, m;
    926:     const PetscScalar *x;
    927:     PetscCall(VecNorm(X, NORM_2, &nrm2));
    928:     PetscCall(VecGetLocalSize(X, &m));
    929:     PetscCall(VecGetArrayRead(X, &x));
    930:     for (i = 0; i < m; i += 2) {
    931:       PetscReal u = PetscRealPart(x[i]), v = PetscRealPart(x[i + 1]), c = PetscSqrtReal(u * u + v * v);
    932:       tmin[0] = PetscMin(u, tmin[0]);
    933:       tmin[1] = PetscMin(v, tmin[1]);
    934:       tmin[2] = PetscMin(c, tmin[2]);
    935:       tmax[0] = PetscMax(u, tmax[0]);
    936:       tmax[1] = PetscMax(v, tmax[1]);
    937:       tmax[2] = PetscMax(c, tmax[2]);
    938:     }
    939:     PetscCall(VecRestoreArrayRead(X, &x));
    940:     PetscCallMPI(MPIU_Allreduce(tmin, min, 3, MPIU_REAL, MPIU_MIN, PetscObjectComm((PetscObject)thi)));
    941:     PetscCallMPI(MPIU_Allreduce(tmax, max, 3, MPIU_REAL, MPIU_MAX, PetscObjectComm((PetscObject)thi)));
    942:     /* Dimensionalize to meters/year */
    943:     nrm2 *= thi->units->year / thi->units->meter;
    944:     for (j = 0; j < 3; j++) {
    945:       min[j] *= thi->units->year / thi->units->meter;
    946:       max[j] *= thi->units->year / thi->units->meter;
    947:     }
    948:     if (min[0] == 0.0) min[0] = 0.0;
    949:     PetscCall(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]));
    950:     {
    951:       PetscReal umin, umax, umean;
    952:       PetscCall(THISurfaceStatistics(dm, X, &umin, &umax, &umean));
    953:       umin *= thi->units->year / thi->units->meter;
    954:       umax *= thi->units->year / thi->units->meter;
    955:       umean *= thi->units->year / thi->units->meter;
    956:       PetscCall(PetscPrintf(comm, "Surface statistics: u in [%12.6e, %12.6e] mean %12.6e\n", (double)umin, (double)umax, (double)umean));
    957:     }
    958:     /* These values stay nondimensional */
    959:     PetscCall(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));
    960:     PetscCall(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));
    961:   }
    962:   PetscFunctionReturn(PETSC_SUCCESS);
    963: }

    965: static PetscErrorCode THIJacobianLocal_2D(DMDALocalInfo *info, Node **x, Mat J, Mat B, THI thi)
    966: {
    967:   PetscInt  xs, ys, xm, ym, i, j, q, l, ll;
    968:   PetscReal hx, hy;
    969:   PrmNode **prm;

    971:   PetscFunctionBeginUser;
    972:   xs = info->ys;
    973:   ys = info->xs;
    974:   xm = info->ym;
    975:   ym = info->xm;
    976:   hx = thi->Lx / info->my;
    977:   hy = thi->Ly / info->mx;

    979:   PetscCall(MatZeroEntries(B));
    980:   PetscCall(THIDAGetPrm(info->da, &prm));

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

    1036:   PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    1037:   PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
    1038:   PetscCall(MatSetOption(B, MAT_SYMMETRIC, PETSC_TRUE));
    1039:   if (thi->verbose) PetscCall(THIMatrixStatistics(thi, B, PETSC_VIEWER_STDOUT_WORLD));
    1040:   PetscFunctionReturn(PETSC_SUCCESS);
    1041: }

    1043: static PetscErrorCode THIJacobianLocal_3D(DMDALocalInfo *info, Node ***x, Mat B, THI thi, THIAssemblyMode amode)
    1044: {
    1045:   PetscInt  xs, ys, xm, ym, zm, i, j, k, q, l, ll;
    1046:   PetscReal hx, hy;
    1047:   PrmNode **prm;

    1049:   PetscFunctionBeginUser;
    1050:   xs = info->zs;
    1051:   ys = info->ys;
    1052:   xm = info->zm;
    1053:   ym = info->ym;
    1054:   zm = info->xm;
    1055:   hx = thi->Lx / info->mz;
    1056:   hy = thi->Ly / info->my;

    1058:   PetscCall(MatZeroEntries(B));
    1059:   PetscCall(MatSetOption(B, MAT_SUBSET_OFF_PROC_ENTRIES, PETSC_TRUE));
    1060:   PetscCall(THIDAGetPrm(info->da, &prm));

    1062:   for (i = xs; i < xs + xm; i++) {
    1063:     for (j = ys; j < ys + ym; j++) {
    1064:       PrmNode pn[4];
    1065:       QuadExtract(prm, i, j, pn);
    1066:       for (k = 0; k < zm - 1; k++) {
    1067:         Node        n[8];
    1068:         PetscReal   zn[8], etabase = 0;
    1069:         PetscScalar Ke[8 * 2][8 * 2];
    1070:         PetscInt    ls = 0;

    1072:         PrmHexGetZ(pn, k, zm, zn);
    1073:         HexExtract(x, i, j, k, n);
    1074:         PetscCall(PetscMemzero(Ke, sizeof(Ke)));
    1075:         if (thi->no_slip && k == 0) {
    1076:           for (l = 0; l < 4; l++) n[l].u = n[l].v = 0;
    1077:           ls = 4;
    1078:         }
    1079:         for (q = 0; q < 8; q++) {
    1080:           PetscReal   dz[3], phi[8], dphi[8][3], jw, eta, deta;
    1081:           PetscScalar du[3], dv[3], u, v;
    1082:           HexGrad(HexQDeriv[q], zn, dz);
    1083:           HexComputeGeometry(q, hx, hy, dz, phi, dphi, &jw);
    1084:           PointwiseNonlinearity(thi, n, phi, dphi, &u, &v, du, dv, &eta, &deta);
    1085:           jw /= thi->rhog; /* residuals are scaled by this factor */
    1086:           if (q == 0) etabase = eta;
    1087:           for (l = ls; l < 8; l++) { /* test functions */
    1088:             const PetscReal *PETSC_RESTRICT dp = dphi[l];
    1089: #if USE_SSE2_KERNELS
    1090:             /* gcc (up to my 4.5 snapshot) is really bad at hoisting intrinsics so we do it manually */
    1091:             __m128d p4 = _mm_set1_pd(4), p2 = _mm_set1_pd(2), p05 = _mm_set1_pd(0.5), p42 = _mm_setr_pd(4, 2), p24 = _mm_shuffle_pd(p42, p42, _MM_SHUFFLE2(0, 1)), du0 = _mm_set1_pd(du[0]), du1 = _mm_set1_pd(du[1]), du2 = _mm_set1_pd(du[2]), dv0 = _mm_set1_pd(dv[0]), dv1 = _mm_set1_pd(dv[1]), dv2 = _mm_set1_pd(dv[2]), jweta = _mm_set1_pd(jw * eta), jwdeta = _mm_set1_pd(jw * deta), dp0 = _mm_set1_pd(dp[0]), dp1 = _mm_set1_pd(dp[1]), dp2 = _mm_set1_pd(dp[2]), dp0jweta = _mm_mul_pd(dp0, jweta), dp1jweta = _mm_mul_pd(dp1, jweta), dp2jweta = _mm_mul_pd(dp2, jweta), p4du0p2dv1 = _mm_add_pd(_mm_mul_pd(p4, du0), _mm_mul_pd(p2, dv1)), /* 4 du0 + 2 dv1 */
    1092:               p4dv1p2du0 = _mm_add_pd(_mm_mul_pd(p4, dv1), _mm_mul_pd(p2, du0)), /* 4 dv1 + 2 du0 */
    1093:               pdu2dv2    = _mm_unpacklo_pd(du2, dv2),                            /* [du2, dv2] */
    1094:               du1pdv0    = _mm_add_pd(du1, dv0),                                 /* du1 + dv0 */
    1095:               t1         = _mm_mul_pd(dp0, p4du0p2dv1),                          /* dp0 (4 du0 + 2 dv1) */
    1096:               t2         = _mm_mul_pd(dp1, p4dv1p2du0);                          /* dp1 (4 dv1 + 2 du0) */

    1098: #endif
    1099: #if defined COMPUTE_LOWER_TRIANGULAR      /* The element matrices are always symmetric so computing the lower-triangular part is not necessary */
    1100:             for (ll = ls; ll < 8; ll++) { /* trial functions */
    1101: #else
    1102:             for (ll = l; ll < 8; ll++) {
    1103: #endif
    1104:               const PetscReal *PETSC_RESTRICT dpl = dphi[ll];
    1105:               if (amode == THIASSEMBLY_TRIDIAGONAL && (l - ll) % 4) continue; /* these entries would not be inserted */
    1106: #if !USE_SSE2_KERNELS
    1107:               /* The analytic Jacobian in nice, easy-to-read form */
    1108:               {
    1109:                 PetscScalar dgdu, dgdv;
    1110:                 dgdu = 2. * du[0] * dpl[0] + dv[1] * dpl[0] + 0.5 * (du[1] + dv[0]) * dpl[1] + 0.5 * du[2] * dpl[2];
    1111:                 dgdv = 2. * dv[1] * dpl[1] + du[0] * dpl[1] + 0.5 * (du[1] + dv[0]) * dpl[0] + 0.5 * dv[2] * dpl[2];
    1112:                 /* Picard part */
    1113:                 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];
    1114:                 Ke[l * 2 + 0][ll * 2 + 1] += dp[0] * jw * eta * 2. * dpl[1] + dp[1] * jw * eta * dpl[0];
    1115:                 Ke[l * 2 + 1][ll * 2 + 0] += dp[1] * jw * eta * 2. * dpl[0] + dp[0] * jw * eta * dpl[1];
    1116:                 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];
    1117:                 /* extra Newton terms */
    1118:                 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];
    1119:                 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];
    1120:                 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];
    1121:                 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];
    1122:               }
    1123: #else
    1124:               /* This SSE2 code is an exact replica of above, but uses explicit packed instructions for some speed
    1125:               * benefit.  On my hardware, these intrinsics are almost twice as fast as above, reducing total assembly cost
    1126:               * by 25 to 30 percent. */
    1127:               {
    1128:                 __m128d keu = _mm_loadu_pd(&Ke[l * 2 + 0][ll * 2 + 0]), kev = _mm_loadu_pd(&Ke[l * 2 + 1][ll * 2 + 0]), dpl01 = _mm_loadu_pd(&dpl[0]), dpl10 = _mm_shuffle_pd(dpl01, dpl01, _MM_SHUFFLE2(0, 1)), dpl2 = _mm_set_sd(dpl[2]), t0, t3, pdgduv;
    1129:                 keu    = _mm_add_pd(keu, _mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp0jweta, p42), dpl01), _mm_add_pd(_mm_mul_pd(dp1jweta, dpl10), _mm_mul_pd(dp2jweta, dpl2))));
    1130:                 kev    = _mm_add_pd(kev, _mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp1jweta, p24), dpl01), _mm_add_pd(_mm_mul_pd(dp0jweta, dpl10), _mm_mul_pd(dp2jweta, _mm_shuffle_pd(dpl2, dpl2, _MM_SHUFFLE2(0, 1))))));
    1131:                 pdgduv = _mm_mul_pd(p05, _mm_add_pd(_mm_add_pd(_mm_mul_pd(p42, _mm_mul_pd(du0, dpl01)), _mm_mul_pd(p24, _mm_mul_pd(dv1, dpl01))), _mm_add_pd(_mm_mul_pd(du1pdv0, dpl10), _mm_mul_pd(pdu2dv2, _mm_set1_pd(dpl[2]))))); /* [dgdu, dgdv] */
    1132:                 t0     = _mm_mul_pd(jwdeta, pdgduv); /* jw deta [dgdu, dgdv] */
    1133:                 t3     = _mm_mul_pd(t0, du1pdv0);    /* t0 (du1 + dv0) */
    1134:                 _mm_storeu_pd(&Ke[l * 2 + 0][ll * 2 + 0], _mm_add_pd(keu, _mm_add_pd(_mm_mul_pd(t1, t0), _mm_add_pd(_mm_mul_pd(dp1, t3), _mm_mul_pd(t0, _mm_mul_pd(dp2, du2))))));
    1135:                 _mm_storeu_pd(&Ke[l * 2 + 1][ll * 2 + 0], _mm_add_pd(kev, _mm_add_pd(_mm_mul_pd(t2, t0), _mm_add_pd(_mm_mul_pd(dp0, t3), _mm_mul_pd(t0, _mm_mul_pd(dp2, dv2))))));
    1136:               }
    1137: #endif
    1138:             }
    1139:           }
    1140:         }
    1141:         if (k == 0) { /* on a bottom face */
    1142:           if (thi->no_slip) {
    1143:             const PetscReal   hz    = PetscRealPart(pn[0].h) / (zm - 1);
    1144:             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);
    1145:             Ke[0][0] = thi->dirichlet_scale * diagu;
    1146:             Ke[1][1] = thi->dirichlet_scale * diagv;
    1147:           } else {
    1148:             for (q = 0; q < 4; q++) {
    1149:               const PetscReal jw = 0.25 * hx * hy / thi->rhog, *phi = QuadQInterp[q];
    1150:               PetscScalar     u = 0, v = 0, rbeta2 = 0;
    1151:               PetscReal       beta2, dbeta2;
    1152:               for (l = 0; l < 4; l++) {
    1153:                 u += phi[l] * n[l].u;
    1154:                 v += phi[l] * n[l].v;
    1155:                 rbeta2 += phi[l] * pn[l].beta2;
    1156:               }
    1157:               THIFriction(thi, PetscRealPart(rbeta2), PetscRealPart(u * u + v * v) / 2, &beta2, &dbeta2);
    1158:               for (l = 0; l < 4; l++) {
    1159:                 const PetscReal pp = phi[l];
    1160:                 for (ll = 0; ll < 4; ll++) {
    1161:                   const PetscReal ppl = phi[ll];
    1162:                   Ke[l * 2 + 0][ll * 2 + 0] += pp * jw * beta2 * ppl + pp * jw * dbeta2 * u * u * ppl;
    1163:                   Ke[l * 2 + 0][ll * 2 + 1] += pp * jw * dbeta2 * u * v * ppl;
    1164:                   Ke[l * 2 + 1][ll * 2 + 0] += pp * jw * dbeta2 * v * u * ppl;
    1165:                   Ke[l * 2 + 1][ll * 2 + 1] += pp * jw * beta2 * ppl + pp * jw * dbeta2 * v * v * ppl;
    1166:                 }
    1167:               }
    1168:             }
    1169:           }
    1170:         }
    1171:         {
    1172:           const MatStencil rc[8] = {
    1173:             {i,     j,     k,     0},
    1174:             {i + 1, j,     k,     0},
    1175:             {i + 1, j + 1, k,     0},
    1176:             {i,     j + 1, k,     0},
    1177:             {i,     j,     k + 1, 0},
    1178:             {i + 1, j,     k + 1, 0},
    1179:             {i + 1, j + 1, k + 1, 0},
    1180:             {i,     j + 1, k + 1, 0}
    1181:           };
    1182:           if (amode == THIASSEMBLY_TRIDIAGONAL) {
    1183:             for (l = 0; l < 4; l++) { /* Copy out each of the blocks, discarding horizontal coupling */
    1184:               const PetscInt   l4     = l + 4;
    1185:               const MatStencil rcl[2] = {
    1186:                 {rc[l].k,  rc[l].j,  rc[l].i,  0},
    1187:                 {rc[l4].k, rc[l4].j, rc[l4].i, 0}
    1188:               };
    1189: #if defined COMPUTE_LOWER_TRIANGULAR
    1190:               const PetscScalar Kel[4][4] = {
    1191:                 {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] },
    1192:                 {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] },
    1193:                 {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]},
    1194:                 {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]}
    1195:               };
    1196: #else
    1197:               /* Same as above except for the lower-left block */
    1198:               const PetscScalar Kel[4][4] = {
    1199:                 {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] },
    1200:                 {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] },
    1201:                 {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]},
    1202:                 {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]}
    1203:               };
    1204: #endif
    1205:               PetscCall(MatSetValuesBlockedStencil(B, 2, rcl, 2, rcl, &Kel[0][0], ADD_VALUES));
    1206:             }
    1207:           } else {
    1208: #if !defined COMPUTE_LOWER_TRIANGULAR /* fill in lower-triangular part, this is really cheap compared to computing the entries */
    1209:             for (l = 0; l < 8; l++) {
    1210:               for (ll = l + 1; ll < 8; ll++) {
    1211:                 Ke[ll * 2 + 0][l * 2 + 0] = Ke[l * 2 + 0][ll * 2 + 0];
    1212:                 Ke[ll * 2 + 1][l * 2 + 0] = Ke[l * 2 + 0][ll * 2 + 1];
    1213:                 Ke[ll * 2 + 0][l * 2 + 1] = Ke[l * 2 + 1][ll * 2 + 0];
    1214:                 Ke[ll * 2 + 1][l * 2 + 1] = Ke[l * 2 + 1][ll * 2 + 1];
    1215:               }
    1216:             }
    1217: #endif
    1218:             PetscCall(MatSetValuesBlockedStencil(B, 8, rc, 8, rc, &Ke[0][0], ADD_VALUES));
    1219:           }
    1220:         }
    1221:       }
    1222:     }
    1223:   }
    1224:   PetscCall(THIDARestorePrm(info->da, &prm));

    1226:   PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    1227:   PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
    1228:   PetscCall(MatSetOption(B, MAT_SYMMETRIC, PETSC_TRUE));
    1229:   if (thi->verbose) PetscCall(THIMatrixStatistics(thi, B, PETSC_VIEWER_STDOUT_WORLD));
    1230:   PetscFunctionReturn(PETSC_SUCCESS);
    1231: }

    1233: static PetscErrorCode THIJacobianLocal_3D_Full(DMDALocalInfo *info, Node ***x, Mat A, Mat B, THI thi)
    1234: {
    1235:   PetscFunctionBeginUser;
    1236:   PetscCall(THIJacobianLocal_3D(info, x, B, thi, THIASSEMBLY_FULL));
    1237:   PetscFunctionReturn(PETSC_SUCCESS);
    1238: }

    1240: static PetscErrorCode THIJacobianLocal_3D_Tridiagonal(DMDALocalInfo *info, Node ***x, Mat A, Mat B, THI thi)
    1241: {
    1242:   PetscFunctionBeginUser;
    1243:   PetscCall(THIJacobianLocal_3D(info, x, B, thi, THIASSEMBLY_TRIDIAGONAL));
    1244:   PetscFunctionReturn(PETSC_SUCCESS);
    1245: }

    1247: static PetscErrorCode DMRefineHierarchy_THI(DM dac0, PetscInt nlevels, DM hierarchy[])
    1248: {
    1249:   THI             thi;
    1250:   PetscInt        dim, M, N, m, n, s, dof;
    1251:   DM              dac, daf;
    1252:   DMDAStencilType st;
    1253:   DM_DA          *ddf, *ddc;

    1255:   PetscFunctionBeginUser;
    1256:   PetscCall(PetscObjectQuery((PetscObject)dac0, "THI", (PetscObject *)&thi));
    1257:   PetscCheck(thi, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Cannot refine this DMDA, missing composed THI instance");
    1258:   if (nlevels > 1) {
    1259:     PetscCall(DMRefineHierarchy(dac0, nlevels - 1, hierarchy));
    1260:     dac = hierarchy[nlevels - 2];
    1261:   } else {
    1262:     dac = dac0;
    1263:   }
    1264:   PetscCall(DMDAGetInfo(dac, &dim, &N, &M, 0, &n, &m, 0, &dof, &s, 0, 0, 0, &st));
    1265:   PetscCheck(dim == 2, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "This function can only refine 2D DMDAs");

    1267:   /* Creates a 3D DMDA with the same map-plane layout as the 2D one, with contiguous columns */
    1268:   PetscCall(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));
    1269:   PetscCall(DMSetUp(daf));

    1271:   daf->ops->creatematrix        = dac->ops->creatematrix;
    1272:   daf->ops->createinterpolation = dac->ops->createinterpolation;
    1273:   daf->ops->getcoloring         = dac->ops->getcoloring;
    1274:   ddf                           = (DM_DA *)daf->data;
    1275:   ddc                           = (DM_DA *)dac->data;
    1276:   ddf->interptype               = ddc->interptype;

    1278:   PetscCall(DMDASetFieldName(daf, 0, "x-velocity"));
    1279:   PetscCall(DMDASetFieldName(daf, 1, "y-velocity"));

    1281:   hierarchy[nlevels - 1] = daf;
    1282:   PetscFunctionReturn(PETSC_SUCCESS);
    1283: }

    1285: static PetscErrorCode DMCreateInterpolation_DA_THI(DM dac, DM daf, Mat *A, Vec *scale)
    1286: {
    1287:   PetscInt dim;

    1289:   PetscFunctionBeginUser;
    1292:   PetscAssertPointer(A, 3);
    1293:   if (scale) PetscAssertPointer(scale, 4);
    1294:   PetscCall(DMDAGetInfo(daf, &dim, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    1295:   if (dim == 2) {
    1296:     /* We are in the 2D problem and use normal DMDA interpolation */
    1297:     PetscCall(DMCreateInterpolation(dac, daf, A, scale));
    1298:   } else {
    1299:     PetscInt i, j, k, xs, ys, zs, xm, ym, zm, mx, my, mz, rstart, cstart;
    1300:     Mat      B;

    1302:     PetscCall(DMDAGetInfo(daf, 0, &mz, &my, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    1303:     PetscCall(DMDAGetCorners(daf, &zs, &ys, &xs, &zm, &ym, &xm));
    1304:     PetscCheck(!zs, PETSC_COMM_SELF, PETSC_ERR_PLIB, "unexpected");
    1305:     PetscCall(MatCreate(PetscObjectComm((PetscObject)daf), &B));
    1306:     PetscCall(MatSetSizes(B, xm * ym * zm, xm * ym, mx * my * mz, mx * my));

    1308:     PetscCall(MatSetType(B, MATAIJ));
    1309:     PetscCall(MatSeqAIJSetPreallocation(B, 1, NULL));
    1310:     PetscCall(MatMPIAIJSetPreallocation(B, 1, NULL, 0, NULL));
    1311:     PetscCall(MatGetOwnershipRange(B, &rstart, NULL));
    1312:     PetscCall(MatGetOwnershipRangeColumn(B, &cstart, NULL));
    1313:     for (i = xs; i < xs + xm; i++) {
    1314:       for (j = ys; j < ys + ym; j++) {
    1315:         for (k = zs; k < zs + zm; k++) {
    1316:           PetscInt    i2 = i * ym + j, i3 = i2 * zm + k;
    1317:           PetscScalar val = ((k == 0 || k == mz - 1) ? 0.5 : 1.) / (mz - 1.); /* Integration using trapezoid rule */
    1318:           PetscCall(MatSetValue(B, cstart + i3, rstart + i2, val, INSERT_VALUES));
    1319:         }
    1320:       }
    1321:     }
    1322:     PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    1323:     PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
    1324:     PetscCall(MatCreateMAIJ(B, sizeof(Node) / sizeof(PetscScalar), A));
    1325:     PetscCall(MatDestroy(&B));
    1326:   }
    1327:   PetscFunctionReturn(PETSC_SUCCESS);
    1328: }

    1330: static PetscErrorCode DMCreateMatrix_THI_Tridiagonal(DM da, Mat *J)
    1331: {
    1332:   Mat                    A;
    1333:   PetscInt               xm, ym, zm, dim, dof = 2, starts[3], dims[3];
    1334:   ISLocalToGlobalMapping ltog;

    1336:   PetscFunctionBeginUser;
    1337:   PetscCall(DMDAGetInfo(da, &dim, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    1338:   PetscCheck(dim == 3, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Expected DMDA to be 3D");
    1339:   PetscCall(DMDAGetCorners(da, 0, 0, 0, &zm, &ym, &xm));
    1340:   PetscCall(DMGetLocalToGlobalMapping(da, &ltog));
    1341:   PetscCall(MatCreate(PetscObjectComm((PetscObject)da), &A));
    1342:   PetscCall(MatSetSizes(A, dof * xm * ym * zm, dof * xm * ym * zm, PETSC_DETERMINE, PETSC_DETERMINE));
    1343:   PetscCall(MatSetType(A, da->mattype));
    1344:   PetscCall(MatSetFromOptions(A));
    1345:   PetscCall(MatSeqAIJSetPreallocation(A, 3 * 2, NULL));
    1346:   PetscCall(MatMPIAIJSetPreallocation(A, 3 * 2, NULL, 0, NULL));
    1347:   PetscCall(MatSeqBAIJSetPreallocation(A, 2, 3, NULL));
    1348:   PetscCall(MatMPIBAIJSetPreallocation(A, 2, 3, NULL, 0, NULL));
    1349:   PetscCall(MatSeqSBAIJSetPreallocation(A, 2, 2, NULL));
    1350:   PetscCall(MatMPISBAIJSetPreallocation(A, 2, 2, NULL, 0, NULL));
    1351:   PetscCall(MatSetLocalToGlobalMapping(A, ltog, ltog));
    1352:   PetscCall(DMDAGetGhostCorners(da, &starts[0], &starts[1], &starts[2], &dims[0], &dims[1], &dims[2]));
    1353:   PetscCall(MatSetStencil(A, dim, dims, starts, dof));
    1354:   *J = A;
    1355:   PetscFunctionReturn(PETSC_SUCCESS);
    1356: }

    1358: static PetscErrorCode THIDAVecView_VTK_XML(THI thi, DM da, Vec X, const char filename[])
    1359: {
    1360:   const PetscInt     dof   = 2;
    1361:   Units              units = thi->units;
    1362:   MPI_Comm           comm;
    1363:   PetscViewer        viewer;
    1364:   PetscMPIInt        rank, size, tag, nn, nmax;
    1365:   PetscInt           mx, my, mz, range[6];
    1366:   const PetscScalar *x;

    1368:   PetscFunctionBeginUser;
    1369:   PetscCall(PetscObjectGetComm((PetscObject)thi, &comm));
    1370:   PetscCall(DMDAGetInfo(da, 0, &mz, &my, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    1371:   PetscCallMPI(MPI_Comm_size(comm, &size));
    1372:   PetscCallMPI(MPI_Comm_rank(comm, &rank));
    1373:   PetscCall(PetscViewerASCIIOpen(comm, filename, &viewer));
    1374:   PetscCall(PetscViewerASCIIPrintf(viewer, "<VTKFile type=\"StructuredGrid\" version=\"0.1\" byte_order=\"LittleEndian\">\n"));
    1375:   PetscCall(PetscViewerASCIIPrintf(viewer, "  <StructuredGrid WholeExtent=\"%d %" PetscInt_FMT " %d %" PetscInt_FMT " %d %" PetscInt_FMT "\">\n", 0, mz - 1, 0, my - 1, 0, mx - 1));

    1377:   PetscCall(DMDAGetCorners(da, range, range + 1, range + 2, range + 3, range + 4, range + 5));
    1378:   PetscCall(PetscMPIIntCast(range[3] * range[4] * range[5] * dof, &nn));
    1379:   PetscCallMPI(MPI_Reduce(&nn, &nmax, 1, MPI_INT, MPI_MAX, 0, comm));
    1380:   tag = ((PetscObject)viewer)->tag;
    1381:   PetscCall(VecGetArrayRead(X, &x));
    1382:   if (rank == 0) {
    1383:     PetscScalar *array;
    1384:     PetscCall(PetscMalloc1(nmax, &array));
    1385:     for (PetscMPIInt r = 0; r < size; r++) {
    1386:       PetscInt           i, j, k, xs, xm, ys, ym, zs, zm;
    1387:       const PetscScalar *ptr;
    1388:       MPI_Status         status;
    1389:       if (r) PetscCallMPI(MPIU_Recv(range, 6, MPIU_INT, r, tag, comm, MPI_STATUS_IGNORE));
    1390:       zs = range[0];
    1391:       ys = range[1];
    1392:       xs = range[2];
    1393:       zm = range[3];
    1394:       ym = range[4];
    1395:       xm = range[5];
    1396:       PetscCheck(xm * ym * zm * dof <= nmax, PETSC_COMM_SELF, PETSC_ERR_PLIB, "should not happen");
    1397:       if (r) {
    1398:         PetscCallMPI(MPIU_Recv(array, nmax, MPIU_SCALAR, r, tag, comm, &status));
    1399:         PetscCallMPI(MPI_Get_count(&status, MPIU_SCALAR, &nn));
    1400:         PetscCheck(nn == xm * ym * zm * dof, PETSC_COMM_SELF, PETSC_ERR_PLIB, "should not happen");
    1401:         ptr = array;
    1402:       } else ptr = x;
    1403:       PetscCall(PetscViewerASCIIPrintf(viewer, "    <Piece Extent=\"%" PetscInt_FMT " %" PetscInt_FMT " %" PetscInt_FMT " %" PetscInt_FMT " %" PetscInt_FMT " %" PetscInt_FMT "\">\n", zs, zs + zm - 1, ys, ys + ym - 1, xs, xs + xm - 1));

    1405:       PetscCall(PetscViewerASCIIPrintf(viewer, "      <Points>\n"));
    1406:       PetscCall(PetscViewerASCIIPrintf(viewer, "        <DataArray type=\"Float32\" NumberOfComponents=\"3\" format=\"ascii\">\n"));
    1407:       for (i = xs; i < xs + xm; i++) {
    1408:         for (j = ys; j < ys + ym; j++) {
    1409:           for (k = zs; k < zs + zm; k++) {
    1410:             PrmNode   p;
    1411:             PetscReal xx = thi->Lx * i / mx, yy = thi->Ly * j / my, zz;
    1412:             thi->initialize(thi, xx, yy, &p);
    1413:             zz = PetscRealPart(p.b) + PetscRealPart(p.h) * k / (mz - 1);
    1414:             PetscCall(PetscViewerASCIIPrintf(viewer, "%f %f %f\n", (double)xx, (double)yy, (double)zz));
    1415:           }
    1416:         }
    1417:       }
    1418:       PetscCall(PetscViewerASCIIPrintf(viewer, "        </DataArray>\n"));
    1419:       PetscCall(PetscViewerASCIIPrintf(viewer, "      </Points>\n"));

    1421:       PetscCall(PetscViewerASCIIPrintf(viewer, "      <PointData>\n"));
    1422:       PetscCall(PetscViewerASCIIPrintf(viewer, "        <DataArray type=\"Float32\" Name=\"velocity\" NumberOfComponents=\"3\" format=\"ascii\">\n"));
    1423:       for (i = 0; i < nn; i += dof) PetscCall(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));
    1424:       PetscCall(PetscViewerASCIIPrintf(viewer, "        </DataArray>\n"));

    1426:       PetscCall(PetscViewerASCIIPrintf(viewer, "        <DataArray type=\"Int32\" Name=\"rank\" NumberOfComponents=\"1\" format=\"ascii\">\n"));
    1427:       for (i = 0; i < nn; i += dof) PetscCall(PetscViewerASCIIPrintf(viewer, "%d\n", r));
    1428:       PetscCall(PetscViewerASCIIPrintf(viewer, "        </DataArray>\n"));
    1429:       PetscCall(PetscViewerASCIIPrintf(viewer, "      </PointData>\n"));

    1431:       PetscCall(PetscViewerASCIIPrintf(viewer, "    </Piece>\n"));
    1432:     }
    1433:     PetscCall(PetscFree(array));
    1434:   } else {
    1435:     PetscCallMPI(MPIU_Send(range, 6, MPIU_INT, 0, tag, comm));
    1436:     PetscCallMPI(MPIU_Send((PetscScalar *)x, nn, MPIU_SCALAR, 0, tag, comm));
    1437:   }
    1438:   PetscCall(VecRestoreArrayRead(X, &x));
    1439:   PetscCall(PetscViewerASCIIPrintf(viewer, "  </StructuredGrid>\n"));
    1440:   PetscCall(PetscViewerASCIIPrintf(viewer, "</VTKFile>\n"));
    1441:   PetscCall(PetscViewerDestroy(&viewer));
    1442:   PetscFunctionReturn(PETSC_SUCCESS);
    1443: }

    1445: int main(int argc, char *argv[])
    1446: {
    1447:   MPI_Comm comm;
    1448:   THI      thi;
    1449:   DM       da;
    1450:   SNES     snes;

    1452:   PetscFunctionBeginUser;
    1453:   PetscCall(PetscInitialize(&argc, &argv, 0, help));
    1454:   comm = PETSC_COMM_WORLD;

    1456:   PetscCall(THICreate(comm, &thi));
    1457:   {
    1458:     PetscInt M = 3, N = 3, P = 2;
    1459:     PetscOptionsBegin(comm, NULL, "Grid resolution options", "");
    1460:     {
    1461:       PetscCall(PetscOptionsInt("-M", "Number of elements in x-direction on coarse level", "", M, &M, NULL));
    1462:       N = M;
    1463:       PetscCall(PetscOptionsInt("-N", "Number of elements in y-direction on coarse level (if different from M)", "", N, &N, NULL));
    1464:       if (thi->coarse2d) {
    1465:         PetscCall(PetscOptionsInt("-zlevels", "Number of elements in z-direction on fine level", "", thi->zlevels, &thi->zlevels, NULL));
    1466:       } else {
    1467:         PetscCall(PetscOptionsInt("-P", "Number of elements in z-direction on coarse level", "", P, &P, NULL));
    1468:       }
    1469:     }
    1470:     PetscOptionsEnd();
    1471:     if (thi->coarse2d) {
    1472:       PetscCall(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));
    1473:       PetscCall(DMSetFromOptions(da));
    1474:       PetscCall(DMSetUp(da));
    1475:       da->ops->refinehierarchy     = DMRefineHierarchy_THI;
    1476:       da->ops->createinterpolation = DMCreateInterpolation_DA_THI;

    1478:       PetscCall(PetscObjectCompose((PetscObject)da, "THI", (PetscObject)thi));
    1479:     } else {
    1480:       PetscCall(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));
    1481:       PetscCall(DMSetFromOptions(da));
    1482:       PetscCall(DMSetUp(da));
    1483:     }
    1484:     PetscCall(DMDASetFieldName(da, 0, "x-velocity"));
    1485:     PetscCall(DMDASetFieldName(da, 1, "y-velocity"));
    1486:   }
    1487:   PetscCall(THISetUpDM(thi, da));
    1488:   if (thi->tridiagonal) da->ops->creatematrix = DMCreateMatrix_THI_Tridiagonal;

    1490:   { /* Set the fine level matrix type if -da_refine */
    1491:     PetscInt rlevel, clevel;
    1492:     PetscCall(DMGetRefineLevel(da, &rlevel));
    1493:     PetscCall(DMGetCoarsenLevel(da, &clevel));
    1494:     if (rlevel - clevel > 0) PetscCall(DMSetMatType(da, thi->mattype));
    1495:   }

    1497:   PetscCall(DMDASNESSetFunctionLocal(da, ADD_VALUES, (DMDASNESFunctionFn *)THIFunctionLocal, thi));
    1498:   if (thi->tridiagonal) {
    1499:     PetscCall(DMDASNESSetJacobianLocal(da, (DMDASNESJacobianFn *)THIJacobianLocal_3D_Tridiagonal, thi));
    1500:   } else {
    1501:     PetscCall(DMDASNESSetJacobianLocal(da, (DMDASNESJacobianFn *)THIJacobianLocal_3D_Full, thi));
    1502:   }
    1503:   PetscCall(DMCoarsenHookAdd(da, DMCoarsenHook_THI, NULL, thi));
    1504:   PetscCall(DMRefineHookAdd(da, DMRefineHook_THI, NULL, thi));

    1506:   PetscCall(DMSetApplicationContext(da, thi));

    1508:   PetscCall(SNESCreate(comm, &snes));
    1509:   PetscCall(SNESSetDM(snes, da));
    1510:   PetscCall(DMDestroy(&da));
    1511:   PetscCall(SNESSetComputeInitialGuess(snes, THIInitial, NULL));
    1512:   PetscCall(SNESSetFromOptions(snes));

    1514:   PetscCall(SNESSolve(snes, NULL, NULL));

    1516:   PetscCall(THISolveStatistics(thi, snes, 0, "Full"));

    1518:   {
    1519:     PetscBool flg;
    1520:     char      filename[PETSC_MAX_PATH_LEN] = "";
    1521:     PetscCall(PetscOptionsGetString(NULL, NULL, "-o", filename, sizeof(filename), &flg));
    1522:     if (flg) {
    1523:       Vec X;
    1524:       DM  dm;
    1525:       PetscCall(SNESGetSolution(snes, &X));
    1526:       PetscCall(SNESGetDM(snes, &dm));
    1527:       PetscCall(THIDAVecView_VTK_XML(thi, dm, X, filename));
    1528:     }
    1529:   }

    1531:   PetscCall(DMDestroy(&da));
    1532:   PetscCall(SNESDestroy(&snes));
    1533:   PetscCall(THIDestroy(&thi));
    1534:   PetscCall(PetscFinalize());
    1535:   return 0;
    1536: }

    1538: /*TEST

    1540:    build:
    1541:       requires: !single

    1543:    test:
    1544:       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

    1546:    test:
    1547:       suffix: 2
    1548:       nsize: 2
    1549:       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 0

    1551:    test:
    1552:       suffix: 3
    1553:       nsize: 3
    1554:       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

    1556:    test:
    1557:       suffix: 4
    1558:       nsize: 6
    1559:       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 0

    1561:    test:
    1562:       suffix: 5
    1563:       nsize: 6
    1564:       args: -M 12 -P 5 -snes_monitor_short -ksp_converged_reason -pc_type asm -pc_asm_type restrict -dm_mat_type {{aij baij sbaij}}

    1566: TEST*/