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smd_math.h
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/* ----------------------------------------------------------------------
*
* *** Smooth Mach Dynamics ***
*
* This file is part of the USER-SMD 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.
*
* ----------------------------------------------------------------------- */
//test
#ifndef SMD_MATH_H_
#define SMD_MATH_H_
#include <Eigen/Eigen>
#include <iostream>
using namespace Eigen;
using namespace std;
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, x);
}
}
/*
* deviator of a tensor
*/
static inline Matrix3d Deviator(const Matrix3d M) {
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(Matrix3d M, Matrix3d &R, Matrix3d &T, bool scaleF) {
JacobiSVD<Matrix3d> svd(M, ComputeFullU | ComputeFullV); // SVD(A) = U S V*
Vector3d S_eigenvalues = svd.singularValues();
Matrix3d S = svd.singularValues().asDiagonal();
Matrix3d U = svd.matrixU();
Matrix3d V = svd.matrixV();
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(Matrix3d &M) {
//JacobiSVD < Matrix3d > svd(M, ComputeFullU | ComputeFullV);
JacobiSVD<Matrix3d> svd(M, ComputeFullU); // one Eigevector base is sufficient because matrix is square and symmetric
Vector3d singularValuesInv;
Vector3d singularValues = svd.singularValues();
//cout << "Here is the matrix V:" << endl << V * singularValues.asDiagonal() * U << endl;
//cout << "Its singular values are:" << endl << singularValues << endl;
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();
// JacobiSVD < Matrix3d > svd(M, ComputeFullU | ComputeFullV);
//
// Vector3d singularValuesInv;
// Vector3d singularValues = svd.singularValues();
//
// //cout << "Here is the matrix V:" << endl << V * singularValues.asDiagonal() * U << endl;
// //cout << "Its singular values are:" << endl << singularValues << endl;
//
// 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.matrixV().transpose();
}
/*
* test if two matrices are equal
*/
static inline double TestMatricesEqual(Matrix3d A, Matrix3d B, double eps) {
Matrix3d diff;
diff = A - B;
double norm = diff.norm();
if (norm > eps) {
printf("Matrices A and B are not equal! The L2-norm difference is: %g\n", norm);
cout << "Here is matrix A:" << endl << A << endl;
cout << "Here is matrix B:" << endl << B << endl;
}
return norm;
}
/* ----------------------------------------------------------------------
Limit eigenvalues of a matrix to upper and lower bounds.
------------------------------------------------------------------------- */
static inline Matrix3d LimitEigenvalues(Matrix3d S, double limitEigenvalue) {
/*
* compute Eigenvalues of matrix S
*/
SelfAdjointEigenSolver < 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;
Matrix3d V = es.eigenvectors();
Matrix3d S_diag = V.inverse() * S * V; // diagonalized input matrix
S_diag /= scale;
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;
Matrix3d V = es.eigenvectors();
Matrix3d S_diag = V.inverse() * S * V; // diagonalized input matrix
S_diag /= scale;
Matrix3d S_scaled = V * S_diag * V.inverse(); // undiagonalize matrix
return S_scaled;
}
} else { // limiting does not apply
return S;
}
}
static inline bool LimitMinMaxEigenvalues(Matrix3d &S, double min, double max) {
/*
* compute Eigenvalues of matrix S
*/
SelfAdjointEigenSolver < Matrix3d > es;
es.compute(S);
if ((es.eigenvalues().maxCoeff() > max) || (es.eigenvalues().minCoeff() < min)) {
Matrix3d S_diag = es.eigenvalues().asDiagonal();
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(Matrix3d &K) {
JacobiSVD<Matrix3d> svd(K, ComputeFullU | ComputeFullV);
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));
}
}
#endif /* SMD_MATH_H_ */

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