367 lines
9.7 KiB
C++
367 lines
9.7 KiB
C++
/* ----------------------------------------------------------------------
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LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
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http://lammps.sandia.gov, Sandia National Laboratories
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Steve Plimpton, sjplimp@sandia.gov
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Copyright (2003) Sandia Corporation. Under the terms of Contract
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DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
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certain rights in this software. This software is distributed under
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the GNU General Public License.
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See the README file in the top-level LAMMPS directory.
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Contributing author: David Nicholson (MIT)
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-------------------------------------------------------------------------
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This class contains functions to calculate the evolution of the periodic
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simulation box under elongational flow as described by Matthew Dobson
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in the arXiv preprint at http://arxiv.org/abs/1408.7078
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Additionally, there are methods to do a lattice reduction to further
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reduce the simulation box using the method of Igor Semaev at
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http://link.springer.com/chapter/10.1007%2F3-540-44670-2_13
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*/
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#include <math.h>
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#include "uef_utils.h"
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namespace LAMMPS_NS {
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namespace UEF_utils{
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UEFBox::UEFBox()
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{
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// initial box (also an inverse eigenvector matrix of automorphisms)
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double x = 0.327985277605681;
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double y = 0.591009048506103;
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double z = 0.736976229099578;
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l0[0][0]= z; l0[0][1]= y; l0[0][2]= x;
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l0[1][0]=-x; l0[1][1]= z; l0[1][2]=-y;
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l0[2][0]=-y; l0[2][1]= x; l0[2][2]= z;
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// spectra of the two automorpisms (log of eigenvalues)
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w1[0]=-1.177725211523360;
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w1[1]=-0.441448620566067;
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w1[2]= 1.619173832089425;
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w2[0]= w1[1];
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w2[1]= w1[2];
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w2[2]= w1[0];
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// initialize theta
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// strain = w1 * theta1 + w2 * theta2
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theta[0]=theta[1]=0;
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//set up the initial box l and change of basis matrix r
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for (int k=0;k<3;k++)
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for (int j=0;j<3;j++)
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{
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l[k][j] = l0[k][j];
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r[j][k]=(j==k);
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}
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// get the initial rotation and upper triangular matrix
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rotation_matrix(rot, lrot ,l);
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// this is just a way to calculate the automorphisms
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// themselves, which play a minor role in the calculations
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// it's overkill, but only called once
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double t1[3][3];
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double t1i[3][3];
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double t2[3][3];
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double t2i[3][3];
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double l0t[3][3];
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for (int k=0; k<3; ++k)
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for (int j=0; j<3; ++j)
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{
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t1[k][j] = exp(w1[k])*l0[k][j];
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t1i[k][j] = exp(-w1[k])*l0[k][j];
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t2[k][j] = exp(w2[k])*l0[k][j];
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t2i[k][j] = exp(-w2[k])*l0[k][j];
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l0t[k][j] = l0[j][k];
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}
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mul_m2(l0t,t1);
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mul_m2(l0t,t1i);
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mul_m2(l0t,t2);
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mul_m2(l0t,t2i);
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for (int k=0; k<3; ++k)
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for (int j=0; j<3; ++j)
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{
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a1[k][j] = round(t1[k][j]);
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a1i[k][j] = round(t1i[k][j]);
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a2[k][j] = round(t2[k][j]);
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a2i[k][j] = round(t2i[k][j]);
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}
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// winv used to transform between
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// strain increments and theta increments
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winv[0][0] = w2[1];
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winv[0][1] = -w2[0];
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winv[1][0] = -w1[1];
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winv[1][1] = w1[0];
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double d = w1[0]*w2[1] - w2[0]*w1[1];
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for (int k=0;k<2;k++)
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for (int j=0;j<2;j++)
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winv[k][j] /= d;
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}
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// get volume-correct r basis in: basis*cbrt(vol) = q*r
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void UEFBox::get_box(double x[3][3], double v)
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{
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v = cbrtf(v);
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for (int k=0;k<3;k++)
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for (int j=0;j<3;j++)
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x[k][j] = lrot[k][j]*v;
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}
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// get rotation matrix q in: basis = q*r
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void UEFBox::get_rot(double x[3][3])
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{
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for (int k=0;k<3;k++)
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for (int j=0;j<3;j++)
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x[k][j]=rot[k][j];
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}
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// diagonal, incompressible deformation
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void UEFBox::step_deform(const double ex, const double ey)
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{
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// increment theta values used in the reduction
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theta[0] +=winv[0][0]*ex + winv[0][1]*ey;
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theta[1] +=winv[1][0]*ex + winv[1][1]*ey;
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// deformation of the box. reduce() needs to
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// be called regularly or calculation will become
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// unstable
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double eps[3];
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eps[0]=ex; eps[1] = ey; eps[2] = -ex-ey;
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for (int k=0;k<3;k++)
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{
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eps[k] = exp(eps[k]);
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l[k][0] = eps[k]*l[k][0];
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l[k][1] = eps[k]*l[k][1];
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l[k][2] = eps[k]*l[k][2];
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}
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rotation_matrix(rot,lrot, l);
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}
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// reuduce the current basis
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bool UEFBox::reduce()
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{
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// determine how many times to apply the automorphisms
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// and find new theta values
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int f1 = round(theta[0]);
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int f2 = round(theta[1]);
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theta[0] -= f1;
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theta[1] -= f2;
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// store old change or basis matrix to determine if it
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// changes
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int r0[3][3];
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for (int k=0;k<3;k++)
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for (int j=0;j<3;j++)
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r0[k][j]=r[k][j];
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// this modifies the old change basis matrix to
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// handle the case where the automorphism transforms
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// the box but the reduced basis doesn't change
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// (r0 should still equal r at the end)
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if (f1 > 0) for (int k=0;k<f1;k++) mul_m2 (a1,r0);
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if (f1 < 0) for (int k=0;k<-f1;k++) mul_m2 (a1i,r0);
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if (f2 > 0) for (int k=0;k<f2;k++) mul_m2 (a2,r0);
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if (f2 < 0) for (int k=0;k<-f2;k++) mul_m2 (a2i,r0);
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// robust reduction to the box defined by Dobson
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for (int k=0;k<3;k++)
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{
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double eps = exp(theta[0]*w1[k]+theta[1]*w2[k]);
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l[k][0] = eps*l0[k][0];
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l[k][1] = eps*l0[k][1];
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l[k][2] = eps*l0[k][2];
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}
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// further reduce the box using greedy reduction and check
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// if it changed from the last step using the change of basis
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// matrices r and r0
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greedy(l,r);
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rotation_matrix(rot,lrot, l);
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return !mat_same(r,r0);
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}
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void UEFBox::set_strain(const double ex, const double ey)
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{
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theta[0] =winv[0][0]*ex + winv[0][1]*ey;
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theta[1] =winv[1][0]*ex + winv[1][1]*ey;
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theta[0] -= round(theta[0]);
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theta[1] -= round(theta[1]);
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for (int k=0;k<3;k++)
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{
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double eps = exp(theta[0]*w1[k]+theta[1]*w2[k]);
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l[k][0] = eps*l0[k][0];
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l[k][1] = eps*l0[k][1];
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l[k][2] = eps*l0[k][2];
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}
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greedy(l,r);
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rotation_matrix(rot,lrot, l);
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}
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// this is just qr reduction using householder reflections
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// m is input matrix, q is a rotation, r is upper triangular
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// q*m = r
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void rotation_matrix(double q[3][3], double r[3][3], const double m[3][3])
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{
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for (int k=0;k<3;k++)
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for (int j=0;j<3;j++)
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r[k][j] = m[k][j];
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double a = -sqrt(col_prod(r,0,0))*r[0][0]/fabs(r[0][0]);
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double v[3];
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v[0] = r[0][0]-a;
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v[1] = r[1][0];
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v[2] = r[2][0];
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a = sqrt(v[0]*v[0]+v[1]*v[1]+v[2]*v[2]);
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v[0] /= a; v[1] /= a; v[2] /= a;
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double qt[3][3];
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for (int k=0;k<3;k++)
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for (int j=0;j<3;j++)
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{
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qt[k][j] = (k==j) - 2*v[k]*v[j];
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q[k][j]= qt[k][j];
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}
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mul_m2(qt,r);
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a = -sqrt(r[1][1]*r[1][1] + r[2][1]*r[2][1])*r[1][1]/fabs(r[1][1]);
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v[0] = 0;
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v[1] = r[1][1] - a;
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v[2] = r[2][1];
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a = sqrt(v[1]*v[1]+v[2]*v[2]);
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v[1] /= a;
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v[2] /= a;
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for (int k=0;k<3;k++)
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for (int j=0;j<3;j++)
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qt[k][j] = (k==j) - 2*v[k]*v[j];
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mul_m2(qt,r);
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mul_m2(qt,q);
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// this makes r have positive diagonals
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// q*m = r <==> (-q)*m = (-r) will hold row-wise
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if (r[0][0] < 0){ neg_row(q,0); neg_row(r,0); }
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if (r[1][1] < 0){ neg_row(q,1); neg_row(r,1); }
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if (r[2][2] < 0){ neg_row(q,2); neg_row(r,2); }
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}
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//sort columns in order of increasing length
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void col_sort(double b[3][3],int r[3][3])
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{
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if (col_prod(b,0,0)>col_prod(b,1,1))
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{
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col_swap(b,0,1);
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col_swap(r,0,1);
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}
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if (col_prod(b,0,0)>col_prod(b,2,2))
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{
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col_swap(b,0,2);
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col_swap(r,0,2);
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}
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if (col_prod(b,1,1)>col_prod(b,2,2))
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{
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col_swap(b,1,2);
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col_swap(r,1,2);
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}
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}
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// 1-2 reduction (Graham-Schmidt)
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void red12(double b[3][3],int r[3][3])
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{
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int y = round(col_prod(b,0,1)/col_prod(b,0,0));
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b[0][1] -= y*b[0][0];
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b[1][1] -= y*b[1][0];
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b[2][1] -= y*b[2][0];
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r[0][1] -= y*r[0][0];
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r[1][1] -= y*r[1][0];
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r[2][1] -= y*r[2][0];
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if (col_prod(b,1,1) < col_prod(b,0,0))
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{
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col_swap(b,0,1);
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col_swap(r,0,1);
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red12(b,r);
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}
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}
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// The Semaev condition for a 3-reduced basis
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void red3(double b[3][3], int r[3][3])
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{
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double b11 = col_prod(b,0,0);
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double b22 = col_prod(b,1,1);
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double b12 = col_prod(b,0,1);
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double b13 = col_prod(b,0,2);
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double b23 = col_prod(b,1,2);
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double y2 =-(b23/b22-b12/b22*b13/b11)/(1-b12/b11*b12/b22);
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double y1 =-(b13/b11-b12/b11*b23/b22)/(1-b12/b11*b12/b22);
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int x1=0;
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int x2=0;
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double min = col_prod(b,2,2);
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int x1v[2];
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int x2v[2];
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x1v[0] = floor(y1); x1v[1] = x1v[0]+1;
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x2v[0] = floor(y2); x2v[1] = x2v[0]+1;
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for (int k=0;k<2;k++)
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for (int j=0;j<2;j++)
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{
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double a[3];
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a[0] = b[0][2] + x1v[k]*b[0][0] + x2v[j]*b[0][1];
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a[1] = b[1][2] + x1v[k]*b[1][0] + x2v[j]*b[1][1];
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a[2] = b[2][2] + x1v[k]*b[2][0] + x2v[j]*b[2][1];
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double val=a[0]*a[0]+a[1]*a[1]+a[2]*a[2];
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if (val<min)
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{
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min = val;
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x1 = x1v[k];
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x2 = x2v[j];
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}
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}
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if (x1 || x2)
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{
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b[0][2] += x1*b[0][0] + x2*b[0][1];
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b[1][2] += x1*b[1][0] + x2*b[1][1];
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b[2][2] += x1*b[2][0] + x2*b[2][1];
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r[0][2] += x1*r[0][0] + x2*r[0][1];
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r[1][2] += x1*r[1][0] + x2*r[1][1];
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r[2][2] += x1*r[2][0] + x2*r[2][1];
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greedy_recurse(b,r); // note the recursion step is here
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}
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}
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// the meat of the greedy reduction algorithm
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void greedy_recurse(double b[3][3], int r[3][3])
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{
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col_sort(b,r);
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red12(b,r);
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red3(b,r); // recursive caller
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}
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// set r (change of basis) to be identity then reduce basis and make it unique
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void greedy(double b[3][3],int r[3][3])
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{
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r[0][1]=r[0][2]=r[1][0]=r[1][2]=r[2][0]=r[2][1]=0;
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r[0][0]=r[1][1]=r[2][2]=1;
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greedy_recurse(b,r);
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make_unique(b,r);
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}
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// A reduced basis isn't unique. This procedure will make it
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// "more" unique. Degenerate cases are possible, but unlikely
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// with floating point math.
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void make_unique(double b[3][3], int r[3][3])
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{
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if (fabs(b[0][0]) < fabs(b[0][1]))
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{ col_swap(b,0,1); col_swap(r,0,1); }
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if (fabs(b[0][0]) < fabs(b[0][2]))
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{ col_swap(b,0,2); col_swap(r,0,2); }
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if (fabs(b[1][1]) < fabs(b[1][2]))
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{ col_swap(b,1,2); col_swap(r,1,2); }
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if (b[0][0] < 0){ neg_col(b,0); neg_col(r,0); }
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if (b[1][1] < 0){ neg_col(b,1); neg_col(r,1); }
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if (det(b) < 0){ neg_col(b,2); neg_col(r,2); }
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}
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}}
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