Actual source code: ex48.c

petsc-master 2016-06-23
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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
 62: #include <petscsnes.h>
 63: #include <petsc/private/dmdaimpl.h>     /* There is not yet a public interface to manipulate dm->ops */
 64: #endif
 65: #include <ctype.h>              /* toupper() */

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

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

 77: /* The SSE2 kernels are only for PetscScalar=double on architectures that support it */
 78: #define USE_SSE2_KERNELS (!defined NO_SSE2                              \
 79:                           && !defined PETSC_USE_COMPLEX                 \
 80:                           && !defined PETSC_USE_REAL_SINGLE             \
 81:                           && !defined PETSC_USE_REAL___FLOAT128         \
 82:                           && defined __SSE2__)

 84: static PetscClassId THI_CLASSID;

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

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

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

170: #define QuadExtract(x,i,j,n) do {               \
171:     (n)[0] = (x)[i][j];                         \
172:     (n)[1] = (x)[i+1][j];                       \
173:     (n)[2] = (x)[i+1][j+1];                     \
174:     (n)[3] = (x)[i][j+1];                       \
175:   } while (0)

177: static void HexGrad(const PetscReal dphi[][3],const PetscReal zn[],PetscReal dz[])
178: {
179:   PetscInt i;
180:   dz[0] = dz[1] = dz[2] = 0;
181:   for (i=0; i<8; i++) {
182:     dz[0] += dphi[i][0] * zn[i];
183:     dz[1] += dphi[i][1] * zn[i];
184:     dz[2] += dphi[i][2] * zn[i];
185:   }
186: }

188: 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)
189: {
190:   const PetscReal jac[3][3]  = {{hx/2,0,0}, {0,hy/2,0}, {dz[0],dz[1],dz[2]}};
191:   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]}};
192:   const PetscReal jdet       = jac[0][0]*jac[1][1]*jac[2][2];
193:   PetscInt        i;

195:   for (i=0; i<8; i++) {
196:     const PetscReal *dphir = HexQDeriv[q][i];
197:     phi[i]     = HexQInterp[q][i];
198:     dphi[i][0] = dphir[0]*ijac[0][0] + dphir[1]*ijac[1][0] + dphir[2]*ijac[2][0];
199:     dphi[i][1] = dphir[0]*ijac[0][1] + dphir[1]*ijac[1][1] + dphir[2]*ijac[2][1];
200:     dphi[i][2] = dphir[0]*ijac[0][2] + dphir[1]*ijac[1][2] + dphir[2]*ijac[2][2];
201:   }
202:   *jw = 1.0 * jdet;
203: }

205: typedef struct _p_THI   *THI;
206: typedef struct _n_Units *Units;

208: typedef struct {
209:   PetscScalar u,v;
210: } Node;

212: typedef struct {
213:   PetscScalar b;                /* bed */
214:   PetscScalar h;                /* thickness */
215:   PetscScalar beta2;            /* friction */
216: } PrmNode;

218: typedef struct {
219:   PetscReal min,max,cmin,cmax;
220: } PRange;

222: typedef enum {THIASSEMBLY_TRIDIAGONAL,THIASSEMBLY_FULL} THIAssemblyMode;

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

249: struct _n_Units {
250:   /* fundamental */
251:   PetscReal meter;
252:   PetscReal kilogram;
253:   PetscReal second;
254:   /* derived */
255:   PetscReal Pascal;
256:   PetscReal year;
257: };

259: static PetscErrorCode THIJacobianLocal_3D_Full(DMDALocalInfo*,Node***,Mat,Mat,THI);
260: static PetscErrorCode THIJacobianLocal_3D_Tridiagonal(DMDALocalInfo*,Node***,Mat,Mat,THI);
261: static PetscErrorCode THIJacobianLocal_2D(DMDALocalInfo*,Node**,Mat,Mat,THI);

263: static void PrmHexGetZ(const PrmNode pn[],PetscInt k,PetscInt zm,PetscReal zn[])
264: {
265:   const PetscScalar zm1    = zm-1,
266:                     znl[8] = {pn[0].b + pn[0].h*(PetscScalar)k/zm1,
267:                               pn[1].b + pn[1].h*(PetscScalar)k/zm1,
268:                               pn[2].b + pn[2].h*(PetscScalar)k/zm1,
269:                               pn[3].b + pn[3].h*(PetscScalar)k/zm1,
270:                               pn[0].b + pn[0].h*(PetscScalar)(k+1)/zm1,
271:                               pn[1].b + pn[1].h*(PetscScalar)(k+1)/zm1,
272:                               pn[2].b + pn[2].h*(PetscScalar)(k+1)/zm1,
273:                               pn[3].b + pn[3].h*(PetscScalar)(k+1)/zm1};
274:   PetscInt i;
275:   for (i=0; i<8; i++) zn[i] = PetscRealPart(znl[i]);
276: }

278: /* Tests A and C are from the ISMIP-HOM paper (Pattyn et al. 2008) */
279: static void THIInitialize_HOM_A(THI thi,PetscReal x,PetscReal y,PrmNode *p)
280: {
281:   Units     units = thi->units;
282:   PetscReal s     = -x*PetscSinReal(thi->alpha);

284:   p->b     = s - 1000*units->meter + 500*units->meter * PetscSinReal(x*2*PETSC_PI/thi->Lx) * PetscSinReal(y*2*PETSC_PI/thi->Ly);
285:   p->h     = s - p->b;
286:   p->beta2 = 1e30;
287: }

289: static void THIInitialize_HOM_C(THI thi,PetscReal x,PetscReal y,PrmNode *p)
290: {
291:   Units     units = thi->units;
292:   PetscReal s     = -x*PetscSinReal(thi->alpha);

294:   p->b = s - 1000*units->meter;
295:   p->h = s - p->b;
296:   /* tau_b = beta2 v   is a stress (Pa) */
297:   p->beta2 = 1000 * (1 + PetscSinReal(x*2*PETSC_PI/thi->Lx)*PetscSinReal(y*2*PETSC_PI/thi->Ly)) * units->Pascal * units->year / units->meter;
298: }

300: /* These are just toys */

302: /* Same bed as test A, free slip everywhere except for a discontinuous jump to a circular sticky region in the middle. */
303: static void THIInitialize_HOM_X(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
304: {
305:   Units     units = thi->units;
306:   PetscReal x     = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
307:   PetscReal r     = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);
308:   p->b     = s - 1000*units->meter + 500*units->meter*PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
309:   p->h     = s - p->b;
310:   p->beta2 = 1000 * (r < 1 ? 2 : 0) * units->Pascal * units->year / units->meter;
311: }

313: /* Like Z, but with 200 meter cliffs */
314: static void THIInitialize_HOM_Y(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
315: {
316:   Units     units = thi->units;
317:   PetscReal x     = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
318:   PetscReal r     = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);

320:   p->b = s - 1000*units->meter + 500*units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
321:   if (PetscRealPart(p->b) > -700*units->meter) p->b += 200*units->meter;
322:   p->h     = s - p->b;
323:   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;
324: }

326: /* Same bed as A, smoothly varying slipperiness, similar to MATLAB's "sombrero" (uncorrelated with bathymetry) */
327: static void THIInitialize_HOM_Z(THI thi,PetscReal xx,PetscReal yy,PrmNode *p)
328: {
329:   Units     units = thi->units;
330:   PetscReal x     = xx*2*PETSC_PI/thi->Lx - PETSC_PI,y = yy*2*PETSC_PI/thi->Ly - PETSC_PI; /* [-pi,pi] */
331:   PetscReal r     = PetscSqrtReal(x*x + y*y),s = -x*PetscSinReal(thi->alpha);

333:   p->b     = s - 1000*units->meter + 500*units->meter * PetscSinReal(x + PETSC_PI) * PetscSinReal(y + PETSC_PI);
334:   p->h     = s - p->b;
335:   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;
336: }

338: static void THIFriction(THI thi,PetscReal rbeta2,PetscReal gam,PetscReal *beta2,PetscReal *dbeta2)
339: {
340:   if (thi->friction.irefgam == 0) {
341:     Units units = thi->units;
342:     thi->friction.irefgam = 1./(0.5*PetscSqr(thi->friction.refvel * units->meter / units->year));
343:     thi->friction.eps2    = 0.5*PetscSqr(thi->friction.epsvel * units->meter / units->year) * thi->friction.irefgam;
344:   }
345:   if (thi->friction.exponent == 0) {
346:     *beta2  = rbeta2;
347:     *dbeta2 = 0;
348:   } else {
349:     *beta2  = rbeta2 * PetscPowReal(thi->friction.eps2 + gam*thi->friction.irefgam,thi->friction.exponent);
350:     *dbeta2 = thi->friction.exponent * *beta2 / (thi->friction.eps2 + gam*thi->friction.irefgam) * thi->friction.irefgam;
351:   }
352: }

354: static void THIViscosity(THI thi,PetscReal gam,PetscReal *eta,PetscReal *deta)
355: {
356:   PetscReal Bd2,eps,exponent;
357:   if (thi->viscosity.Bd2 == 0) {
358:     Units units = thi->units;
359:     const PetscReal
360:       n = 3.,                                           /* Glen exponent */
361:       p = 1. + 1./n,                                    /* for Stokes */
362:       A = 1.e-16 * PetscPowReal(units->Pascal,-n) / units->year, /* softness parameter (Pa^{-n}/s) */
363:       B = PetscPowReal(A,-1./n);                                 /* hardness parameter */
364:     thi->viscosity.Bd2      = B/2;
365:     thi->viscosity.exponent = (p-2)/2;
366:     thi->viscosity.eps      = 0.5*PetscSqr(1e-5 / units->year);
367:   }
368:   Bd2      = thi->viscosity.Bd2;
369:   exponent = thi->viscosity.exponent;
370:   eps      = thi->viscosity.eps;
371:   *eta     = Bd2 * PetscPowReal(eps + gam,exponent);
372:   *deta    = exponent * (*eta) / (eps + gam);
373: }

375: static void RangeUpdate(PetscReal *min,PetscReal *max,PetscReal x)
376: {
377:   if (x < *min) *min = x;
378:   if (x > *max) *max = x;
379: }

381: static void PRangeClear(PRange *p)
382: {
383:   p->cmin = p->min = 1e100;
384:   p->cmax = p->max = -1e100;
385: }

389: static PetscErrorCode PRangeMinMax(PRange *p,PetscReal min,PetscReal max)
390: {

393:   p->cmin = min;
394:   p->cmax = max;
395:   if (min < p->min) p->min = min;
396:   if (max > p->max) p->max = max;
397:   return(0);
398: }

402: static PetscErrorCode THIDestroy(THI *thi)
403: {

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

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: }

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

568:   DMDAGetGhostCorners(da2prm,&ys,&xs,0,&ym,&xm,0);
569:   DMDAGetInfo(da2prm,0, &my,&mx,0, 0,0,0, 0,0,0,0,0,0);
570:   DMDAVecGetArray(da2prm,prm,&p);
571:   for (i=xs; i<xs+xm; i++) {
572:     for (j=ys; j<ys+ym; j++) {
573:       PetscReal xx = thi->Lx*i/mx,yy = thi->Ly*j/my;
574:       thi->initialize(thi,xx,yy,&p[i][j]);
575:     }
576:   }
577:   DMDAVecRestoreArray(da2prm,prm,&p);
578:   return(0);
579: }

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

592:   DMDAGetInfo(dm,&dim, &Mz,&My,&Mx, 0,&my,&mx, 0,&s,0,0,0,&st);
593:   if (dim == 2) {
594:     DMDAGetInfo(dm,&dim, &My,&Mx,0, &my,&mx,0, 0,&s,0,0,0,&st);
595:   }
596:   DMGetRefineLevel(dm,&refinelevel);
597:   DMGetCoarsenLevel(dm,&coarsenlevel);
598:   level = refinelevel - coarsenlevel;
599:   DMDACreate2d(PetscObjectComm((PetscObject)thi),DM_BOUNDARY_PERIODIC,DM_BOUNDARY_PERIODIC,st,My,Mx,my,mx,sizeof(PrmNode)/sizeof(PetscScalar),s,0,0,&da2prm);
600:   DMCreateLocalVector(da2prm,&X);
601:   {
602:     PetscReal Lx = thi->Lx / thi->units->meter,Ly = thi->Ly / thi->units->meter,Lz = thi->Lz / thi->units->meter;
603:     if (dim == 2) {
604:       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));
605:     } else {
606:       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)));
607:     }
608:   }
609:   THIInitializePrm(thi,da2prm,X);
610:   if (thi->tridiagonal) {       /* Reset coarse Jacobian evaluation */
611:     DMDASNESSetJacobianLocal(dm,(DMDASNESJacobian)THIJacobianLocal_3D_Full,thi);
612:   }
613:   if (thi->coarse2d) {
614:     DMDASNESSetJacobianLocal(dm,(DMDASNESJacobian)THIJacobianLocal_2D,thi);
615:   }
616:   PetscObjectCompose((PetscObject)dm,"DMDA2Prm",(PetscObject)da2prm);
617:   PetscObjectCompose((PetscObject)dm,"DMDA2Prm_Vec",(PetscObject)X);
618:   DMDestroy(&da2prm);
619:   VecDestroy(&X);
620:   return(0);
621: }

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

632:   THISetUpDM(thi,dmc);
633:   DMGetRefineLevel(dmc,&rlevel);
634:   DMGetCoarsenLevel(dmc,&clevel);
635:   if (rlevel-clevel == 0) {DMSetMatType(dmc,MATAIJ);}
636:   DMCoarsenHookAdd(dmc,DMCoarsenHook_THI,NULL,thi);
637:   return(0);
638: }

642: static PetscErrorCode DMRefineHook_THI(DM dmc,DM dmf,void *ctx)
643: {
644:   THI            thi = (THI)ctx;

648:   THISetUpDM(thi,dmf);
649:   DMSetMatType(dmf,thi->mattype);
650:   DMRefineHookAdd(dmf,DMRefineHook_THI,NULL,thi);
651:   /* With grid sequencing, a formerly-refined DM will later be coarsened by PCSetUp_MG */
652:   DMCoarsenHookAdd(dmf,DMCoarsenHook_THI,NULL,thi);
653:   return(0);
654: }

658: static PetscErrorCode THIDAGetPrm(DM da,PrmNode ***prm)
659: {
661:   DM             da2prm;
662:   Vec            X;

665:   PetscObjectQuery((PetscObject)da,"DMDA2Prm",(PetscObject*)&da2prm);
666:   if (!da2prm) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm composed with given DMDA");
667:   PetscObjectQuery((PetscObject)da,"DMDA2Prm_Vec",(PetscObject*)&X);
668:   if (!X) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm_Vec composed with given DMDA");
669:   DMDAVecGetArray(da2prm,X,prm);
670:   return(0);
671: }

675: static PetscErrorCode THIDARestorePrm(DM da,PrmNode ***prm)
676: {
678:   DM             da2prm;
679:   Vec            X;

682:   PetscObjectQuery((PetscObject)da,"DMDA2Prm",(PetscObject*)&da2prm);
683:   if (!da2prm) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm composed with given DMDA");
684:   PetscObjectQuery((PetscObject)da,"DMDA2Prm_Vec",(PetscObject*)&X);
685:   if (!X) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No DMDA2Prm_Vec composed with given DMDA");
686:   DMDAVecRestoreArray(da2prm,X,prm);
687:   return(0);
688: }

692: static PetscErrorCode THIInitial(SNES snes,Vec X,void *ctx)
693: {
694:   THI            thi;
695:   PetscInt       i,j,k,xs,xm,ys,ym,zs,zm,mx,my;
696:   PetscReal      hx,hy;
697:   PrmNode        **prm;
698:   Node           ***x;
700:   DM             da;

703:   SNESGetDM(snes,&da);
704:   DMGetApplicationContext(da,&thi);
705:   DMDAGetInfo(da,0, 0,&my,&mx, 0,0,0, 0,0,0,0,0,0);
706:   DMDAGetCorners(da,&zs,&ys,&xs,&zm,&ym,&xm);
707:   DMDAVecGetArray(da,X,&x);
708:   THIDAGetPrm(da,&prm);
709:   hx   = thi->Lx / mx;
710:   hy   = thi->Ly / my;
711:   for (i=xs; i<xs+xm; i++) {
712:     for (j=ys; j<ys+ym; j++) {
713:       for (k=zs; k<zs+zm; k++) {
714:         const PetscScalar zm1      = zm-1,
715:                           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),
716:                           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);
717:         x[i][j][k].u = 0. * drivingx * prm[i][j].h*(PetscScalar)k/zm1;
718:         x[i][j][k].v = 0. * drivingy * prm[i][j].h*(PetscScalar)k/zm1;
719:       }
720:     }
721:   }
722:   DMDAVecRestoreArray(da,X,&x);
723:   THIDARestorePrm(da,&prm);
724:   return(0);
725: }

727: 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)
728: {
729:   PetscInt    l,ll;
730:   PetscScalar gam;

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

748: 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)
749: {
750:   PetscInt    l,ll;
751:   PetscScalar gam;

753:   du[0] = du[1] = 0;
754:   dv[0] = dv[1] = 0;
755:   *u    = 0;
756:   *v    = 0;
757:   for (l=0; l<4; l++) {
758:     *u += phi[l] * n[l].u;
759:     *v += phi[l] * n[l].v;
760:     for (ll=0; ll<2; ll++) {
761:       du[ll] += dphi[l][ll] * n[l].u;
762:       dv[ll] += dphi[l][ll] * n[l].v;
763:     }
764:   }
765:   gam = PetscSqr(du[0]) + PetscSqr(dv[1]) + du[0]*dv[1] + 0.25*PetscSqr(du[1]+dv[0]);
766:   THIViscosity(thi,PetscRealPart(gam),eta,deta);
767: }

771: static PetscErrorCode THIFunctionLocal(DMDALocalInfo *info,Node ***x,Node ***f,THI thi)
772: {
773:   PetscInt       xs,ys,xm,ym,zm,i,j,k,q,l;
774:   PetscReal      hx,hy,etamin,etamax,beta2min,beta2max;
775:   PrmNode        **prm;

779:   xs = info->zs;
780:   ys = info->ys;
781:   xm = info->zm;
782:   ym = info->ym;
783:   zm = info->xm;
784:   hx = thi->Lx / info->mz;
785:   hy = thi->Ly / info->my;

787:   etamin   = 1e100;
788:   etamax   = 0;
789:   beta2min = 1e100;
790:   beta2max = 0;

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

794:   for (i=xs; i<xs+xm; i++) {
795:     for (j=ys; j<ys+ym; j++) {
796:       PrmNode pn[4];
797:       QuadExtract(prm,i,j,pn);
798:       for (k=0; k<zm-1; k++) {
799:         PetscInt  ls = 0;
800:         Node      n[8],*fn[8];
801:         PetscReal zn[8],etabase = 0;
802:         PrmHexGetZ(pn,k,zm,zn);
803:         HexExtract(x,i,j,k,n);
804:         HexExtractRef(f,i,j,k,fn);
805:         if (thi->no_slip && k == 0) {
806:           for (l=0; l<4; l++) n[l].u = n[l].v = 0;
807:           /* The first 4 basis functions lie on the bottom layer, so their contribution is exactly 0, hence we can skip them */
808:           ls = 4;
809:         }
810:         for (q=0; q<8; q++) {
811:           PetscReal   dz[3],phi[8],dphi[8][3],jw,eta,deta;
812:           PetscScalar du[3],dv[3],u,v;
813:           HexGrad(HexQDeriv[q],zn,dz);
814:           HexComputeGeometry(q,hx,hy,dz,phi,dphi,&jw);
815:           PointwiseNonlinearity(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
816:           jw /= thi->rhog;      /* scales residuals to be O(1) */
817:           if (q == 0) etabase = eta;
818:           RangeUpdate(&etamin,&etamax,eta);
819:           for (l=ls; l<8; l++) { /* test functions */
820:             const PetscReal ds[2] = {-PetscSinReal(thi->alpha),0};
821:             const PetscReal pp    = phi[l],*dp = dphi[l];
822:             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];
823:             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];
824:           }
825:         }
826:         if (k == 0) { /* we are on a bottom face */
827:           if (thi->no_slip) {
828:             /* Note: Non-Galerkin coarse grid operators are very sensitive to the scaling of Dirichlet boundary
829:             * conditions.  After shenanigans above, etabase contains the effective viscosity at the closest quadrature
830:             * point to the bed.  We want the diagonal entry in the Dirichlet condition to have similar magnitude to the
831:             * diagonal entry corresponding to the adjacent node.  The fundamental scaling of the viscous part is in
832:             * diagu, diagv below.  This scaling is easy to recognize by considering the finite difference operator after
833:             * scaling by element size.  The no-slip Dirichlet condition is scaled by this factor, and also in the
834:             * assembled matrix (see the similar block in THIJacobianLocal).
835:             *
836:             * Note that the residual at this Dirichlet node is linear in the state at this node, but also depends
837:             * (nonlinearly in general) on the neighboring interior nodes through the local viscosity.  This will make
838:             * a matrix-free Jacobian have extra entries in the corresponding row.  We assemble only the diagonal part,
839:             * so the solution will exactly satisfy the boundary condition after the first linear iteration.
840:             */
841:             const PetscReal   hz    = PetscRealPart(pn[0].h)/(zm-1.);
842:             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);
843:             fn[0]->u = thi->dirichlet_scale*diagu*x[i][j][k].u;
844:             fn[0]->v = thi->dirichlet_scale*diagv*x[i][j][k].v;
845:           } else {              /* Integrate over bottom face to apply boundary condition */
846:             for (q=0; q<4; q++) {
847:               const PetscReal jw = 0.25*hx*hy/thi->rhog,*phi = QuadQInterp[q];
848:               PetscScalar     u  =0,v=0,rbeta2=0;
849:               PetscReal       beta2,dbeta2;
850:               for (l=0; l<4; l++) {
851:                 u      += phi[l]*n[l].u;
852:                 v      += phi[l]*n[l].v;
853:                 rbeta2 += phi[l]*pn[l].beta2;
854:               }
855:               THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
856:               RangeUpdate(&beta2min,&beta2max,beta2);
857:               for (l=0; l<4; l++) {
858:                 const PetscReal pp = phi[l];
859:                 fn[ls+l]->u += pp*jw*beta2*u;
860:                 fn[ls+l]->v += pp*jw*beta2*v;
861:               }
862:             }
863:           }
864:         }
865:       }
866:     }
867:   }

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

871:   PRangeMinMax(&thi->eta,etamin,etamax);
872:   PRangeMinMax(&thi->beta2,beta2min,beta2max);
873:   return(0);
874: }

878: static PetscErrorCode THIMatrixStatistics(THI thi,Mat B,PetscViewer viewer)
879: {
881:   PetscReal      nrm;
882:   PetscInt       m;
883:   PetscMPIInt    rank;

886:   MatNorm(B,NORM_FROBENIUS,&nrm);
887:   MatGetSize(B,&m,0);
888:   MPI_Comm_rank(PetscObjectComm((PetscObject)B),&rank);
889:   if (!rank) {
890:     PetscScalar val0,val2;
891:     MatGetValue(B,0,0,&val0);
892:     MatGetValue(B,2,2,&val2);
893:     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);
894:   }
895:   return(0);
896: }

900: static PetscErrorCode THISurfaceStatistics(DM da,Vec X,PetscReal *min,PetscReal *max,PetscReal *mean)
901: {
903:   Node           ***x;
904:   PetscInt       i,j,xs,ys,zs,xm,ym,zm,mx,my,mz;
905:   PetscReal      umin = 1e100,umax=-1e100;
906:   PetscScalar    usum = 0.0,gusum;

909:   *min = *max = *mean = 0;
910:   DMDAGetInfo(da,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
911:   DMDAGetCorners(da,&zs,&ys,&xs,&zm,&ym,&xm);
912:   if (zs != 0 || zm != mz) SETERRQ(PETSC_COMM_SELF,1,"Unexpected decomposition");
913:   DMDAVecGetArray(da,X,&x);
914:   for (i=xs; i<xs+xm; i++) {
915:     for (j=ys; j<ys+ym; j++) {
916:       PetscReal u = PetscRealPart(x[i][j][zm-1].u);
917:       RangeUpdate(&umin,&umax,u);
918:       usum += u;
919:     }
920:   }
921:   DMDAVecRestoreArray(da,X,&x);
922:   MPI_Allreduce(&umin,min,1,MPIU_REAL,MPIU_MIN,PetscObjectComm((PetscObject)da));
923:   MPI_Allreduce(&umax,max,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)da));
924:   MPI_Allreduce(&usum,&gusum,1,MPIU_SCALAR,MPIU_SUM,PetscObjectComm((PetscObject)da));
925:   *mean = PetscRealPart(gusum) / (mx*my);
926:   return(0);
927: }

931: static PetscErrorCode THISolveStatistics(THI thi,SNES snes,PetscInt coarsened,const char name[])
932: {
933:   MPI_Comm       comm;
934:   Vec            X;
935:   DM             dm;

939:   PetscObjectGetComm((PetscObject)thi,&comm);
940:   SNESGetSolution(snes,&X);
941:   SNESGetDM(snes,&dm);
942:   PetscPrintf(comm,"Solution statistics after solve: %s\n",name);
943:   {
944:     PetscInt            its,lits;
945:     SNESConvergedReason reason;
946:     SNESGetIterationNumber(snes,&its);
947:     SNESGetConvergedReason(snes,&reason);
948:     SNESGetLinearSolveIterations(snes,&lits);
949:     PetscPrintf(comm,"%s: Number of SNES iterations = %D, total linear iterations = %D\n",SNESConvergedReasons[reason],its,lits);
950:   }
951:   {
952:     PetscReal         nrm2,tmin[3]={1e100,1e100,1e100},tmax[3]={-1e100,-1e100,-1e100},min[3],max[3];
953:     PetscInt          i,j,m;
954:     const PetscScalar *x;
955:     VecNorm(X,NORM_2,&nrm2);
956:     VecGetLocalSize(X,&m);
957:     VecGetArrayRead(X,&x);
958:     for (i=0; i<m; i+=2) {
959:       PetscReal u = PetscRealPart(x[i]),v = PetscRealPart(x[i+1]),c = PetscSqrtReal(u*u+v*v);
960:       tmin[0] = PetscMin(u,tmin[0]);
961:       tmin[1] = PetscMin(v,tmin[1]);
962:       tmin[2] = PetscMin(c,tmin[2]);
963:       tmax[0] = PetscMax(u,tmax[0]);
964:       tmax[1] = PetscMax(v,tmax[1]);
965:       tmax[2] = PetscMax(c,tmax[2]);
966:     }
967:     VecRestoreArrayRead(X,&x);
968:     MPI_Allreduce(tmin,min,3,MPIU_REAL,MPIU_MIN,PetscObjectComm((PetscObject)thi));
969:     MPI_Allreduce(tmax,max,3,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)thi));
970:     /* Dimensionalize to meters/year */
971:     nrm2 *= thi->units->year / thi->units->meter;
972:     for (j=0; j<3; j++) {
973:       min[j] *= thi->units->year / thi->units->meter;
974:       max[j] *= thi->units->year / thi->units->meter;
975:     }
976:     if (min[0] == 0.0) min[0] = 0.0;
977:     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]);
978:     {
979:       PetscReal umin,umax,umean;
980:       THISurfaceStatistics(dm,X,&umin,&umax,&umean);
981:       umin  *= thi->units->year / thi->units->meter;
982:       umax  *= thi->units->year / thi->units->meter;
983:       umean *= thi->units->year / thi->units->meter;
984:       PetscPrintf(comm,"Surface statistics: u in [%12.6e, %12.6e] mean %12.6e\n",(double)umin,(double)umax,(double)umean);
985:     }
986:     /* These values stay nondimensional */
987:     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);
988:     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);
989:   }
990:   return(0);
991: }

995: static PetscErrorCode THIJacobianLocal_2D(DMDALocalInfo *info,Node **x,Mat J,Mat B,THI thi)
996: {
997:   PetscInt       xs,ys,xm,ym,i,j,q,l,ll;
998:   PetscReal      hx,hy;
999:   PrmNode        **prm;

1003:   xs = info->ys;
1004:   ys = info->xs;
1005:   xm = info->ym;
1006:   ym = info->xm;
1007:   hx = thi->Lx / info->my;
1008:   hy = thi->Ly / info->mx;

1010:   MatZeroEntries(B);
1011:   THIDAGetPrm(info->da,&prm);

1013:   for (i=xs; i<xs+xm; i++) {
1014:     for (j=ys; j<ys+ym; j++) {
1015:       Node        n[4];
1016:       PrmNode     pn[4];
1017:       PetscScalar Ke[4*2][4*2];
1018:       QuadExtract(prm,i,j,pn);
1019:       QuadExtract(x,i,j,n);
1020:       PetscMemzero(Ke,sizeof(Ke));
1021:       for (q=0; q<4; q++) {
1022:         PetscReal   phi[4],dphi[4][2],jw,eta,deta,beta2,dbeta2;
1023:         PetscScalar u,v,du[2],dv[2],h = 0,rbeta2 = 0;
1024:         for (l=0; l<4; l++) {
1025:           phi[l]     = QuadQInterp[q][l];
1026:           dphi[l][0] = QuadQDeriv[q][l][0]*2./hx;
1027:           dphi[l][1] = QuadQDeriv[q][l][1]*2./hy;
1028:           h         += phi[l] * pn[l].h;
1029:           rbeta2    += phi[l] * pn[l].beta2;
1030:         }
1031:         jw = 0.25*hx*hy / thi->rhog; /* rhog is only scaling */
1032:         PointwiseNonlinearity2D(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
1033:         THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
1034:         for (l=0; l<4; l++) {
1035:           const PetscReal pp = phi[l],*dp = dphi[l];
1036:           for (ll=0; ll<4; ll++) {
1037:             const PetscReal ppl = phi[ll],*dpl = dphi[ll];
1038:             PetscScalar     dgdu,dgdv;
1039:             dgdu = 2.*du[0]*dpl[0] + dv[1]*dpl[0] + 0.5*(du[1]+dv[0])*dpl[1];
1040:             dgdv = 2.*dv[1]*dpl[1] + du[0]*dpl[1] + 0.5*(du[1]+dv[0])*dpl[0];
1041:             /* Picard part */
1042:             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;
1043:             Ke[l*2+0][ll*2+1] += dp[0]*jw*eta*2.*dpl[1] + dp[1]*jw*eta*dpl[0];
1044:             Ke[l*2+1][ll*2+0] += dp[1]*jw*eta*2.*dpl[0] + dp[0]*jw*eta*dpl[1];
1045:             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;
1046:             /* extra Newton terms */
1047:             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;
1048:             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;
1049:             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;
1050:             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;
1051:           }
1052:         }
1053:       }
1054:       {
1055:         const MatStencil rc[4] = {{0,i,j,0},{0,i+1,j,0},{0,i+1,j+1,0},{0,i,j+1,0}};
1056:         MatSetValuesBlockedStencil(B,4,rc,4,rc,&Ke[0][0],ADD_VALUES);
1057:       }
1058:     }
1059:   }
1060:   THIDARestorePrm(info->da,&prm);

1062:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
1063:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
1064:   MatSetOption(B,MAT_SYMMETRIC,PETSC_TRUE);
1065:   if (thi->verbose) {THIMatrixStatistics(thi,B,PETSC_VIEWER_STDOUT_WORLD);}
1066:   return(0);
1067: }

1071: static PetscErrorCode THIJacobianLocal_3D(DMDALocalInfo *info,Node ***x,Mat B,THI thi,THIAssemblyMode amode)
1072: {
1073:   PetscInt       xs,ys,xm,ym,zm,i,j,k,q,l,ll;
1074:   PetscReal      hx,hy;
1075:   PrmNode        **prm;

1079:   xs = info->zs;
1080:   ys = info->ys;
1081:   xm = info->zm;
1082:   ym = info->ym;
1083:   zm = info->xm;
1084:   hx = thi->Lx / info->mz;
1085:   hy = thi->Ly / info->my;

1087:   MatZeroEntries(B);
1088:   MatSetOption(B,MAT_SUBSET_OFF_PROC_ENTRIES,PETSC_TRUE);
1089:   THIDAGetPrm(info->da,&prm);

1091:   for (i=xs; i<xs+xm; i++) {
1092:     for (j=ys; j<ys+ym; j++) {
1093:       PrmNode pn[4];
1094:       QuadExtract(prm,i,j,pn);
1095:       for (k=0; k<zm-1; k++) {
1096:         Node        n[8];
1097:         PetscReal   zn[8],etabase = 0;
1098:         PetscScalar Ke[8*2][8*2];
1099:         PetscInt    ls = 0;

1101:         PrmHexGetZ(pn,k,zm,zn);
1102:         HexExtract(x,i,j,k,n);
1103:         PetscMemzero(Ke,sizeof(Ke));
1104:         if (thi->no_slip && k == 0) {
1105:           for (l=0; l<4; l++) n[l].u = n[l].v = 0;
1106:           ls = 4;
1107:         }
1108:         for (q=0; q<8; q++) {
1109:           PetscReal   dz[3],phi[8],dphi[8][3],jw,eta,deta;
1110:           PetscScalar du[3],dv[3],u,v;
1111:           HexGrad(HexQDeriv[q],zn,dz);
1112:           HexComputeGeometry(q,hx,hy,dz,phi,dphi,&jw);
1113:           PointwiseNonlinearity(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
1114:           jw /= thi->rhog;      /* residuals are scaled by this factor */
1115:           if (q == 0) etabase = eta;
1116:           for (l=ls; l<8; l++) { /* test functions */
1117:             const PetscReal *PETSC_RESTRICT dp = dphi[l];
1118: #if USE_SSE2_KERNELS
1119:             /* gcc (up to my 4.5 snapshot) is really bad at hoisting intrinsics so we do it manually */
1120:             __m128d
1121:               p4         = _mm_set1_pd(4),p2 = _mm_set1_pd(2),p05 = _mm_set1_pd(0.5),
1122:               p42        = _mm_setr_pd(4,2),p24 = _mm_shuffle_pd(p42,p42,_MM_SHUFFLE2(0,1)),
1123:               du0        = _mm_set1_pd(du[0]),du1 = _mm_set1_pd(du[1]),du2 = _mm_set1_pd(du[2]),
1124:               dv0        = _mm_set1_pd(dv[0]),dv1 = _mm_set1_pd(dv[1]),dv2 = _mm_set1_pd(dv[2]),
1125:               jweta      = _mm_set1_pd(jw*eta),jwdeta = _mm_set1_pd(jw*deta),
1126:               dp0        = _mm_set1_pd(dp[0]),dp1 = _mm_set1_pd(dp[1]),dp2 = _mm_set1_pd(dp[2]),
1127:               dp0jweta   = _mm_mul_pd(dp0,jweta),dp1jweta = _mm_mul_pd(dp1,jweta),dp2jweta = _mm_mul_pd(dp2,jweta),
1128:               p4du0p2dv1 = _mm_add_pd(_mm_mul_pd(p4,du0),_mm_mul_pd(p2,dv1)), /* 4 du0 + 2 dv1 */
1129:               p4dv1p2du0 = _mm_add_pd(_mm_mul_pd(p4,dv1),_mm_mul_pd(p2,du0)), /* 4 dv1 + 2 du0 */
1130:               pdu2dv2    = _mm_unpacklo_pd(du2,dv2),                          /* [du2, dv2] */
1131:               du1pdv0    = _mm_add_pd(du1,dv0),                               /* du1 + dv0 */
1132:               t1         = _mm_mul_pd(dp0,p4du0p2dv1),                        /* dp0 (4 du0 + 2 dv1) */
1133:               t2         = _mm_mul_pd(dp1,p4dv1p2du0);                        /* dp1 (4 dv1 + 2 du0) */

1135: #endif
1136: #if defined COMPUTE_LOWER_TRIANGULAR  /* The element matrices are always symmetric so computing the lower-triangular part is not necessary */
1137:             for (ll=ls; ll<8; ll++) { /* trial functions */
1138: #else
1139:             for (ll=l; ll<8; ll++) {
1140: #endif
1141:               const PetscReal *PETSC_RESTRICT dpl = dphi[ll];
1142:               if (amode == THIASSEMBLY_TRIDIAGONAL && (l-ll)%4) continue; /* these entries would not be inserted */
1143: #if !USE_SSE2_KERNELS
1144:               /* The analytic Jacobian in nice, easy-to-read form */
1145:               {
1146:                 PetscScalar dgdu,dgdv;
1147:                 dgdu = 2.*du[0]*dpl[0] + dv[1]*dpl[0] + 0.5*(du[1]+dv[0])*dpl[1] + 0.5*du[2]*dpl[2];
1148:                 dgdv = 2.*dv[1]*dpl[1] + du[0]*dpl[1] + 0.5*(du[1]+dv[0])*dpl[0] + 0.5*dv[2]*dpl[2];
1149:                 /* Picard part */
1150:                 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];
1151:                 Ke[l*2+0][ll*2+1] += dp[0]*jw*eta*2.*dpl[1] + dp[1]*jw*eta*dpl[0];
1152:                 Ke[l*2+1][ll*2+0] += dp[1]*jw*eta*2.*dpl[0] + dp[0]*jw*eta*dpl[1];
1153:                 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];
1154:                 /* extra Newton terms */
1155:                 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];
1156:                 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];
1157:                 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];
1158:                 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];
1159:               }
1160: #else
1161:               /* This SSE2 code is an exact replica of above, but uses explicit packed instructions for some speed
1162:               * benefit.  On my hardware, these intrinsics are almost twice as fast as above, reducing total assembly cost
1163:               * by 25 to 30 percent. */
1164:               {
1165:                 __m128d
1166:                   keu   = _mm_loadu_pd(&Ke[l*2+0][ll*2+0]),
1167:                   kev   = _mm_loadu_pd(&Ke[l*2+1][ll*2+0]),
1168:                   dpl01 = _mm_loadu_pd(&dpl[0]),dpl10 = _mm_shuffle_pd(dpl01,dpl01,_MM_SHUFFLE2(0,1)),dpl2 = _mm_set_sd(dpl[2]),
1169:                   t0,t3,pdgduv;
1170:                 keu = _mm_add_pd(keu,_mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp0jweta,p42),dpl01),
1171:                                                 _mm_add_pd(_mm_mul_pd(dp1jweta,dpl10),
1172:                                                            _mm_mul_pd(dp2jweta,dpl2))));
1173:                 kev = _mm_add_pd(kev,_mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp1jweta,p24),dpl01),
1174:                                                 _mm_add_pd(_mm_mul_pd(dp0jweta,dpl10),
1175:                                                            _mm_mul_pd(dp2jweta,_mm_shuffle_pd(dpl2,dpl2,_MM_SHUFFLE2(0,1))))));
1176:                 pdgduv = _mm_mul_pd(p05,_mm_add_pd(_mm_add_pd(_mm_mul_pd(p42,_mm_mul_pd(du0,dpl01)),
1177:                                                               _mm_mul_pd(p24,_mm_mul_pd(dv1,dpl01))),
1178:                                                    _mm_add_pd(_mm_mul_pd(du1pdv0,dpl10),
1179:                                                               _mm_mul_pd(pdu2dv2,_mm_set1_pd(dpl[2]))))); /* [dgdu, dgdv] */
1180:                 t0 = _mm_mul_pd(jwdeta,pdgduv);  /* jw deta [dgdu, dgdv] */
1181:                 t3 = _mm_mul_pd(t0,du1pdv0);     /* t0 (du1 + dv0) */
1182:                 _mm_storeu_pd(&Ke[l*2+0][ll*2+0],_mm_add_pd(keu,_mm_add_pd(_mm_mul_pd(t1,t0),
1183:                                                                            _mm_add_pd(_mm_mul_pd(dp1,t3),
1184:                                                                                       _mm_mul_pd(t0,_mm_mul_pd(dp2,du2))))));
1185:                 _mm_storeu_pd(&Ke[l*2+1][ll*2+0],_mm_add_pd(kev,_mm_add_pd(_mm_mul_pd(t2,t0),
1186:                                                                            _mm_add_pd(_mm_mul_pd(dp0,t3),
1187:                                                                                       _mm_mul_pd(t0,_mm_mul_pd(dp2,dv2))))));
1188:               }
1189: #endif
1190:             }
1191:           }
1192:         }
1193:         if (k == 0) { /* on a bottom face */
1194:           if (thi->no_slip) {
1195:             const PetscReal   hz    = PetscRealPart(pn[0].h)/(zm-1);
1196:             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);
1197:             Ke[0][0] = thi->dirichlet_scale*diagu;
1198:             Ke[1][1] = thi->dirichlet_scale*diagv;
1199:           } else {
1200:             for (q=0; q<4; q++) {
1201:               const PetscReal jw = 0.25*hx*hy/thi->rhog,*phi = QuadQInterp[q];
1202:               PetscScalar     u  =0,v=0,rbeta2=0;
1203:               PetscReal       beta2,dbeta2;
1204:               for (l=0; l<4; l++) {
1205:                 u      += phi[l]*n[l].u;
1206:                 v      += phi[l]*n[l].v;
1207:                 rbeta2 += phi[l]*pn[l].beta2;
1208:               }
1209:               THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
1210:               for (l=0; l<4; l++) {
1211:                 const PetscReal pp = phi[l];
1212:                 for (ll=0; ll<4; ll++) {
1213:                   const PetscReal ppl = phi[ll];
1214:                   Ke[l*2+0][ll*2+0] += pp*jw*beta2*ppl + pp*jw*dbeta2*u*u*ppl;
1215:                   Ke[l*2+0][ll*2+1] +=                   pp*jw*dbeta2*u*v*ppl;
1216:                   Ke[l*2+1][ll*2+0] +=                   pp*jw*dbeta2*v*u*ppl;
1217:                   Ke[l*2+1][ll*2+1] += pp*jw*beta2*ppl + pp*jw*dbeta2*v*v*ppl;
1218:                 }
1219:               }
1220:             }
1221:           }
1222:         }
1223:         {
1224:           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}};
1225:           if (amode == THIASSEMBLY_TRIDIAGONAL) {
1226:             for (l=0; l<4; l++) { /* Copy out each of the blocks, discarding horizontal coupling */
1227:               const PetscInt   l4     = l+4;
1228:               const MatStencil rcl[2] = {{rc[l].k,rc[l].j,rc[l].i,0},{rc[l4].k,rc[l4].j,rc[l4].i,0}};
1229: #if defined COMPUTE_LOWER_TRIANGULAR
1230:               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]},
1231:                                              {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]},
1232:                                              {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]},
1233:                                              {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]}};
1234: #else
1235:               /* Same as above except for the lower-left block */
1236:               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]},
1237:                                              {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]},
1238:                                              {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]},
1239:                                              {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]}};
1240: #endif
1241:               MatSetValuesBlockedStencil(B,2,rcl,2,rcl,&Kel[0][0],ADD_VALUES);
1242:             }
1243:           } else {
1244: #if !defined COMPUTE_LOWER_TRIANGULAR /* fill in lower-triangular part, this is really cheap compared to computing the entries */
1245:             for (l=0; l<8; l++) {
1246:               for (ll=l+1; ll<8; ll++) {
1247:                 Ke[ll*2+0][l*2+0] = Ke[l*2+0][ll*2+0];
1248:                 Ke[ll*2+1][l*2+0] = Ke[l*2+0][ll*2+1];
1249:                 Ke[ll*2+0][l*2+1] = Ke[l*2+1][ll*2+0];
1250:                 Ke[ll*2+1][l*2+1] = Ke[l*2+1][ll*2+1];
1251:               }
1252:             }
1253: #endif
1254:             MatSetValuesBlockedStencil(B,8,rc,8,rc,&Ke[0][0],ADD_VALUES);
1255:           }
1256:         }
1257:       }
1258:     }
1259:   }
1260:   THIDARestorePrm(info->da,&prm);

1262:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
1263:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
1264:   MatSetOption(B,MAT_SYMMETRIC,PETSC_TRUE);
1265:   if (thi->verbose) {THIMatrixStatistics(thi,B,PETSC_VIEWER_STDOUT_WORLD);}
1266:   return(0);
1267: }

1271: static PetscErrorCode THIJacobianLocal_3D_Full(DMDALocalInfo *info,Node ***x,Mat A,Mat B,THI thi)
1272: {

1276:   THIJacobianLocal_3D(info,x,B,thi,THIASSEMBLY_FULL);
1277:   return(0);
1278: }

1282: static PetscErrorCode THIJacobianLocal_3D_Tridiagonal(DMDALocalInfo *info,Node ***x,Mat A,Mat B,THI thi)
1283: {

1287:   THIJacobianLocal_3D(info,x,B,thi,THIASSEMBLY_TRIDIAGONAL);
1288:   return(0);
1289: }

1293: static PetscErrorCode DMRefineHierarchy_THI(DM dac0,PetscInt nlevels,DM hierarchy[])
1294: {
1295:   PetscErrorCode  ierr;
1296:   THI             thi;
1297:   PetscInt        dim,M,N,m,n,s,dof;
1298:   DM              dac,daf;
1299:   DMDAStencilType st;
1300:   DM_DA           *ddf,*ddc;

1303:   PetscObjectQuery((PetscObject)dac0,"THI",(PetscObject*)&thi);
1304:   if (!thi) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Cannot refine this DMDA, missing composed THI instance");
1305:   if (nlevels > 1) {
1306:     DMRefineHierarchy(dac0,nlevels-1,hierarchy);
1307:     dac  = hierarchy[nlevels-2];
1308:   } else {
1309:     dac = dac0;
1310:   }
1311:   DMDAGetInfo(dac,&dim, &N,&M,0, &n,&m,0, &dof,&s,0,0,0,&st);
1312:   if (dim != 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"This function can only refine 2D DMDAs");

1314:   /* Creates a 3D DMDA with the same map-plane layout as the 2D one, with contiguous columns */
1315:   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);

1317:   daf->ops->creatematrix        = dac->ops->creatematrix;
1318:   daf->ops->createinterpolation = dac->ops->createinterpolation;
1319:   daf->ops->getcoloring         = dac->ops->getcoloring;
1320:   ddf                           = (DM_DA*)daf->data;
1321:   ddc                           = (DM_DA*)dac->data;
1322:   ddf->interptype               = ddc->interptype;

1324:   DMDASetFieldName(daf,0,"x-velocity");
1325:   DMDASetFieldName(daf,1,"y-velocity");

1327:   hierarchy[nlevels-1] = daf;
1328:   return(0);
1329: }

1333: static PetscErrorCode DMCreateInterpolation_DA_THI(DM dac,DM daf,Mat *A,Vec *scale)
1334: {
1336:   PetscInt       dim;

1343:   DMDAGetInfo(daf,&dim,0,0,0,0,0,0,0,0,0,0,0,0);
1344:   if (dim  == 2) {
1345:     /* We are in the 2D problem and use normal DMDA interpolation */
1346:     DMCreateInterpolation(dac,daf,A,scale);
1347:   } else {
1348:     PetscInt i,j,k,xs,ys,zs,xm,ym,zm,mx,my,mz,rstart,cstart;
1349:     Mat      B;

1351:     DMDAGetInfo(daf,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
1352:     DMDAGetCorners(daf,&zs,&ys,&xs,&zm,&ym,&xm);
1353:     if (zs != 0) SETERRQ(PETSC_COMM_SELF,1,"unexpected");
1354:     MatCreate(PetscObjectComm((PetscObject)daf),&B);
1355:     MatSetSizes(B,xm*ym*zm,xm*ym,mx*my*mz,mx*my);

1357:     MatSetType(B,MATAIJ);
1358:     MatSeqAIJSetPreallocation(B,1,NULL);
1359:     MatMPIAIJSetPreallocation(B,1,NULL,0,NULL);
1360:     MatGetOwnershipRange(B,&rstart,NULL);
1361:     MatGetOwnershipRangeColumn(B,&cstart,NULL);
1362:     for (i=xs; i<xs+xm; i++) {
1363:       for (j=ys; j<ys+ym; j++) {
1364:         for (k=zs; k<zs+zm; k++) {
1365:           PetscInt    i2  = i*ym+j,i3 = i2*zm+k;
1366:           PetscScalar val = ((k == 0 || k == mz-1) ? 0.5 : 1.) / (mz-1.); /* Integration using trapezoid rule */
1367:           MatSetValue(B,cstart+i3,rstart+i2,val,INSERT_VALUES);
1368:         }
1369:       }
1370:     }
1371:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
1372:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
1373:     MatCreateMAIJ(B,sizeof(Node)/sizeof(PetscScalar),A);
1374:     MatDestroy(&B);
1375:   }
1376:   return(0);
1377: }

1381: static PetscErrorCode DMCreateMatrix_THI_Tridiagonal(DM da,Mat *J)
1382: {
1383:   PetscErrorCode         ierr;
1384:   Mat                    A;
1385:   PetscInt               xm,ym,zm,dim,dof = 2,starts[3],dims[3];
1386:   ISLocalToGlobalMapping ltog;

1389:   DMDAGetInfo(da,&dim, 0,0,0, 0,0,0, 0,0,0,0,0,0);
1390:   if (dim != 3) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Expected DMDA to be 3D");
1391:   DMDAGetCorners(da,0,0,0,&zm,&ym,&xm);
1392:   DMGetLocalToGlobalMapping(da,&ltog);
1393:   MatCreate(PetscObjectComm((PetscObject)da),&A);
1394:   MatSetSizes(A,dof*xm*ym*zm,dof*xm*ym*zm,PETSC_DETERMINE,PETSC_DETERMINE);
1395:   MatSetType(A,da->mattype);
1396:   MatSetFromOptions(A);
1397:   MatSeqAIJSetPreallocation(A,3*2,NULL);
1398:   MatMPIAIJSetPreallocation(A,3*2,NULL,0,NULL);
1399:   MatSeqBAIJSetPreallocation(A,2,3,NULL);
1400:   MatMPIBAIJSetPreallocation(A,2,3,NULL,0,NULL);
1401:   MatSeqSBAIJSetPreallocation(A,2,2,NULL);
1402:   MatMPISBAIJSetPreallocation(A,2,2,NULL,0,NULL);
1403:   MatSetLocalToGlobalMapping(A,ltog,ltog);
1404:   DMDAGetGhostCorners(da,&starts[0],&starts[1],&starts[2],&dims[0],&dims[1],&dims[2]);
1405:   MatSetStencil(A,dim,dims,starts,dof);
1406:   *J   = A;
1407:   return(0);
1408: }

1412: static PetscErrorCode THIDAVecView_VTK_XML(THI thi,DM da,Vec X,const char filename[])
1413: {
1414:   const PetscInt    dof   = 2;
1415:   Units             units = thi->units;
1416:   MPI_Comm          comm;
1417:   PetscErrorCode    ierr;
1418:   PetscViewer       viewer;
1419:   PetscMPIInt       rank,size,tag,nn,nmax;
1420:   PetscInt          mx,my,mz,r,range[6];
1421:   const PetscScalar *x;

1424:   PetscObjectGetComm((PetscObject)thi,&comm);
1425:   DMDAGetInfo(da,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
1426:   MPI_Comm_size(comm,&size);
1427:   MPI_Comm_rank(comm,&rank);
1428:   PetscViewerASCIIOpen(comm,filename,&viewer);
1429:   PetscViewerASCIIPrintf(viewer,"<VTKFile type=\"StructuredGrid\" version=\"0.1\" byte_order=\"LittleEndian\">\n");
1430:   PetscViewerASCIIPrintf(viewer,"  <StructuredGrid WholeExtent=\"%d %D %d %D %d %D\">\n",0,mz-1,0,my-1,0,mx-1);

1432:   DMDAGetCorners(da,range,range+1,range+2,range+3,range+4,range+5);
1433:   PetscMPIIntCast(range[3]*range[4]*range[5]*dof,&nn);
1434:   MPI_Reduce(&nn,&nmax,1,MPI_INT,MPI_MAX,0,comm);
1435:   tag  = ((PetscObject) viewer)->tag;
1436:   VecGetArrayRead(X,&x);
1437:   if (!rank) {
1438:     PetscScalar *array;
1439:     PetscMalloc1(nmax,&array);
1440:     for (r=0; r<size; r++) {
1441:       PetscInt          i,j,k,xs,xm,ys,ym,zs,zm;
1442:       const PetscScalar *ptr;
1443:       MPI_Status        status;
1444:       if (r) {
1445:         MPI_Recv(range,6,MPIU_INT,r,tag,comm,MPI_STATUS_IGNORE);
1446:       }
1447:       zs = range[0];ys = range[1];xs = range[2];zm = range[3];ym = range[4];xm = range[5];
1448:       if (xm*ym*zm*dof > nmax) SETERRQ(PETSC_COMM_SELF,1,"should not happen");
1449:       if (r) {
1450:         MPI_Recv(array,nmax,MPIU_SCALAR,r,tag,comm,&status);
1451:         MPI_Get_count(&status,MPIU_SCALAR,&nn);
1452:         if (nn != xm*ym*zm*dof) SETERRQ(PETSC_COMM_SELF,1,"should not happen");
1453:         ptr = array;
1454:       } else ptr = x;
1455:       PetscViewerASCIIPrintf(viewer,"    <Piece Extent=\"%D %D %D %D %D %D\">\n",zs,zs+zm-1,ys,ys+ym-1,xs,xs+xm-1);

1457:       PetscViewerASCIIPrintf(viewer,"      <Points>\n");
1458:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Float32\" NumberOfComponents=\"3\" format=\"ascii\">\n");
1459:       for (i=xs; i<xs+xm; i++) {
1460:         for (j=ys; j<ys+ym; j++) {
1461:           for (k=zs; k<zs+zm; k++) {
1462:             PrmNode   p;
1463:             PetscReal xx = thi->Lx*i/mx,yy = thi->Ly*j/my,zz;
1464:             thi->initialize(thi,xx,yy,&p);
1465:             zz   = PetscRealPart(p.b) + PetscRealPart(p.h)*k/(mz-1);
1466:             PetscViewerASCIIPrintf(viewer,"%f %f %f\n",(double)xx,(double)yy,(double)zz);
1467:           }
1468:         }
1469:       }
1470:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");
1471:       PetscViewerASCIIPrintf(viewer,"      </Points>\n");

1473:       PetscViewerASCIIPrintf(viewer,"      <PointData>\n");
1474:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Float32\" Name=\"velocity\" NumberOfComponents=\"3\" format=\"ascii\">\n");
1475:       for (i=0; i<nn; i+=dof) {
1476:         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);
1477:       }
1478:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");

1480:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Int32\" Name=\"rank\" NumberOfComponents=\"1\" format=\"ascii\">\n");
1481:       for (i=0; i<nn; i+=dof) {
1482:         PetscViewerASCIIPrintf(viewer,"%D\n",r);
1483:       }
1484:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");
1485:       PetscViewerASCIIPrintf(viewer,"      </PointData>\n");

1487:       PetscViewerASCIIPrintf(viewer,"    </Piece>\n");
1488:     }
1489:     PetscFree(array);
1490:   } else {
1491:     MPI_Send(range,6,MPIU_INT,0,tag,comm);
1492:     MPI_Send((PetscScalar*)x,nn,MPIU_SCALAR,0,tag,comm);
1493:   }
1494:   VecRestoreArrayRead(X,&x);
1495:   PetscViewerASCIIPrintf(viewer,"  </StructuredGrid>\n");
1496:   PetscViewerASCIIPrintf(viewer,"</VTKFile>\n");
1497:   PetscViewerDestroy(&viewer);
1498:   return(0);
1499: }

1503: int main(int argc,char *argv[])
1504: {
1505:   MPI_Comm       comm;
1506:   THI            thi;
1508:   DM             da;
1509:   SNES           snes;

1511:   PetscInitialize(&argc,&argv,0,help);
1512:   comm = PETSC_COMM_WORLD;

1514:   THICreate(comm,&thi);
1515:   {
1516:     PetscInt M = 3,N = 3,P = 2;
1517:     PetscOptionsBegin(comm,NULL,"Grid resolution options","");
1518:     {
1519:       PetscOptionsInt("-M","Number of elements in x-direction on coarse level","",M,&M,NULL);
1520:       N    = M;
1521:       PetscOptionsInt("-N","Number of elements in y-direction on coarse level (if different from M)","",N,&N,NULL);
1522:       if (thi->coarse2d) {
1523:         PetscOptionsInt("-zlevels","Number of elements in z-direction on fine level","",thi->zlevels,&thi->zlevels,NULL);
1524:       } else {
1525:         PetscOptionsInt("-P","Number of elements in z-direction on coarse level","",P,&P,NULL);
1526:       }
1527:     }
1528:     PetscOptionsEnd();
1529:     if (thi->coarse2d) {
1530:       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);

1532:       da->ops->refinehierarchy     = DMRefineHierarchy_THI;
1533:       da->ops->createinterpolation = DMCreateInterpolation_DA_THI;

1535:       PetscObjectCompose((PetscObject)da,"THI",(PetscObject)thi);
1536:     } else {
1537:       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);
1538:     }
1539:     DMDASetFieldName(da,0,"x-velocity");
1540:     DMDASetFieldName(da,1,"y-velocity");
1541:   }
1542:   THISetUpDM(thi,da);
1543:   if (thi->tridiagonal) da->ops->creatematrix = DMCreateMatrix_THI_Tridiagonal;

1545:   {                             /* Set the fine level matrix type if -da_refine */
1546:     PetscInt rlevel,clevel;
1547:     DMGetRefineLevel(da,&rlevel);
1548:     DMGetCoarsenLevel(da,&clevel);
1549:     if (rlevel - clevel > 0) {DMSetMatType(da,thi->mattype);}
1550:   }

1552:   DMDASNESSetFunctionLocal(da,ADD_VALUES,(DMDASNESFunction)THIFunctionLocal,thi);
1553:   if (thi->tridiagonal) {
1554:     DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)THIJacobianLocal_3D_Tridiagonal,thi);
1555:   } else {
1556:     DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)THIJacobianLocal_3D_Full,thi);
1557:   }
1558:   DMCoarsenHookAdd(da,DMCoarsenHook_THI,NULL,thi);
1559:   DMRefineHookAdd(da,DMRefineHook_THI,NULL,thi);

1561:   DMSetApplicationContext(da,thi);

1563:   SNESCreate(comm,&snes);
1564:   SNESSetDM(snes,da);
1565:   DMDestroy(&da);
1566:   SNESSetComputeInitialGuess(snes,THIInitial,NULL);
1567:   SNESSetFromOptions(snes);

1569:   SNESSolve(snes,NULL,NULL);

1571:   THISolveStatistics(thi,snes,0,"Full");

1573:   {
1574:     PetscBool flg;
1575:     char      filename[PETSC_MAX_PATH_LEN] = "";
1576:     PetscOptionsGetString(NULL,NULL,"-o",filename,sizeof(filename),&flg);
1577:     if (flg) {
1578:       Vec X;
1579:       DM  dm;
1580:       SNESGetSolution(snes,&X);
1581:       SNESGetDM(snes,&dm);
1582:       THIDAVecView_VTK_XML(thi,dm,X,filename);
1583:     }
1584:   }

1586:   DMDestroy(&da);
1587:   SNESDestroy(&snes);
1588:   THIDestroy(&thi);
1589:   PetscFinalize();
1590:   return ierr;
1591: }