xref: /petsc/src/tao/bound/tutorials/plate2f.F90 (revision 4820e4ea99a084ae862a8c395f732bc7c0e1a6d0)

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

module plate2fmodule
#include "petsc/finclude/petscdmda.h"
#include "petsc/finclude/petsctao.h"
  use petscdmda
  use petsctao

  Vec localX, localV
  Vec Top, Left
  Vec Right, Bottom
  DM dm
  PetscReal bheight
  PetscInt bmx, bmy
  PetscInt mx, my, Nx, Ny, N
end module

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!                   Variable declarations
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
!  Variables:
!    (common from plate2f.h):
!    Nx, Ny           number of processors in x- and y- directions
!    mx, my           number of grid points in x,y directions
!    N    global dimension of vector
use plate2fmodule
implicit none

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

external FormFunctionGradient
external FormHessian
external MSA_BoundaryConditions
external MSA_Plate
external MSA_InitialPoint
! Initialize Tao

i1 = 1
i3 = 3
i7 = 7

PetscCallA(PetscInitialize(ierr))

! Specify default dimensions of the problem
mx = 10
my = 10
bheight = 0.1

! Check for any command line arguments that override defaults

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

bmx = mx/2
bmy = my/2

PetscCallA(PetscOptionsGetInt(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-bmx', bmx, flg, ierr))
PetscCallA(PetscOptionsGetInt(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-bmy', bmy, flg, ierr))
PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-bheight', bheight, flg, ierr))

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

! Let PETSc determine the dimensions of the local vectors
Nx = PETSC_DECIDE
NY = PETSC_DECIDE

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

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))
PetscCallA(DMSetFromOptions(dm, ierr))
PetscCallA(DMSetUp(dm, ierr))

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

PetscCallA(DMCreateGlobalVector(dm, x, ierr))
PetscCallA(DMCreateLocalVector(dm, localX, ierr))
PetscCallA(VecDuplicate(localX, localV, ierr))

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

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

PetscCallA(MatSetOption(H, MAT_SYMMETRIC, PETSC_TRUE, ierr))
PetscCallA(DMGetLocalToGlobalMapping(dm, isltog, ierr))
PetscCallA(MatSetLocalToGlobalMapping(H, isltog, isltog, ierr))

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

PetscCallA(TaoCreate(PETSC_COMM_WORLD, ta, ierr))
PetscCallA(TaoSetType(ta, TAOBLMVM, ierr))

!     Set minimization function and gradient, hessian evaluation functions

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

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

! Set Variable bounds
PetscCallA(MSA_BoundaryConditions(ierr))
PetscCallA(TaoSetVariableBoundsRoutine(ta, MSA_Plate, 0, ierr))

! Set the initial solution guess
PetscCallA(MSA_InitialPoint(x, ierr))
PetscCallA(TaoSetSolution(ta, x, ierr))

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

! Solve the application
PetscCallA(TaoSolve(ta, ierr))

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

! Free PETSc data structures
PetscCallA(VecDestroy(x, ierr))
PetscCallA(VecDestroy(Top, ierr))
PetscCallA(VecDestroy(Bottom, ierr))
PetscCallA(VecDestroy(Left, ierr))
PetscCallA(VecDestroy(Right, ierr))
PetscCallA(MatDestroy(H, ierr))
PetscCallA(VecDestroy(localX, ierr))
PetscCallA(VecDestroy(localV, ierr))
PetscCallA(DMDestroy(dm, ierr))

! Finalize TAO

PetscCallA(PetscFinalize(ierr))

end

! ---------------------------------------------------------------------
!
!  FormFunctionGradient - Evaluates function f(X).
!
!  Input Parameters:
!  tao   - the Tao context
!  X     - the input vector
!  dummy - optional user-defined context, as set by TaoSetFunction()
!          (not used here)
!
!  Output Parameters:
!  fcn     - the newly evaluated function
!  G       - the gradient vector
!  info  - error code
!

subroutine FormFunctionGradient(ta, X, fcn, G, dummy, ierr)
  use plate2fmodule
  implicit none

! Input/output variables

  Tao ta
  PetscReal fcn
  Vec X, G
  PetscErrorCode ierr
  PetscInt dummy

  PetscInt i, j, row
  PetscInt xs, xm
  PetscInt gxs, gxm
  PetscInt ys, ym
  PetscInt gys, gym
  PetscReal ft, zero, hx, hy, hydhx, hxdhy
  PetscReal area, rhx, rhy
  PetscReal f1, f2, f3, f4, f5, f6, d1, d2, d3
  PetscReal d4, d5, d6, d7, d8
  PetscReal df1dxc, df2dxc, df3dxc, df4dxc
  PetscReal df5dxc, df6dxc
  PetscReal xc, xl, xr, xt, xb, xlt, xrb

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

  ft = 0.0
  zero = 0.0
  hx = 1.0/real(mx + 1)
  hy = 1.0/real(my + 1)
  hydhx = hy/hx
  hxdhy = hx/hy
  area = 0.5*hx*hy
  rhx = real(mx) + 1.0
  rhy = real(my) + 1.0

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

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

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

! Get arrays to vector data (See note above about using VecGetArray in Fortran)
  PetscCall(VecGetArray(localX, x_v, ierr))
  PetscCall(VecGetArray(localV, g_v, ierr))
  PetscCall(VecGetArray(Top, top_v, ierr))
  PetscCall(VecGetArray(Bottom, bottom_v, ierr))
  PetscCall(VecGetArray(Left, left_v, ierr))
  PetscCall(VecGetArray(Right, right_v, ierr))

! Compute function over the locally owned part of the mesh
  do j = ys, ys + ym - 1
    do i = xs, xs + xm - 1
      row = (j - gys)*gxm + (i - gxs)
      xc = x_v(1 + row)
      xt = xc
      xb = xc
      xr = xc
      xl = xc
      xrb = xc
      xlt = xc

      if (i == 0) then !left side
        xl = left_v(1 + j - ys + 1)
        xlt = left_v(1 + j - ys + 2)
      else
        xl = x_v(1 + row - 1)
      end if

      if (j == 0) then !bottom side
        xb = bottom_v(1 + i - xs + 1)
        xrb = bottom_v(1 + i - xs + 2)
      else
        xb = x_v(1 + row - gxm)
      end if

      if (i + 1 == gxs + gxm) then !right side
        xr = right_v(1 + j - ys + 1)
        xrb = right_v(1 + j - ys)
      else
        xr = x_v(1 + row + 1)
      end if

      if (j + 1 == gys + gym) then !top side
        xt = top_v(1 + i - xs + 1)
        xlt = top_v(1 + i - xs)
      else
        xt = x_v(1 + row + gxm)
      end if

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

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

      d1 = xc - xl
      d2 = xc - xr
      d3 = xc - xt
      d4 = xc - xb
      d5 = xr - xrb
      d6 = xrb - xb
      d7 = xlt - xl
      d8 = xt - xlt

      df1dxc = d1*hydhx
      df2dxc = d1*hydhx + d4*hxdhy
      df3dxc = d3*hxdhy
      df4dxc = d2*hydhx + d3*hxdhy
      df5dxc = d2*hydhx
      df6dxc = d4*hxdhy

      d1 = d1*rhx
      d2 = d2*rhx
      d3 = d3*rhy
      d4 = d4*rhy
      d5 = d5*rhy
      d6 = d6*rhx
      d7 = d7*rhy
      d8 = d8*rhx

      f1 = sqrt(1.0 + d1*d1 + d7*d7)
      f2 = sqrt(1.0 + d1*d1 + d4*d4)
      f3 = sqrt(1.0 + d3*d3 + d8*d8)
      f4 = sqrt(1.0 + d3*d3 + d2*d2)
      f5 = sqrt(1.0 + d2*d2 + d5*d5)
      f6 = sqrt(1.0 + d4*d4 + d6*d6)

      ft = ft + f2 + f4

      df1dxc = df1dxc/f1
      df2dxc = df2dxc/f2
      df3dxc = df3dxc/f3
      df4dxc = df4dxc/f4
      df5dxc = df5dxc/f5
      df6dxc = df6dxc/f6

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

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

  if (ys == 0) then !bottom side
    do i = xs, xs + xm - 1
      d2 = (bottom_v(1 + i + 1 - xs) - bottom_v(1 + i - xs + 2))*rhx
      d3 = (bottom_v(1 + i - xs + 1) - x_v(1 + i - gxs))*rhy
      ft = ft + sqrt(1.0 + d3*d3 + d2*d2)
    end do
  end if

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

  if (ys + ym == my) then
    do i = xs, xs + xm - 1
      d1 = (x_v(1 + (gym - 1)*gxm + i - gxs) - top_v(1 + i - xs + 1))*rhy
      d4 = (top_v(1 + i - xs + 1) - top_v(1 + i - xs))*rhx
      ft = ft + sqrt(1.0 + d1*d1 + d4*d4)
    end do
  end if

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

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

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

! Restore vectors
  PetscCall(VecRestoreArray(localX, x_v, ierr))
  PetscCall(VecRestoreArray(localV, g_v, ierr))
  PetscCall(VecRestoreArray(Left, left_v, ierr))
  PetscCall(VecRestoreArray(Top, top_v, ierr))
  PetscCall(VecRestoreArray(Bottom, bottom_v, ierr))
  PetscCall(VecRestoreArray(Right, right_v, ierr))

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

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

end  !FormFunctionGradient

! ----------------------------------------------------------------------------
!
!
!   FormHessian - Evaluates Hessian matrix.
!
!   Input Parameters:
!.  tao  - the Tao context
!.  X    - input vector
!.  dummy  - not used
!
!   Output Parameters:
!.  Hessian    - Hessian matrix
!.  Hpc    - optionally different matrix used to construct the preconditioner
!
!   Notes:
!   Due to mesh point reordering with DMs, we must always work
!   with the local mesh points, and then transform them to the new
!   global numbering with the local-to-global mapping.  We cannot work
!   directly with the global numbers for the original uniprocessor mesh!
!
!      MatSetValuesLocal(), using the local ordering (including
!         ghost points!)
!         - Then set matrix entries using the local ordering
!           by calling MatSetValuesLocal()

subroutine FormHessian(ta, X, Hessian, Hpc, dummy, ierr)
  use plate2fmodule
  implicit none

  Tao ta
  Vec X
  Mat Hessian, Hpc
  PetscInt dummy
  PetscErrorCode ierr

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

  PetscReal, pointer ::  right_v(:), left_v(:)
  PetscReal, pointer ::  bottom_v(:), top_v(:)
  PetscReal, pointer ::  x_v(:)
  PetscReal v(0:6)
  PetscBool assembled
  PetscInt i1

  i1 = 1

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

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

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

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

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

  rhx = real(mx) + 1.0
  rhy = real(my) + 1.0
  hx = 1.0/rhx
  hy = 1.0/rhy
  hydhx = hy/hx
  hxdhy = hx/hy
! compute Hessian over the locally owned part of the mesh

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

      xc = x_v(1 + row)
      xt = xc
      xb = xc
      xr = xc
      xl = xc
      xrb = xc
      xlt = xc

      if (i == gxs) then   ! Left side
        xl = left_v(1 + j - ys + 1)
        xlt = left_v(1 + j - ys + 2)
      else
        xl = x_v(1 + row - 1)
      end if

      if (j == gys) then ! bottom side
        xb = bottom_v(1 + i - xs + 1)
        xrb = bottom_v(1 + i - xs + 2)
      else
        xb = x_v(1 + row - gxm)
      end if

      if (i + 1 == gxs + gxm) then !right side
        xr = right_v(1 + j - ys + 1)
        xrb = right_v(1 + j - ys)
      else
        xr = x_v(1 + row + 1)
      end if

      if (j + 1 == gym + gys) then !top side
        xt = top_v(1 + i - xs + 1)
        xlt = top_v(1 + i - xs)
      else
        xt = x_v(1 + row + gxm)
      end if

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

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

      d1 = (xc - xl)*rhx
      d2 = (xc - xr)*rhx
      d3 = (xc - xt)*rhy
      d4 = (xc - xb)*rhy
      d5 = (xrb - xr)*rhy
      d6 = (xrb - xb)*rhx
      d7 = (xlt - xl)*rhy
      d8 = (xlt - xt)*rhx

      f1 = sqrt(1.0 + d1*d1 + d7*d7)
      f2 = sqrt(1.0 + d1*d1 + d4*d4)
      f3 = sqrt(1.0 + d3*d3 + d8*d8)
      f4 = sqrt(1.0 + d3*d3 + d2*d2)
      f5 = sqrt(1.0 + d2*d2 + d5*d5)
      f6 = sqrt(1.0 + d4*d4 + d6*d6)

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

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

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

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

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

      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) +                      &
&           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) +   &
&           hydhx*(1.0 + d3*d3) - 2*d2*d3)/(f4*f4*f4)

      hl = hl*0.5
      hr = hr*0.5
      ht = ht*0.5
      hb = hb*0.5
      hbr = hbr*0.5
      htl = htl*0.5
      hc = hc*0.5

      k = 0

      if (j > 0) then
        v(k) = hb
        col(k) = row - gxm
        k = k + 1
      end if

      if ((j > 0) .and. (i < mx - 1)) then
        v(k) = hbr
        col(k) = row - gxm + 1
        k = k + 1
      end if

      if (i > 0) then
        v(k) = hl
        col(k) = row - 1
        k = k + 1
      end if

      v(k) = hc
      col(k) = row
      k = k + 1

      if (i < mx - 1) then
        v(k) = hr
        col(k) = row + 1
        k = k + 1
      end if

      if ((i > 0) .and. (j < my - 1)) then
        v(k) = htl
        col(k) = row + gxm - 1
        k = k + 1
      end if

      if (j < my - 1) then
        v(k) = ht
        col(k) = row + gxm
        k = k + 1
      end if

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

    end do
  end do

! restore vectors
  PetscCall(VecRestoreArray(localX, x_v, ierr))
  PetscCall(VecRestoreArray(Left, left_v, ierr))
  PetscCall(VecRestoreArray(Right, right_v, ierr))
  PetscCall(VecRestoreArray(Top, top_v, ierr))
  PetscCall(VecRestoreArray(Bottom, bottom_v, ierr))

! Assemble the matrix
  PetscCall(MatAssemblyBegin(Hessian, MAT_FINAL_ASSEMBLY, ierr))
  PetscCall(MatAssemblyEnd(Hessian, MAT_FINAL_ASSEMBLY, ierr))

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

end

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

! ----------------------------------------------------------------------------
!
!/*
!     MSA_BoundaryConditions - calculates the boundary conditions for the region
!
!
!*/

subroutine MSA_BoundaryConditions(ierr)
  use plate2fmodule
  implicit none

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

  logical exitloop
  PetscBool flg

  limit = 0
  maxits = 5
  tol = 1e-10
  b = -0.5
  t = 0.5
  l = -0.5
  r = 0.5
  xt = 0
  yt = 0
  one = 1.0
  two = 2.0
  three = 3.0

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

  bsize = xm + 2
  lsize = ym + 2
  rsize = ym + 2
  tsize = xm + 2

  PetscCall(VecCreateMPI(PETSC_COMM_WORLD, bsize, PETSC_DECIDE, Bottom, ierr))
  PetscCall(VecCreateMPI(PETSC_COMM_WORLD, tsize, PETSC_DECIDE, Top, ierr))
  PetscCall(VecCreateMPI(PETSC_COMM_WORLD, lsize, PETSC_DECIDE, Left, ierr))
  PetscCall(VecCreateMPI(PETSC_COMM_WORLD, rsize, PETSC_DECIDE, Right, ierr))

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

  do j = 0, 3

    if (j == 0) then
      yt = b
      xt = l + hx*xs
      limit = bsize
      PetscCall(VecGetArray(Bottom, boundary_v, ierr))

    elseif (j == 1) then
      yt = t
      xt = l + hx*xs
      limit = tsize
      PetscCall(VecGetArray(Top, boundary_v, ierr))

    elseif (j == 2) then
      yt = b + hy*ys
      xt = l
      limit = lsize
      PetscCall(VecGetArray(Left, boundary_v, ierr))

    elseif (j == 3) then
      yt = b + hy*ys
      xt = r
      limit = rsize
      PetscCall(VecGetArray(Right, boundary_v, ierr))
    end if

    do i = 0, limit - 1

      u1 = xt
      u2 = -yt
      k = 0
      exitloop = .false.
      do while (k < maxits .and. (.not. exitloop))

        nf1 = u1 + u1*u2*u2 - u1*u1*u1/three - xt
        nf2 = -u2 - u1*u1*u2 + u2*u2*u2/three - yt
        fnorm = sqrt(nf1*nf1 + nf2*nf2)
        if (fnorm > tol) then
          njac11 = one + u2*u2 - u1*u1
          njac12 = two*u1*u2
          njac21 = -two*u1*u2
          njac22 = -one - u1*u1 + u2*u2
          det = njac11*njac22 - njac21*njac12
          u1 = u1 - (njac22*nf1 - njac12*nf2)/det
          u2 = u2 - (njac11*nf2 - njac21*nf1)/det
        else
          exitloop = .true.
        end if
        k = k + 1
      end do

      boundary_v(1 + i) = u1*u1 - u2*u2
      if ((j == 0) .or. (j == 1)) then
        xt = xt + hx
      else
        yt = yt + hy
      end if

    end do

    if (j == 0) then
      PetscCall(VecRestoreArray(Bottom, boundary_v, ierr))
    elseif (j == 1) then
      PetscCall(VecRestoreArray(Top, boundary_v, ierr))
    elseif (j == 2) then
      PetscCall(VecRestoreArray(Left, boundary_v, ierr))
    elseif (j == 3) then
      PetscCall(VecRestoreArray(Right, boundary_v, ierr))
    end if

  end do

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

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

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

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

end

! ----------------------------------------------------------------------------
!
!/*
!     MSA_Plate - Calculates an obstacle for surface to stretch over
!
!     Output Parameter:
!.    xl - lower bound vector
!.    xu - upper bound vector
!
!*/

subroutine MSA_Plate(ta, xl, xu, dummy, ierr)
  use plate2fmodule
  implicit none

  Tao ta
  Vec xl, xu
  PetscErrorCode ierr
  PetscInt i, j, row
  PetscInt xs, xm, ys, ym
  PetscReal lb, ub
  PetscInt dummy
  PetscReal, pointer :: xl_v(:)

  lb = PETSC_NINFINITY
  ub = PETSC_INFINITY

  if (bmy < 0) bmy = 0
  if (bmy > my) bmy = my
  if (bmx < 0) bmx = 0
  if (bmx > mx) bmx = mx

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

  PetscCall(VecSet(xl, lb, ierr))
  PetscCall(VecSet(xu, ub, ierr))

  PetscCall(VecGetArray(xl, xl_v, ierr))

  do i = xs, xs + xm - 1

    do j = ys, ys + ym - 1

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

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

      end if

    end do
  end do

  PetscCall(VecRestoreArray(xl, xl_v, ierr))

end

! ----------------------------------------------------------------------------
!
!/*
!     MSA_InitialPoint - Calculates an obstacle for surface to stretch over
!
!     Output Parameter:
!.    X - vector for initial guess
!
!*/

subroutine MSA_InitialPoint(X, ierr)
  use plate2fmodule
  implicit none

  Vec X
  PetscErrorCode ierr
  PetscInt start, i, j
  PetscInt row
  PetscInt xs, xm, gxs, gxm
  PetscInt ys, ym, gys, gym
  PetscReal zero, np5

  PetscReal, pointer :: left_v(:), right_v(:)
  PetscReal, pointer :: bottom_v(:), top_v(:)
  PetscReal, pointer :: x_v(:)
  PetscBool flg
  PetscRandom rctx

  zero = 0.0
  np5 = -0.5

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

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

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

    PetscCall(PetscRandomDestroy(rctx, ierr))
    PetscCall(VecShift(X, np5, ierr))

  else   ! average of boundary conditions

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

!        Get pointers to vector data
    PetscCall(VecGetArray(Top, top_v, ierr))
    PetscCall(VecGetArray(Bottom, bottom_v, ierr))
    PetscCall(VecGetArray(Left, left_v, ierr))
    PetscCall(VecGetArray(Right, right_v, ierr))

    PetscCall(VecGetArray(localX, x_v, ierr))

!        Perform local computations
    do j = ys, ys + ym - 1
      do i = xs, xs + xm - 1
        row = (j - gys)*gxm + (i - gxs)
        x_v(1 + row) = ((j + 1)*bottom_v(1 + i - xs + 1)/my + (my - j + 1)*top_v(1 + i - xs + 1)/(my + 2) +                  &
&                          (i + 1)*left_v(1 + j - ys + 1)/mx + (mx - i + 1)*right_v(1 + j - ys + 1)/(mx + 2))*0.5
      end do
    end do

!        Restore vectors
    PetscCall(VecRestoreArray(localX, x_v, ierr))

    PetscCall(VecRestoreArray(Left, left_v, ierr))
    PetscCall(VecRestoreArray(Top, top_v, ierr))
    PetscCall(VecRestoreArray(Bottom, bottom_v, ierr))
    PetscCall(VecRestoreArray(Right, right_v, ierr))

    PetscCall(DMLocalToGlobalBegin(dm, localX, INSERT_VALUES, X, ierr))
    PetscCall(DMLocalToGlobalEnd(dm, localX, INSERT_VALUES, X, ierr))

  end if

end

!
!/*TEST
!
!   build:
!      requires: !complex
!
!   test:
!      args: -tao_monitor_short -mx 8 -my 6 -bmx 3 -bmy 3 -bheight 0.2 -tao_type bqnls -tao_gatol 1.e-4
!      filter: sort -b
!      filter_output: sort -b
!      requires: !single
!
!   test:
!      suffix: 2
!      nsize: 2
!      args: -tao_monitor_short -mx 8 -my 6 -bmx 3 -bmy 3 -bheight 0.2 -tao_type bqnls -tao_gatol 1.e-4
!      filter: sort -b
!      filter_output: sort -b
!      requires: !single
!
!TEST*/