Actual source code: plate2f.F90

  1: !  Program usage: mpiexec -n <proc> plate2f [all TAO options]
  2: !
  3: !  This example demonstrates use of the TAO package to solve a bound constrained
  4: !  minimization problem.  This example is based on a problem from the
  5: !  MINPACK-2 test suite.  Given a rectangular 2-D domain and boundary values
  6: !  along the edges of the domain, the objective is to find the surface
  7: !  with the minimal area that satisfies the boundary conditions.
  8: !  The command line options are:
  9: !    -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction
 10: !    -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction
 11: !    -bmx <bxg>, where <bxg> = number of grid points under plate in 1st direction
 12: !    -bmy <byg>, where <byg> = number of grid points under plate in 2nd direction
 13: !    -bheight <ht>, where <ht> = height of the plate
 14: !

 16: module plate2fmodule
 17: #include "petsc/finclude/petscdmda.h"
 18: #include "petsc/finclude/petsctao.h"
 19:   use petscdmda
 20:   use petsctao

 22:   Vec localX, localV
 23:   Vec Top, Left
 24:   Vec Right, Bottom
 25:   DM dm
 26:   PetscReal bheight
 27:   PetscInt bmx, bmy
 28:   PetscInt mx, my, Nx, Ny, N
 29: end module

 31: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 32: !                   Variable declarations
 33: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 34: !
 35: !  Variables:
 36: !    (common from plate2f.h):
 37: !    Nx, Ny           number of processors in x- and y- directions
 38: !    mx, my           number of grid points in x,y directions
 39: !    N    global dimension of vector
 40: use plate2fmodule
 41: implicit none

 43: PetscErrorCode ierr          ! used to check for functions returning nonzeros
 44: Vec x             ! solution vector
 45: PetscInt m             ! number of local elements in vector
 46: Tao ta           ! Tao solver context
 47: Mat H             ! Hessian matrix
 48: ISLocalToGlobalMapping isltog  ! local to global mapping object
 49: PetscBool flg
 50: PetscInt i1, i3, i7

 52: external FormFunctionGradient
 53: external FormHessian
 54: external MSA_BoundaryConditions
 55: external MSA_Plate
 56: external MSA_InitialPoint
 57: ! Initialize Tao

 59: i1 = 1
 60: i3 = 3
 61: i7 = 7

 63: PetscCallA(PetscInitialize(ierr))

 65: ! Specify default dimensions of the problem
 66: mx = 10
 67: my = 10
 68: bheight = 0.1

 70: ! Check for any command line arguments that override defaults

 72: PetscCallA(PetscOptionsGetInt(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-mx', mx, flg, ierr))
 73: PetscCallA(PetscOptionsGetInt(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-my', my, flg, ierr))

 75: bmx = mx/2
 76: bmy = my/2

 78: PetscCallA(PetscOptionsGetInt(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-bmx', bmx, flg, ierr))
 79: PetscCallA(PetscOptionsGetInt(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-bmy', bmy, flg, ierr))
 80: PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-bheight', bheight, flg, ierr))

 82: ! Calculate any derived values from parameters
 83: N = mx*my

 85: ! Let PETSc determine the dimensions of the local vectors
 86: Nx = PETSC_DECIDE
 87: NY = PETSC_DECIDE

 89: ! A two dimensional distributed array will help define this problem, which
 90: ! derives from an elliptic PDE on a two-dimensional domain.  From the
 91: ! distributed array, create the vectors

 93: PetscCallA(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_BOX, mx, my, Nx, Ny, i1, i1, PETSC_NULL_INTEGER_ARRAY, PETSC_NULL_INTEGER_ARRAY, dm, ierr))
 94: PetscCallA(DMSetFromOptions(dm, ierr))
 95: PetscCallA(DMSetUp(dm, ierr))

 97: ! Extract global and local vectors from DM; The local vectors are
 98: ! used solely as work space for the evaluation of the function,
 99: ! gradient, and Hessian.  Duplicate for remaining vectors that are
100: ! the same types.

102: PetscCallA(DMCreateGlobalVector(dm, x, ierr))
103: PetscCallA(DMCreateLocalVector(dm, localX, ierr))
104: PetscCallA(VecDuplicate(localX, localV, ierr))

106: ! Create a matrix data structure to store the Hessian.
107: ! Here we (optionally) also associate the local numbering scheme
108: ! with the matrix so that later we can use local indices for matrix
109: ! assembly

111: PetscCallA(VecGetLocalSize(x, m, ierr))
112: PetscCallA(MatCreateAIJ(PETSC_COMM_WORLD, m, m, N, N, i7, PETSC_NULL_INTEGER_ARRAY, i3, PETSC_NULL_INTEGER_ARRAY, H, ierr))

114: PetscCallA(MatSetOption(H, MAT_SYMMETRIC, PETSC_TRUE, ierr))
115: PetscCallA(DMGetLocalToGlobalMapping(dm, isltog, ierr))
116: PetscCallA(MatSetLocalToGlobalMapping(H, isltog, isltog, ierr))

118: ! The Tao code begins here
119: ! Create TAO solver and set desired solution method.
120: ! This problems uses bounded variables, so the
121: ! method must either be 'tao_tron' or 'tao_blmvm'

123: PetscCallA(TaoCreate(PETSC_COMM_WORLD, ta, ierr))
124: PetscCallA(TaoSetType(ta, TAOBLMVM, ierr))

126: !     Set minimization function and gradient, hessian evaluation functions

128: PetscCallA(TaoSetObjectiveAndGradient(ta, PETSC_NULL_VEC, FormFunctionGradient, 0, ierr))

130: PetscCallA(TaoSetHessian(ta, H, H, FormHessian, 0, ierr))

132: ! Set Variable bounds
133: PetscCallA(MSA_BoundaryConditions(ierr))
134: PetscCallA(TaoSetVariableBoundsRoutine(ta, MSA_Plate, 0, ierr))

136: ! Set the initial solution guess
137: PetscCallA(MSA_InitialPoint(x, ierr))
138: PetscCallA(TaoSetSolution(ta, x, ierr))

140: ! Check for any tao command line options
141: PetscCallA(TaoSetFromOptions(ta, ierr))

143: ! Solve the application
144: PetscCallA(TaoSolve(ta, ierr))

146: ! Free TAO data structures
147: PetscCallA(TaoDestroy(ta, ierr))

149: ! Free PETSc data structures
150: PetscCallA(VecDestroy(x, ierr))
151: PetscCallA(VecDestroy(Top, ierr))
152: PetscCallA(VecDestroy(Bottom, ierr))
153: PetscCallA(VecDestroy(Left, ierr))
154: PetscCallA(VecDestroy(Right, ierr))
155: PetscCallA(MatDestroy(H, ierr))
156: PetscCallA(VecDestroy(localX, ierr))
157: PetscCallA(VecDestroy(localV, ierr))
158: PetscCallA(DMDestroy(dm, ierr))

160: ! Finalize TAO

162: PetscCallA(PetscFinalize(ierr))

164: end

166: ! ---------------------------------------------------------------------
167: !
168: !  FormFunctionGradient - Evaluates function f(X).
169: !
170: !  Input Parameters:
171: !  tao   - the Tao context
172: !  X     - the input vector
173: !  dummy - optional user-defined context, as set by TaoSetFunction()
174: !          (not used here)
175: !
176: !  Output Parameters:
177: !  fcn     - the newly evaluated function
178: !  G       - the gradient vector
179: !  info  - error code
180: !

182: subroutine FormFunctionGradient(ta, X, fcn, G, dummy, ierr)
183:   use plate2fmodule
184:   implicit none

186: ! Input/output variables

188:   Tao ta
189:   PetscReal fcn
190:   Vec X, G
191:   PetscErrorCode ierr
192:   PetscInt dummy

194:   PetscInt i, j, row
195:   PetscInt xs, xm
196:   PetscInt gxs, gxm
197:   PetscInt ys, ym
198:   PetscInt gys, gym
199:   PetscReal ft, zero, hx, hy, hydhx, hxdhy
200:   PetscReal area, rhx, rhy
201:   PetscReal f1, f2, f3, f4, f5, f6, d1, d2, d3
202:   PetscReal d4, d5, d6, d7, d8
203:   PetscReal df1dxc, df2dxc, df3dxc, df4dxc
204:   PetscReal df5dxc, df6dxc
205:   PetscReal xc, xl, xr, xt, xb, xlt, xrb

207:   PetscReal, pointer :: g_v(:), x_v(:)
208:   PetscReal, pointer :: top_v(:), left_v(:)
209:   PetscReal, pointer :: right_v(:), bottom_v(:)

211:   ft = 0.0
212:   zero = 0.0
213:   hx = 1.0/real(mx + 1)
214:   hy = 1.0/real(my + 1)
215:   hydhx = hy/hx
216:   hxdhy = hx/hy
217:   area = 0.5*hx*hy
218:   rhx = real(mx) + 1.0
219:   rhy = real(my) + 1.0

221: ! Get local mesh boundaries
222:   PetscCall(DMDAGetCorners(dm, xs, ys, PETSC_NULL_INTEGER, xm, ym, PETSC_NULL_INTEGER, ierr))
223:   PetscCall(DMDAGetGhostCorners(dm, gxs, gys, PETSC_NULL_INTEGER, gxm, gym, PETSC_NULL_INTEGER, ierr))

225: ! Scatter ghost points to local vector
226:   PetscCall(DMGlobalToLocalBegin(dm, X, INSERT_VALUES, localX, ierr))
227:   PetscCall(DMGlobalToLocalEnd(dm, X, INSERT_VALUES, localX, ierr))

229: ! Initialize the vector to zero
230:   PetscCall(VecSet(localV, zero, ierr))

232: ! Get arrays to vector data (See note above about using VecGetArray in Fortran)
233:   PetscCall(VecGetArray(localX, x_v, ierr))
234:   PetscCall(VecGetArray(localV, g_v, ierr))
235:   PetscCall(VecGetArray(Top, top_v, ierr))
236:   PetscCall(VecGetArray(Bottom, bottom_v, ierr))
237:   PetscCall(VecGetArray(Left, left_v, ierr))
238:   PetscCall(VecGetArray(Right, right_v, ierr))

240: ! Compute function over the locally owned part of the mesh
241:   do j = ys, ys + ym - 1
242:     do i = xs, xs + xm - 1
243:       row = (j - gys)*gxm + (i - gxs)
244:       xc = x_v(1 + row)
245:       xt = xc
246:       xb = xc
247:       xr = xc
248:       xl = xc
249:       xrb = xc
250:       xlt = xc

252:       if (i == 0) then !left side
253:         xl = left_v(1 + j - ys + 1)
254:         xlt = left_v(1 + j - ys + 2)
255:       else
256:         xl = x_v(1 + row - 1)
257:       end if

259:       if (j == 0) then !bottom side
260:         xb = bottom_v(1 + i - xs + 1)
261:         xrb = bottom_v(1 + i - xs + 2)
262:       else
263:         xb = x_v(1 + row - gxm)
264:       end if

266:       if (i + 1 == gxs + gxm) then !right side
267:         xr = right_v(1 + j - ys + 1)
268:         xrb = right_v(1 + j - ys)
269:       else
270:         xr = x_v(1 + row + 1)
271:       end if

273:       if (j + 1 == gys + gym) then !top side
274:         xt = top_v(1 + i - xs + 1)
275:         xlt = top_v(1 + i - xs)
276:       else
277:         xt = x_v(1 + row + gxm)
278:       end if

280:       if ((i > gxs) .and. (j + 1 < gys + gym)) then
281:         xlt = x_v(1 + row - 1 + gxm)
282:       end if

284:       if ((j > gys) .and. (i + 1 < gxs + gxm)) then
285:         xrb = x_v(1 + row + 1 - gxm)
286:       end if

288:       d1 = xc - xl
289:       d2 = xc - xr
290:       d3 = xc - xt
291:       d4 = xc - xb
292:       d5 = xr - xrb
293:       d6 = xrb - xb
294:       d7 = xlt - xl
295:       d8 = xt - xlt

297:       df1dxc = d1*hydhx
298:       df2dxc = d1*hydhx + d4*hxdhy
299:       df3dxc = d3*hxdhy
300:       df4dxc = d2*hydhx + d3*hxdhy
301:       df5dxc = d2*hydhx
302:       df6dxc = d4*hxdhy

304:       d1 = d1*rhx
305:       d2 = d2*rhx
306:       d3 = d3*rhy
307:       d4 = d4*rhy
308:       d5 = d5*rhy
309:       d6 = d6*rhx
310:       d7 = d7*rhy
311:       d8 = d8*rhx

313:       f1 = sqrt(1.0 + d1*d1 + d7*d7)
314:       f2 = sqrt(1.0 + d1*d1 + d4*d4)
315:       f3 = sqrt(1.0 + d3*d3 + d8*d8)
316:       f4 = sqrt(1.0 + d3*d3 + d2*d2)
317:       f5 = sqrt(1.0 + d2*d2 + d5*d5)
318:       f6 = sqrt(1.0 + d4*d4 + d6*d6)

320:       ft = ft + f2 + f4

322:       df1dxc = df1dxc/f1
323:       df2dxc = df2dxc/f2
324:       df3dxc = df3dxc/f3
325:       df4dxc = df4dxc/f4
326:       df5dxc = df5dxc/f5
327:       df6dxc = df6dxc/f6

329:       g_v(1 + row) = 0.5*(df1dxc + df2dxc + df3dxc + df4dxc + df5dxc + df6dxc)
330:     end do
331:   end do

333: ! Compute triangular areas along the border of the domain.
334:   if (xs == 0) then  ! left side
335:     do j = ys, ys + ym - 1
336:       d3 = (left_v(1 + j - ys + 1) - left_v(1 + j - ys + 2))*rhy
337:       d2 = (left_v(1 + j - ys + 1) - x_v(1 + (j - gys)*gxm))*rhx
338:       ft = ft + sqrt(1.0 + d3*d3 + d2*d2)
339:     end do
340:   end if

342:   if (ys == 0) then !bottom side
343:     do i = xs, xs + xm - 1
344:       d2 = (bottom_v(1 + i + 1 - xs) - bottom_v(1 + i - xs + 2))*rhx
345:       d3 = (bottom_v(1 + i - xs + 1) - x_v(1 + i - gxs))*rhy
346:       ft = ft + sqrt(1.0 + d3*d3 + d2*d2)
347:     end do
348:   end if

350:   if (xs + xm == mx) then ! right side
351:     do j = ys, ys + ym - 1
352:       d1 = (x_v(1 + (j + 1 - gys)*gxm - 1) - right_v(1 + j - ys + 1))*rhx
353:       d4 = (right_v(1 + j - ys) - right_v(1 + j - ys + 1))*rhy
354:       ft = ft + sqrt(1.0 + d1*d1 + d4*d4)
355:     end do
356:   end if

358:   if (ys + ym == my) then
359:     do i = xs, xs + xm - 1
360:       d1 = (x_v(1 + (gym - 1)*gxm + i - gxs) - top_v(1 + i - xs + 1))*rhy
361:       d4 = (top_v(1 + i - xs + 1) - top_v(1 + i - xs))*rhx
362:       ft = ft + sqrt(1.0 + d1*d1 + d4*d4)
363:     end do
364:   end if

366:   if ((ys == 0) .and. (xs == 0)) then
367:     d1 = (left_v(1 + 0) - left_v(1 + 1))*rhy
368:     d2 = (bottom_v(1 + 0) - bottom_v(1 + 1))*rhx
369:     ft = ft + sqrt(1.0 + d1*d1 + d2*d2)
370:   end if

372:   if ((ys + ym == my) .and. (xs + xm == mx)) then
373:     d1 = (right_v(1 + ym + 1) - right_v(1 + ym))*rhy
374:     d2 = (top_v(1 + xm + 1) - top_v(1 + xm))*rhx
375:     ft = ft + sqrt(1.0 + d1*d1 + d2*d2)
376:   end if

378:   ft = ft*area
379:   PetscCallMPI(MPI_Allreduce(ft, fcn, 1, MPIU_SCALAR, MPIU_SUM, PETSC_COMM_WORLD, ierr))

381: ! Restore vectors
382:   PetscCall(VecRestoreArray(localX, x_v, ierr))
383:   PetscCall(VecRestoreArray(localV, g_v, ierr))
384:   PetscCall(VecRestoreArray(Left, left_v, ierr))
385:   PetscCall(VecRestoreArray(Top, top_v, ierr))
386:   PetscCall(VecRestoreArray(Bottom, bottom_v, ierr))
387:   PetscCall(VecRestoreArray(Right, right_v, ierr))

389: ! Scatter values to global vector
390:   PetscCall(DMLocalToGlobalBegin(dm, localV, INSERT_VALUES, G, ierr))
391:   PetscCall(DMLocalToGlobalEnd(dm, localV, INSERT_VALUES, G, ierr))

393:   PetscCall(PetscLogFlops(70.0d0*xm*ym, ierr))

395: end  !FormFunctionGradient

397: ! ----------------------------------------------------------------------------
398: !
399: !
400: !   FormHessian - Evaluates Hessian matrix.
401: !
402: !   Input Parameters:
403: !.  tao  - the Tao context
404: !.  X    - input vector
405: !.  dummy  - not used
406: !
407: !   Output Parameters:
408: !.  Hessian    - Hessian matrix
409: !.  Hpc    - optionally different matrix used to construct the preconditioner
410: !
411: !   Notes:
412: !   Due to mesh point reordering with DMs, we must always work
413: !   with the local mesh points, and then transform them to the new
414: !   global numbering with the local-to-global mapping.  We cannot work
415: !   directly with the global numbers for the original uniprocessor mesh!
416: !
417: !      MatSetValuesLocal(), using the local ordering (including
418: !         ghost points!)
419: !         - Then set matrix entries using the local ordering
420: !           by calling MatSetValuesLocal()

422: subroutine FormHessian(ta, X, Hessian, Hpc, dummy, ierr)
423:   use plate2fmodule
424:   implicit none

426:   Tao ta
427:   Vec X
428:   Mat Hessian, Hpc
429:   PetscInt dummy
430:   PetscErrorCode ierr

432:   PetscInt i, j, k, row
433:   PetscInt xs, xm, gxs, gxm
434:   PetscInt ys, ym, gys, gym
435:   PetscInt col(0:6)
436:   PetscReal hx, hy, hydhx, hxdhy, rhx, rhy
437:   PetscReal f1, f2, f3, f4, f5, f6, d1, d2, d3
438:   PetscReal d4, d5, d6, d7, d8
439:   PetscReal xc, xl, xr, xt, xb, xlt, xrb
440:   PetscReal hl, hr, ht, hb, hc, htl, hbr

442:   PetscReal, pointer ::  right_v(:), left_v(:)
443:   PetscReal, pointer ::  bottom_v(:), top_v(:)
444:   PetscReal, pointer ::  x_v(:)
445:   PetscReal v(0:6)
446:   PetscBool assembled
447:   PetscInt i1

449:   i1 = 1

451: ! Set various matrix options
452:   PetscCall(MatSetOption(Hessian, MAT_IGNORE_OFF_PROC_ENTRIES, PETSC_TRUE, ierr))

454: ! Get local mesh boundaries
455:   PetscCall(DMDAGetCorners(dm, xs, ys, PETSC_NULL_INTEGER, xm, ym, PETSC_NULL_INTEGER, ierr))
456:   PetscCall(DMDAGetGhostCorners(dm, gxs, gys, PETSC_NULL_INTEGER, gxm, gym, PETSC_NULL_INTEGER, ierr))

458: ! Scatter ghost points to local vectors
459:   PetscCall(DMGlobalToLocalBegin(dm, X, INSERT_VALUES, localX, ierr))
460:   PetscCall(DMGlobalToLocalEnd(dm, X, INSERT_VALUES, localX, ierr))

462: ! Get pointers to vector data (see note on Fortran arrays above)
463:   PetscCall(VecGetArray(localX, x_v, ierr))
464:   PetscCall(VecGetArray(Top, top_v, ierr))
465:   PetscCall(VecGetArray(Bottom, bottom_v, ierr))
466:   PetscCall(VecGetArray(Left, left_v, ierr))
467:   PetscCall(VecGetArray(Right, right_v, ierr))

469: ! Initialize matrix entries to zero
470:   PetscCall(MatAssembled(Hessian, assembled, ierr))
471:   if (assembled .eqv. PETSC_TRUE) PetscCall(MatZeroEntries(Hessian, ierr))

473:   rhx = real(mx) + 1.0
474:   rhy = real(my) + 1.0
475:   hx = 1.0/rhx
476:   hy = 1.0/rhy
477:   hydhx = hy/hx
478:   hxdhy = hx/hy
479: ! compute Hessian over the locally owned part of the mesh

481:   do i = xs, xs + xm - 1
482:     do j = ys, ys + ym - 1
483:       row = (j - gys)*gxm + (i - gxs)

485:       xc = x_v(1 + row)
486:       xt = xc
487:       xb = xc
488:       xr = xc
489:       xl = xc
490:       xrb = xc
491:       xlt = xc

493:       if (i == gxs) then   ! Left side
494:         xl = left_v(1 + j - ys + 1)
495:         xlt = left_v(1 + j - ys + 2)
496:       else
497:         xl = x_v(1 + row - 1)
498:       end if

500:       if (j == gys) then ! bottom side
501:         xb = bottom_v(1 + i - xs + 1)
502:         xrb = bottom_v(1 + i - xs + 2)
503:       else
504:         xb = x_v(1 + row - gxm)
505:       end if

507:       if (i + 1 == gxs + gxm) then !right side
508:         xr = right_v(1 + j - ys + 1)
509:         xrb = right_v(1 + j - ys)
510:       else
511:         xr = x_v(1 + row + 1)
512:       end if

514:       if (j + 1 == gym + gys) then !top side
515:         xt = top_v(1 + i - xs + 1)
516:         xlt = top_v(1 + i - xs)
517:       else
518:         xt = x_v(1 + row + gxm)
519:       end if

521:       if ((i > gxs) .and. (j + 1 < gys + gym)) then
522:         xlt = x_v(1 + row - 1 + gxm)
523:       end if

525:       if ((i + 1 < gxs + gxm) .and. (j > gys)) then
526:         xrb = x_v(1 + row + 1 - gxm)
527:       end if

529:       d1 = (xc - xl)*rhx
530:       d2 = (xc - xr)*rhx
531:       d3 = (xc - xt)*rhy
532:       d4 = (xc - xb)*rhy
533:       d5 = (xrb - xr)*rhy
534:       d6 = (xrb - xb)*rhx
535:       d7 = (xlt - xl)*rhy
536:       d8 = (xlt - xt)*rhx

538:       f1 = sqrt(1.0 + d1*d1 + d7*d7)
539:       f2 = sqrt(1.0 + d1*d1 + d4*d4)
540:       f3 = sqrt(1.0 + d3*d3 + d8*d8)
541:       f4 = sqrt(1.0 + d3*d3 + d2*d2)
542:       f5 = sqrt(1.0 + d2*d2 + d5*d5)
543:       f6 = sqrt(1.0 + d4*d4 + d6*d6)

545:       hl = (-hydhx*(1.0 + d7*d7) + d1*d7)/(f1*f1*f1) + (-hydhx*(1.0 + d4*d4) + d1*d4)/(f2*f2*f2)

547:       hr = (-hydhx*(1.0 + d5*d5) + d2*d5)/(f5*f5*f5) + (-hydhx*(1.0 + d3*d3) + d2*d3)/(f4*f4*f4)

549:       ht = (-hxdhy*(1.0 + d8*d8) + d3*d8)/(f3*f3*f3) + (-hxdhy*(1.0 + d2*d2) + d2*d3)/(f4*f4*f4)

551:       hb = (-hxdhy*(1.0 + d6*d6) + d4*d6)/(f6*f6*f6) + (-hxdhy*(1.0 + d1*d1) + d1*d4)/(f2*f2*f2)

553:       hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6)
554:       htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3)

556:       hc = hydhx*(1.0 + d7*d7)/(f1*f1*f1) + hxdhy*(1.0 + d8*d8)/(f3*f3*f3) + hydhx*(1.0 + d5*d5)/(f5*f5*f5) +                      &
557: &           hxdhy*(1.0 + d6*d6)/(f6*f6*f6) + (hxdhy*(1.0 + d1*d1) + hydhx*(1.0 + d4*d4) - 2*d1*d4)/(f2*f2*f2) + (hxdhy*(1.0 + d2*d2) +   &
558: &           hydhx*(1.0 + d3*d3) - 2*d2*d3)/(f4*f4*f4)

560:       hl = hl*0.5
561:       hr = hr*0.5
562:       ht = ht*0.5
563:       hb = hb*0.5
564:       hbr = hbr*0.5
565:       htl = htl*0.5
566:       hc = hc*0.5

568:       k = 0

570:       if (j > 0) then
571:         v(k) = hb
572:         col(k) = row - gxm
573:         k = k + 1
574:       end if

576:       if ((j > 0) .and. (i < mx - 1)) then
577:         v(k) = hbr
578:         col(k) = row - gxm + 1
579:         k = k + 1
580:       end if

582:       if (i > 0) then
583:         v(k) = hl
584:         col(k) = row - 1
585:         k = k + 1
586:       end if

588:       v(k) = hc
589:       col(k) = row
590:       k = k + 1

592:       if (i < mx - 1) then
593:         v(k) = hr
594:         col(k) = row + 1
595:         k = k + 1
596:       end if

598:       if ((i > 0) .and. (j < my - 1)) then
599:         v(k) = htl
600:         col(k) = row + gxm - 1
601:         k = k + 1
602:       end if

604:       if (j < my - 1) then
605:         v(k) = ht
606:         col(k) = row + gxm
607:         k = k + 1
608:       end if

610: ! Set matrix values using local numbering, defined earlier in main routine
611:       PetscCall(MatSetValuesLocal(Hessian, i1, [row], k, col, v, INSERT_VALUES, ierr))

613:     end do
614:   end do

616: ! restore vectors
617:   PetscCall(VecRestoreArray(localX, x_v, ierr))
618:   PetscCall(VecRestoreArray(Left, left_v, ierr))
619:   PetscCall(VecRestoreArray(Right, right_v, ierr))
620:   PetscCall(VecRestoreArray(Top, top_v, ierr))
621:   PetscCall(VecRestoreArray(Bottom, bottom_v, ierr))

623: ! Assemble the matrix
624:   PetscCall(MatAssemblyBegin(Hessian, MAT_FINAL_ASSEMBLY, ierr))
625:   PetscCall(MatAssemblyEnd(Hessian, MAT_FINAL_ASSEMBLY, ierr))

627:   PetscCall(PetscLogFlops(199.0d0*xm*ym, ierr))

629: end

631: ! Top,Left,Right,Bottom,bheight,mx,my,bmx,bmy,H, defined in plate2f.h

633: ! ----------------------------------------------------------------------------
634: !
635: !/*
636: !     MSA_BoundaryConditions - calculates the boundary conditions for the region
637: !
638: !
639: !*/

641: subroutine MSA_BoundaryConditions(ierr)
642:   use plate2fmodule
643:   implicit none

645:   PetscErrorCode ierr
646:   PetscInt i, j, k, limit, maxits
647:   PetscInt xs, xm, gxs, gxm
648:   PetscInt ys, ym, gys, gym
649:   PetscInt bsize, lsize
650:   PetscInt tsize, rsize
651:   PetscReal one, two, three, tol
652:   PetscReal scl, fnorm, det, xt
653:   PetscReal yt, hx, hy, u1, u2, nf1, nf2
654:   PetscReal njac11, njac12, njac21, njac22
655:   PetscReal b, t, l, r
656:   PetscReal, pointer :: boundary_v(:)

658:   logical exitloop
659:   PetscBool flg

661:   limit = 0
662:   maxits = 5
663:   tol = 1e-10
664:   b = -0.5
665:   t = 0.5
666:   l = -0.5
667:   r = 0.5
668:   xt = 0
669:   yt = 0
670:   one = 1.0
671:   two = 2.0
672:   three = 3.0

674:   PetscCall(DMDAGetCorners(dm, xs, ys, PETSC_NULL_INTEGER, xm, ym, PETSC_NULL_INTEGER, ierr))
675:   PetscCall(DMDAGetGhostCorners(dm, gxs, gys, PETSC_NULL_INTEGER, gxm, gym, PETSC_NULL_INTEGER, ierr))

677:   bsize = xm + 2
678:   lsize = ym + 2
679:   rsize = ym + 2
680:   tsize = xm + 2

682:   PetscCall(VecCreateMPI(PETSC_COMM_WORLD, bsize, PETSC_DECIDE, Bottom, ierr))
683:   PetscCall(VecCreateMPI(PETSC_COMM_WORLD, tsize, PETSC_DECIDE, Top, ierr))
684:   PetscCall(VecCreateMPI(PETSC_COMM_WORLD, lsize, PETSC_DECIDE, Left, ierr))
685:   PetscCall(VecCreateMPI(PETSC_COMM_WORLD, rsize, PETSC_DECIDE, Right, ierr))

687:   hx = (r - l)/(mx + 1)
688:   hy = (t - b)/(my + 1)

690:   do j = 0, 3

692:     if (j == 0) then
693:       yt = b
694:       xt = l + hx*xs
695:       limit = bsize
696:       PetscCall(VecGetArray(Bottom, boundary_v, ierr))

698:     elseif (j == 1) then
699:       yt = t
700:       xt = l + hx*xs
701:       limit = tsize
702:       PetscCall(VecGetArray(Top, boundary_v, ierr))

704:     elseif (j == 2) then
705:       yt = b + hy*ys
706:       xt = l
707:       limit = lsize
708:       PetscCall(VecGetArray(Left, boundary_v, ierr))

710:     elseif (j == 3) then
711:       yt = b + hy*ys
712:       xt = r
713:       limit = rsize
714:       PetscCall(VecGetArray(Right, boundary_v, ierr))
715:     end if

717:     do i = 0, limit - 1

719:       u1 = xt
720:       u2 = -yt
721:       k = 0
722:       exitloop = .false.
723:       do while (k < maxits .and. (.not. exitloop))

725:         nf1 = u1 + u1*u2*u2 - u1*u1*u1/three - xt
726:         nf2 = -u2 - u1*u1*u2 + u2*u2*u2/three - yt
727:         fnorm = sqrt(nf1*nf1 + nf2*nf2)
728:         if (fnorm > tol) then
729:           njac11 = one + u2*u2 - u1*u1
730:           njac12 = two*u1*u2
731:           njac21 = -two*u1*u2
732:           njac22 = -one - u1*u1 + u2*u2
733:           det = njac11*njac22 - njac21*njac12
734:           u1 = u1 - (njac22*nf1 - njac12*nf2)/det
735:           u2 = u2 - (njac11*nf2 - njac21*nf1)/det
736:         else
737:           exitloop = .true.
738:         end if
739:         k = k + 1
740:       end do

742:       boundary_v(1 + i) = u1*u1 - u2*u2
743:       if ((j == 0) .or. (j == 1)) then
744:         xt = xt + hx
745:       else
746:         yt = yt + hy
747:       end if

749:     end do

751:     if (j == 0) then
752:       PetscCall(VecRestoreArray(Bottom, boundary_v, ierr))
753:     elseif (j == 1) then
754:       PetscCall(VecRestoreArray(Top, boundary_v, ierr))
755:     elseif (j == 2) then
756:       PetscCall(VecRestoreArray(Left, boundary_v, ierr))
757:     elseif (j == 3) then
758:       PetscCall(VecRestoreArray(Right, boundary_v, ierr))
759:     end if

761:   end do

763: ! Scale the boundary if desired
764:   PetscCall(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-bottom', scl, flg, ierr))
765:   if (flg .eqv. PETSC_TRUE) then
766:     PetscCall(VecScale(Bottom, scl, ierr))
767:   end if

769:   PetscCall(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-top', scl, flg, ierr))
770:   if (flg .eqv. PETSC_TRUE) then
771:     PetscCall(VecScale(Top, scl, ierr))
772:   end if

774:   PetscCall(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-right', scl, flg, ierr))
775:   if (flg .eqv. PETSC_TRUE) then
776:     PetscCall(VecScale(Right, scl, ierr))
777:   end if

779:   PetscCall(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-left', scl, flg, ierr))
780:   if (flg .eqv. PETSC_TRUE) then
781:     PetscCall(VecScale(Left, scl, ierr))
782:   end if

784: end

786: ! ----------------------------------------------------------------------------
787: !
788: !/*
789: !     MSA_Plate - Calculates an obstacle for surface to stretch over
790: !
791: !     Output Parameter:
792: !.    xl - lower bound vector
793: !.    xu - upper bound vector
794: !
795: !*/

797: subroutine MSA_Plate(ta, xl, xu, dummy, ierr)
798:   use plate2fmodule
799:   implicit none

801:   Tao ta
802:   Vec xl, xu
803:   PetscErrorCode ierr
804:   PetscInt i, j, row
805:   PetscInt xs, xm, ys, ym
806:   PetscReal lb, ub
807:   PetscInt dummy
808:   PetscReal, pointer :: xl_v(:)

810:   lb = PETSC_NINFINITY
811:   ub = PETSC_INFINITY

813:   if (bmy < 0) bmy = 0
814:   if (bmy > my) bmy = my
815:   if (bmx < 0) bmx = 0
816:   if (bmx > mx) bmx = mx

818:   PetscCall(DMDAGetCorners(dm, xs, ys, PETSC_NULL_INTEGER, xm, ym, PETSC_NULL_INTEGER, ierr))

820:   PetscCall(VecSet(xl, lb, ierr))
821:   PetscCall(VecSet(xu, ub, ierr))

823:   PetscCall(VecGetArray(xl, xl_v, ierr))

825:   do i = xs, xs + xm - 1

827:     do j = ys, ys + ym - 1

829:       row = (j - ys)*xm + (i - xs)

831:       if (i >= ((mx - bmx)/2) .and. i < (mx - (mx - bmx)/2) .and.           &
832: &          j >= ((my - bmy)/2) .and. j < (my - (my - bmy)/2)) then
833:         xl_v(1 + row) = bheight

835:       end if

837:     end do
838:   end do

840:   PetscCall(VecRestoreArray(xl, xl_v, ierr))

842: end

844: ! ----------------------------------------------------------------------------
845: !
846: !/*
847: !     MSA_InitialPoint - Calculates an obstacle for surface to stretch over
848: !
849: !     Output Parameter:
850: !.    X - vector for initial guess
851: !
852: !*/

854: subroutine MSA_InitialPoint(X, ierr)
855:   use plate2fmodule
856:   implicit none

858:   Vec X
859:   PetscErrorCode ierr
860:   PetscInt start, i, j
861:   PetscInt row
862:   PetscInt xs, xm, gxs, gxm
863:   PetscInt ys, ym, gys, gym
864:   PetscReal zero, np5

866:   PetscReal, pointer :: left_v(:), right_v(:)
867:   PetscReal, pointer :: bottom_v(:), top_v(:)
868:   PetscReal, pointer :: x_v(:)
869:   PetscBool flg
870:   PetscRandom rctx

872:   zero = 0.0
873:   np5 = -0.5

875:   PetscCall(PetscOptionsGetInt(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-start', start, flg, ierr))

877:   if ((flg .eqv. PETSC_TRUE) .and. (start == 0)) then  ! the zero vector is reasonable
878:     PetscCall(VecSet(X, zero, ierr))

880:   elseif ((flg .eqv. PETSC_TRUE) .and. (start > 0)) then  ! random start -0.5 < xi < 0.5
881:     PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, rctx, ierr))
882:     do i = 0, start - 1
883:       PetscCall(VecSetRandom(X, rctx, ierr))
884:     end do

886:     PetscCall(PetscRandomDestroy(rctx, ierr))
887:     PetscCall(VecShift(X, np5, ierr))

889:   else   ! average of boundary conditions

891: !        Get Local mesh boundaries
892:     PetscCall(DMDAGetCorners(dm, xs, ys, PETSC_NULL_INTEGER, xm, ym, PETSC_NULL_INTEGER, ierr))
893:     PetscCall(DMDAGetGhostCorners(dm, gxs, gys, PETSC_NULL_INTEGER, gxm, gym, PETSC_NULL_INTEGER, ierr))

895: !        Get pointers to vector data
896:     PetscCall(VecGetArray(Top, top_v, ierr))
897:     PetscCall(VecGetArray(Bottom, bottom_v, ierr))
898:     PetscCall(VecGetArray(Left, left_v, ierr))
899:     PetscCall(VecGetArray(Right, right_v, ierr))

901:     PetscCall(VecGetArray(localX, x_v, ierr))

903: !        Perform local computations
904:     do j = ys, ys + ym - 1
905:       do i = xs, xs + xm - 1
906:         row = (j - gys)*gxm + (i - gxs)
907:         x_v(1 + row) = ((j + 1)*bottom_v(1 + i - xs + 1)/my + (my - j + 1)*top_v(1 + i - xs + 1)/(my + 2) +                  &
908: &                          (i + 1)*left_v(1 + j - ys + 1)/mx + (mx - i + 1)*right_v(1 + j - ys + 1)/(mx + 2))*0.5
909:       end do
910:     end do

912: !        Restore vectors
913:     PetscCall(VecRestoreArray(localX, x_v, ierr))

915:     PetscCall(VecRestoreArray(Left, left_v, ierr))
916:     PetscCall(VecRestoreArray(Top, top_v, ierr))
917:     PetscCall(VecRestoreArray(Bottom, bottom_v, ierr))
918:     PetscCall(VecRestoreArray(Right, right_v, ierr))

920:     PetscCall(DMLocalToGlobalBegin(dm, localX, INSERT_VALUES, X, ierr))
921:     PetscCall(DMLocalToGlobalEnd(dm, localX, INSERT_VALUES, X, ierr))

923:   end if

925: end

927: !
928: !/*TEST
929: !
930: !   build:
931: !      requires: !complex
932: !
933: !   test:
934: !      args: -tao_monitor_short -mx 8 -my 6 -bmx 3 -bmy 3 -bheight 0.2 -tao_type bqnls -tao_gatol 1.e-4
935: !      filter: sort -b
936: !      filter_output: sort -b
937: !      requires: !single
938: !
939: !   test:
940: !      suffix: 2
941: !      nsize: 2
942: !      args: -tao_monitor_short -mx 8 -my 6 -bmx 3 -bmy 3 -bheight 0.2 -tao_type bqnls -tao_gatol 1.e-4
943: !      filter: sort -b
944: !      filter_output: sort -b
945: !      requires: !single
946: !
947: !TEST*/