現在(2014-06-11 (水) 22:33:33)作成中です。 既に書いている内容も大幅に変わる可能性が高いので注意。


神戸大学 大学院システム情報学研究科 計算科学専攻 陰山 聡



前回のレポート(並列化)の解説

問題設定の復習

#ref(): File not found: "problem_thermal.jpg" at page "6.データ可視化"

  • 2次元正方形領域 [0,1]×[0,1] での熱伝導を考える
  • 境界をすべて0℃に固定
  • 領域全体に一定の熱を加える
  • ⇒ このとき,十分な時間が経った後での温度分布はどうなるか?

非並列版コードの復習

熱伝導とは注目する一点が、「近所」の温度と等しくなろうという傾向である。 ⇒周囲の温度の平均値になろうとする。 熱源があれば、一定の割合で温度が上がろうとする。

ヤコビ法アルゴリズム。

#ref(): File not found: "jacobi_method.jpg" at page "6.データ可視化"

do j=1, m
    do i=1, m
        uij(n+1) = (ui-1,j(n) + ui+1,j(n) + ui,j-1(n) + ui,j+1(n) ) / 4 + fij
    end do
end do

このヤコビ法に基づいたコード heat1.f90を前回の演習で見た。

heat1_wi_comments.f90 (詳細コメント付きバージョン)

今回は、まずこのheat1.f90を少し丁寧に見る。

! 
! heat1_wi_comments.f90
!
!-----------------------------------------------------------------------
! Time development form of the thermal diffusion equation is
!    \partial T(x,y,t) / \partial t = \nabla^2 T(x,y,t) + heat_source.
! In the stationary state, 
!    \nabla^2 T(x,y) + heat_source = 0.
! The finite difference method with grid spacings dx and dy leads to
!    (T(i+1,j)-2*T(i,j)+T(i-1,j))/dx^2 
!  + (T(i,j+1)-2*T(i,j)+T(i,j-1))/dy^2 + heat_source = 0.
! When dx=dy=h,
!    T(i,j) = (T(i+1,j)+T(i-1,j)+T(i,j+1)+T(i,j-1))/4+heat_source*h^2/4.
! This suggests a relaxation method called Jacobi method adopted in
! this code.
!-----------------------------------------------------------------------

! 
! heat1_wi_comments.f90
!
!-----------------------------------------------------------------------
! Time development form of the thermal diffusion equation is
!    \partial T(x,y,t) / \partial t = \nabla^2 T(x,y,t) + heat_source.
! In the stationary state, 
!    \nabla^2 T(x,y) + heat_source = 0.
! The finite difference method with grid spacings dx and dy leads to
!    (T(i+1,j)-2*T(i,j)+T(i-1,j))/dx^2 
!  + (T(i,j+1)-2*T(i,j)+T(i,j-1))/dy^2 + heat_source = 0.
! When dx=dy=h,
!    T(i,j) = (T(i+1,j)+T(i-1,j)+T(i,j+1)+T(i,j-1))/4+heat_source*h^2/4.
! This suggests a relaxation method called Jacobi method adopted in
! this code.
!-----------------------------------------------------------------------

program heat1_wi_comments
  implicit none
  integer, parameter :: m=31            ! mesh size
  integer            :: nmax=20000      ! max loop counter
  integer :: i,j,n
  integer, parameter :: SP = kind(1.0)
  integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
  real(DP), dimension(:,:), allocatable :: u    ! temperature field
  real(DP), dimension(:,:), allocatable :: un   ! work array
  real(DP) :: heat=1.0_DP       ! heat source term; uniform distribution.
  real(DP) :: h                 ! grid size

  !                    |<----- 1.0 ----->|
  !               j=m+1+-------u=0-------+  ---
  !               j=m  |                 |   ^
  !                .   |                 |   | 
  !                .   |     uiform      |   | 
  !                   u=0    heat       u=0 1.0   
  ! j-direction    .   |     source      |   | 
  !(y-direction)  j=3  |                 |   | 
  !     ^         j=2  |                 |   v 
  !     |         j=1  +-------u=0-------+  ---
  !     |             i=0 1 2 ...      i=m+1
  !     |
  !     +-------> i-direction (x-direction)
  !
  ! when m = 7, 
  !       |<-------------------- 1.0 -------------------->|
  !       |                                               |
  !       |  h     h     h     h     h     h     h     h  |
  !       |<--->|<--->|<--->|<--->|<--->|<--->|<--->|<--->|
  !       +-----+-----+-----+-----+-----+-----+-----+-----+
  !     i=0     1     2     3     4     5     6     7     8 
  !

  if ( mod(m,2)==0 ) then
     print *, 'm must be odd to have a grid on the center.'
     stop
  end if

  allocate( u(0:m+1,0:m+1) )    ! memory allocation of 2-D array.
                                ! you can access each element of u
                                ! as u(i,j), where 0<=i<=m+1 
                                ! and  0<=j<=m+1.

  allocate( un(m,m) )           ! another memory allocation. in
                                ! this case un(i,j) with
                                ! 1<=i<=m and 1<=j<=m.

  h = 1.0_DP/(m+1)              ! grid spacing.

  u(:,:) = 0.0_DP  ! initial temperature is zero all over the square.

  do n = 1 , nmax     ! relaxation loop 
    do j = 1 , m      ! in the y-direction
      do i = 1 , m
        un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
      end do
    end do                      ! same as the following do-loops:
    u(1:m,1:m) = un(1:m,1:m)    ! do j = 1 , m
                                !   do i = 1 , m
                                !     u(i,j) = un(i,j)
                                !   end do
                                ! end do
    if ( mod(n,100)==0 ) print *, n, u(m/2+1,m/2+1)  ! temperature at
  end do                                             ! the center
end program heat1_wi_comments

再びheat1.f90(コメントなしコード)

次に詳しいコメントをはずしてもう一度もとのheat1.f90を見てみる。 (ただし、先週のコードから本質的でないところを少しだけ修正している。)

program heat1
  implicit none
  integer, parameter :: m=31, nmax=20000
  integer :: i,j,n
  integer, parameter :: SP = kind(1.0)
  integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
  real(DP), dimension(:,:), allocatable :: u, un
  real(DP) :: h, heat=1.0_DP

  if ( mod(m,2)==0 ) then
     print *, 'm must be odd to have a grid on the center.'
     stop
  end if

  allocate(u(0:m+1,0:m+1))
  allocate(un(m,m))

  h=1.0_DP/(m+1)
  u(:,:) = 0.0_DP

  do n=1, nmax
    do j=1, m
      do i=1, m
        un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
      end do
    end do
    u(1:m,1:m)=un(1:m,1:m)
    if (mod(n,100)==0) print *, n, u(m/2+1,m/2+1)
  end do
end program heat1

先週の課題(演習3-3)

  • heat1.f90 を MPI を用いて並列化せよ

解答例(heat2.f90)

program heat2
  use mpi
  implicit none
  integer, parameter :: m=31, nmax=20000
  integer :: i,j,jstart,jend,n
  integer, parameter :: SP = kind(1.0)
  integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
  real(DP), dimension(:,:), allocatable :: u, un
  real(DP) :: h, heat=1.0_DP
  integer ::  nprocs,myrank,ierr,left,right
  integer, dimension(MPI_STATUS_SIZE) :: istat

  if ( mod(m,2)==0 ) then
     print *, 'm must be odd to have a grid on the center.'
     stop
  end if

  call mpi_init(ierr)
  call mpi_comm_size(MPI_COMM_WORLD,nprocs,ierr)
  call mpi_comm_rank(MPI_COMM_WORLD,myrank,ierr)

  jstart=m*myrank/nprocs+1
  jend=m*(myrank+1)/nprocs
  allocate(u(0:m+1,jstart-1:jend+1))
  allocate(un(m,jstart:jend))
!
! when m = 7, nprocs = 3
! 
!     |<-------------------- 1.0 -------------------->|      
!     |                                               |
!     |  h  |  h     h     h     h     h     h     h  |
!     |<--->|<--->|<--->|<--->|<--->|<--->|<--->|<--->|
!     |     |     |     |     |     |     |     |     |
!     +-----+-----+-----+-----+-----+-----+-----+-----+
!   j=0     1     2     3     4     5     6     7     8
!     +-----+-----+-----+-----+-----+-----+-----+-----+
!     |  jstart   |     |           |     |     |     | 
!     |     |   jend    |           |   jstart  |     | 
!     o-----rank=0------o           |     |    jend   |
!                 |     |           |     |     |     |
!                 |   jstart      jend    |     |     |
!                 |     |           |     |     |     |
!                 o------- rank=1---------o     |     |
!                                   |     |     |     | 
!                                   o------rank=3-----o
!
  h = 1.0_DP/(m+1)
  u(:,:) = 0.0_DP

  left=myrank-1
  if (myrank==0) left=nprocs-1
  right=myrank+1
  if (myrank==nprocs-1) right=0

  do n=1, nmax
    call mpi_sendrecv(u(1,jend),m,MPI_DOUBLE_PRECISION,right,100, &
&                     u(1,jstart-1),m,MPI_DOUBLE_PRECISION,left,100, &
&                     MPI_COMM_WORLD,istat,ierr)
    call mpi_sendrecv(u(1,jstart),m,MPI_DOUBLE_PRECISION,left,100, &
&                     u(1,jend+1),m,MPI_DOUBLE_PRECISION,right,100, &
&                     MPI_COMM_WORLD,istat,ierr)

    if (myrank==0) u(1:m,0)=0.0_DP
    if (myrank==nprocs-1) u(1:m,m+1)=0.0_DP

    do j=jstart, jend
      do i=1, m
        un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
      end do
    end do
    u(1:m,jstart:jend)=un(1:m,jstart:jend)

    if (jstart<=m/2+1 .and. jend>=m/2+1) then
      if (mod(n,100)==0) print *, n, u(m/2+1,m/2+1)
    end if
  end do
  call mpi_finalize(ierr)
end program heat2

コード解説 (コメント付きソースコード heat2_wi_comments.f90)

上のheat2.f90を解説するために詳しいコメントをつけたソースコードheat2_wi_comments.f90を以下に示す。

program heat2_wi_comments
  use mpi
  implicit none
  integer, parameter :: m=31            ! mesh size
  integer, parameter :: nmax=20000      ! max loop counter
  integer :: i,j,jstart,jend,n
  integer, parameter :: SP = kind(1.0)
  integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
  real(DP), dimension(:,:), allocatable :: u  ! temperature field
  real(DP), dimension(:,:), allocatable :: un ! work array
  real(DP) :: heat=1.0_DP   ! heat source term. uniform distribution.
  real(DP) :: h             ! grid spacing; dx=dy=h
  integer  :: nprocs        ! number of MPI processes
  integer  :: myrank        ! my rank number
  integer  :: left, right   ! nearest neighbor processes
  integer, dimension(MPI_STATUS_SIZE) :: istat ! used for MPI
  integer  :: ierr          ! used for mpi routines

  if ( mod(m,2)==0 ) then   ! to print out the centeral temperature.
     print *, 'm must be odd to have a grid on the center.'
     stop
  end if

  call mpi_init(ierr)
  call mpi_comm_size(MPI_COMM_WORLD,nprocs,ierr)
  call mpi_comm_rank(MPI_COMM_WORLD,myrank,ierr)

  h = 1.0_DP/(m+1)          ! grid spacings. see the comment fig below.

  jstart = m*myrank/nprocs+1   ! you are in charge of this point
  jend   = m*(myrank+1)/nprocs ! ...to this points. see the fig below.
!
! when m = 7, nprocs = 3
! 
!     |<-------------------- 1.0 -------------------->|      
!     |                                               |
!     |  h  |  h     h     h     h     h     h     h  |
!     |<--->|<--->|<--->|<--->|<--->|<--->|<--->|<--->|
!     |     |     |     |     |     |     |     |     |
!     +-----+-----+-----+-----+-----+-----+-----+-----+
!   j=0     1     2     3     4     5     6     7     8
!     +-----+-----+-----+-----+-----+-----+-----+-----+
!     |  jstart   |     |           |     |     |     | 
!     |     |   jend    |           |   jstart  |     | 
!     o-----rank=0------o           |     |    jend   |
!                 |     |           |     |     |     |
!                 |   jstart      jend    |     |     |
!                 |     |           |     |     |     |
!                 o------- rank=1---------o     |     |
!                                   |     |     |     | 
!                                   o------rank=3-----o
!

  allocate( u(0:m+1,jstart-1:jend+1)) ! 2-D array for temperature.
  allocate(un(m,jstart:jend))         ! 2-D work array.
  u(:,:) = 0.0_DP

  left  = myrank-1  ! left neighbor.
  if (myrank==0)        left = nprocs-1  ! a dummy periodic distribution
  right = myrank+1  ! right neighbor.    ! for code simplicity. this has
  if (myrank==nprocs-1) right = 0        ! actually no effect.

  do n = 1 , nmax   ! main loop for the relaxation, to equilibrium.
    call mpi_sendrecv(u(1,jend),    m,MPI_DOUBLE_PRECISION,right,100,   &
                      u(1,jstart-1),m,MPI_DOUBLE_PRECISION,left,100,    &
                      MPI_COMM_WORLD,istat,ierr)
    call mpi_sendrecv(u(1,jstart),m,MPI_DOUBLE_PRECISION,left, 100,     &
                      u(1,jend+1),m,MPI_DOUBLE_PRECISION,right,100,     &
                      MPI_COMM_WORLD,istat,ierr)
    if (myrank==0)        u(1:m,0  ) = 0.0_DP   ! to keep the boundary 
    if (myrank==nprocs-1) u(1:m,m+1) = 0.0_DP   ! condition (temp=0).

    do j = jstart , jend   ! Jacobi method. 
      do i = 1 , m         ! no need to calculate the boundary i=0 and m+1.
        un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
      end do
    end do
    u(1:m,jstart:jend)=un(1:m,jstart:jend) ! this is actually doubled do-loops.

    if (jstart<=m/2+1 .and. jend>=m/2+1) then         ! print out the temperature
      if (mod(n,100)==0) print *, n, u(m/2+1,m/2+1)   ! at the central grid point.
    end if
  end do

  deallocate(un,u)        ! or you can call deallocate for two times for un and u.
  call mpi_finalize(ierr)    
end program heat2_wi_comments

可視化の必要性

可視化とは

ここではスカラーデータの可視化のみ。

1次元グラフ

2次元等高線アルゴリズム

3次元等値面アルゴリズム

コードのリファクタリング

今後の演習で行うコードの改訂

  • 1次元可視化機能の追加(中心点の温度の収束の様子を示す)
  • 2次元可視化機能の追加(温度場全体の収束の様子をアニメーションで示す)
  • 2次元領域分割による並列化
  • 問題の3次元化

gnuplot入門

1次元可視化(中心点の温度の収束の様子を示す)

【演習】

2次元可視化(温度場全体の収束の様子のアニメーション)

【演習】


as of 2019-08-25 (日) 19:03:38 (4738)