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1 I

2 MPI OpenMP

4 gnuplot Xming Tera term

5 cp -r /tmp/ chmod

10 0

12 L x L y 0 k

13 T (x, t) k: T t = k 2 T x 2

14 T t = s s : heat source

15 1D T (x, t) t = k 2 T (x, t) x 2 + s(x) 2D T (x, y, t) t = k { 2 } T (x, y, t) x T (x, y, t) y 2 + s(x, y)

16 T (x, y, t) t 2 : ( 2 = k T (x, y, t) t x y 2 ) T (x, y, t) + s(x, y) = k 2 T (x, y, t) + s(x, y)

17 (0, 0) (x, y) (L x, L y ) T (0, y) = T (L x, y) = T (x, 0) = T (x, L y ) = 0 T (x, y, t) t T t = 0 T (x, y) = k 2 T (x, y, t) + s(x, y)

20 T (x, y, t) t = k 2 T (x, y, t) + s(x, y)

21 T(t+Δt) T(t) T t t t+δt T (t + t) T (t) t 1 1 1/ t

22 2 T (x i, y j ) x 2 T (x i+1, y j ) 2 T (x i, y j ) + T (x i 1, y j ) x 2 2 T (x i, y j ) y 2 T (x i, y j+1 ) 2 T (x i, y j ) + T (x i, y j 1 ) y 2

24 T (x, y, t) t ( 2 = k T (x i, y j, t) T i,j (t) x y 2 ) T (x, y, t) + s(x, y) T i,j (t + t) T i,j (t) t ( Ti+1,j (t) 2 T i,j (t) + T i 1,j (t) = k x 2 + T ) i,j+1(t) 2 T i,j (t) + T i,j 1 (t) y 2 +s i,j

25 T (x i+1,y j ) 2 T (x i,y j )+T (x i 1,y j ) x 2, T (x i,y j+1 ) 2 T (x i,y j )+T (x i,y j 1 ) y 2

26 T i,j (t + t) = T i,j (t) + k t x 2 {T i+1,j(t) 2 T i,j (t) + T i 1,j (t)} + k t y 2 {T i,j+1(t) 2 T i,j (t) + T i,j 1 (t)} + t s i,j t =

27 { k T i,j (t + t) = t x 2 (T i+1,j + T i 1,j ) + k } y 2 (T i,j+1 + T i,j 1 ) + s i,j { ( k + 1 t 2 x 2 + k )} y 2 T i,j T i,j t

28 t = α x = α y = k x 2 k y 2 1 2(α x + α y )

29 T i,j (t + t) = t {α x (T i+1,j (t) + T i 1,j (t)) +α y (T i,j+1 (t) + T i,j 1 (t)) +s i,j } t Jacobi

30 k = 1 s(x, y) = 4 x = y h α x = α y = 1/h 2, t = h 2 /4 T i,j (t + t) = T i+1,j(t) + T i 1,j (t) + T i,j+1 (t) + T i,j 1 (t) 4 + h 2

31 T i,j (t + t) = T i+1,j(t) + T i 1,j (t) + T i,j+1 (t) + T i,j 1 (t) 4 + h 2

32 T i,j (t + t) = T i+1,j(t) + T i 1,j (t) + T i,j+1 (t) + T i,j 1 (t) 4 + h 2

34 : heat2.f90 100

35 : heat2.f90 do n=1, LOOP_MAX! do j=1, NGRID do i=1, NGRID un(i,j)=(u(i-1,j)+u(i+1,j)+...! end do end do u...= un...! un u end do

36 heat2.f90 π computer : gfortran heat2.f90 &&./a.out > heat2.data gnuplot heat2.gp

38 1 x y 1 MPI sendrecv y

39 1 y x

40 1 y(j) コードのコメント y(j) x(i) x(i)

41 : heat3.f90!! When NGRID=11, nprocs=3!! j= ! i= ! 1! 2! 3!.!.!.!.! 11! ! rank0 rank2! jstt-----jend jstt jend! rank1! jstt jend

42 heat3.f90 MPI mpifrtpx heat3.f90 pjsub -o heat3.data heat3.sh heat3 bug.f90 heat3.f90 diff heat3_bug.f90 heat3b.f90 mail kage 7/27 23:59

43 heat3.sh #!/bin/bash #PJM -N "heat3" #PJM -L "rscgrp=small" #PJM -L "node=4" #PJM -L "elapse=02:00" #PJM -j drawline() { echo "#"{1..50} sed s/[ 0-9]//g } drawline mpiexec./a.out drawline

44 heat3.data ################################################## # myrank= 1 jstart & jend = # myrank= 2 jstart & jend = # myrank= 3 jstart & jend = # myrank= 0 jstart & jend = E E

45 heat3.data more / less / head / tail gnuplot gnuplot> plot heat3.data w lp gnuplot heat3.gp

47 heat3.f90 gnuplot y=0.5 x i x i x x i = h (i n mid )

48 data../data

49 heat3.f90 heat3 print final x prof.f90 diff heat3.f90 heat3 print final x prof.f90! heat3_print_final_x_prof.f90! + open/close file 10! + print out cross section 1D data.! + integer jcut (cross section for output)! + function this_process_has()! usage (on pi-computer)! 1) mkdir../data (unless there is already.)! 2) mpifrtpx heat3_print_final_x_prof.f90! 3) pjsub heat3.sh (share the jobscript with heat3.f90.

50 (1) heat3 print final x prof.f90 - mpifrtpx heat3 print final x prof.f90 - pjsub heat3.sh (2)../data/ temp.final profile x (3) more / less / head / tail (4) gnuplot (5) plot "../data/temp.final_profile_x" w lp 2 gnuplot heat3_print_final_x_prof.gp OK.

51 gnuplot src/heat3 print final x prof.gp # heat3_print_final_x_prof.gp # # final temperature profile at y=0.5 as a function of x # set xrange [-0.5:0.5] set yrange [0:0.5] set xlabel "x" set ylabel "temp at y=0.5" plot "../data/temp.final_profile_x" w lp pause -1

53 heat3 print x prof for animation.f90 heat3.sh data ls -l

54 # # gnuplot script generated by heat3_animation_x_prof_gp_ge # set xlabel "x" # x-axis set ylabel "temperature" # y-axis set xrange [-0.5:0.5] # x-coordinate set yrange [0.0:0.5] # temperature min & max plot "../data/temp.j=middle.0000" w lp pause 5 plot "../data/temp.j=middle.0001" w lp pause 1 plot "../data/temp.j=middle.0002" w lp pause 1..

55 1 heat3 print x prof for animation plotscript generator.f90 NGRID counter end gfortran heat3 print x prof for animation plotscript generator.f90./a.out > automatically generated.gp automatically generated.gp gnuplot automatically generated.gp

56 2

57 diff heat3 print x prof for animation.f90 heat4 print final 2d prof.f90! heat4_print_final_2d_prof.f90! + subroutine print profile_2d! c module constants --> module common! + type ranks_t :: p! + type span_t :: jj! - myrank, nprocs, left, right (combined into "p")! - jstart, jend (combined into "jj")! + function adjust_jstart_and_jend! + function set_prof_2d

58 2D x,y subroutine print profile_2d(p,jj,f) type(ranks_t), intent(in) :: p type(span_t), intent(in) :: jj real(dp), dimension(0:ngrid+1, & jj%stt-1:jj%end+1), intent(in) :: f real(dp), dimension(0:ngrid+1,0:ngrid+1) & :: f_global! 2d prof to be saved integer :: counter = 0! has sav type(span_t) :: jj2! used fo character(len=4) :: serial_num! put on character(len=*), parameter :: base = "../data/temp.2d." integer :: i, j

59 2D jj2 = adjust_jstart_and_jend(p,jj) write(serial_num, (i4.4) ) counter f_global(:,:) = set_prof_2d(jj,jj2,f) if ( p%myrank==0 ) then open(10,file=base//serial_num) do j = 0, NGRID+1 do i = 0, NGRID+1 write(10,*) i, j, f_global(i,j) end do write(10,*)! gnuplot requires a blank line here. end do close(10) end if counter = counter + 1 end subroutine print profile_2d

60 heat4 print final 2d prof.f90../data/temp.2d.0000

61 E E E E-002../data temp.2d E E E E E

62 gnuplot heat4_plot_contour_lines # # A sample gnuplot script: heat4_plot_contour_lines.gp # # [ line contours ] # # set size square # same side lengths for x and y set size 0.65, 1 # same side lengths for x and y set xlabel "i" # x-axis set ylabel "j" # y-axis set xrange[0:50] # i-grid min & max set yrange[0:50] # j-grid min & max set nosurface # do not show surface plot unset ztics # do not show z-tics set contour base # enables contour lines set cntrparam levels 10 # draw 10 contours set view 0,0 # view from the due north set title "Temperature"

63 2 data/temp.2d.0000 gnuplot heat4 plot contour lines.gp gnuplot heat4 plot contour lines.gp

65 gnuplot

66 heat4 plot contour colors.gp # # A sample gnuplot script: heat4_plot_contour_colors.gp # # [ color contours ] # # set size square # same side lengths for x and y set size 0.65, 1 # same side lengths for x and y set xlabel "i" # x-axis set ylabel "j" # y-axis set xrange[0:50] # i-grid min & max set yrange[0:50] # j-grid min & max set palette defined (0 "blue", 0.15 "red", 0.3 "yellow") set nosurface # do not show surface plot unset ztics # do not show z-tics set pm3d at b # draw with colored contour set view 0,0 # view from the due north set title "Temperature " splot "../data/temp.2d.0000" using 1:2:3

68 gnuplot 2 T(x,y) (z) height plot bird s eye view

69 gnuplot plot4_plot_birdseyeview.gp # # a sample gnuplot script: plot4_plot_birdseyeview.gp # # [ Bird"s Eye View ] # # set size square # same side lengths for x and y set size 0.65, 1 set xlabel "i" # x-axis set ylabel "j" # y-axis set xrange[0:50] # i-grid min & max set yrange[0:50] # j-grid min & max set contour base # enables contour lines set cntrparam levels 10 # draw 10 contours # set palette defined (0 "blue", 0.15 "red", 0.3 "yellow") # set pm3d # draw with colored contour set title "Temperature " splot "../data/temp.2d.0000" using 1:2:3 w l

71 gnuplot gnuplot view

72 gnuplot heat4 plot rotating birdseyeview generator.f90

74 #PJM -L "rscgrp=small" π-computer small: 1 12 medium: 1 48 large: = 1344 large

76 MPI + OpenMP = MPI MPI OpenMP do-loop OpenMP π-computer 1 16 fork

Join Fork & Join OpenMP スレッド 00 OpenMP スレッド 01 OpenMP スレッド 15 プロセッサ 1 OpenMP スレッド 00

78 プロセッサ 0 プロセッサ 0 プロセッサ 0 コア 00 コア 01 コア 15 プロセッサ 1 コア 00 コア 01 コア 15 プロセッサ N-1 コア 00 コア 01 コア 15 MPI プロセス 0 プロセッサ 1 MPI プロセス 1 プロセッサ N-1 MPI プロセス N-1 Fork & Join Fork & Join Fork & Join OpenMP スレッド 00 OpenMP スレッド 01 OpenMP スレッド 15 プロセッサ 1 OpenMP スレッド 00 OpenMP スレッド 01 OpenMP スレッド 15 プロセッサ N-1 OpenMP スレッド 00 OpenMP スレッド 01 OpenMP スレッド 15

80 CPU Fortran95 cpu time() (wall clock time) MPI MPI WTIME() OpenMP omp get wtime() Fortran90 system clock()

81 heat5.f90 heat4...f90 system clock() stopwatch m!! heat5.f90! + module stopwatch, to monitor time.! + many calls to stopwatch stt and.. stp.! - data output calls for profile 1d and 2d (commented out.)!! usage (on pi-computer)!! 1) mkdir../data (unless there is already.)!! 2) mpifrtpx -O3 heat5.f90 (copy un to u is slow in defaul!! 3) pjsub heat5.sh

82 heat5.f90 usage -O3 stopwatch copy un to u u(1:ngrid,jj%stt:jj%end)=un(1:ngrid,jj%stt:jj%end)

83 ################################################## job start at Tue Jul 16 21:07:29 JST 2013 ################################################## # myrank= 3 jj%stt & jj%end = # myrank= 0 jj%stt & jj%end = # myrank= 2 jj%stt & jj%end = # myrank= 1 jj%stt & jj%end = //=============<stop watch>===============\\ profile 1d: sec main loop: sec mpi sendrecv: sec jacobi: sec copy un to u: sec Total: sec \\=============<stop watch>===============// ################################################## job end at Tue Jul 16 21:07:39 JST 2013

84 heat5.f90 do-loop OpenMP heat6.f90 heat5.f90 u(1:ngrid,jj%stt:jj%end)=un(1:ngrid,jj%stt:jj%end) omp parallel do OpenMP 2 do loop

85 mpifrtpx -Kopenmp heat6.f90 heat6.sh medium pjsub heat6.sh

86 heat6.sh #!/bin/bash #PJM -N "heat6" #PJM -L "rscgrp=small" #PJM -L "node=4" #PJM -L "elapse=02:00" #PJM -j export FLIB_CNTL_BARRIER_ERR=FALSE.. for opn in do export OMP_NUM_THREADS=$opn echo "# omp_num_threads = " $opn mpiexec -n 4./a.out done.

87 heat6.f90 1 M ( 16) N ( 84) P (= M N) v.s. S S stopwatch module Total gnuplot NGRID

2 A I / 58

2 A I / 58 2 A 2018.07.12 I 2 2018.07.12 1 / 58 I 2 2018.07.12 2 / 58 π-computer gnuplot 5/31 1 π-computer -X ssh π-computer gnuplot I 2 2018.07.12 3 / 58 gnuplot> gnuplot> plot sin(x) I 2 2018.07.12 4 / 58 cp -r