Actual source code: ex48.c

petsc-master 2017-10-20
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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) || defined (PETSC_HAVE_WINDOWS_COMPILERS)
 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: #if !defined NO_SSE2                           \
 79:      && !defined PETSC_USE_COMPLEX             \
 80:      && !defined PETSC_USE_REAL_SINGLE         \
 81:      && !defined PETSC_USE_REAL___FLOAT128     \
 82:      && !defined PETSC_USE_REAL___FP16         \
 83:      && defined __SSE2__
 84: #define USE_SSE2_KERNELS 1
 85: #else
 86: #define USE_SSE2_KERNELS 0
 87: #endif

 89: static PetscClassId THI_CLASSID;

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

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

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

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

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

193: 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)
194: {
195:   const PetscReal jac[3][3]  = {{hx/2,0,0}, {0,hy/2,0}, {dz[0],dz[1],dz[2]}};
196:   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]}};
197:   const PetscReal jdet       = jac[0][0]*jac[1][1]*jac[2][2];
198:   PetscInt        i;

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

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

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

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

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

227: typedef enum {THIASSEMBLY_TRIDIAGONAL,THIASSEMBLY_FULL} THIAssemblyMode;

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

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

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

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

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

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

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

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

305: /* These are just toys */

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

537:   {
538:     PetscReal u       = 1000*units->meter/(3e7*units->second),
539:               gradu   = u / (100*units->meter),eta,deta,
540:               rho     = 910 * units->kilogram/PetscPowReal(units->meter,3),
541:               grav    = 9.81 * units->meter/PetscSqr(units->second),
542:               driving = rho * grav * PetscSinReal(thi->alpha) * 1000*units->meter;
543:     THIViscosity(thi,0.5*gradu*gradu,&eta,&deta);
544:     thi->rhog = rho * grav;
545:     if (thi->verbose) {
546:       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);
547:       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);
548:       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));
549:       THIViscosity(thi,0.5*PetscSqr(1e-3*gradu),&eta,&deta);
550:       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));
551:     }
552:   }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

713: 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)
714: {
715:   PetscInt    l,ll;
716:   PetscScalar gam;

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

734: 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)
735: {
736:   PetscInt    l,ll;
737:   PetscScalar gam;

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

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

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

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

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

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

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

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

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

868:   MatNorm(B,NORM_FROBENIUS,&nrm);
869:   MatGetSize(B,&m,0);
870:   MPI_Comm_rank(PetscObjectComm((PetscObject)B),&rank);
871:   if (!rank) {
872:     PetscScalar val0,val2;
873:     MatGetValue(B,0,0,&val0);
874:     MatGetValue(B,2,2,&val2);
875:     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);
876:   }
877:   return(0);
878: }

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

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

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

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

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

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

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

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

1038:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
1039:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
1040:   MatSetOption(B,MAT_SYMMETRIC,PETSC_TRUE);
1041:   if (thi->verbose) {THIMatrixStatistics(thi,B,PETSC_VIEWER_STDOUT_WORLD);}
1042:   return(0);
1043: }

1045: static PetscErrorCode THIJacobianLocal_3D(DMDALocalInfo *info,Node ***x,Mat B,THI thi,THIAssemblyMode amode)
1046: {
1047:   PetscInt       xs,ys,xm,ym,zm,i,j,k,q,l,ll;
1048:   PetscReal      hx,hy;
1049:   PrmNode        **prm;

1053:   xs = info->zs;
1054:   ys = info->ys;
1055:   xm = info->zm;
1056:   ym = info->ym;
1057:   zm = info->xm;
1058:   hx = thi->Lx / info->mz;
1059:   hy = thi->Ly / info->my;

1061:   MatZeroEntries(B);
1062:   MatSetOption(B,MAT_SUBSET_OFF_PROC_ENTRIES,PETSC_TRUE);
1063:   THIDAGetPrm(info->da,&prm);

1065:   for (i=xs; i<xs+xm; i++) {
1066:     for (j=ys; j<ys+ym; j++) {
1067:       PrmNode pn[4];
1068:       QuadExtract(prm,i,j,pn);
1069:       for (k=0; k<zm-1; k++) {
1070:         Node        n[8];
1071:         PetscReal   zn[8],etabase = 0;
1072:         PetscScalar Ke[8*2][8*2];
1073:         PetscInt    ls = 0;

1075:         PrmHexGetZ(pn,k,zm,zn);
1076:         HexExtract(x,i,j,k,n);
1077:         PetscMemzero(Ke,sizeof(Ke));
1078:         if (thi->no_slip && k == 0) {
1079:           for (l=0; l<4; l++) n[l].u = n[l].v = 0;
1080:           ls = 4;
1081:         }
1082:         for (q=0; q<8; q++) {
1083:           PetscReal   dz[3],phi[8],dphi[8][3],jw,eta,deta;
1084:           PetscScalar du[3],dv[3],u,v;
1085:           HexGrad(HexQDeriv[q],zn,dz);
1086:           HexComputeGeometry(q,hx,hy,dz,phi,dphi,&jw);
1087:           PointwiseNonlinearity(thi,n,phi,dphi,&u,&v,du,dv,&eta,&deta);
1088:           jw /= thi->rhog;      /* residuals are scaled by this factor */
1089:           if (q == 0) etabase = eta;
1090:           for (l=ls; l<8; l++) { /* test functions */
1091:             const PetscReal *PETSC_RESTRICT dp = dphi[l];
1092: #if USE_SSE2_KERNELS
1093:             /* gcc (up to my 4.5 snapshot) is really bad at hoisting intrinsics so we do it manually */
1094:             __m128d
1095:               p4         = _mm_set1_pd(4),p2 = _mm_set1_pd(2),p05 = _mm_set1_pd(0.5),
1096:               p42        = _mm_setr_pd(4,2),p24 = _mm_shuffle_pd(p42,p42,_MM_SHUFFLE2(0,1)),
1097:               du0        = _mm_set1_pd(du[0]),du1 = _mm_set1_pd(du[1]),du2 = _mm_set1_pd(du[2]),
1098:               dv0        = _mm_set1_pd(dv[0]),dv1 = _mm_set1_pd(dv[1]),dv2 = _mm_set1_pd(dv[2]),
1099:               jweta      = _mm_set1_pd(jw*eta),jwdeta = _mm_set1_pd(jw*deta),
1100:               dp0        = _mm_set1_pd(dp[0]),dp1 = _mm_set1_pd(dp[1]),dp2 = _mm_set1_pd(dp[2]),
1101:               dp0jweta   = _mm_mul_pd(dp0,jweta),dp1jweta = _mm_mul_pd(dp1,jweta),dp2jweta = _mm_mul_pd(dp2,jweta),
1102:               p4du0p2dv1 = _mm_add_pd(_mm_mul_pd(p4,du0),_mm_mul_pd(p2,dv1)), /* 4 du0 + 2 dv1 */
1103:               p4dv1p2du0 = _mm_add_pd(_mm_mul_pd(p4,dv1),_mm_mul_pd(p2,du0)), /* 4 dv1 + 2 du0 */
1104:               pdu2dv2    = _mm_unpacklo_pd(du2,dv2),                          /* [du2, dv2] */
1105:               du1pdv0    = _mm_add_pd(du1,dv0),                               /* du1 + dv0 */
1106:               t1         = _mm_mul_pd(dp0,p4du0p2dv1),                        /* dp0 (4 du0 + 2 dv1) */
1107:               t2         = _mm_mul_pd(dp1,p4dv1p2du0);                        /* dp1 (4 dv1 + 2 du0) */

1109: #endif
1110: #if defined COMPUTE_LOWER_TRIANGULAR  /* The element matrices are always symmetric so computing the lower-triangular part is not necessary */
1111:             for (ll=ls; ll<8; ll++) { /* trial functions */
1112: #else
1113:             for (ll=l; ll<8; ll++) {
1114: #endif
1115:               const PetscReal *PETSC_RESTRICT dpl = dphi[ll];
1116:               if (amode == THIASSEMBLY_TRIDIAGONAL && (l-ll)%4) continue; /* these entries would not be inserted */
1117: #if !USE_SSE2_KERNELS
1118:               /* The analytic Jacobian in nice, easy-to-read form */
1119:               {
1120:                 PetscScalar dgdu,dgdv;
1121:                 dgdu = 2.*du[0]*dpl[0] + dv[1]*dpl[0] + 0.5*(du[1]+dv[0])*dpl[1] + 0.5*du[2]*dpl[2];
1122:                 dgdv = 2.*dv[1]*dpl[1] + du[0]*dpl[1] + 0.5*(du[1]+dv[0])*dpl[0] + 0.5*dv[2]*dpl[2];
1123:                 /* Picard part */
1124:                 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];
1125:                 Ke[l*2+0][ll*2+1] += dp[0]*jw*eta*2.*dpl[1] + dp[1]*jw*eta*dpl[0];
1126:                 Ke[l*2+1][ll*2+0] += dp[1]*jw*eta*2.*dpl[0] + dp[0]*jw*eta*dpl[1];
1127:                 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];
1128:                 /* extra Newton terms */
1129:                 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];
1130:                 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];
1131:                 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];
1132:                 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];
1133:               }
1134: #else
1135:               /* This SSE2 code is an exact replica of above, but uses explicit packed instructions for some speed
1136:               * benefit.  On my hardware, these intrinsics are almost twice as fast as above, reducing total assembly cost
1137:               * by 25 to 30 percent. */
1138:               {
1139:                 __m128d
1140:                   keu   = _mm_loadu_pd(&Ke[l*2+0][ll*2+0]),
1141:                   kev   = _mm_loadu_pd(&Ke[l*2+1][ll*2+0]),
1142:                   dpl01 = _mm_loadu_pd(&dpl[0]),dpl10 = _mm_shuffle_pd(dpl01,dpl01,_MM_SHUFFLE2(0,1)),dpl2 = _mm_set_sd(dpl[2]),
1143:                   t0,t3,pdgduv;
1144:                 keu = _mm_add_pd(keu,_mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp0jweta,p42),dpl01),
1145:                                                 _mm_add_pd(_mm_mul_pd(dp1jweta,dpl10),
1146:                                                            _mm_mul_pd(dp2jweta,dpl2))));
1147:                 kev = _mm_add_pd(kev,_mm_add_pd(_mm_mul_pd(_mm_mul_pd(dp1jweta,p24),dpl01),
1148:                                                 _mm_add_pd(_mm_mul_pd(dp0jweta,dpl10),
1149:                                                            _mm_mul_pd(dp2jweta,_mm_shuffle_pd(dpl2,dpl2,_MM_SHUFFLE2(0,1))))));
1150:                 pdgduv = _mm_mul_pd(p05,_mm_add_pd(_mm_add_pd(_mm_mul_pd(p42,_mm_mul_pd(du0,dpl01)),
1151:                                                               _mm_mul_pd(p24,_mm_mul_pd(dv1,dpl01))),
1152:                                                    _mm_add_pd(_mm_mul_pd(du1pdv0,dpl10),
1153:                                                               _mm_mul_pd(pdu2dv2,_mm_set1_pd(dpl[2]))))); /* [dgdu, dgdv] */
1154:                 t0 = _mm_mul_pd(jwdeta,pdgduv);  /* jw deta [dgdu, dgdv] */
1155:                 t3 = _mm_mul_pd(t0,du1pdv0);     /* t0 (du1 + dv0) */
1156:                 _mm_storeu_pd(&Ke[l*2+0][ll*2+0],_mm_add_pd(keu,_mm_add_pd(_mm_mul_pd(t1,t0),
1157:                                                                            _mm_add_pd(_mm_mul_pd(dp1,t3),
1158:                                                                                       _mm_mul_pd(t0,_mm_mul_pd(dp2,du2))))));
1159:                 _mm_storeu_pd(&Ke[l*2+1][ll*2+0],_mm_add_pd(kev,_mm_add_pd(_mm_mul_pd(t2,t0),
1160:                                                                            _mm_add_pd(_mm_mul_pd(dp0,t3),
1161:                                                                                       _mm_mul_pd(t0,_mm_mul_pd(dp2,dv2))))));
1162:               }
1163: #endif
1164:             }
1165:           }
1166:         }
1167:         if (k == 0) { /* on a bottom face */
1168:           if (thi->no_slip) {
1169:             const PetscReal   hz    = PetscRealPart(pn[0].h)/(zm-1);
1170:             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);
1171:             Ke[0][0] = thi->dirichlet_scale*diagu;
1172:             Ke[1][1] = thi->dirichlet_scale*diagv;
1173:           } else {
1174:             for (q=0; q<4; q++) {
1175:               const PetscReal jw = 0.25*hx*hy/thi->rhog,*phi = QuadQInterp[q];
1176:               PetscScalar     u  =0,v=0,rbeta2=0;
1177:               PetscReal       beta2,dbeta2;
1178:               for (l=0; l<4; l++) {
1179:                 u      += phi[l]*n[l].u;
1180:                 v      += phi[l]*n[l].v;
1181:                 rbeta2 += phi[l]*pn[l].beta2;
1182:               }
1183:               THIFriction(thi,PetscRealPart(rbeta2),PetscRealPart(u*u+v*v)/2,&beta2,&dbeta2);
1184:               for (l=0; l<4; l++) {
1185:                 const PetscReal pp = phi[l];
1186:                 for (ll=0; ll<4; ll++) {
1187:                   const PetscReal ppl = phi[ll];
1188:                   Ke[l*2+0][ll*2+0] += pp*jw*beta2*ppl + pp*jw*dbeta2*u*u*ppl;
1189:                   Ke[l*2+0][ll*2+1] +=                   pp*jw*dbeta2*u*v*ppl;
1190:                   Ke[l*2+1][ll*2+0] +=                   pp*jw*dbeta2*v*u*ppl;
1191:                   Ke[l*2+1][ll*2+1] += pp*jw*beta2*ppl + pp*jw*dbeta2*v*v*ppl;
1192:                 }
1193:               }
1194:             }
1195:           }
1196:         }
1197:         {
1198:           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}};
1199:           if (amode == THIASSEMBLY_TRIDIAGONAL) {
1200:             for (l=0; l<4; l++) { /* Copy out each of the blocks, discarding horizontal coupling */
1201:               const PetscInt   l4     = l+4;
1202:               const MatStencil rcl[2] = {{rc[l].k,rc[l].j,rc[l].i,0},{rc[l4].k,rc[l4].j,rc[l4].i,0}};
1203: #if defined COMPUTE_LOWER_TRIANGULAR
1204:               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]},
1205:                                              {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]},
1206:                                              {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]},
1207:                                              {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]}};
1208: #else
1209:               /* Same as above except for the lower-left block */
1210:               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]},
1211:                                              {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]},
1212:                                              {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]},
1213:                                              {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]}};
1214: #endif
1215:               MatSetValuesBlockedStencil(B,2,rcl,2,rcl,&Kel[0][0],ADD_VALUES);
1216:             }
1217:           } else {
1218: #if !defined COMPUTE_LOWER_TRIANGULAR /* fill in lower-triangular part, this is really cheap compared to computing the entries */
1219:             for (l=0; l<8; l++) {
1220:               for (ll=l+1; ll<8; ll++) {
1221:                 Ke[ll*2+0][l*2+0] = Ke[l*2+0][ll*2+0];
1222:                 Ke[ll*2+1][l*2+0] = Ke[l*2+0][ll*2+1];
1223:                 Ke[ll*2+0][l*2+1] = Ke[l*2+1][ll*2+0];
1224:                 Ke[ll*2+1][l*2+1] = Ke[l*2+1][ll*2+1];
1225:               }
1226:             }
1227: #endif
1228:             MatSetValuesBlockedStencil(B,8,rc,8,rc,&Ke[0][0],ADD_VALUES);
1229:           }
1230:         }
1231:       }
1232:     }
1233:   }
1234:   THIDARestorePrm(info->da,&prm);

1236:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
1237:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
1238:   MatSetOption(B,MAT_SYMMETRIC,PETSC_TRUE);
1239:   if (thi->verbose) {THIMatrixStatistics(thi,B,PETSC_VIEWER_STDOUT_WORLD);}
1240:   return(0);
1241: }

1243: static PetscErrorCode THIJacobianLocal_3D_Full(DMDALocalInfo *info,Node ***x,Mat A,Mat B,THI thi)
1244: {

1248:   THIJacobianLocal_3D(info,x,B,thi,THIASSEMBLY_FULL);
1249:   return(0);
1250: }

1252: static PetscErrorCode THIJacobianLocal_3D_Tridiagonal(DMDALocalInfo *info,Node ***x,Mat A,Mat B,THI thi)
1253: {

1257:   THIJacobianLocal_3D(info,x,B,thi,THIASSEMBLY_TRIDIAGONAL);
1258:   return(0);
1259: }

1261: static PetscErrorCode DMRefineHierarchy_THI(DM dac0,PetscInt nlevels,DM hierarchy[])
1262: {
1263:   PetscErrorCode  ierr;
1264:   THI             thi;
1265:   PetscInt        dim,M,N,m,n,s,dof;
1266:   DM              dac,daf;
1267:   DMDAStencilType st;
1268:   DM_DA           *ddf,*ddc;

1271:   PetscObjectQuery((PetscObject)dac0,"THI",(PetscObject*)&thi);
1272:   if (!thi) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Cannot refine this DMDA, missing composed THI instance");
1273:   if (nlevels > 1) {
1274:     DMRefineHierarchy(dac0,nlevels-1,hierarchy);
1275:     dac  = hierarchy[nlevels-2];
1276:   } else {
1277:     dac = dac0;
1278:   }
1279:   DMDAGetInfo(dac,&dim, &N,&M,0, &n,&m,0, &dof,&s,0,0,0,&st);
1280:   if (dim != 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"This function can only refine 2D DMDAs");

1282:   /* Creates a 3D DMDA with the same map-plane layout as the 2D one, with contiguous columns */
1283:   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);
1284:   DMSetUp(daf);

1286:   daf->ops->creatematrix        = dac->ops->creatematrix;
1287:   daf->ops->createinterpolation = dac->ops->createinterpolation;
1288:   daf->ops->getcoloring         = dac->ops->getcoloring;
1289:   ddf                           = (DM_DA*)daf->data;
1290:   ddc                           = (DM_DA*)dac->data;
1291:   ddf->interptype               = ddc->interptype;

1293:   DMDASetFieldName(daf,0,"x-velocity");
1294:   DMDASetFieldName(daf,1,"y-velocity");

1296:   hierarchy[nlevels-1] = daf;
1297:   return(0);
1298: }

1300: static PetscErrorCode DMCreateInterpolation_DA_THI(DM dac,DM daf,Mat *A,Vec *scale)
1301: {
1303:   PetscInt       dim;

1310:   DMDAGetInfo(daf,&dim,0,0,0,0,0,0,0,0,0,0,0,0);
1311:   if (dim  == 2) {
1312:     /* We are in the 2D problem and use normal DMDA interpolation */
1313:     DMCreateInterpolation(dac,daf,A,scale);
1314:   } else {
1315:     PetscInt i,j,k,xs,ys,zs,xm,ym,zm,mx,my,mz,rstart,cstart;
1316:     Mat      B;

1318:     DMDAGetInfo(daf,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
1319:     DMDAGetCorners(daf,&zs,&ys,&xs,&zm,&ym,&xm);
1320:     if (zs != 0) SETERRQ(PETSC_COMM_SELF,1,"unexpected");
1321:     MatCreate(PetscObjectComm((PetscObject)daf),&B);
1322:     MatSetSizes(B,xm*ym*zm,xm*ym,mx*my*mz,mx*my);

1324:     MatSetType(B,MATAIJ);
1325:     MatSeqAIJSetPreallocation(B,1,NULL);
1326:     MatMPIAIJSetPreallocation(B,1,NULL,0,NULL);
1327:     MatGetOwnershipRange(B,&rstart,NULL);
1328:     MatGetOwnershipRangeColumn(B,&cstart,NULL);
1329:     for (i=xs; i<xs+xm; i++) {
1330:       for (j=ys; j<ys+ym; j++) {
1331:         for (k=zs; k<zs+zm; k++) {
1332:           PetscInt    i2  = i*ym+j,i3 = i2*zm+k;
1333:           PetscScalar val = ((k == 0 || k == mz-1) ? 0.5 : 1.) / (mz-1.); /* Integration using trapezoid rule */
1334:           MatSetValue(B,cstart+i3,rstart+i2,val,INSERT_VALUES);
1335:         }
1336:       }
1337:     }
1338:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
1339:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
1340:     MatCreateMAIJ(B,sizeof(Node)/sizeof(PetscScalar),A);
1341:     MatDestroy(&B);
1342:   }
1343:   return(0);
1344: }

1346: static PetscErrorCode DMCreateMatrix_THI_Tridiagonal(DM da,Mat *J)
1347: {
1348:   PetscErrorCode         ierr;
1349:   Mat                    A;
1350:   PetscInt               xm,ym,zm,dim,dof = 2,starts[3],dims[3];
1351:   ISLocalToGlobalMapping ltog;

1354:   DMDAGetInfo(da,&dim, 0,0,0, 0,0,0, 0,0,0,0,0,0);
1355:   if (dim != 3) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Expected DMDA to be 3D");
1356:   DMDAGetCorners(da,0,0,0,&zm,&ym,&xm);
1357:   DMGetLocalToGlobalMapping(da,&ltog);
1358:   MatCreate(PetscObjectComm((PetscObject)da),&A);
1359:   MatSetSizes(A,dof*xm*ym*zm,dof*xm*ym*zm,PETSC_DETERMINE,PETSC_DETERMINE);
1360:   MatSetType(A,da->mattype);
1361:   MatSetFromOptions(A);
1362:   MatSeqAIJSetPreallocation(A,3*2,NULL);
1363:   MatMPIAIJSetPreallocation(A,3*2,NULL,0,NULL);
1364:   MatSeqBAIJSetPreallocation(A,2,3,NULL);
1365:   MatMPIBAIJSetPreallocation(A,2,3,NULL,0,NULL);
1366:   MatSeqSBAIJSetPreallocation(A,2,2,NULL);
1367:   MatMPISBAIJSetPreallocation(A,2,2,NULL,0,NULL);
1368:   MatSetLocalToGlobalMapping(A,ltog,ltog);
1369:   DMDAGetGhostCorners(da,&starts[0],&starts[1],&starts[2],&dims[0],&dims[1],&dims[2]);
1370:   MatSetStencil(A,dim,dims,starts,dof);
1371:   *J   = A;
1372:   return(0);
1373: }

1375: static PetscErrorCode THIDAVecView_VTK_XML(THI thi,DM da,Vec X,const char filename[])
1376: {
1377:   const PetscInt    dof   = 2;
1378:   Units             units = thi->units;
1379:   MPI_Comm          comm;
1380:   PetscErrorCode    ierr;
1381:   PetscViewer       viewer;
1382:   PetscMPIInt       rank,size,tag,nn,nmax;
1383:   PetscInt          mx,my,mz,r,range[6];
1384:   const PetscScalar *x;

1387:   PetscObjectGetComm((PetscObject)thi,&comm);
1388:   DMDAGetInfo(da,0, &mz,&my,&mx, 0,0,0, 0,0,0,0,0,0);
1389:   MPI_Comm_size(comm,&size);
1390:   MPI_Comm_rank(comm,&rank);
1391:   PetscViewerASCIIOpen(comm,filename,&viewer);
1392:   PetscViewerASCIIPrintf(viewer,"<VTKFile type=\"StructuredGrid\" version=\"0.1\" byte_order=\"LittleEndian\">\n");
1393:   PetscViewerASCIIPrintf(viewer,"  <StructuredGrid WholeExtent=\"%d %D %d %D %d %D\">\n",0,mz-1,0,my-1,0,mx-1);

1395:   DMDAGetCorners(da,range,range+1,range+2,range+3,range+4,range+5);
1396:   PetscMPIIntCast(range[3]*range[4]*range[5]*dof,&nn);
1397:   MPI_Reduce(&nn,&nmax,1,MPI_INT,MPI_MAX,0,comm);
1398:   tag  = ((PetscObject) viewer)->tag;
1399:   VecGetArrayRead(X,&x);
1400:   if (!rank) {
1401:     PetscScalar *array;
1402:     PetscMalloc1(nmax,&array);
1403:     for (r=0; r<size; r++) {
1404:       PetscInt          i,j,k,xs,xm,ys,ym,zs,zm;
1405:       const PetscScalar *ptr;
1406:       MPI_Status        status;
1407:       if (r) {
1408:         MPI_Recv(range,6,MPIU_INT,r,tag,comm,MPI_STATUS_IGNORE);
1409:       }
1410:       zs = range[0];ys = range[1];xs = range[2];zm = range[3];ym = range[4];xm = range[5];
1411:       if (xm*ym*zm*dof > nmax) SETERRQ(PETSC_COMM_SELF,1,"should not happen");
1412:       if (r) {
1413:         MPI_Recv(array,nmax,MPIU_SCALAR,r,tag,comm,&status);
1414:         MPI_Get_count(&status,MPIU_SCALAR,&nn);
1415:         if (nn != xm*ym*zm*dof) SETERRQ(PETSC_COMM_SELF,1,"should not happen");
1416:         ptr = array;
1417:       } else ptr = x;
1418:       PetscViewerASCIIPrintf(viewer,"    <Piece Extent=\"%D %D %D %D %D %D\">\n",zs,zs+zm-1,ys,ys+ym-1,xs,xs+xm-1);

1420:       PetscViewerASCIIPrintf(viewer,"      <Points>\n");
1421:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Float32\" NumberOfComponents=\"3\" format=\"ascii\">\n");
1422:       for (i=xs; i<xs+xm; i++) {
1423:         for (j=ys; j<ys+ym; j++) {
1424:           for (k=zs; k<zs+zm; k++) {
1425:             PrmNode   p;
1426:             PetscReal xx = thi->Lx*i/mx,yy = thi->Ly*j/my,zz;
1427:             thi->initialize(thi,xx,yy,&p);
1428:             zz   = PetscRealPart(p.b) + PetscRealPart(p.h)*k/(mz-1);
1429:             PetscViewerASCIIPrintf(viewer,"%f %f %f\n",(double)xx,(double)yy,(double)zz);
1430:           }
1431:         }
1432:       }
1433:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");
1434:       PetscViewerASCIIPrintf(viewer,"      </Points>\n");

1436:       PetscViewerASCIIPrintf(viewer,"      <PointData>\n");
1437:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Float32\" Name=\"velocity\" NumberOfComponents=\"3\" format=\"ascii\">\n");
1438:       for (i=0; i<nn; i+=dof) {
1439:         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);
1440:       }
1441:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");

1443:       PetscViewerASCIIPrintf(viewer,"        <DataArray type=\"Int32\" Name=\"rank\" NumberOfComponents=\"1\" format=\"ascii\">\n");
1444:       for (i=0; i<nn; i+=dof) {
1445:         PetscViewerASCIIPrintf(viewer,"%D\n",r);
1446:       }
1447:       PetscViewerASCIIPrintf(viewer,"        </DataArray>\n");
1448:       PetscViewerASCIIPrintf(viewer,"      </PointData>\n");

1450:       PetscViewerASCIIPrintf(viewer,"    </Piece>\n");
1451:     }
1452:     PetscFree(array);
1453:   } else {
1454:     MPI_Send(range,6,MPIU_INT,0,tag,comm);
1455:     MPI_Send((PetscScalar*)x,nn,MPIU_SCALAR,0,tag,comm);
1456:   }
1457:   VecRestoreArrayRead(X,&x);
1458:   PetscViewerASCIIPrintf(viewer,"  </StructuredGrid>\n");
1459:   PetscViewerASCIIPrintf(viewer,"</VTKFile>\n");
1460:   PetscViewerDestroy(&viewer);
1461:   return(0);
1462: }

1464: int main(int argc,char *argv[])
1465: {
1466:   MPI_Comm       comm;
1467:   THI            thi;
1469:   DM             da;
1470:   SNES           snes;

1472:   PetscInitialize(&argc,&argv,0,help);
1473:   comm = PETSC_COMM_WORLD;

1475:   THICreate(comm,&thi);
1476:   {
1477:     PetscInt M = 3,N = 3,P = 2;
1478:     PetscOptionsBegin(comm,NULL,"Grid resolution options","");
1479:     {
1480:       PetscOptionsInt("-M","Number of elements in x-direction on coarse level","",M,&M,NULL);
1481:       N    = M;
1482:       PetscOptionsInt("-N","Number of elements in y-direction on coarse level (if different from M)","",N,&N,NULL);
1483:       if (thi->coarse2d) {
1484:         PetscOptionsInt("-zlevels","Number of elements in z-direction on fine level","",thi->zlevels,&thi->zlevels,NULL);
1485:       } else {
1486:         PetscOptionsInt("-P","Number of elements in z-direction on coarse level","",P,&P,NULL);
1487:       }
1488:     }
1489:     PetscOptionsEnd();
1490:     if (thi->coarse2d) {
1491:       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);
1492:       DMSetFromOptions(da);
1493:       DMSetUp(da);
1494:       da->ops->refinehierarchy     = DMRefineHierarchy_THI;
1495:       da->ops->createinterpolation = DMCreateInterpolation_DA_THI;

1497:       PetscObjectCompose((PetscObject)da,"THI",(PetscObject)thi);
1498:     } else {
1499:       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);
1500:       DMSetFromOptions(da);
1501:       DMSetUp(da);
1502:     }
1503:     DMDASetFieldName(da,0,"x-velocity");
1504:     DMDASetFieldName(da,1,"y-velocity");
1505:   }
1506:   THISetUpDM(thi,da);
1507:   if (thi->tridiagonal) da->ops->creatematrix = DMCreateMatrix_THI_Tridiagonal;

1509:   {                             /* Set the fine level matrix type if -da_refine */
1510:     PetscInt rlevel,clevel;
1511:     DMGetRefineLevel(da,&rlevel);
1512:     DMGetCoarsenLevel(da,&clevel);
1513:     if (rlevel - clevel > 0) {DMSetMatType(da,thi->mattype);}
1514:   }

1516:   DMDASNESSetFunctionLocal(da,ADD_VALUES,(DMDASNESFunction)THIFunctionLocal,thi);
1517:   if (thi->tridiagonal) {
1518:     DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)THIJacobianLocal_3D_Tridiagonal,thi);
1519:   } else {
1520:     DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)THIJacobianLocal_3D_Full,thi);
1521:   }
1522:   DMCoarsenHookAdd(da,DMCoarsenHook_THI,NULL,thi);
1523:   DMRefineHookAdd(da,DMRefineHook_THI,NULL,thi);

1525:   DMSetApplicationContext(da,thi);

1527:   SNESCreate(comm,&snes);
1528:   SNESSetDM(snes,da);
1529:   DMDestroy(&da);
1530:   SNESSetComputeInitialGuess(snes,THIInitial,NULL);
1531:   SNESSetFromOptions(snes);

1533:   SNESSolve(snes,NULL,NULL);

1535:   THISolveStatistics(thi,snes,0,"Full");

1537:   {
1538:     PetscBool flg;
1539:     char      filename[PETSC_MAX_PATH_LEN] = "";
1540:     PetscOptionsGetString(NULL,NULL,"-o",filename,sizeof(filename),&flg);
1541:     if (flg) {
1542:       Vec X;
1543:       DM  dm;
1544:       SNESGetSolution(snes,&X);
1545:       SNESGetDM(snes,&dm);
1546:       THIDAVecView_VTK_XML(thi,dm,X,filename);
1547:     }
1548:   }

1550:   DMDestroy(&da);
1551:   SNESDestroy(&snes);
1552:   THIDestroy(&thi);
1553:   PetscFinalize();
1554:   return ierr;
1555: }