xref: /libCEED/examples/solids/elasticity.c (revision a3c02c40dbaabd6e273b5f0c924b4ea9631cf171)
1 // Copyright (c) 2017, Lawrence Livermore National Security, LLC. Produced at
2 // the Lawrence Livermore National Laboratory. LLNL-CODE-734707. All Rights
3 // reserved. See files LICENSE and NOTICE for details.
4 //
5 // This file is part of CEED, a collection of benchmarks, miniapps, software
6 // libraries and APIs for efficient high-order finite element and spectral
7 // element discretizations for exascale applications. For more information and
8 // source code availability see http://github.com/ceed.
9 //
10 // The CEED research is supported by the Exascale Computing Project 17-SC-20-SC,
11 // a collaborative effort of two U.S. Department of Energy organizations (Office
12 // of Science and the National Nuclear Security Administration) responsible for
13 // the planning and preparation of a capable exascale ecosystem, including
14 // software, applications, hardware, advanced system engineering and early
15 // testbed platforms, in support of the nation's exascale computing imperative.
16 
17 //                        libCEED + PETSc Example: Elasticity
18 //
19 // This example demonstrates a simple usage of libCEED with PETSc to solve
20 //   elasticity problems.
21 //
22 // The code uses higher level communication protocols in DMPlex.
23 //
24 // Build with:
25 //
26 //     make elasticity [PETSC_DIR=</path/to/petsc>] [CEED_DIR=</path/to/libceed>]
27 //
28 // Sample runs:
29 //
30 //     ./elasticity -mesh [.exo file] -degree 2 -E 1 -nu 0.3 -problem linElas -forcing mms
31 //     ./elasticity -mesh [.exo file] -degree 2 -E 1 -nu 0.3 -bc_clamp 998,999 -bc_clamp_998_translate 0.1,0.2,0.3 -problem hyperSS -forcing none -ceed /cpu/self
32 //     ./elasticity -mesh [.exo file] -degree 2 -E 1 -nu 0.3 -bc_clamp 998,999 -bc_clamp_998 rotate 1,0,0,0.2 -problem hyperFS -forcing none -ceed /gpu/occa
33 //
34 // Sample meshes can be found at https://github.com/jeremylt/ceedSampleMeshes
35 //
36 //TESTARGS -ceed {ceed_resource} -test -degree 2 -nu 0.3 -E 1 -dm_plex_box_faces 3,3,3
37 
38 /// @file
39 /// CEED elasticity example using PETSc with DMPlex
40 
41 const char help[] = "Solve solid Problems with CEED and PETSc DMPlex\n";
42 
43 #include "elasticity.h"
44 
45 int main(int argc, char **argv) {
46   PetscInt       ierr;
47   MPI_Comm       comm;
48   // Context structs
49   AppCtx         appCtx;                 // Contains problem options
50   Physics        phys;                   // Contains physical constants
51   Units          units;                  // Contains units scaling
52   // PETSc objects
53   PetscLogStage  stageDMSetup, stageLibceedSetup,
54                  stageSnesSetup, stageSnesSolve;
55   DM             dmOrig;                 // Distributed DM to clone
56   DM             dmEnergy, dmDiagnostic; // DMs for postprocessing
57   DM             *levelDMs;
58   Vec            U, *Ug, *Uloc;          // U: solution, R: residual, F: forcing
59   Vec            R, Rloc, F, Floc;       // g: global, loc: local
60   SNES           snes, snesCoarse = NULL;
61   Mat            *jacobMat, jacobMatCoarse, *prolongRestrMat;
62   // PETSc data
63   UserMult       resCtx, jacobCoarseCtx = NULL, *jacobCtx;
64   FormJacobCtx   formJacobCtx;
65   UserMultProlongRestr *prolongRestrCtx;
66   PCMGCycleType  pcmgCycleType = PC_MG_CYCLE_V;
67   // libCEED objects
68   Ceed           ceed, ceedFine = NULL;
69   CeedData       *ceedData;
70   CeedQFunction  qfRestrict = NULL, qfProlong = NULL;
71   // Parameters
72   PetscInt       ncompu = 3;             // 3 DoFs in 3D
73   PetscInt       numLevels = 1, fineLevel = 0;
74   PetscInt       *Ugsz, *Ulsz, *Ulocsz;  // sz: size
75   PetscInt       snesIts = 0;
76   // Timing
77   double         startTime, elapsedTime, minTime, maxTime;
78 
79   ierr = PetscInitialize(&argc, &argv, NULL, help);
80   if (ierr)
81     return ierr;
82 
83   // ---------------------------------------------------------------------------
84   // Process command line options
85   // ---------------------------------------------------------------------------
86   comm = PETSC_COMM_WORLD;
87 
88   // -- Set mesh file, polynomial degree, problem type
89   ierr = PetscCalloc1(1, &appCtx); CHKERRQ(ierr);
90   ierr = ProcessCommandLineOptions(comm, appCtx); CHKERRQ(ierr);
91   numLevels = appCtx->numLevels;
92   fineLevel = numLevels - 1;
93 
94   // -- Set Poison's ratio, Young's Modulus
95   ierr = PetscMalloc1(1, &phys); CHKERRQ(ierr);
96   ierr = PetscMalloc1(1, &units); CHKERRQ(ierr);
97   ierr = ProcessPhysics(comm, phys, units); CHKERRQ(ierr);
98 
99   // ---------------------------------------------------------------------------
100   // Setup DM
101   // ---------------------------------------------------------------------------
102   // Performance logging
103   ierr = PetscLogStageRegister("DM and Vector Setup Stage", &stageDMSetup);
104   CHKERRQ(ierr);
105   ierr = PetscLogStagePush(stageDMSetup); CHKERRQ(ierr);
106 
107   // -- Create distributed DM from mesh file
108   ierr = CreateDistributedDM(comm, appCtx, &dmOrig); CHKERRQ(ierr);
109 
110   // -- Setup DM by polynomial degree
111   ierr = PetscMalloc1(numLevels, &levelDMs); CHKERRQ(ierr);
112   for (PetscInt level = 0; level < numLevels; level++) {
113     ierr = DMClone(dmOrig, &levelDMs[level]); CHKERRQ(ierr);
114     ierr = SetupDMByDegree(levelDMs[level], appCtx, appCtx->levelDegrees[level],
115                            PETSC_TRUE, ncompu); CHKERRQ(ierr);
116     // -- Label field components for viewing
117     // Empty name for conserved field (because there is only one field)
118     PetscSection section;
119     ierr = DMGetLocalSection(levelDMs[level], &section); CHKERRQ(ierr);
120     ierr = PetscSectionSetFieldName(section, 0, "Displacement"); CHKERRQ(ierr);
121     ierr = PetscSectionSetComponentName(section, 0, 0, "DisplacementX");
122     CHKERRQ(ierr);
123     ierr = PetscSectionSetComponentName(section, 0, 1, "DisplacementY");
124     CHKERRQ(ierr);
125     ierr = PetscSectionSetComponentName(section, 0, 2, "DisplacementZ");
126     CHKERRQ(ierr);
127   }
128 
129   // -- Setup postprocessing DMs
130   ierr = DMClone(dmOrig, &dmEnergy); CHKERRQ(ierr);
131   ierr = SetupDMByDegree(dmEnergy, appCtx, appCtx->levelDegrees[fineLevel],
132                          PETSC_FALSE, 1); CHKERRQ(ierr);
133   ierr = DMClone(dmOrig, &dmDiagnostic); CHKERRQ(ierr);
134   ierr = SetupDMByDegree(dmDiagnostic, appCtx, appCtx->levelDegrees[fineLevel],
135                          PETSC_FALSE, 2); CHKERRQ(ierr);
136   {
137     // -- Label field components for viewing
138     // Empty name for conserved field (because there is only one field)
139     PetscSection section;
140     ierr = DMGetLocalSection(dmDiagnostic, &section); CHKERRQ(ierr);
141     ierr = PetscSectionSetFieldName(section, 0, "Diagnostics"); CHKERRQ(ierr);
142     ierr = PetscSectionSetComponentName(section, 0, 0, "Stress");
143     ierr = PetscSectionSetComponentName(section, 0, 0, "StrainEnergyDensity");
144     CHKERRQ(ierr);
145   }
146 
147   // ---------------------------------------------------------------------------
148   // Setup solution and work vectors
149   // ---------------------------------------------------------------------------
150   // Allocate arrays
151   ierr = PetscMalloc1(numLevels, &Ug); CHKERRQ(ierr);
152   ierr = PetscMalloc1(numLevels, &Uloc); CHKERRQ(ierr);
153   ierr = PetscMalloc1(numLevels, &Ugsz); CHKERRQ(ierr);
154   ierr = PetscMalloc1(numLevels, &Ulsz); CHKERRQ(ierr);
155   ierr = PetscMalloc1(numLevels, &Ulocsz); CHKERRQ(ierr);
156 
157   // -- Setup solution vectors for each level
158   for (PetscInt level = 0; level < numLevels; level++) {
159     // -- Create global unknown vector U
160     ierr = DMCreateGlobalVector(levelDMs[level], &Ug[level]); CHKERRQ(ierr);
161     ierr = VecGetSize(Ug[level], &Ugsz[level]); CHKERRQ(ierr);
162     // Note: Local size for matShell
163     ierr = VecGetLocalSize(Ug[level], &Ulsz[level]); CHKERRQ(ierr);
164 
165     // -- Create local unknown vector Uloc
166     ierr = DMCreateLocalVector(levelDMs[level], &Uloc[level]); CHKERRQ(ierr);
167     // Note: local size for libCEED
168     ierr = VecGetSize(Uloc[level], &Ulocsz[level]); CHKERRQ(ierr);
169   }
170 
171   // -- Create residual and forcing vectors
172   ierr = VecDuplicate(Ug[fineLevel], &U); CHKERRQ(ierr);
173   ierr = VecDuplicate(Ug[fineLevel], &R); CHKERRQ(ierr);
174   ierr = VecDuplicate(Ug[fineLevel], &F); CHKERRQ(ierr);
175   ierr = VecDuplicate(Uloc[fineLevel], &Rloc); CHKERRQ(ierr);
176   ierr = VecDuplicate(Uloc[fineLevel], &Floc); CHKERRQ(ierr);
177 
178   // Performance logging
179   ierr = PetscLogStagePop();
180 
181   // ---------------------------------------------------------------------------
182   // Set up libCEED
183   // ---------------------------------------------------------------------------
184   // Performance logging
185   ierr = PetscLogStageRegister("libCEED Setup Stage", &stageLibceedSetup);
186   CHKERRQ(ierr);
187   ierr = PetscLogStagePush(stageLibceedSetup); CHKERRQ(ierr);
188 
189   // Initalize backend
190   CeedInit(appCtx->ceedResource, &ceed);
191   if (appCtx->degree > 4 && appCtx->ceedResourceFine[0])
192     CeedInit(appCtx->ceedResourceFine, &ceedFine);
193 
194   // -- Create libCEED local forcing vector
195   CeedVector forceCeed;
196   CeedScalar *f;
197   if (appCtx->forcingChoice != FORCE_NONE) {
198     ierr = VecGetArray(Floc, &f); CHKERRQ(ierr);
199     CeedVectorCreate(ceed, Ulocsz[fineLevel], &forceCeed);
200     CeedVectorSetArray(forceCeed, CEED_MEM_HOST, CEED_USE_POINTER, f);
201   }
202 
203   // -- Restriction and prolongation QFunction
204   if (appCtx->multigridChoice != MULTIGRID_NONE) {
205     CeedQFunctionCreateIdentity(ceed, ncompu, CEED_EVAL_NONE, CEED_EVAL_INTERP,
206                                 &qfRestrict);
207     CeedQFunctionCreateIdentity(ceed, ncompu, CEED_EVAL_INTERP, CEED_EVAL_NONE,
208                                 &qfProlong);
209   }
210 
211   // -- Setup libCEED objects
212   ierr = PetscMalloc1(numLevels, &ceedData); CHKERRQ(ierr);
213   // ---- Setup residual evaluator and geometric information
214   ierr = PetscCalloc1(1, &ceedData[fineLevel]); CHKERRQ(ierr);
215   {
216     bool highOrder = (appCtx->levelDegrees[fineLevel] > 4 && ceedFine);
217     Ceed levelCeed = highOrder ? ceedFine : ceed;
218     ierr = SetupLibceedFineLevel(levelDMs[fineLevel], dmEnergy, dmDiagnostic,
219                                  levelCeed, appCtx, phys, ceedData, fineLevel,
220                                  ncompu, Ugsz[fineLevel], Ulocsz[fineLevel],
221                                  forceCeed, qfRestrict, qfProlong);
222     CHKERRQ(ierr);
223   }
224   // ---- Setup Jacobian evaluator and prolongation/restriction
225   for (PetscInt level = 0; level < numLevels; level++) {
226     if (level != fineLevel) {
227       ierr = PetscCalloc1(1, &ceedData[level]); CHKERRQ(ierr);
228     }
229 
230     // Note: use high order ceed, if specified and degree > 4
231     bool highOrder = (appCtx->levelDegrees[level] > 4 && ceedFine);
232     Ceed levelCeed = highOrder ? ceedFine : ceed;
233     ierr = SetupLibceedLevel(levelDMs[level], levelCeed, appCtx, phys,
234                              ceedData,  level, ncompu, Ugsz[level],
235                              Ulocsz[level], forceCeed, qfRestrict,
236                              qfProlong); CHKERRQ(ierr);
237   }
238 
239   // Performance logging
240   ierr = PetscLogStagePop();
241 
242   // ---------------------------------------------------------------------------
243   // Setup global forcing vector
244   // ---------------------------------------------------------------------------
245   ierr = VecZeroEntries(F); CHKERRQ(ierr);
246 
247   if (appCtx->forcingChoice != FORCE_NONE) {
248     ierr = VecRestoreArray(Floc, &f); CHKERRQ(ierr);
249     ierr = DMLocalToGlobal(levelDMs[fineLevel], Floc, ADD_VALUES, F);
250     CHKERRQ(ierr);
251     CeedVectorDestroy(&forceCeed);
252   }
253 
254   // ---------------------------------------------------------------------------
255   // Print problem summary
256   // ---------------------------------------------------------------------------
257   if (!appCtx->testMode) {
258     const char *usedresource;
259     CeedGetResource(ceed, &usedresource);
260 
261     ierr = PetscPrintf(comm,
262                        "\n-- Elastisticy Example - libCEED + PETSc --\n"
263                        "  libCEED:\n"
264                        "    libCEED Backend                    : %s\n",
265                        usedresource); CHKERRQ(ierr);
266 
267     if (ceedFine) {
268       ierr = PetscPrintf(comm,
269                          "    libCEED Backend - high order       : %s\n",
270                          appCtx->ceedResourceFine); CHKERRQ(ierr);
271     }
272 
273     ierr = PetscPrintf(comm,
274                        "  Problem:\n"
275                        "    Problem Name                       : %s\n"
276                        "    Forcing Function                   : %s\n"
277                        "  Mesh:\n"
278                        "    File                               : %s\n"
279                        "    Number of 1D Basis Nodes (p)       : %d\n"
280                        "    Number of 1D Quadrature Points (q) : %d\n"
281                        "    Global nodes                       : %D\n"
282                        "    Owned nodes                        : %D\n"
283                        "    DoF per node                       : %D\n"
284                        "  Multigrid:\n"
285                        "    Type                               : %s\n"
286                        "    Number of Levels                   : %d\n",
287                        problemTypesForDisp[appCtx->problemChoice],
288                        forcingTypesForDisp[appCtx->forcingChoice],
289                        appCtx->meshFile[0] ? appCtx->meshFile : "Box Mesh",
290                        appCtx->degree + 1, appCtx->degree + 1,
291                        Ugsz[fineLevel]/ncompu, Ulsz[fineLevel]/ncompu, ncompu,
292                        (appCtx->degree == 1 &&
293                         appCtx->multigridChoice != MULTIGRID_NONE) ?
294                        "Algebraic multigrid" :
295                        multigridTypesForDisp[appCtx->multigridChoice],
296                        (appCtx->degree == 1 ||
297                         appCtx->multigridChoice == MULTIGRID_NONE) ?
298                        0 : numLevels); CHKERRQ(ierr);
299 
300     if (appCtx->multigridChoice != MULTIGRID_NONE) {
301       for (PetscInt i = 0; i < 2; i++) {
302         CeedInt level = i ? fineLevel : 0;
303         ierr = PetscPrintf(comm,
304                            "    Level %D (%s):\n"
305                            "      Number of 1D Basis Nodes (p)     : %d\n"
306                            "      Global Nodes                     : %D\n"
307                            "      Owned Nodes                      : %D\n",
308                            level, i ? "fine" : "coarse",
309                            appCtx->levelDegrees[level] + 1,
310                            Ugsz[level]/ncompu, Ulsz[level]/ncompu);
311         CHKERRQ(ierr);
312       }
313     }
314   }
315 
316   // ---------------------------------------------------------------------------
317   // Setup SNES
318   // ---------------------------------------------------------------------------
319   // Performance logging
320   ierr = PetscLogStageRegister("SNES Setup Stage", &stageSnesSetup);
321   CHKERRQ(ierr);
322   ierr = PetscLogStagePush(stageSnesSetup); CHKERRQ(ierr);
323 
324   // Create SNES
325   ierr = SNESCreate(comm, &snes); CHKERRQ(ierr);
326   ierr = SNESSetDM(snes, levelDMs[fineLevel]); CHKERRQ(ierr);
327 
328   // -- Jacobian evaluators
329   ierr = PetscMalloc1(numLevels, &jacobCtx); CHKERRQ(ierr);
330   ierr = PetscMalloc1(numLevels, &jacobMat); CHKERRQ(ierr);
331   for (PetscInt level = 0; level < numLevels; level++) {
332     // -- Jacobian context for level
333     ierr = PetscMalloc1(1, &jacobCtx[level]); CHKERRQ(ierr);
334     ierr = SetupJacobianCtx(comm, appCtx, levelDMs[level], Ug[level],
335                             Uloc[level], ceedData[level], ceed,
336                             jacobCtx[level]); CHKERRQ(ierr);
337 
338     // -- Form Action of Jacobian on delta_u
339     ierr = MatCreateShell(comm, Ulsz[level], Ulsz[level], Ugsz[level],
340                           Ugsz[level], jacobCtx[level], &jacobMat[level]);
341     CHKERRQ(ierr);
342     ierr = MatShellSetOperation(jacobMat[level], MATOP_MULT,
343                                 (void (*)(void))ApplyJacobian_Ceed);
344     CHKERRQ(ierr);
345     ierr = MatShellSetOperation(jacobMat[level], MATOP_GET_DIAGONAL,
346                                 (void(*)(void))GetDiag_Ceed);
347 
348   }
349   // Note: FormJacobian updates Jacobian matrices on each level
350   //   and assembles the Jpre matrix, if needed
351   ierr = PetscMalloc1(1, &formJacobCtx); CHKERRQ(ierr);
352   formJacobCtx->jacobCtx = jacobCtx;
353   formJacobCtx->numLevels = numLevels;
354   formJacobCtx->jacobMat = jacobMat;
355 
356   // -- Residual evaluation function
357   ierr = PetscMalloc1(1, &resCtx); CHKERRQ(ierr);
358   ierr = PetscMemcpy(resCtx, jacobCtx[fineLevel],
359                      sizeof(*jacobCtx[fineLevel])); CHKERRQ(ierr);
360   resCtx->op = ceedData[fineLevel]->opApply;
361   ierr = SNESSetFunction(snes, R, FormResidual_Ceed, resCtx); CHKERRQ(ierr);
362 
363   // -- Prolongation/Restriction evaluation
364   ierr = PetscMalloc1(numLevels, &prolongRestrCtx); CHKERRQ(ierr);
365   ierr = PetscMalloc1(numLevels, &prolongRestrMat); CHKERRQ(ierr);
366   for (PetscInt level = 1; level < numLevels; level++) {
367     // ---- Prolongation/restriction context for level
368     ierr = PetscMalloc1(1, &prolongRestrCtx[level]); CHKERRQ(ierr);
369     ierr = SetupProlongRestrictCtx(comm, levelDMs[level-1], levelDMs[level],
370                                    Ug[level], Uloc[level-1], Uloc[level],
371                                    ceedData[level-1], ceedData[level], ceed,
372                                    prolongRestrCtx[level]); CHKERRQ(ierr);
373 
374     // ---- Form Action of Jacobian on delta_u
375     ierr = MatCreateShell(comm, Ulsz[level], Ulsz[level-1], Ugsz[level],
376                           Ugsz[level-1], prolongRestrCtx[level],
377                           &prolongRestrMat[level]); CHKERRQ(ierr);
378     // Note: In PCMG, restriction is the transpose of prolongation
379     ierr = MatShellSetOperation(prolongRestrMat[level], MATOP_MULT,
380                                 (void (*)(void))Prolong_Ceed);
381     ierr = MatShellSetOperation(prolongRestrMat[level], MATOP_MULT_TRANSPOSE,
382                                 (void (*)(void))Restrict_Ceed);
383     CHKERRQ(ierr);
384   }
385 
386   // ---------------------------------------------------------------------------
387   // Setup dummy SNES for AMG coarse solve
388   // ---------------------------------------------------------------------------
389   if (appCtx->multigridChoice != MULTIGRID_NONE) {
390     // -- Jacobian Matrix
391     ierr = DMSetMatType(levelDMs[0], MATAIJ); CHKERRQ(ierr);
392     ierr = DMCreateMatrix(levelDMs[0], &jacobMatCoarse); CHKERRQ(ierr);
393 
394     if (appCtx->degree > 1) {
395       ierr = SNESCreate(comm, &snesCoarse); CHKERRQ(ierr);
396       ierr = SNESSetDM(snesCoarse, levelDMs[0]); CHKERRQ(ierr);
397       ierr = SNESSetSolution(snesCoarse, Ug[0]); CHKERRQ(ierr);
398 
399       // -- Jacobian function
400       ierr = SNESSetJacobian(snesCoarse, jacobMatCoarse, jacobMatCoarse, NULL,
401                              NULL); CHKERRQ(ierr);
402 
403       // -- Residual evaluation function
404       ierr = PetscMalloc1(1, &jacobCoarseCtx); CHKERRQ(ierr);
405       ierr = PetscMemcpy(jacobCoarseCtx, jacobCtx[0], sizeof(*jacobCtx[0]));
406       CHKERRQ(ierr);
407       ierr = SNESSetFunction(snesCoarse, Ug[0], ApplyJacobianCoarse_Ceed,
408                              jacobCoarseCtx); CHKERRQ(ierr);
409 
410       // -- Update formJacobCtx
411       formJacobCtx->Ucoarse = Ug[0];
412       formJacobCtx->snesCoarse = snesCoarse;
413       formJacobCtx->jacobMatCoarse = jacobMatCoarse;
414     }
415   }
416 
417   // Set Jacobian function
418   if (appCtx->degree > 1) {
419     ierr = SNESSetJacobian(snes, jacobMat[fineLevel], jacobMat[fineLevel],
420                            FormJacobian, formJacobCtx); CHKERRQ(ierr);
421   } else {
422     ierr = SNESSetJacobian(snes, jacobMat[0], jacobMatCoarse,
423                            SNESComputeJacobianDefaultColor, NULL);
424     CHKERRQ(ierr);
425   }
426 
427   // ---------------------------------------------------------------------------
428   // Setup KSP
429   // ---------------------------------------------------------------------------
430   {
431     PC pc;
432     KSP ksp;
433 
434     // -- KSP
435     ierr = SNESGetKSP(snes, &ksp); CHKERRQ(ierr);
436     ierr = KSPSetType(ksp, KSPCG); CHKERRQ(ierr);
437     ierr = KSPSetNormType(ksp, KSP_NORM_NATURAL); CHKERRQ(ierr);
438     ierr = KSPSetTolerances(ksp, 1e-10, PETSC_DEFAULT, PETSC_DEFAULT,
439                             PETSC_DEFAULT); CHKERRQ(ierr);
440     ierr = KSPSetOptionsPrefix(ksp, "outer_"); CHKERRQ(ierr);
441 
442     // -- Preconditioning
443     ierr = KSPGetPC(ksp, &pc); CHKERRQ(ierr);
444     ierr = PCSetDM(pc, levelDMs[fineLevel]); CHKERRQ(ierr);
445     ierr = PCSetOptionsPrefix(pc, "outer_"); CHKERRQ(ierr);
446 
447     if (appCtx->multigridChoice == MULTIGRID_NONE) {
448       // ---- No Multigrid
449       ierr = PCSetType(pc, PCJACOBI); CHKERRQ(ierr);
450       ierr = PCJacobiSetType(pc, PC_JACOBI_DIAGONAL); CHKERRQ(ierr);
451     } else if (appCtx->degree == 1) {
452       // ---- AMG for degree 1
453       ierr = PCSetType(pc, PCGAMG); CHKERRQ(ierr);
454     } else {
455       // ---- PCMG
456       ierr = PCSetType(pc, PCMG); CHKERRQ(ierr);
457 
458       // ------ PCMG levels
459       ierr = PCMGSetLevels(pc, numLevels, NULL); CHKERRQ(ierr);
460       for (PetscInt level = 0; level < numLevels; level++) {
461         // -------- Smoother
462         KSP kspSmoother, kspEst;
463         PC pcSmoother;
464 
465         // ---------- Smoother KSP
466         ierr = PCMGGetSmoother(pc, level, &kspSmoother); CHKERRQ(ierr);
467         ierr = KSPSetDM(kspSmoother, levelDMs[level]); CHKERRQ(ierr);
468         ierr = KSPSetDMActive(kspSmoother, PETSC_FALSE); CHKERRQ(ierr);
469 
470         // ---------- Chebyshev options
471         ierr = KSPSetType(kspSmoother, KSPCHEBYSHEV); CHKERRQ(ierr);
472         ierr = KSPChebyshevEstEigSet(kspSmoother, 0, 0.1, 0, 1.1);
473         CHKERRQ(ierr);
474         ierr = KSPChebyshevEstEigGetKSP(kspSmoother, &kspEst); CHKERRQ(ierr);
475         ierr = KSPSetType(kspEst, KSPCG); CHKERRQ(ierr);
476         ierr = KSPChebyshevEstEigSetUseNoisy(kspSmoother, PETSC_TRUE);
477         CHKERRQ(ierr);
478         ierr = KSPSetOperators(kspSmoother, jacobMat[level], jacobMat[level]);
479         CHKERRQ(ierr);
480 
481         // ---------- Smoother preconditioner
482         ierr = KSPGetPC(kspSmoother, &pcSmoother); CHKERRQ(ierr);
483         ierr = PCSetType(pcSmoother, PCJACOBI); CHKERRQ(ierr);
484         ierr = PCJacobiSetType(pcSmoother, PC_JACOBI_DIAGONAL); CHKERRQ(ierr);
485 
486         // -------- Work vector
487         if (level != fineLevel) {
488           ierr = PCMGSetX(pc, level, Ug[level]); CHKERRQ(ierr);
489         }
490 
491         // -------- Level prolongation/restriction operator
492         if (level > 0) {
493           ierr = PCMGSetInterpolation(pc, level, prolongRestrMat[level]);
494           CHKERRQ(ierr);
495           ierr = PCMGSetRestriction(pc, level, prolongRestrMat[level]);
496           CHKERRQ(ierr);
497         }
498       }
499 
500       // ------ PCMG coarse solve
501       KSP kspCoarse;
502       PC pcCoarse;
503 
504       // -------- Coarse KSP
505       ierr = PCMGGetCoarseSolve(pc, &kspCoarse); CHKERRQ(ierr);
506       ierr = KSPSetType(kspCoarse, KSPPREONLY); CHKERRQ(ierr);
507       ierr = KSPSetOperators(kspCoarse, jacobMatCoarse, jacobMatCoarse);
508       CHKERRQ(ierr);
509       ierr = KSPSetOptionsPrefix(kspCoarse, "coarse_"); CHKERRQ(ierr);
510 
511       // -------- Coarse preconditioner
512       ierr = KSPGetPC(kspCoarse, &pcCoarse); CHKERRQ(ierr);
513       ierr = PCSetType(pcCoarse, PCGAMG); CHKERRQ(ierr);
514       ierr = PCSetOptionsPrefix(pcCoarse, "coarse_"); CHKERRQ(ierr);
515 
516       ierr = KSPSetFromOptions(kspCoarse); CHKERRQ(ierr);
517       ierr = PCSetFromOptions(pcCoarse); CHKERRQ(ierr);
518 
519       // ------ PCMG options
520       ierr = PCMGSetType(pc, PC_MG_MULTIPLICATIVE); CHKERRQ(ierr);
521       ierr = PCMGSetNumberSmooth(pc, 3); CHKERRQ(ierr);
522       ierr = PCMGSetCycleType(pc, pcmgCycleType); CHKERRQ(ierr);
523     }
524     ierr = KSPSetFromOptions(ksp);
525     ierr = PCSetFromOptions(pc);
526   }
527   {
528     // Default to critical-point (CP) line search (related to Wolfe's curvature condition)
529     SNESLineSearch linesearch;
530 
531     ierr = SNESGetLineSearch(snes, &linesearch); CHKERRQ(ierr);
532     ierr = SNESLineSearchSetType(linesearch, SNESLINESEARCHCP); CHKERRQ(ierr);
533   }
534 
535   ierr = SNESSetFromOptions(snes); CHKERRQ(ierr);
536 
537   // Performance logging
538   ierr = PetscLogStagePop();
539 
540   // ---------------------------------------------------------------------------
541   // Set initial guess
542   // ---------------------------------------------------------------------------
543   ierr = VecSet(U, 0.0); CHKERRQ(ierr);
544 
545   // View solution
546   if (appCtx->viewSoln) {
547     ierr = ViewSolution(comm, U, 0, 0.0); CHKERRQ(ierr);
548   }
549 
550   // ---------------------------------------------------------------------------
551   // Solve SNES
552   // ---------------------------------------------------------------------------
553   PetscBool snesMonitor = PETSC_FALSE;
554   ierr = PetscOptionsHasName(NULL, NULL, "-snes_monitor", &snesMonitor);
555   CHKERRQ(ierr);
556 
557   // Performance logging
558   ierr = PetscLogStageRegister("SNES Solve Stage", &stageSnesSolve);
559   CHKERRQ(ierr);
560   ierr = PetscLogStagePush(stageSnesSolve); CHKERRQ(ierr);
561 
562   // Timing
563   ierr = PetscBarrier((PetscObject)snes); CHKERRQ(ierr);
564   startTime = MPI_Wtime();
565 
566   // Solve for each load increment
567   PetscInt increment;
568   for (increment = 1; increment <= appCtx->numIncrements; increment++) {
569     // -- Log increment count
570     if (snesMonitor) {
571       ierr = PetscPrintf(comm, "%d Load Increment\n", increment - 1);
572       CHKERRQ(ierr);
573     }
574 
575     // -- Scale the problem
576     PetscScalar loadIncrement = 1.0*increment / appCtx->numIncrements,
577                 scalingFactor = loadIncrement /
578                                 (increment == 1 ? 1 : resCtx->loadIncrement);
579     resCtx->loadIncrement = loadIncrement;
580     if (appCtx->numIncrements > 1 && appCtx->forcingChoice != FORCE_NONE) {
581       ierr = VecScale(F, scalingFactor); CHKERRQ(ierr);
582     }
583 
584     // -- Solve
585     ierr = SNESSolve(snes, F, U); CHKERRQ(ierr);
586 
587     // -- View solution
588     if (appCtx->viewSoln) {
589       ierr = ViewSolution(comm, U, increment, loadIncrement); CHKERRQ(ierr);
590     }
591 
592     // -- Update SNES iteration count
593     PetscInt its;
594     ierr = SNESGetIterationNumber(snes, &its); CHKERRQ(ierr);
595     snesIts += its;
596 
597     // -- Check for divergence
598     SNESConvergedReason reason;
599     ierr = SNESGetConvergedReason(snes, &reason); CHKERRQ(ierr);
600     if (reason < 0)
601       break;
602   }
603 
604   // Timing
605   elapsedTime = MPI_Wtime() - startTime;
606 
607   // Performance logging
608   ierr = PetscLogStagePop();
609 
610   // View solution
611   if (appCtx->viewFinalSoln && !appCtx->viewSoln) {
612     ierr = ViewSolution(comm, U, increment - 1, 1.0); CHKERRQ(ierr);
613   }
614 
615   // ---------------------------------------------------------------------------
616   // Output summary
617   // ---------------------------------------------------------------------------
618   if (!appCtx->testMode) {
619     // -- SNES
620     SNESType snesType;
621     SNESConvergedReason reason;
622     PetscReal rnorm;
623     ierr = SNESGetType(snes, &snesType); CHKERRQ(ierr);
624     ierr = SNESGetConvergedReason(snes, &reason); CHKERRQ(ierr);
625     ierr = SNESGetFunctionNorm(snes, &rnorm); CHKERRQ(ierr);
626     ierr = PetscPrintf(comm,
627                        "  SNES:\n"
628                        "    SNES Type                          : %s\n"
629                        "    SNES Convergence                   : %s\n"
630                        "    Number of Load Increments          : %d\n"
631                        "    Completed Load Increments          : %d\n"
632                        "    Total SNES Iterations              : %D\n"
633                        "    Final rnorm                        : %e\n",
634                        snesType, SNESConvergedReasons[reason],
635                        appCtx->numIncrements, increment - 1,
636                        snesIts, (double)rnorm); CHKERRQ(ierr);
637 
638     // -- KSP
639     KSP ksp;
640     KSPType kspType;
641     ierr = SNESGetKSP(snes, &ksp); CHKERRQ(ierr);
642     ierr = KSPGetType(ksp, &kspType); CHKERRQ(ierr);
643     ierr = PetscPrintf(comm,
644                        "  Linear Solver:\n"
645                        "    KSP Type                           : %s\n",
646                        kspType); CHKERRQ(ierr);
647 
648     // -- PC
649     PC pc;
650     PCType pcType;
651     ierr = KSPGetPC(ksp, &pc); CHKERRQ(ierr);
652     ierr = PCGetType(pc, &pcType); CHKERRQ(ierr);
653     ierr = PetscPrintf(comm,
654                        "    PC Type                            : %s\n",
655                        pcType); CHKERRQ(ierr);
656 
657     if (!strcmp(pcType, PCMG)) {
658       PCMGType pcmgType;
659       ierr = PCMGGetType(pc, &pcmgType); CHKERRQ(ierr);
660       ierr = PetscPrintf(comm,
661                          "  P-Multigrid:\n"
662                          "    PCMG Type                          : %s\n"
663                          "    PCMG Cycle Type                    : %s\n",
664                          PCMGTypes[pcmgType],
665                          PCMGCycleTypes[pcmgCycleType]); CHKERRQ(ierr);
666 
667       // -- Coarse Solve
668       KSP kspCoarse;
669       PC pcCoarse;
670       PCType pcType;
671 
672       ierr = PCMGGetCoarseSolve(pc, &kspCoarse); CHKERRQ(ierr);
673       ierr = KSPGetType(kspCoarse, &kspType); CHKERRQ(ierr);
674       ierr = KSPGetPC(kspCoarse, &pcCoarse); CHKERRQ(ierr);
675       ierr = PCGetType(pcCoarse, &pcType); CHKERRQ(ierr);
676       ierr = PetscPrintf(comm,
677                          "    Coarse Solve:\n"
678                          "      KSP Type                         : %s\n"
679                          "      PC Type                          : %s\n",
680                          kspType, pcType); CHKERRQ(ierr);
681     }
682   }
683 
684   // ---------------------------------------------------------------------------
685   // Compute solve time
686   // ---------------------------------------------------------------------------
687   if (!appCtx->testMode) {
688     ierr = MPI_Allreduce(&elapsedTime, &minTime, 1, MPI_DOUBLE, MPI_MIN, comm);
689     CHKERRQ(ierr);
690     ierr = MPI_Allreduce(&elapsedTime, &maxTime, 1, MPI_DOUBLE, MPI_MAX, comm);
691     CHKERRQ(ierr);
692     ierr = PetscPrintf(comm,
693                        "  Performance:\n"
694                        "    SNES Solve Time                    : %g (%g) sec\n",
695                        maxTime, minTime); CHKERRQ(ierr);
696   }
697 
698   // ---------------------------------------------------------------------------
699   // Compute error
700   // ---------------------------------------------------------------------------
701   if (appCtx->forcingChoice == FORCE_MMS) {
702     CeedScalar l2Error = 1., l2Unorm = 1.;
703     const CeedScalar *truearray;
704     Vec errorVec, trueVec;
705 
706     // -- Work vectors
707     ierr = VecDuplicate(U, &errorVec); CHKERRQ(ierr);
708     ierr = VecSet(errorVec, 0.0); CHKERRQ(ierr);
709     ierr = VecDuplicate(U, &trueVec); CHKERRQ(ierr);
710     ierr = VecSet(trueVec, 0.0); CHKERRQ(ierr);
711 
712     // -- Assemble global true solution vector
713     CeedVectorGetArrayRead(ceedData[fineLevel]->truesoln, CEED_MEM_HOST,
714                            &truearray);
715     ierr = VecPlaceArray(resCtx->Yloc, truearray); CHKERRQ(ierr);
716     ierr = DMLocalToGlobal(resCtx->dm, resCtx->Yloc, INSERT_VALUES, trueVec);
717     CHKERRQ(ierr);
718     ierr = VecResetArray(resCtx->Yloc); CHKERRQ(ierr);
719     CeedVectorRestoreArrayRead(ceedData[fineLevel]->truesoln, &truearray);
720 
721     // -- Compute L2 error
722     ierr = VecWAXPY(errorVec, -1.0, U, trueVec); CHKERRQ(ierr);
723     ierr = VecNorm(errorVec, NORM_2, &l2Error); CHKERRQ(ierr);
724     ierr = VecNorm(U, NORM_2, &l2Unorm); CHKERRQ(ierr);
725     l2Error /= l2Unorm;
726 
727     // -- Output
728     if (!appCtx->testMode || l2Error > 0.05) {
729       ierr = PetscPrintf(comm,
730                          "    L2 Error                           : %e\n",
731                          l2Error); CHKERRQ(ierr);
732     }
733 
734     // -- Cleanup
735     ierr = VecDestroy(&errorVec); CHKERRQ(ierr);
736     ierr = VecDestroy(&trueVec); CHKERRQ(ierr);
737   }
738 
739   // ---------------------------------------------------------------------------
740   // Compute energy
741   // ---------------------------------------------------------------------------
742   if (!appCtx->testMode) {
743     // -- Compute L2 error
744     CeedScalar energy;
745     ierr = ComputeStrainEnergy(dmEnergy, resCtx, ceedData[fineLevel]->opEnergy,
746                                U, &energy); CHKERRQ(ierr);
747 
748     // -- Output
749     ierr = PetscPrintf(comm,
750                        "    Strain Energy                      : %e\n",
751                        energy); CHKERRQ(ierr);
752   }
753 
754   // ---------------------------------------------------------------------------
755   // Free objects
756   // ---------------------------------------------------------------------------
757   // Data in arrays per level
758   for (PetscInt level = 0; level < numLevels; level++) {
759     // Vectors
760     ierr = VecDestroy(&Ug[level]); CHKERRQ(ierr);
761     ierr = VecDestroy(&Uloc[level]); CHKERRQ(ierr);
762 
763     // Jacobian matrix and data
764     ierr = VecDestroy(&jacobCtx[level]->Yloc); CHKERRQ(ierr);
765     ierr = MatDestroy(&jacobMat[level]); CHKERRQ(ierr);
766     ierr = PetscFree(jacobCtx[level]); CHKERRQ(ierr);
767 
768     // Prolongation/Restriction matrix and data
769     if (level > 0) {
770       ierr = VecDestroy(&prolongRestrCtx[level]->multVec); CHKERRQ(ierr);
771       ierr = PetscFree(prolongRestrCtx[level]); CHKERRQ(ierr);
772       ierr = MatDestroy(&prolongRestrMat[level]); CHKERRQ(ierr);
773     }
774 
775     // DM
776     ierr = DMDestroy(&levelDMs[level]); CHKERRQ(ierr);
777 
778     // libCEED objects
779     ierr = CeedDataDestroy(level, ceedData[level]); CHKERRQ(ierr);
780   }
781 
782   // Arrays
783   ierr = PetscFree(Ug); CHKERRQ(ierr);
784   ierr = PetscFree(Uloc); CHKERRQ(ierr);
785   ierr = PetscFree(Ugsz); CHKERRQ(ierr);
786   ierr = PetscFree(Ulsz); CHKERRQ(ierr);
787   ierr = PetscFree(Ulocsz); CHKERRQ(ierr);
788   ierr = PetscFree(jacobCtx); CHKERRQ(ierr);
789   ierr = PetscFree(jacobMat); CHKERRQ(ierr);
790   ierr = PetscFree(prolongRestrCtx); CHKERRQ(ierr);
791   ierr = PetscFree(prolongRestrMat); CHKERRQ(ierr);
792   ierr = PetscFree(appCtx->levelDegrees); CHKERRQ(ierr);
793   ierr = PetscFree(ceedData); CHKERRQ(ierr);
794 
795   // libCEED objects
796   CeedQFunctionDestroy(&qfRestrict);
797   CeedQFunctionDestroy(&qfProlong);
798   CeedDestroy(&ceed);
799   CeedDestroy(&ceedFine);
800 
801   // PETSc objects
802   ierr = VecDestroy(&U); CHKERRQ(ierr);
803   ierr = VecDestroy(&R); CHKERRQ(ierr);
804   ierr = VecDestroy(&Rloc); CHKERRQ(ierr);
805   ierr = VecDestroy(&F); CHKERRQ(ierr);
806   ierr = VecDestroy(&Floc); CHKERRQ(ierr);
807   ierr = MatDestroy(&jacobMatCoarse); CHKERRQ(ierr);
808   ierr = SNESDestroy(&snes); CHKERRQ(ierr);
809   ierr = SNESDestroy(&snesCoarse); CHKERRQ(ierr);
810   ierr = DMDestroy(&dmOrig); CHKERRQ(ierr);
811   ierr = DMDestroy(&dmEnergy); CHKERRQ(ierr);
812   ierr = DMDestroy(&dmDiagnostic); CHKERRQ(ierr);
813   ierr = PetscFree(levelDMs); CHKERRQ(ierr);
814 
815   // Structs
816   ierr = PetscFree(resCtx); CHKERRQ(ierr);
817   ierr = PetscFree(formJacobCtx); CHKERRQ(ierr);
818   ierr = PetscFree(jacobCoarseCtx); CHKERRQ(ierr);
819   ierr = PetscFree(appCtx); CHKERRQ(ierr);
820   ierr = PetscFree(phys); CHKERRQ(ierr);
821   ierr = PetscFree(units); CHKERRQ(ierr);
822 
823   return PetscFinalize();
824 }
825