src/MACHDYN/smd_math.h

/* -*- c++ -*- ----------------------------------------------------------
 *
 *                    *** Smooth Mach Dynamics ***
 *
 * This file is part of the MACHDYN package for LAMMPS.
 * Copyright (2014) Georg C. Ganzenmueller, georg.ganzenmueller@emi.fhg.de
 * Fraunhofer Ernst-Mach Institute for High-Speed Dynamics, EMI,
 * Eckerstrasse 4, D-79104 Freiburg i.Br, Germany.
 *
 * ----------------------------------------------------------------------- */

#ifndef SMD_MATH_H
#define SMD_MATH_H

#include <Eigen/Eigen>
#include <iostream>

namespace SMD_Math {
static inline void LimitDoubleMagnitude(double &x, const double limit)
{
  /*
         * if |x| exceeds limit, set x to limit with the sign of x
         */
  if (fabs(x) > limit) {    // limit delVdotDelR to a fraction of speed of sound
    x = limit * copysign(1.0, x);
  }
}

/*
 * deviator of a tensor
 */
static inline Eigen::Matrix3d Deviator(const Eigen::Matrix3d M)
{
  Eigen::Matrix3d eye;
  eye.setIdentity();
  eye *= M.trace() / 3.0;
  return M - eye;
}

/*
 * Polar Decomposition M = R * T
 * where R is a rotation and T a pure translation/stretch matrix.
 *
 * The decomposition is achieved using SVD, i.e. M = U S V^T,
 * where U = R V and S is diagonal.
 *
 *
 * For any physically admissible deformation gradient, the determinant of R must equal +1.
 * However, scenerios can arise, where the particles interpenetrate and cause inversion, leading to a determinant of R equal to -1.
 * In this case, the inversion direction is heuristically identified with the eigenvector of the smallest entry of S, which should work for most cases.
 * The sign of this corresponding eigenvalue is flipped, the original matrix M is recomputed using the flipped S, and the rotation and translation matrices are
 * obtained again from an SVD. The rotation should proper now, i.e., det(R) = +1.
 */

static inline bool PolDec(Eigen::Matrix3d M, Eigen::Matrix3d &R, Eigen::Matrix3d &T, bool scaleF)
{

  Eigen::JacobiSVD<Eigen::Matrix3d> svd(
      M, Eigen::ComputeFullU | Eigen::ComputeFullV);    // SVD(A) = U S V*
  Eigen::Vector3d S_eigenvalues = svd.singularValues();
  Eigen::Matrix3d S = svd.singularValues().asDiagonal();
  Eigen::Matrix3d U = svd.matrixU();
  Eigen::Matrix3d V = svd.matrixV();
  Eigen::Matrix3d eye;
  eye.setIdentity();

  // now do polar decomposition into M = R * T, where R is rotation
  // and T is translation matrix
  R = U * V.transpose();
  T = V * S * V.transpose();

  if (R.determinant() < 0.0) {    // this is an improper rotation
    // identify the smallest entry in S and flip its sign
    int imin;
    S_eigenvalues.minCoeff(&imin);
    S(imin, imin) *= -1.0;

    R = M * V * S.inverse() * V.transpose();    // recompute R using flipped stretch eigenvalues
  }

  /*
         * scale S to avoid small principal strains
         */

  if (scaleF) {
    double min = 0.3;    // 0.3^2 = 0.09, should suffice for most problems
    double max = 2.0;
    for (int i = 0; i < 3; i++) {
      if (S(i, i) < min) {
        S(i, i) = min;
      } else if (S(i, i) > max) {
        S(i, i) = max;
      }
    }
    T = V * S * V.transpose();
  }

  if (R.determinant() > 0.0) {
    return true;
  } else {
    return false;
  }
}

/*
 * Pseudo-inverse via SVD
 */

static inline void pseudo_inverse_SVD(Eigen::Matrix3d &M)
{

  Eigen::JacobiSVD<Eigen::Matrix3d> svd(
      M,
      Eigen::
          ComputeFullU);    // one Eigevector base is sufficient because matrix is square and symmetric

  Eigen::Vector3d singularValuesInv;
  Eigen::Vector3d singularValues = svd.singularValues();

  double pinvtoler =
      1.0e-16;    // 2d machining example goes unstable if this value is increased (1.0e-16).
  for (int row = 0; row < 3; row++) {
    if (singularValues(row) > pinvtoler) {
      singularValuesInv(row) = 1.0 / singularValues(row);
    } else {
      singularValuesInv(row) = 0.0;
    }
  }

  M = svd.matrixU() * singularValuesInv.asDiagonal() * svd.matrixU().transpose();
}

/*
 * test if two matrices are equal
 */
static inline double TestMatricesEqual(Eigen::Matrix3d A, Eigen::Matrix3d B, double eps)
{
  Eigen::Matrix3d diff;
  diff = A - B;
  double norm = diff.norm();
  if (norm > eps) {
    std::cout << "Matrices A and B are not equal! The L2-norm difference is: " << norm << "\n"
              << "Here is matrix A:\n"
              << A << "\n"
              << "Here is matrix B:\n"
              << B << std::endl;
  }
  return norm;
}

/* ----------------------------------------------------------------------
 Limit eigenvalues of a matrix to upper and lower bounds.
 ------------------------------------------------------------------------- */

static inline Eigen::Matrix3d LimitEigenvalues(Eigen::Matrix3d S, double limitEigenvalue)
{

  /*
         * compute Eigenvalues of matrix S
         */
  Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> es;
  es.compute(S);

  double max_eigenvalue = es.eigenvalues().maxCoeff();
  double min_eigenvalue = es.eigenvalues().minCoeff();
  double amax_eigenvalue = fabs(max_eigenvalue);
  double amin_eigenvalue = fabs(min_eigenvalue);

  if ((amax_eigenvalue > limitEigenvalue) || (amin_eigenvalue > limitEigenvalue)) {
    if (amax_eigenvalue > amin_eigenvalue) {    // need to scale with max_eigenvalue
      double scale = amax_eigenvalue / limitEigenvalue;
      Eigen::Matrix3d V = es.eigenvectors();
      Eigen::Matrix3d S_diag = V.inverse() * S * V;    // diagonalized input matrix
      S_diag /= scale;
      Eigen::Matrix3d S_scaled = V * S_diag * V.inverse();    // undiagonalize matrix
      return S_scaled;
    } else {    // need to scale using min_eigenvalue
      double scale = amin_eigenvalue / limitEigenvalue;
      Eigen::Matrix3d V = es.eigenvectors();
      Eigen::Matrix3d S_diag = V.inverse() * S * V;    // diagonalized input matrix
      S_diag /= scale;
      Eigen::Matrix3d S_scaled = V * S_diag * V.inverse();    // undiagonalize matrix
      return S_scaled;
    }
  } else {    // limiting does not apply
    return S;
  }
}

static inline bool LimitMinMaxEigenvalues(Eigen::Matrix3d &S, double min, double max)
{

  /*
         * compute Eigenvalues of matrix S
         */
  Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> es;
  es.compute(S);

  if ((es.eigenvalues().maxCoeff() > max) || (es.eigenvalues().minCoeff() < min)) {
    Eigen::Matrix3d S_diag = es.eigenvalues().asDiagonal();
    Eigen::Matrix3d V = es.eigenvectors();
    for (int i = 0; i < 3; i++) {
      if (S_diag(i, i) < min) {
        //printf("limiting eigenvalue %f --> %f\n", S_diag(i, i), min);
        //printf("these are the eigenvalues of U: %f %f %f\n", es.eigenvalues()(0), es.eigenvalues()(1), es.eigenvalues()(2));
        S_diag(i, i) = min;
      } else if (S_diag(i, i) > max) {
        //printf("limiting eigenvalue %f --> %f\n", S_diag(i, i), max);
        S_diag(i, i) = max;
      }
    }
    S = V * S_diag * V.inverse();    // undiagonalize matrix
    return true;
  } else {
    return false;
  }
}

static inline void reconstruct_rank_deficient_shape_matrix(Eigen::Matrix3d &K)
{

  Eigen::JacobiSVD<Eigen::Matrix3d> svd(K, Eigen::ComputeFullU | Eigen::ComputeFullV);
  Eigen::Vector3d singularValues = svd.singularValues();

  for (int i = 0; i < 3; i++) {
    if (singularValues(i) < 1.0e-8) { singularValues(i) = 1.0; }
  }

  //              int imin;
  //              double minev = singularValues.minCoeff(&imin);
  //
  //              printf("min eigenvalue=%f has index %d\n", minev, imin);
  //              Vector3d singularVec = U.col(0).cross(U.col(1));
  //              cout << "the eigenvalues are " << endl << singularValues << endl;
  //              cout << "the singular vector is " << endl << singularVec << endl;
  //
  //              // reconstruct original K
  //
  //              singularValues(2) = 1.0;

  K = svd.matrixU() * singularValues.asDiagonal() * svd.matrixV().transpose();
  //cout << "the reconstructed K is " << endl << K << endl;
  //exit(1);
}

/* ----------------------------------------------------------------------
 helper functions for crack_exclude
 ------------------------------------------------------------------------- */
static inline bool IsOnSegment(double xi, double yi, double xj, double yj, double xk, double yk)
{
  return (xi <= xk || xj <= xk) && (xk <= xi || xk <= xj) && (yi <= yk || yj <= yk) &&
      (yk <= yi || yk <= yj);
}

static inline char ComputeDirection(double xi, double yi, double xj, double yj, double xk,
                                    double yk)
{
  double a = (xk - xi) * (yj - yi);
  double b = (xj - xi) * (yk - yi);
  return a < b ? -1.0 : a > b ? 1.0 : 0;
}

/** Do line segments (x1, y1)--(x2, y2) and (x3, y3)--(x4, y4) intersect? */
static inline bool DoLineSegmentsIntersect(double x1, double y1, double x2, double y2, double x3,
                                           double y3, double x4, double y4)
{
  char d1 = ComputeDirection(x3, y3, x4, y4, x1, y1);
  char d2 = ComputeDirection(x3, y3, x4, y4, x2, y2);
  char d3 = ComputeDirection(x1, y1, x2, y2, x3, y3);
  char d4 = ComputeDirection(x1, y1, x2, y2, x4, y4);
  return (((d1 > 0 && d2 < 0) || (d1 < 0 && d2 > 0)) &&
          ((d3 > 0 && d4 < 0) || (d3 < 0 && d4 > 0))) ||
      (d1 == 0 && IsOnSegment(x3, y3, x4, y4, x1, y1)) ||
      (d2 == 0 && IsOnSegment(x3, y3, x4, y4, x2, y2)) ||
      (d3 == 0 && IsOnSegment(x1, y1, x2, y2, x3, y3)) ||
      (d4 == 0 && IsOnSegment(x1, y1, x2, y2, x4, y4));
}

}    // namespace SMD_Math

#endif /* SMD_MATH_H_ */