[021] 分子動力学プログラムの高速化

 投稿者：mike  投稿日：2017年 7月16日(日)10時21分52秒
  　Morseポテンシャルによる分子動力学法（2次元）の粒子登録法による高速化プログラム[015]のさらなる高速化
プログラム(021fasterMMD2D.bas)を公開します。
　単純な粒子登録法はO(N^2)なので、他の部分がO(N)でも粒子数Nが大きくなるにつれて粒子登録法の部分が効いてきて、
やがて、実行時間はO(N^2)に近づきます。
　粒子登録の前に、全体を格子状の領域に区分し、粒子に近い領域から近い粒子を登録することにより、粒子登録をO(N)に
することができます。

実行時間の実測：
decimal BASIC 6.6.3.3 /MacOS 10.7.5/ corei7(2.7GHz)
fast   :従来の単純な粒子登録法(EXTERNAL SUB registerNearMolec)
faster :前処理+粒子登録法(EXTERNAL SUB preRegistration, EXTERNAL SUB registration)
      Molec   fast    (time/N)   faster  (time/N)
Fe   N= 535    10.02s (0.0187)    10.00s (0.0187)
Fe   N= 952    18.03s (0.0189)    17.01s (0.0179)
Fe   N=2139    48.00s (0.0224)    36.04s (0.0168)
Fe   N=3802   104.61s (0.0275)    63.99s (0.0168)
Fe   N=7187   272.17s (0.0379)   120.93s (0.0168)

前処理付きの粒子登録法（faster）では、ほぼ粒子数Nに比例した実行時間になっています。

試験環境：
本プログラムは十進BASIC 6.6.3.3 / macOS 10.7.5 でテストしました。


-------------

!
! ========= molecular dynamics 2D - Morse potential ==========
!
! 021fasterMMD2D.bas
!   Copyright(C) 2017 Mitsuru Ikeuchi
!
! ver 0.0.1   2017.07.16  created
!
OPTION ARITHMETIC NATIVE
DECLARE EXTERNAL SUB mmd2d.setInitialCondition, mmd2d.moveParticles, mmd2d.drawParticles
DECLARE EXTERNAL FUNCTION INKEY$
LET tempMode = 0    !tempMode: 0:adiabatic  1:constant temperature
LET contTemp = 300  !contTemp: controled constant temperature(K)
LET drawMode = 1    !drawMode: 0:bond  1:circle+bond 2:velocitySpace


!setInitialCondition(molKind,boxSizeInNM,xtalSizeInNM,contTemp)
! decimal BASIC 6.6.2.2 /MacOS 10.7.5/ corei7(2.7GHz)  nMolec   fast     faster
!CALL setInitialCondition(3, 8.0, 6.0,contTemp) !3:Fe  N= 535    10.02s   10.00s
!CALL setInitialCondition(3,12.0, 8.0,contTemp) !3:Fe  N= 952    18.03s   17.01s
CALL setInitialCondition(3,16.0,12.0,contTemp)  !3:Fe  N=2139    48.00s   36.04s
!CALL setInitialCondition(3,20.0,16.0,contTemp) !3:Fe  N=3802   104.61s   63.99s
!CALL setInitialCondition(3,24.0,22.0,contTemp) !3:Fe  N=7187   272.17s  120.93s

INPUT  PROMPT "choice 0:fast 1:faster ":a$
IF VAL(a$)=1 THEN
   LET fasterSW=1
   PRINT "faster mode"
ELSE
   LET fasterSW=0
   PRINT "fast mode"
END IF

LET t0 = TIME
FOR it=1 TO 20
   CALL moveParticles(tempMode,contTemp,fasterSW)
   CALL drawParticles(tempMode,contTemp,drawMode)
NEXT it
PRINT TIME-t0
stop

END

! ---------- molecular dynamics 2D - Morse potential ----------
!
!  method: velocity Verlet ( F=m*d^2r/dt^2 -> r(t+dt)=r(t)+v*dt,v=v+(F/m)*dt )
!    (1) vi = vi + (Fi/mi)*(0.5dt)
!    (2) ri = ri + vi*dt
!    (3) calculation Fi <- {r1,r2,...,rn} Fi=sum(Fij,j=1 to n),Fij=F(ri-rj)
!    (4) vi = vi + (Fi/mi)*(0.5dt)
!    (6) goto (1)
!  potential: Morse V(r) = D*((1-EXP(-A*(r-r0)))^2-1)
!                        = D*(EXP(-2*A*(r-r0))-2*EXP(-A*(r-r0)))
!    (D:dissociation energy, r0:bond length, A:width parameter { A=SQR(k/(2*D)) }
!             force F(r) = -dV(r)/dr
!                        = 2*D*A*y*(y-1), y=EXP(-A*(r-r0))
MODULE mmd2d
MODULE OPTION ARITHMETIC NATIVE
PUBLIC SUB setInitialCondition !(molKind,boxSizeInNM,xtalSizeInNM,contTemp)
PUBLIC SUB moveParticles !(tempMode,contTemp)
PUBLIC SUB drawParticles !(drawMode:0:realSpace 1:velocitySpace)
SHARE NUMERIC molecKind,sysTime,dt,nMolec,xMax,yMax, Nsx,Nsy, mass,Dmrs,Amrs,r0mrs, rCutoff, hh
SHARE NUMERIC xx(8000),yy(8000)           ! (xx(i),yy(i))  : position of i-th particle
SHARE NUMERIC vx(8000),vy(8000)           ! (vx(i),vy(i))  : velocity of i-th particle
SHARE NUMERIC ffx(8000),ffy(8000)         ! (ffx(i),ffy(i)): total force of i-th particle
SHARE NUMERIC reg(8000,0 TO 100)         ! register near molec reg(i,0):number of near i-th molec
SHARE NUMERIC section(0 TO 101,0 TO 101,0 TO 20) !use pre-registration
SHARE NUMERIC molecData(0 TO 18,0 TO 3) ! molecule 0:mass, 1:epsilon, 2:sigma, 3:dt
SHARE NUMERIC forceTable(0 TO 1200)     ! force table
SHARE STRING molecSTR$(0 TO 18)         ! molec string
LET molecKind = 3          ! 3:Fe
LET sysTime = 0.0          ! system time (s) in the module
LET dt = 5.0*1.0e-15       ! time step (s)
LET nMolec = 100           ! number of particles
LET xMax = 6.0E-9          ! x-Box size (m)
LET yMax = 6.0E-9          ! y-Box size (m)
LET Nsx = 80
LET Nsy = 80
LET mass = 55.847*1.661e-27! mass of Fe (kg)
LET Dmrs = 0.4174*1.602e-19! D of Morse potential (J)   : energy of dissociation
LET Amrs = 1.3885e10       ! A of Morse potential (1/m) : width parameter
LET r0mrs = 2.845e-10      ! r0 of Morse potential (m)  : bond length
LET rCutoff = 1.0e-9       ! force cutoff radius (m)
LET hh = 1e-12             ! force table step (m)
LET molecData(0,0)=0       ! if molecData(0,0)=0 then set molecData(,)

! ---------- set initial condition

EXTERNAL SUB setInitialCondition(molKind,boxSizeInNM,xtalSizeInNM,contTemp)
   DECLARE EXTERNAL SUB setMolecData,setMolecules,ajustVelocity,setForceTable
   DECLARE EXTERNAL FUNCTION setCrystalBlock
   RANDOMIZE
   CALL setMolecData
   LET molecKind = molKind
   LET mass = molecData(molecKind,0)
   LET Dmrs = molecData(molecKind,1)
   LET Amrs = molecData(molecKind,2)
   LET r0mrs = molecData(molecKind,3)
   LET sysTime = 0.0
   LET xMax = boxSizeInNM*1.0e-9
   LET yMax = boxSizeInNM*1.0e-9
   ! set particles
   LET s = 0.5*(boxSizeInNM-xtalSizeInNM)*1.0e-9
   LET nMolec = setCrystalBlock(1, s, s, xtalSizeInNM*1.0e-9, xtalSizeInNM*1.0e-9, PI/4)
   CALL ajustVelocity(contTemp)
   LET rCutoff = MIN(1.0e-9, 3.0*r0mrs)
   CALL setForceTable
   ! set window
   SET WINDOW 0,500,0,500
END SUB

EXTERNAL SUB setMolecData
! Morse potential data
!       0:mass(AU) 1:D(eV)  2:A(m^-1)  3:r0(m)
   DATA   183.85 , 0.9906, 1.4116e10, 3.032e-10 !  0 W
   DATA    95.94 , 0.8032, 1.5079e10, 2.976e-10 !  1 Mo
   DATA    51.996, 0.4414, 1.5721e10, 2.754e-10 !  2 Cr
   DATA    55.847, 0.4174, 1.3885e10, 2.845e-10 !  3 Fe
   DATA    58.71 , 0.4205, 1.4199e10, 2.780e-10 !  4 Ni
   DATA    26.98 , 0.2703, 1.1646e10, 3.253e-10 !  5 Al
   DATA   207.19 , 0.2348, 1.1836e10, 3.733e-10 !  6 Pb
   DATA    63.54 , 0.3429, 1.3588e10, 2.866e-10 !  7 Cu
   DATA   107.87 , 0.3323, 1.3690e10, 3.115e-10 !  8 Ag
   DATA    39.948, 0.0104, 1.3400e10, 3.816e-10 !  9 Ar
   DATA   200.59 , 0.0734, 1.4900e10, 3.255e-10 ! 10 Hg
   DATA    40.08 , 0.1623, 0.8054e10, 4.569e-10 ! 11 Ca
   DATA    87.62 , 0.1513, 0.7878e10, 4.988e-10 ! 12 Sr
   DATA   137.34 , 0.1416, 0.6570e10, 5.373e-10 ! 13 Ba
   DATA    22.99 , 0.0633, 0.5900e10, 5.336e-10 ! 14 Na
   DATA    39.102, 0.0542, 0.4977e10, 6.369e-10 ! 15 K
   DATA    20.183, 0.0031, 1.6500e10, 3.076e-10 ! 16 Ne
   DATA    83.80 , 0.0141, 1.2500e10, 4.097e-10 ! 17 Kr
   DATA   131.30 , 0.0200, 1.2400e10, 4.467e-10 ! 18 Xe

   DATA "W" ,"Mo","Cr","Fe","Ni","Al","Pb","Cu","Ag","Ar","Hg"
   DATA "Ca","Sr","Ba","Na","K" ,"Ne","Kr","Xe"

   IF molecData(0,0)=0 THEN
      MAT READ molecData
      FOR i=0 TO 18
         LET molecData(i,0) = molecData(i,0)*1.661e-27 !mass(AU)--> (kg)
         LET molecData(i,1) = molecData(i,1)*1.602e-19 !eps(eV) --> (J)
      NEXT i
      MAT READ molecSTR$
   END IF
END SUB

EXTERNAL function setCrystalBlock(ii, x0, y0, xLen, yLen, theta)
   DECLARE EXTERNAL SUB setParticle
   LET iip = ii
   LET a = 0.98*r0mrs
   LET b = 0.866025*a
   LET leng = xLen
   IF (leng<yLen) THEN LET leng = yLen
   LET leng = 1.5*leng
   LET nx = INT(leng/b) + 1
   LET ny = INT(leng/a) + 1
   LET sth = SIN(theta)
   LET cth = COS(theta)
   FOR i=0 TO nx-1
      LET x = b*i - leng/2.0
      FOR j=0 TO ny-1
         LET y = a*j - leng/2.0
         IF MOD(i,2)=1 THEN LET y = y + 0.5*a
         LET xp = x0 + xLen/2.0 + cth*x - sth*y
         LET yp = y0 + yLen/2.0 + sth*x + cth*y
         IF (xp>=x0 AND xp<=x0+xLen AND yp>=y0 AND yp<=y0+yLen) THEN
            CALL setParticle(iip, xp, yp)
            LET iip = iip + 1
         END IF
      NEXT j
   NEXT i
   LET setCrystalBlock = iip - 1
end function

EXTERNAL SUB setParticle(i, x, y)
   LET xx(i) = x
   LET yy(i) = y
   LET vx(i) = 200.0*(RND+RND+RND+RND+RND+RND-3)
   LET vy(i) = 200.0*(RND+RND+RND+RND+RND+RND-3)
   LET ffx(i) = 0.0
   LET ffy(i) = 0.0
END SUB

EXTERNAL SUB setForceTable
   DECLARE EXTERNAL  FUNCTION cutoff
   FOR ir=10 TO 1200
      LET r = ir*hh
      LET y = EXP(-Amrs*(r-r0mrs))
      LET forceTable(ir) = cutoff(r)*2.0*Dmrs*Amrs*y*(y-1)
   NEXT ir
   FOR ir=0 TO 9
      LET forceTable(ir) = forceTable(10)
   NEXT ir
END SUB

EXTERNAL FUNCTION cutoff(r)
   IF r>0 AND r<0.8*rCutoff THEN
      LET ret = 1
   ELSEIF r>=0.8*rCutoff AND r<rCutoff THEN
      LET ret = 0.5+0.5*COS(PI*(r-0.8*rCutoff)/(0.2*rCutoff))
   else
      LET ret = 0
   END IF
   LET cutoff = ret
END FUNCTION

! ---------- move particles

EXTERNAL SUB moveParticles(tempMode,contTemp,fasterSW) !tempMode 0:adiabatic  1:constant-temp
   DECLARE EXTERNAL SUB moveParticlesDT,ajustVelocity,registerNearMolec,registration
   IF (tempMode=1) THEN CALL ajustVelocity(contTemp)
   IF fasterSW=1 THEN
      CALL registration !faster registration
   ELSE
      CALL registerNearMolec !fast registration
   END IF
   FOR i=1 TO 20
      CALL moveParticlesDT
   NEXT i
END SUB

EXTERNAL SUB moveParticlesDT ! velocity Verlet method
   DECLARE EXTERNAL SUB calcForce
   LET a = 0.5*dt/mass
   FOR i=1 TO nMolec
   !LET a = 0.5*dt/mass(i)
      LET vx(i) = vx(i)+a*ffx(i)
      LET vy(i) = vy(i)+a*ffy(i)
      LET xx(i) = xx(i)+vx(i)*dt
      LET yy(i) = yy(i)+vy(i)*dt
   NEXT i
   CALL calcForce
   FOR i=1 TO nMolec
   !LET a = 0.5*dt/mass(i)
      LET vx(i) = vx(i)+a*ffx(i)
      LET vy(i) = vy(i)+a*ffy(i)
   NEXT i
   LET sysTime=sysTime+dt
END SUB

EXTERNAL SUB calcForce
   DECLARE EXTERNAL FUNCTION force,boundaryForce
   LET s = 0.5*3.418e-10
   FOR i=1 TO nMolec
      LET ffx(i) = 0.0
      LET ffy(i) = 0.0
   NEXT i
   FOR i=1 TO nMolec-1
      FOR k=1 TO reg(i,0)-1
         LET j = reg(i,k)
         LET xij = xx(i)-xx(j)
         LET yij = yy(i)-yy(j)
         LET rij = SQR(xij*xij+yij*yij)
         IF rij<rCutoff THEN
            LET f = force(rij)
            LET fxij = f*xij/rij
            LET fyij = f*yij/rij
            LET ffx(i) = ffx(i)+fxij
            LET ffy(i) = ffy(i)+fyij
            LET ffx(j) = ffx(j)-fxij
            LET ffy(j) = ffy(j)-fyij
         END IF
      NEXT k
   NEXT i
   FOR i=1 TO nMolec  ! boundary force
      LET ffx(i) = ffx(i)+boundaryForce(xx(i)+s)+boundaryForce(xx(i)-xMax-s)
      LET ffy(i) = ffy(i)+boundaryForce(yy(i)+s)+boundaryForce(yy(i)-yMax-s)
   NEXT i
END SUB

!EXTERNAL FUNCTION force(r) ! force(r) = -dV(r)/dr
!   LET y = EXP(-Amrs*(r-r0mrs))
!   LET force = 2*Dmrs*Amrs*y*(y-1)
!END FUNCTION

EXTERNAL FUNCTION force(r) !force(r) <-- forceTable - linear interporation
   LET ir = INT(r/hh)
   LET a = r - ir*hh
   LET force = ((hh-a)*forceTable(ir) + a*forceTable(ir+1))/hh
END FUNCTION

EXTERNAL FUNCTION boundaryForce(r)
   LET r6 = (3.418e-10/r)^6
   LET boundaryForce = (24.0*(0.5*1.711e-21)*r6*(2.0*r6-1.0)/r)
END FUNCTION

EXTERNAL SUB registerNearMolec !fast registeration
   LET rCut = rCutoff+20*2000*dt
   LET rcut2 = rCut*rCut
   FOR i=1 TO nMolec-1
      LET k = 1
      FOR j=i+1 TO nMolec
         LET r2 = (xx(i)-xx(j))*(xx(i)-xx(j))+(yy(i)-yy(j))*(yy(i)-yy(j))
         IF (r2<rcut2) THEN
            LET reg(i,k) = j
            LET k = k + 1
         END IF
      NEXT j
      LET reg(i,0) = k
   NEXT i
END SUB

EXTERNAL FUNCTION maxNearMolec
   LET mx = 0
   FOR i=1 TO nMolec-1
      IF mx<reg(i,0) THEN LET mx = reg(i,0)
   NEXT i
   LET maxNearMolec = mx-1
END FUNCTION


EXTERNAL SUB registration !faster registration
   DECLARE EXTERNAL SUB preRegistration
   CALL preRegistration
   LET rreg = rCutoff+20*2000*dt
   LET rreg2 = rreg*rreg
   FOR ipp=1 TO nMolec-1
      LET kp = 1
      LET i0 = INT(Nsx*(xx(ipp)-rreg)/xMax)
      IF (i0<0) THEN LET i0 = 0
      LET i1 = INT(Nsx*(xx(ipp)+rreg)/xMax )
      IF (i1>=Nsx) THEN LET i1 = Nsx-1
      LET j0 = INT(Nsy*(yy(ipp)-rreg)/yMax )
      IF (j0<0) THEN LET j0 = 0
      LET j1 = INT(Nsy*(yy(ipp)+rreg)/yMax )
      IF (j1>=Nsy) THEN LET j1 = Nsy-1
      FOR i=i0 TO i1
         FOR j=j0 TO j1
            FOR iq=1 TO section(i,j,0)
               LET jp = section(i,j,iq)
               IF (jp>ipp) THEN
                  LET r2=(xx(ipp)-xx(jp))*(xx(ipp)-xx(jp))+(yy(ipp)-yy(jp))*(yy(ipp)-yy(jp))
                  IF (r2<rreg2) THEN
                     LET reg(ipp,kp) = jp
                     LET kp = kp + 1
                  END IF
               END IF
            NEXT iq
         NEXT j
      NEXT i
      LET reg(ipp,0) = kp
   NEXT ipp
END SUB

EXTERNAL SUB preRegistration
   FOR i=0 TO Nsx-1
      FOR j=0 TO Nsy-1
         LET section(i,j,0) = 0
      NEXT j
   NEXT i
   FOR ipp=1 TO nMolec
      LET i = INT(Nsx*xx(ipp)/xMax)
      IF i>=Nsx THEN LET i = Nsx-1
      LET j = INT(Nsy*yy(ipp)/yMax)
      IF j>=Nsy THEN LET j = Nsy-1
      LET iq = section(i,j,0) + 1
      LET section(i,j,0) = iq
      LET section(i,j,iq) = ipp
   NEXT ipp
END SUB

! ---------- utility

EXTERNAL FUNCTION systemTemprature
   LET kB = 1.38e-23 ! Boltzman's constant (J/K)
   LET ek= 0.0       !kinetic energy (J)
   FOR i=1 TO nMolec
      LET ek = ek + 0.5*mass*(vx(i)^2+vy(i)^2)
   NEXT i
   LET systemTemprature = ek/(nMolec*kB) !2D: E/N=kT, 3D: E/N=(3/2)kT
END FUNCTION

EXTERNAL SUB ajustVelocity(temp)
   DECLARE EXTERNAL FUNCTION systemTemprature
   LET r = sqr(temp/systemTemprature)
   FOR i=1 TO nMolec
      LET vx(i) = r*vx(i)
      LET vy(i) = r*vy(i)
   NEXT i
END SUB

! ---------- draw particles

EXTERNAL SUB drawParticles(tempMode,contTemp,drawMode)
   DECLARE EXTERNAL FUNCTION systemTemprature,maxNearMolec
   DECLARE EXTERNAL sub realSpace,velocitySpace,plotBond
   SET DRAW MODE HIDDEN
   CLEAR
   IF drawMode=0 OR drawMode=1 THEN !--- 0:circle  1:circle+bond
      call plotBond(drawMode)
   ELSEIF drawMode=2 THEN
      call velocitySpace
   END IF

   !--- draw caption
   SET TEXT HEIGHT 10
   SET TEXT COLOR 1 ! black
   LET tmp$ = "adiabatic   constantTemp  "
   PLOT TEXT, AT  50, 70 ,USING "time =#####.## (ps)   temp =####.## (K)":sysTime*1E12,systemTemprature
   PLOT TEXT, AT  50, 55 ,USING molecSTR$(molecKind)&"  N =####":nMolec
   PLOT TEXT, AT  50, 40 ,USING "tempMode = ## ":tempMode
   PLOT TEXT, AT 200, 40 :tmp$(tempMode*12+1:tempMode*12+12)
   PLOT TEXT, AT  50, 25 ,USING "controled Temperature =####.# (K)":contTemp
   PLOT TEXT, AT  50, 10 :"2-dimensional molecular dynamics"
   SET DRAW MODE EXPLICIT
END SUB

EXTERNAL sub plotBond(drawMode)
   LET boxSize = 300
   LET mag = boxSize/xMax
   LET xp = 100
   LET yp = 100
   SET LINE COLOR 1 ! black : !--- box
   PLOT LINES: xp,yp; boxSize+xp,yp; boxSize+xp,boxSize+yp; xp,boxSize+yp; xp,yp
   SET TEXT HEIGHT 6
   SET TEXT COLOR 1 ! black
   PLOT TEXT, AT xp,boxSize+2+yp ,USING  "box size =##.# x ##.# (nm)":xMax*1e9,yMax*1e9
   FOR i=1 TO nMolec
      SET LINE COLOR 8 ! gray
      DRAW circle WITH SCALE(r0mrs/2.0*mag)*SHIFT(xx(i)*mag+xp,yy(i)*mag+yp)
   NEXT i
   IF drawMode=1 THEN
      FOR i=1 TO nMolec-1
         FOR k=1 TO reg(i,0)-1
            LET j = reg(i,k)
            LET r = SQR((xx(i)-xx(j))*(xx(i)-xx(j))+(yy(i)-yy(j))*(yy(i)-yy(j)))
            IF r<1.2*r0mrs THEN
               IF r<0.93*r0mrs THEN
                  SET LINE COLOR 4 !red 13 !'oleave
               ELSEIF r<1.03*r0mrs THEN
                  SET LINE COLOR 3 !green
               ELSEIF r<1.08*r0mrs THEN
                  SET LINE COLOR 2 !blue
               ELSE
                  SET LINE COLOR 8 !gray
               END IF
               PLOT LINES: xx(i)*mag+xp,yy(i)*mag+yp;xx(j)*mag+xp,yy(j)*mag+yp
            END IF
         NEXT k
      NEXT i
   END IF
END sub

EXTERNAL sub velocitySpace
   LET boxSize = 300
   LET xp = 100
   LET yp = 100
   SET LINE COLOR 1 !black : axis
   PLOT LINES: xp,boxSize/2+yp; boxSize+xp,boxSize/2+yp !vx-axis
   PLOT LINES: boxSize/2+xp,yp; boxSize/2+xp,boxSize+yp !vy-axis
   SET TEXT HEIGHT 6
   SET TEXT COLOR 1 ! black
   PLOT TEXT, AT boxSize+xp,boxSize/2+yp: "vx"
   PLOT TEXT, AT boxSize+xp,boxSize/2-12+yp: "2000m/s"
   PLOT TEXT, AT boxSize/2-12+xp,boxSize+yp: "vy  2000m/s"
   PLOT TEXT, AT boxSize/2-8+xp,boxSize/2-10+yp: "0"
   PLOT TEXT, AT xp,boxSize+8+yp: "velocity space (vx,vy)"
   LET mag = boxSize/4000
   FOR i=1 TO nMolec
      SET LINE COLOR 2 ! blue
      DRAW circle WITH SCALE(5)*SHIFT(vx(i)*mag+boxSize/2+xp,vy(i)*mag+boxSize/2+yp)
   NEXT i
END sub

END MODULE
 

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