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lammps/src/USER-UEF/uef_utils.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: David Nicholson (MIT)
-------------------------------------------------------------------------
This class contains functions to calculate the evolution of the periodic
simulation box under elongational flow as described by Matthew Dobson
in the arXiv preprint at http://arxiv.org/abs/1408.7078
Additionally, there are methods to do a lattice reduction to further
reduce the simulation box using the method of Igor Semaev at
http://link.springer.com/chapter/10.1007%2F3-540-44670-2_13
*/
#include <math.h>
#include "uef_utils.h"
namespace LAMMPS_NS {
namespace UEF_utils{
UEFBox::UEFBox()
{
// initial box (also an inverse eigenvector matrix of automorphisms)
double x = 0.327985277605681;
double y = 0.591009048506103;
double z = 0.736976229099578;
l0[0][0]= z; l0[0][1]= y; l0[0][2]= x;
l0[1][0]=-x; l0[1][1]= z; l0[1][2]=-y;
l0[2][0]=-y; l0[2][1]= x; l0[2][2]= z;
// spectra of the two automorpisms (log of eigenvalues)
w1[0]=-1.177725211523360;
w1[1]=-0.441448620566067;
w1[2]= 1.619173832089425;
w2[0]= w1[1];
w2[1]= w1[2];
w2[2]= w1[0];
// initialize theta
// strain = w1 * theta1 + w2 * theta2
theta[0]=theta[1]=0;
//set up the initial box l and change of basis matrix r
for (int k=0;k<3;k++)
for (int j=0;j<3;j++)
{
l[k][j] = l0[k][j];
r[j][k]=(j==k);
}
// get the initial rotation and upper triangular matrix
rotation_matrix(rot, lrot ,l);
// this is just a way to calculate the automorphisms
// themselves, which play a minor role in the calculations
// it's overkill, but only called once
double t1[3][3];
double t1i[3][3];
double t2[3][3];
double t2i[3][3];
double l0t[3][3];
for (int k=0; k<3; ++k)
for (int j=0; j<3; ++j)
{
t1[k][j] = exp(w1[k])*l0[k][j];
t1i[k][j] = exp(-w1[k])*l0[k][j];
t2[k][j] = exp(w2[k])*l0[k][j];
t2i[k][j] = exp(-w2[k])*l0[k][j];
l0t[k][j] = l0[j][k];
}
mul_m2(l0t,t1);
mul_m2(l0t,t1i);
mul_m2(l0t,t2);
mul_m2(l0t,t2i);
for (int k=0; k<3; ++k)
for (int j=0; j<3; ++j)
{
a1[k][j] = round(t1[k][j]);
a1i[k][j] = round(t1i[k][j]);
a2[k][j] = round(t2[k][j]);
a2i[k][j] = round(t2i[k][j]);
}
// winv used to transform between
// strain increments and theta increments
winv[0][0] = w2[1];
winv[0][1] = -w2[0];
winv[1][0] = -w1[1];
winv[1][1] = w1[0];
double d = w1[0]*w2[1] - w2[0]*w1[1];
for (int k=0;k<2;k++)
for (int j=0;j<2;j++)
winv[k][j] /= d;
}
// get volume-correct r basis in: basis*cbrt(vol) = q*r
void UEFBox::get_box(double x[3][3], double v)
{
v = cbrtf(v);
for (int k=0;k<3;k++)
for (int j=0;j<3;j++)
x[k][j] = lrot[k][j]*v;
}
// get rotation matrix q in: basis = q*r
void UEFBox::get_rot(double x[3][3])
{
for (int k=0;k<3;k++)
for (int j=0;j<3;j++)
x[k][j]=rot[k][j];
}
// diagonal, incompressible deformation
void UEFBox::step_deform(const double ex, const double ey)
{
// increment theta values used in the reduction
theta[0] +=winv[0][0]*ex + winv[0][1]*ey;
theta[1] +=winv[1][0]*ex + winv[1][1]*ey;
// deformation of the box. reduce() needs to
// be called regularly or calculation will become
// unstable
double eps[3];
eps[0]=ex; eps[1] = ey; eps[2] = -ex-ey;
for (int k=0;k<3;k++)
{
eps[k] = exp(eps[k]);
l[k][0] = eps[k]*l[k][0];
l[k][1] = eps[k]*l[k][1];
l[k][2] = eps[k]*l[k][2];
}
rotation_matrix(rot,lrot, l);
}
// reuduce the current basis
bool UEFBox::reduce()
{
// determine how many times to apply the automorphisms
// and find new theta values
int f1 = round(theta[0]);
int f2 = round(theta[1]);
theta[0] -= f1;
theta[1] -= f2;
// store old change or basis matrix to determine if it
// changes
int r0[3][3];
for (int k=0;k<3;k++)
for (int j=0;j<3;j++)
r0[k][j]=r[k][j];
// this modifies the old change basis matrix to
// handle the case where the automorphism transforms
// the box but the reduced basis doesn't change
// (r0 should still equal r at the end)
if (f1 > 0) for (int k=0;k<f1;k++) mul_m2 (a1,r0);
if (f1 < 0) for (int k=0;k<-f1;k++) mul_m2 (a1i,r0);
if (f2 > 0) for (int k=0;k<f2;k++) mul_m2 (a2,r0);
if (f2 < 0) for (int k=0;k<-f2;k++) mul_m2 (a2i,r0);
// robust reduction to the box defined by Dobson
for (int k=0;k<3;k++)
{
double eps = exp(theta[0]*w1[k]+theta[1]*w2[k]);
l[k][0] = eps*l0[k][0];
l[k][1] = eps*l0[k][1];
l[k][2] = eps*l0[k][2];
}
// further reduce the box using greedy reduction and check
// if it changed from the last step using the change of basis
// matrices r and r0
greedy(l,r);
rotation_matrix(rot,lrot, l);
return !mat_same(r,r0);
}
void UEFBox::set_strain(const double ex, const double ey)
{
theta[0] =winv[0][0]*ex + winv[0][1]*ey;
theta[1] =winv[1][0]*ex + winv[1][1]*ey;
theta[0] -= round(theta[0]);
theta[1] -= round(theta[1]);
for (int k=0;k<3;k++)
{
double eps = exp(theta[0]*w1[k]+theta[1]*w2[k]);
l[k][0] = eps*l0[k][0];
l[k][1] = eps*l0[k][1];
l[k][2] = eps*l0[k][2];
}
greedy(l,r);
rotation_matrix(rot,lrot, l);
}
// this is just qr reduction using householder reflections
// m is input matrix, q is a rotation, r is upper triangular
// q*m = r
void rotation_matrix(double q[3][3], double r[3][3], const double m[3][3])
{
for (int k=0;k<3;k++)
for (int j=0;j<3;j++)
r[k][j] = m[k][j];
double a = -sqrt(col_prod(r,0,0))*r[0][0]/fabs(r[0][0]);
double v[3];
v[0] = r[0][0]-a;
v[1] = r[1][0];
v[2] = r[2][0];
a = sqrt(v[0]*v[0]+v[1]*v[1]+v[2]*v[2]);
v[0] /= a; v[1] /= a; v[2] /= a;
double qt[3][3];
for (int k=0;k<3;k++)
for (int j=0;j<3;j++)
{
qt[k][j] = (k==j) - 2*v[k]*v[j];
q[k][j]= qt[k][j];
}
mul_m2(qt,r);
a = -sqrt(r[1][1]*r[1][1] + r[2][1]*r[2][1])*r[1][1]/fabs(r[1][1]);
v[0] = 0;
v[1] = r[1][1] - a;
v[2] = r[2][1];
a = sqrt(v[1]*v[1]+v[2]*v[2]);
v[1] /= a;
v[2] /= a;
for (int k=0;k<3;k++)
for (int j=0;j<3;j++)
qt[k][j] = (k==j) - 2*v[k]*v[j];
mul_m2(qt,r);
mul_m2(qt,q);
// this makes r have positive diagonals
// q*m = r <==> (-q)*m = (-r) will hold row-wise
if (r[0][0] < 0){ neg_row(q,0); neg_row(r,0); }
if (r[1][1] < 0){ neg_row(q,1); neg_row(r,1); }
if (r[2][2] < 0){ neg_row(q,2); neg_row(r,2); }
}
//sort columns in order of increasing length
void col_sort(double b[3][3],int r[3][3])
{
if (col_prod(b,0,0)>col_prod(b,1,1))
{
col_swap(b,0,1);
col_swap(r,0,1);
}
if (col_prod(b,0,0)>col_prod(b,2,2))
{
col_swap(b,0,2);
col_swap(r,0,2);
}
if (col_prod(b,1,1)>col_prod(b,2,2))
{
col_swap(b,1,2);
col_swap(r,1,2);
}
}
// 1-2 reduction (Graham-Schmidt)
void red12(double b[3][3],int r[3][3])
{
int y = round(col_prod(b,0,1)/col_prod(b,0,0));
b[0][1] -= y*b[0][0];
b[1][1] -= y*b[1][0];
b[2][1] -= y*b[2][0];
r[0][1] -= y*r[0][0];
r[1][1] -= y*r[1][0];
r[2][1] -= y*r[2][0];
if (col_prod(b,1,1) < col_prod(b,0,0))
{
col_swap(b,0,1);
col_swap(r,0,1);
red12(b,r);
}
}
// The Semaev condition for a 3-reduced basis
void red3(double b[3][3], int r[3][3])
{
double b11 = col_prod(b,0,0);
double b22 = col_prod(b,1,1);
double b12 = col_prod(b,0,1);
double b13 = col_prod(b,0,2);
double b23 = col_prod(b,1,2);
double y2 =-(b23/b22-b12/b22*b13/b11)/(1-b12/b11*b12/b22);
double y1 =-(b13/b11-b12/b11*b23/b22)/(1-b12/b11*b12/b22);
int x1=0;
int x2=0;
double min = col_prod(b,2,2);
int x1v[2];
int x2v[2];
x1v[0] = floor(y1); x1v[1] = x1v[0]+1;
x2v[0] = floor(y2); x2v[1] = x2v[0]+1;
for (int k=0;k<2;k++)
for (int j=0;j<2;j++)
{
double a[3];
a[0] = b[0][2] + x1v[k]*b[0][0] + x2v[j]*b[0][1];
a[1] = b[1][2] + x1v[k]*b[1][0] + x2v[j]*b[1][1];
a[2] = b[2][2] + x1v[k]*b[2][0] + x2v[j]*b[2][1];
double val=a[0]*a[0]+a[1]*a[1]+a[2]*a[2];
if (val<min)
{
min = val;
x1 = x1v[k];
x2 = x2v[j];
}
}
if (x1 || x2)
{
b[0][2] += x1*b[0][0] + x2*b[0][1];
b[1][2] += x1*b[1][0] + x2*b[1][1];
b[2][2] += x1*b[2][0] + x2*b[2][1];
r[0][2] += x1*r[0][0] + x2*r[0][1];
r[1][2] += x1*r[1][0] + x2*r[1][1];
r[2][2] += x1*r[2][0] + x2*r[2][1];
greedy_recurse(b,r); // note the recursion step is here
}
}
// the meat of the greedy reduction algorithm
void greedy_recurse(double b[3][3], int r[3][3])
{
col_sort(b,r);
red12(b,r);
red3(b,r); // recursive caller
}
// set r (change of basis) to be identity then reduce basis and make it unique
void greedy(double b[3][3],int r[3][3])
{
r[0][1]=r[0][2]=r[1][0]=r[1][2]=r[2][0]=r[2][1]=0;
r[0][0]=r[1][1]=r[2][2]=1;
greedy_recurse(b,r);
make_unique(b,r);
}
// A reduced basis isn't unique. This procedure will make it
// "more" unique. Degenerate cases are possible, but unlikely
// with floating point math.
void make_unique(double b[3][3], int r[3][3])
{
if (fabs(b[0][0]) < fabs(b[0][1]))
{ col_swap(b,0,1); col_swap(r,0,1); }
if (fabs(b[0][0]) < fabs(b[0][2]))
{ col_swap(b,0,2); col_swap(r,0,2); }
if (fabs(b[1][1]) < fabs(b[1][2]))
{ col_swap(b,1,2); col_swap(r,1,2); }
if (b[0][0] < 0){ neg_col(b,0); neg_col(r,0); }
if (b[1][1] < 0){ neg_col(b,1); neg_col(r,1); }
if (det(b) < 0){ neg_col(b,2); neg_col(r,2); }
}
}}
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