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fix_nph.cpp
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Sat, Jun 28, 12:47

fix_nph.cpp
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/* ----------------------------------------------------------------------
LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
http://lammps.sandia.gov, Sandia National Laboratories
Steve Plimpton, sjplimp@sandia.gov
Copyright (2003) Sandia Corporation. Under the terms of Contract
DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
certain rights in this software. This software is distributed under
the GNU General Public License.
See the README file in the top-level LAMMPS directory.
------------------------------------------------------------------------- */
/* ----------------------------------------------------------------------
Contributing author: Mark Stevens (SNL)
------------------------------------------------------------------------- */
#include "string.h"
#include "stdlib.h"
#include "math.h"
#include "fix_nph.h"
#include "atom.h"
#include "force.h"
#include "comm.h"
#include "domain.h"
#include "modify.h"
#include "fix_deform.h"
#include "compute.h"
#include "kspace.h"
#include "update.h"
#include "respa.h"
#include "domain.h"
#include "error.h"
using namespace LAMMPS_NS;
#define MIN(A,B) ((A) < (B)) ? (A) : (B)
#define MAX(A,B) ((A) > (B)) ? (A) : (B)
enum{XYZ,XY,YZ,XZ,ANISO};
/* ---------------------------------------------------------------------- */
FixNPH::FixNPH(LAMMPS *lmp, int narg, char **arg) :
Fix(lmp, narg, arg)
{
if (narg < 4) error->all("Illegal fix nph command");
restart_global = 1;
box_change = 1;
time_integrate = 1;
scalar_flag = 1;
scalar_vector_freq = 1;
extscalar = 1;
double p_period[3];
if (strcmp(arg[3],"xyz") == 0) {
if (narg < 7) error->all("Illegal fix nph command");
press_couple = XYZ;
p_start[0] = p_start[1] = p_start[2] = atof(arg[4]);
p_stop[0] = p_stop[1] = p_stop[2] = atof(arg[5]);
p_period[0] = p_period[1] = p_period[2] = atof(arg[6]);
p_flag[0] = p_flag[1] = p_flag[2] = 1;
if (domain->dimension == 2) {
p_start[2] = p_stop[2] = p_period[2] = 0.0;
p_flag[2] = 0;
}
} else {
if (strcmp(arg[3],"xy") == 0) press_couple = XY;
else if (strcmp(arg[3],"yz") == 0) press_couple = YZ;
else if (strcmp(arg[3],"xz") == 0) press_couple = XZ;
else if (strcmp(arg[3],"aniso") == 0) press_couple = ANISO;
else error->all("Illegal fix nph command");
if (narg < 11) error->all("Illegal fix nph command");
if (domain->dimension == 2 &&
(press_couple == XY || press_couple == YZ || press_couple == XZ))
error->all("Invalid fix nph command for a 2d simulation");
if (strcmp(arg[4],"NULL") == 0) {
p_start[0] = p_stop[0] = p_period[0] = 0.0;
p_flag[0] = 0;
} else {
p_start[0] = atof(arg[4]);
p_stop[0] = atof(arg[5]);
p_flag[0] = 1;
}
if (strcmp(arg[6],"NULL") == 0) {
p_start[1] = p_stop[1] = p_period[1] = 0.0;
p_flag[1] = 0;
} else {
p_start[1] = atof(arg[6]);
p_stop[1] = atof(arg[7]);
p_flag[1] = 1;
}
if (strcmp(arg[8],"NULL") == 0) {
p_start[2] = p_stop[2] = p_period[2] = 0.0;
p_flag[2] = 0;
} else {
if (domain->dimension == 2)
error->all("Invalid fix nph command for a 2d simulation");
p_start[2] = atof(arg[8]);
p_stop[2] = atof(arg[9]);
p_flag[2] = 1;
}
double period = atof(arg[10]);
if (p_flag[0]) p_period[0] = period;
if (p_flag[1]) p_period[1] = period;
if (p_flag[2]) p_period[2] = period;
}
// process extra keywords
drag = 0.0;
allremap = 1;
int iarg;
if (press_couple == XYZ) iarg = 7;
else iarg = 11;
while (iarg < narg) {
if (strcmp(arg[iarg],"drag") == 0) {
if (iarg+2 > narg) error->all("Illegal fix nph command");
drag = atof(arg[iarg+1]);
iarg += 2;
} else if (strcmp(arg[iarg],"dilate") == 0) {
if (iarg+2 > narg) error->all("Illegal fix nph command");
if (strcmp(arg[iarg+1],"all") == 0) allremap = 1;
else if (strcmp(arg[iarg+1],"partial") == 0) allremap = 0;
else error->all("Illegal fix nph command");
iarg += 2;
} else error->all("Illegal fix nph command");
}
// error checks
if (press_couple == XY && (p_flag[0] == 0 || p_flag[1] == 0))
error->all("Invalid fix nph command pressure settings");
if (press_couple == YZ && (p_flag[1] == 0 || p_flag[2] == 0))
error->all("Invalid fix nph command pressure settings");
if (press_couple == XZ && (p_flag[0] == 0 || p_flag[2] == 0))
error->all("Invalid fix nph command pressure settings");
if (press_couple == XY &&
(p_start[0] != p_start[1] || p_stop[0] != p_stop[1]))
error->all("Invalid fix nph command pressure settings");
if (press_couple == YZ &&
(p_start[1] != p_start[2] || p_stop[1] != p_stop[2]))
error->all("Invalid fix nph command pressure settings");
if (press_couple == XZ &&
(p_start[0] != p_start[2] || p_stop[0] != p_stop[2]))
error->all("Invalid fix nph command pressure settings");
if (p_flag[0] && domain->xperiodic == 0)
error->all("Cannot use fix nph on a non-periodic dimension");
if (p_flag[1] && domain->yperiodic == 0)
error->all("Cannot use fix nph on a non-periodic dimension");
if (p_flag[2] && domain->zperiodic == 0)
error->all("Cannot use fix nph on a non-periodic dimension");
// convert input periods to frequencies
if ((p_flag[0] && p_period[0] <= 0.0) ||
(p_flag[1] && p_period[1] <= 0.0) || (p_flag[2] && p_period[2] <= 0.0))
error->all("Fix nph periods must be > 0.0");
p_freq[0] = p_freq[1] = p_freq[2] = 0.0;
if (p_flag[0]) p_freq[0] = 1.0 / p_period[0];
if (p_flag[1]) p_freq[1] = 1.0 / p_period[1];
if (p_flag[2]) p_freq[2] = 1.0 / p_period[2];
// create a new compute temp style
// id = fix-ID + temp
// compute group = all since pressure is always global (group all)
// and thus its KE/temperature contribution should use group all
int n = strlen(id) + 6;
id_temp = new char[n];
strcpy(id_temp,id);
strcat(id_temp,"_temp");
char **newarg = new char*[3];
newarg[0] = id_temp;
newarg[1] = (char *) "all";
newarg[2] = (char *) "temp";
modify->add_compute(3,newarg);
delete [] newarg;
tflag = 1;
// create a new compute pressure style
// id = fix-ID + press, compute group = all
// pass id_temp as 4th arg to pressure constructor
n = strlen(id) + 7;
id_press = new char[n];
strcpy(id_press,id);
strcat(id_press,"_press");
newarg = new char*[4];
newarg[0] = id_press;
newarg[1] = (char *) "all";
newarg[2] = (char *) "pressure";
newarg[3] = id_temp;
modify->add_compute(4,newarg);
delete [] newarg;
pflag = 1;
// Nose/Hoover pressure init
omega[0] = omega[1] = omega[2] = 0.0;
omega_dot[0] = omega_dot[1] = omega_dot[2] = 0.0;
nrigid = 0;
rfix = NULL;
}
/* ---------------------------------------------------------------------- */
FixNPH::~FixNPH()
{
delete [] rfix;
// delete temperature and pressure if fix created them
if (tflag) modify->delete_compute(id_temp);
if (pflag) modify->delete_compute(id_press);
delete [] id_temp;
delete [] id_press;
}
/* ---------------------------------------------------------------------- */
int FixNPH::setmask()
{
int mask = 0;
mask |= INITIAL_INTEGRATE;
mask |= FINAL_INTEGRATE;
mask |= THERMO_ENERGY;
mask |= INITIAL_INTEGRATE_RESPA;
mask |= FINAL_INTEGRATE_RESPA;
return mask;
}
/* ---------------------------------------------------------------------- */
void FixNPH::init()
{
if (domain->triclinic) error->all("Cannot use fix nph with triclinic box");
if (atom->mass == NULL)
error->all("Cannot use fix nph without per-type mass defined");
for (int i = 0; i < modify->nfix; i++)
if (strcmp(modify->fix[i]->style,"deform") == 0) {
int *dimflag = ((FixDeform *) modify->fix[i])->dimflag;
if ((p_flag[0] && dimflag[0]) || (p_flag[1] && dimflag[1]) ||
(p_flag[2] && dimflag[2]))
error->all("Cannot use fix npt and fix deform on same dimension");
}
// set temperature and pressure ptrs
int icompute = modify->find_compute(id_temp);
if (icompute < 0) error->all("Temp ID for fix nph does not exist");
temperature = modify->compute[icompute];
icompute = modify->find_compute(id_press);
if (icompute < 0) error->all("Press ID for fix nph does not exist");
pressure = modify->compute[icompute];
// set timesteps and frequencies
// Nkt = initial value for piston mass and energy conservation
// guesstimate a unit-dependent t_initial if actual T = 0.0
dtv = update->dt;
dtf = 0.5 * update->dt * force->ftm2v;
dthalf = 0.5 * update->dt;
double freq = MAX(p_freq[0],p_freq[1]);
freq = MAX(freq,p_freq[2]);
drag_factor = 1.0 - (update->dt * freq * drag);
boltz = force->boltz;
nktv2p = force->nktv2p;
dimension = domain->dimension;
if (dimension == 3) vol0 = domain->xprd * domain->yprd * domain->zprd;
else vol0 = domain->xprd * domain->yprd;
temperature->init(); // not yet called by Modify::init()
double t_initial = temperature->compute_scalar();
if (t_initial == 0.0) {
if (strcmp(update->unit_style,"lj") == 0) t_initial = 1.0;
else t_initial = 300.0;
}
nkt = atom->natoms * boltz * t_initial;
if (force->kspace) kspace_flag = 1;
else kspace_flag = 0;
if (strcmp(update->integrate_style,"respa") == 0) {
nlevels_respa = ((Respa *) update->integrate)->nlevels;
step_respa = ((Respa *) update->integrate)->step;
}
// detect if any rigid fixes exist so rigid bodies move when box is remapped
// rfix[] = indices to each fix rigid
delete [] rfix;
nrigid = 0;
rfix = NULL;
for (int i = 0; i < modify->nfix; i++)
if (modify->fix[i]->rigid_flag) nrigid++;
if (nrigid) {
rfix = new int[nrigid];
nrigid = 0;
for (int i = 0; i < modify->nfix; i++)
if (modify->fix[i]->rigid_flag) rfix[nrigid++] = i;
}
}
/* ----------------------------------------------------------------------
compute T,P before integrator starts
------------------------------------------------------------------------- */
void FixNPH::setup(int vflag)
{
p_target[0] = p_start[0]; // used by compute_scalar()
p_target[1] = p_start[1];
p_target[2] = p_start[2];
if (press_couple == XYZ) {
double tmp = temperature->compute_scalar();
tmp = pressure->compute_scalar();
} else {
temperature->compute_vector();
pressure->compute_vector();
}
couple();
// trigger virial computation on next timestep
pressure->addstep(update->ntimestep+1);
}
/* ----------------------------------------------------------------------
1st half of Verlet update
------------------------------------------------------------------------- */
void FixNPH::initial_integrate(int vflag)
{
int i;
double delta = update->ntimestep - update->beginstep;
delta /= update->endstep - update->beginstep;
// update omega_dot
// for non-varying dims, p_freq is 0.0, so omega doesn't change
double f_omega,volume;
if (dimension == 3) volume = domain->xprd*domain->yprd*domain->zprd;
else volume = domain->xprd*domain->yprd;
double denskt = nkt / volume * nktv2p;
for (i = 0; i < 3; i++) {
p_target[i] = p_start[i] + delta * (p_stop[i]-p_start[i]);
f_omega = p_freq[i]*p_freq[i] * (p_current[i]-p_target[i])/denskt;
omega_dot[i] += f_omega*dthalf;
omega_dot[i] *= drag_factor;
omega[i] += dtv*omega_dot[i];
factor[i] = exp(-dthalf*omega_dot[i]);
dilation[i] = exp(dthalf*omega_dot[i]);
}
// v update only for atoms in NPH group
double **x = atom->x;
double **v = atom->v;
double **f = atom->f;
double *mass = atom->mass;
int *type = atom->type;
int *mask = atom->mask;
int nlocal = atom->nlocal;
double dtfm;
for (i = 0; i < nlocal; i++) {
if (mask[i] & groupbit) {
dtfm = dtf / mass[type[i]];
v[i][0] = v[i][0]*factor[0] + dtfm*f[i][0];
v[i][1] = v[i][1]*factor[1] + dtfm*f[i][1];
v[i][2] = v[i][2]*factor[2] + dtfm*f[i][2];
}
}
// remap simulation box and all owned atoms by 1/2 step
remap(0);
// x update by full step only for atoms in NPH group
for (i = 0; i < nlocal; i++) {
if (mask[i] & groupbit) {
x[i][0] += dtv * v[i][0];
x[i][1] += dtv * v[i][1];
x[i][2] += dtv * v[i][2];
}
}
// remap simulation box and all owned atoms by 1/2 step
// redo KSpace coeffs since volume has changed
remap(0);
if (kspace_flag) force->kspace->setup();
}
/* ----------------------------------------------------------------------
2nd half of Verlet update
------------------------------------------------------------------------- */
void FixNPH::final_integrate()
{
int i;
// v update only for atoms in NPH group
double **v = atom->v;
double **f = atom->f;
double *mass = atom->mass;
int *type = atom->type;
int *mask = atom->mask;
int nlocal = atom->nlocal;
double dtfm;
for (i = 0; i < nlocal; i++) {
if (mask[i] & groupbit) {
dtfm = dtf / mass[type[i]];
v[i][0] = (v[i][0] + dtfm*f[i][0]) * factor[0];
v[i][1] = (v[i][1] + dtfm*f[i][1]) * factor[1];
v[i][2] = (v[i][2] + dtfm*f[i][2]) * factor[2];
}
}
// compute new pressure
if (press_couple == XYZ) {
double tmp = temperature->compute_scalar();
tmp = pressure->compute_scalar();
} else {
temperature->compute_vector();
pressure->compute_vector();
}
couple();
// trigger virial computation on next timestep
pressure->addstep(update->ntimestep+1);
// update omega_dot
// for non-varying dims, p_freq is 0.0, so omega_dot doesn't change
double f_omega,volume;
if (dimension == 3) volume = domain->xprd*domain->yprd*domain->zprd;
else volume = domain->xprd*domain->yprd;
double denskt = nkt / volume * nktv2p;
for (i = 0; i < 3; i++) {
f_omega = p_freq[i]*p_freq[i] * (p_current[i]-p_target[i])/denskt;
omega_dot[i] += f_omega*dthalf;
omega_dot[i] *= drag_factor;
}
}
/* ---------------------------------------------------------------------- */
void FixNPH::initial_integrate_respa(int vflag, int ilevel, int flag)
{
// if flag = 1, then is 2nd call at outermost level from rRESPA
// perform 2nd half of box remap on own + ghost atoms and return
// redo KSpace coeffs since volume has changed
if (flag == 1) {
remap(1);
if (kspace_flag) force->kspace->setup();
return;
}
// set timesteps by level
double dtfm;
dtv = step_respa[ilevel];
dtf = 0.5 * step_respa[ilevel] * force->ftm2v;
dthalf = 0.5 * step_respa[ilevel];
// atom quantities
double **x = atom->x;
double **v = atom->v;
double **f = atom->f;
double *mass = atom->mass;
int *type = atom->type;
int *mask = atom->mask;
int nlocal = atom->nlocal;
// outermost level - update omega_dot, apply to v, remap box
// all other levels - NVE update of v
// x,v updates only performed for atoms in NPH group
if (ilevel == nlevels_respa-1) {
double delta = update->ntimestep - update->beginstep;
delta /= update->endstep - update->beginstep;
// update omega_dot
// for non-varying dims, p_freq is 0.0, so omega doesn't change
double f_omega,volume;
if (dimension == 3) volume = domain->xprd*domain->yprd*domain->zprd;
else volume = domain->xprd*domain->yprd;
double denskt = nkt / volume * nktv2p;
for (int i = 0; i < 3; i++) {
p_target[i] = p_start[i] + delta * (p_stop[i]-p_start[i]);
f_omega = p_freq[i]*p_freq[i] * (p_current[i]-p_target[i])/denskt;
omega_dot[i] += f_omega*dthalf;
omega_dot[i] *= drag_factor;
omega[i] += dtv*omega_dot[i];
factor[i] = exp(-dthalf*omega_dot[i]);
dilation[i] = exp(dthalf*omega_dot[i]);
}
// v update only for atoms in NPH group
for (int i = 0; i < nlocal; i++) {
if (mask[i] & groupbit) {
dtfm = dtf / mass[type[i]];
v[i][0] = v[i][0]*factor[0] + dtfm*f[i][0];
v[i][1] = v[i][1]*factor[1] + dtfm*f[i][1];
v[i][2] = v[i][2]*factor[2] + dtfm*f[i][2];
}
}
// remap simulation box and all owned atoms by 1/2 step
remap(0);
} else {
// v update only for atoms in NPH group
for (int i = 0; i < nlocal; i++) {
if (mask[i] & groupbit) {
dtfm = dtf / mass[type[i]];
v[i][0] += dtfm*f[i][0];
v[i][1] += dtfm*f[i][1];
v[i][2] += dtfm*f[i][2];
}
}
}
// innermost level - also update x only for atoms in NPH group
if (ilevel == 0) {
for (int i = 0; i < nlocal; i++) {
if (mask[i] & groupbit) {
x[i][0] += dtv * v[i][0];
x[i][1] += dtv * v[i][1];
x[i][2] += dtv * v[i][2];
}
}
}
}
/* ---------------------------------------------------------------------- */
void FixNPH::final_integrate_respa(int ilevel)
{
double dtfm;
// set timesteps by level
dtf = 0.5 * step_respa[ilevel] * force->ftm2v;
dthalf = 0.5 * step_respa[ilevel];
// outermost level - update omega_dot, apply to v via final_integrate()
// all other levels - NVE update of v
// v update only performed for atoms in NPH group
if (ilevel == nlevels_respa-1) final_integrate();
else {
// v update only for atoms in NPH group
double **v = atom->v;
double **f = atom->f;
double *mass = atom->mass;
int *type = atom->type;
int *mask = atom->mask;
int nlocal = atom->nlocal;
for (int i = 0; i < nlocal; i++) {
if (mask[i] & groupbit) {
dtfm = dtf / mass[type[i]];
v[i][0] += dtfm*f[i][0];
v[i][1] += dtfm*f[i][1];
v[i][2] += dtfm*f[i][2];
}
}
}
}
/* ---------------------------------------------------------------------- */
void FixNPH::couple()
{
double *tensor = pressure->vector;
if (press_couple == XYZ)
p_current[0] = p_current[1] = p_current[2] = pressure->scalar;
else if (press_couple == XY) {
double ave = 0.5 * (tensor[0] + tensor[1]);
p_current[0] = p_current[1] = ave;
p_current[2] = tensor[2];
} else if (press_couple == YZ) {
double ave = 0.5 * (tensor[1] + tensor[2]);
p_current[1] = p_current[2] = ave;
p_current[0] = tensor[0];
} else if (press_couple == XZ) {
double ave = 0.5 * (tensor[0] + tensor[2]);
p_current[0] = p_current[2] = ave;
p_current[1] = tensor[1];
} else if (press_couple == ANISO) {
p_current[0] = tensor[0];
p_current[1] = tensor[1];
p_current[2] = tensor[2];
}
}
/* ----------------------------------------------------------------------
change box size
remap owned or owned+ghost atoms depending on flag
remap all atoms or fix group atoms depending on allremap flag
if rigid bodies exist, scale rigid body centers-of-mass
------------------------------------------------------------------------- */
void FixNPH::remap(int flag)
{
int i,n;
double oldlo,oldhi,ctr;
double **x = atom->x;
int *mask = atom->mask;
if (flag) n = atom->nlocal + atom->nghost;
else n = atom->nlocal;
// convert pertinent atoms and rigid bodies to lamda coords
if (allremap) domain->x2lamda(n);
else {
for (i = 0; i < n; i++)
if (mask[i] & groupbit)
domain->x2lamda(x[i],x[i]);
}
if (nrigid)
for (i = 0; i < nrigid; i++)
modify->fix[rfix[i]]->deform(0);
// reset global and local box to new size/shape
for (i = 0; i < 3; i++) {
if (p_flag[i]) {
oldlo = domain->boxlo[i];
oldhi = domain->boxhi[i];
ctr = 0.5 * (oldlo + oldhi);
domain->boxlo[i] = (oldlo-ctr)*dilation[i] + ctr;
domain->boxhi[i] = (oldhi-ctr)*dilation[i] + ctr;
}
}
domain->set_global_box();
domain->set_local_box();
// convert pertinent atoms and rigid bodies back to box coords
if (allremap) domain->lamda2x(n);
else {
for (i = 0; i < n; i++)
if (mask[i] & groupbit)
domain->lamda2x(x[i],x[i]);
}
if (nrigid)
for (i = 0; i < nrigid; i++)
modify->fix[rfix[i]]->deform(1);
}
/* ----------------------------------------------------------------------
pack entire state of Fix into one write
------------------------------------------------------------------------- */
void FixNPH::write_restart(FILE *fp)
{
int n = 0;
double list[6];
list[n++] = omega[0];
list[n++] = omega[1];
list[n++] = omega[2];
list[n++] = omega_dot[0];
list[n++] = omega_dot[1];
list[n++] = omega_dot[2];
if (comm->me == 0) {
int size = n * sizeof(double);
fwrite(&size,sizeof(int),1,fp);
fwrite(&list,sizeof(double),n,fp);
}
}
/* ----------------------------------------------------------------------
use state info from restart file to restart the Fix
------------------------------------------------------------------------- */
void FixNPH::restart(char *buf)
{
int n = 0;
double *list = (double *) buf;
omega[0] = list[n++];
omega[1] = list[n++];
omega[2] = list[n++];
omega_dot[0] = list[n++];
omega_dot[1] = list[n++];
omega_dot[2] = list[n++];
}
/* ---------------------------------------------------------------------- */
int FixNPH::modify_param(int narg, char **arg)
{
if (strcmp(arg[0],"temp") == 0) {
if (narg < 2) error->all("Illegal fix_modify command");
if (tflag) {
modify->delete_compute(id_temp);
tflag = 0;
}
delete [] id_temp;
int n = strlen(arg[1]) + 1;
id_temp = new char[n];
strcpy(id_temp,arg[1]);
int icompute = modify->find_compute(id_temp);
if (icompute < 0) error->all("Could not find fix_modify temp ID");
temperature = modify->compute[icompute];
if (temperature->tempflag == 0)
error->all("Fix_modify temp ID does not compute temperature");
if (temperature->igroup != 0 && comm->me == 0)
error->warning("Temperature for NPH is not for group all");
// reset id_pre of pressure to new temp ID
icompute = modify->find_compute(id_press);
if (icompute < 0) error->all("Press ID for fix npt does not exist");
delete [] modify->compute[icompute]->id_pre;
modify->compute[icompute]->id_pre = new char[n];
strcpy(modify->compute[icompute]->id_pre,id_temp);
return 2;
} else if (strcmp(arg[0],"press") == 0) {
if (narg < 2) error->all("Illegal fix_modify command");
if (pflag) {
modify->delete_compute(id_press);
pflag = 0;
}
delete [] id_press;
int n = strlen(arg[1]) + 1;
id_press = new char[n];
strcpy(id_press,arg[1]);
int icompute = modify->find_compute(id_press);
if (icompute < 0) error->all("Could not find fix_modify press ID");
pressure = modify->compute[icompute];
if (pressure->pressflag == 0)
error->all("Fix_modify press ID does not compute pressure");
return 2;
}
return 0;
}
/* ---------------------------------------------------------------------- */
double FixNPH::compute_scalar()
{
double volume;
if (dimension == 3) volume = domain->xprd * domain->yprd * domain->zprd;
else volume = domain->xprd * domain->yprd;
int pdim = p_flag[0] + p_flag[1] + p_flag[2];
double energy = 0.0;
for (int i = 0; i < 3; i++)
if (p_freq[i] > 0.0)
energy += 0.5*nkt*omega_dot[i]*omega_dot[i] /
(p_freq[i]*p_freq[i]) + p_target[i]*(volume-vol0) / (pdim*nktv2p);
return energy;
}

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