void matrix3::from_rotation(const float angle, const vector3& axis) {
float sin_angle, cos_angle;
fsincos(angle, sin_angle, cos_angle);
const vector3 axis2 = axis * axis;
const float l2 = (axis2.x * axis2.x) + (axis2.y * axis2.y) + (axis2.z * axis2.z);
const float l2_sqrt = sqrtf(l2);
if (approx_ne(l2, 0.f)) {
m[0][0] = (axis2.x + (axis2.y + axis2.z) * cos_angle) / l2;
m[0][1] = (axis.x * axis.y * (1 - cos_angle) - axis.z * l2_sqrt * sin_angle) / l2;
m[0][2] = (axis.x * axis.z * (1 - cos_angle) + axis.y * l2_sqrt * sin_angle) / l2;
m[1][0] = (axis.x * axis.y * (1 - cos_angle) + axis.z * l2_sqrt * sin_angle) / l2;
m[1][1] = (axis2.y + (axis2.x + axis2.z) * cos_angle) / l2;
m[1][2] = (axis.y * axis.z * (1 - cos_angle) - axis.x * l2_sqrt * sin_angle) / l2;
m[2][0] = (axis.x * axis.z * (1 - cos_angle) - axis.y * l2_sqrt * sin_angle) / l2;
m[2][1] = (axis.y * axis.z * (1 - cos_angle) + axis.x * l2_sqrt * sin_angle) / l2;
m[2][2] = (axis2.z + (axis2.x + axis2.y) * cos_angle) / l2;
}
else {
(*this) = matrix3(mth::IDENTITY_INITIALIZATION);
}
}
void UIRenderer::pushTransform(const vn::matrix3 &mat) {
if (m_transforms.empty()) {
m_transforms.push(mat);
} else {
m_transforms.push(matrix3(matrix3::c_nothing).set_by_product(mat, m_transforms.top()));
}
Device::instance().setViewTransform(m_transforms.top());
}
void UIRenderer::popTransform() {
vnassert(!m_transforms.empty());
m_transforms.pop();
if (m_transforms.empty()) {
Device::instance().setViewTransform(matrix3(matrix3::c_identity));
} else {
Device::instance().setViewTransform(m_transforms.top());
}
}
void rdlimitrot(int *t1, int *t2, float *maxangle, float *qx, float *qy, float *qz, float *qw)
{
checkragdoll;
ragdollskel::rotlimit &r = ragdoll->rotlimits.add();
r.tri[0] = *t1;
r.tri[1] = *t2;
r.maxangle = *maxangle * RAD;
r.middle = matrix3(quat(*qx, *qy, *qz, *qw));
}
vector<rigid_sys::body_info> rigid_sys::current_body(vector<dynamic_info> &status)
{
vector<body_info> cb;
for(int i=0;i<bodies.size();i++)
{
cb.push_back(bodies[i]);
for(int j=0;j<bodies[i].pos.size();j++)
cb[i].pos[j]=matrix3(status[i].q)*bodies[i].pos[j];
}
return cb;
}
void StaticMatrixTest::invert()
{
chemkit::StaticMatrix<chemkit::Float, 3, 3> matrix3;
matrix3(0, 0) = 1;
matrix3(0, 1) = 2;
matrix3(0, 2) = 3;
matrix3(1, 0) = 0;
matrix3(1, 1) = 1;
matrix3(1, 2) = 0;
matrix3(2, 0) = 4;
matrix3(2, 1) = 0;
matrix3(2, 2) = 4;
chemkit::StaticMatrix<chemkit::Float, 3, 3> inverse3 = matrix3.inverted();
QCOMPARE(inverse3(0, 0), chemkit::Float(-0.5));
QCOMPARE(inverse3(0, 1), chemkit::Float(1.0));
QCOMPARE(inverse3(0, 2), chemkit::Float(0.375));
QCOMPARE(inverse3(1, 0), chemkit::Float(0.0));
QCOMPARE(inverse3(1, 1), chemkit::Float(1.0));
QCOMPARE(inverse3(1, 2), chemkit::Float(0.0));
QCOMPARE(inverse3(2, 0), chemkit::Float(0.5));
QCOMPARE(inverse3(2, 1), chemkit::Float(-1.0));
QCOMPARE(inverse3(2, 2), chemkit::Float(-0.125));
}
matrix3 operator *(const matrix3& LM, const matrix3& RM)
{
const float L[3][3] =
{
LM[0][0],
LM[0][1],
LM[0][2],
LM[1][0],
LM[1][1],
LM[1][2],
LM[2][0],
LM[2][1],
LM[2][2]
};
const float R[3][3] =
{
RM[0][0],
RM[0][1],
RM[0][2],
RM[1][0],
RM[1][1],
RM[1][2],
RM[2][0],
RM[2][1],
RM[2][2]
};
return matrix3(
L[0][0] * R[0][0] + L[1][0] * R[0][1] + L[2][0] * R[0][2],
L[0][1] * R[0][0] + L[1][1] * R[0][1] + L[2][1] * R[0][2],
L[0][2] * R[0][0] + L[1][2] * R[0][1] + L[2][2] * R[0][2],
L[0][0] * R[1][0] + L[1][0] * R[1][1] + L[2][0] * R[1][2],
L[0][1] * R[1][0] + L[1][1] * R[1][1] + L[2][1] * R[1][2],
L[0][2] * R[1][0] + L[1][2] * R[1][1] + L[2][2] * R[1][2],
L[0][0] * R[2][0] + L[1][0] * R[2][1] + L[2][0] * R[2][2],
L[0][1] * R[2][0] + L[1][1] * R[2][1] + L[2][1] * R[2][2],
L[0][2] * R[2][0] + L[1][2] * R[2][1] + L[2][2] * R[2][2]
);
}
bool squares::apply_impulse(vector<dynamic_info> &status_temp,
vector<dynamic_info> &status,pair curr,vec p,vec n)
{
if(n==vec(0,0,0))
return false;
//set impulse
n=n.normalize();
vec ra=p-status[curr.a].x;
vec rb=p-status[curr.b].x;
matrix3 R(status[curr.a].q);
matrix3 I_inv_a=R*position[curr.a].Ibody_Inv*R.transpose();
vec omega_a=I_inv_a*status[curr.a].L;
vec p_dot_a=status[curr.a].P/position[curr.a].total_mass+cross(omega_a,ra);
R=matrix3(status[curr.b].q);
matrix3 I_inv_b=R*position[curr.b].Ibody_Inv*R.transpose();
vec omega_b=I_inv_b*status[curr.b].L;
vec p_dot_b=status[curr.b].P/position[curr.b].total_mass+cross(omega_b,rb);
number_t v_rel=n*(p_dot_a-p_dot_b);
if(v_rel>-1e-4)
return false;
number_t j=-(1+restitution_coef)*v_rel/(1/position[curr.a].total_mass+1/position[curr.b].total_mass
+n*cross((I_inv_a*cross(ra,n)),ra)
+n*cross((I_inv_b*cross(rb,n)),rb));
cout<<j<<endl;
vec P=j*n;
status_temp[curr.a].P=status_temp[curr.a].P+P;
status_temp[curr.b].P=status_temp[curr.b].P-P;
status_temp[curr.a].L=status_temp[curr.a].L+cross(ra,P);
status_temp[curr.b].L=status_temp[curr.b].L-cross(rb,P);
return true;
}
vector<rigid_sys::dynamic_info> rigid_sys::get_dynamic(number_t t,vector<dynamic_info> start)
{
vector<rigid_sys::dynamic_info> dynamic_dot;
for(int i=0;i<start.size();i++)
{
rigid_sys::dynamic_info change;
rigid_sys::dynamic_info origin;
origin=start[i];
//dx
change.x=origin.P/bodies[i].total_mass;
//dq
matrix3 R(origin.q);
matrix3 I_inv=R*bodies[i].Ibody_Inv*R.transpose();
vec omega=I_inv*origin.L;
change.q=quaternion(omega)*origin.q;
//dP
vec total_force(0,0,0);
for(int j=0;j<bodies[i].mass.size();j++)
total_force=total_force+extern_force(t,i,j);
change.P=total_force;
//dL
vec total_torque(0,0,0);
for(int j=0;j<bodies[i].mass.size();j++)
total_torque=total_torque+cross(matrix3(origin.q)*bodies[i].pos[j]-origin.x,extern_force(t,i,j));
change.L=total_torque;
dynamic_dot.push_back(change);
}
return dynamic_dot;
}
void matrix3x3Test()
{
printf("33¾ØÕó²âÊÔ\n");
S3D::Matrix3x3 matrix1;
S3D::Matrix3x3 matrix2(1,2,3,4,5,6,7,8,9);
S3D::Matrix3x3 matrix3(9,8,7,6,5,4,3,2,1);
S3D::Matrix3x3 matrix4;
S3D::print(&matrix1);
S3D::print(&matrix2);
PRINT_LINE;
S3D::print(&matrix2);
S3D::print(&matrix3);
PRINT_LINE;
matrix4=matrix2*matrix3;
S3D::print(&matrix4);
PRINT_LINE;
S3D::print(&matrix2);
S3D::print(&matrix3);
PRINT_LINE;
matrix2*=matrix3;
matrix2=S3D::Matrix3x3(1,2,3,4,5,6,6,80,9);
S3D::print(&matrix2);
{
printf("---ÇóÄæ¾ØÕó\n");
S3D::Matrix3x3 matrix5;
S3D::Matrix3x3Inverse(&matrix5,&matrix2);
matrix5=matrix5*matrix2;
S3D::print(&matrix5);
}
}
void UIRenderer::_begin() {
Device &device = Device::instance();
device.setViewTransform(matrix3(matrix3::c_identity));
device.setBlendMode(Device::kAlphaBlend);
}
namespace mth {
// ------------------------------------------------------------------------- //
// matrix3 //
// ------------------------------------------------------------------------- //
// static constants
const matrix3 matrix3::IDENTITY = matrix3(mth::IDENTITY_INITIALIZATION);
const matrix3 matrix3::ZERO = matrix3(mth::ZERO_INITIALIZATION);
// linear algebra
//TODO: tidy up
const matrix3 matrix3::inverse() const {
const float det = determinant();
l_assert_msg(approx_ne(det, 0.0f), "singular matrix!");
const float det_inv = 1.0f / det;
matrix3 inv(SKIP_INITIALIZATION);
inv.m[0][0] = (m[1][1] * m[2][2] - m[1][2] * m[2][1]) * det_inv;
inv.m[0][1] = (m[0][2] * m[2][1] - m[0][1] * m[2][2]) * det_inv;
inv.m[0][2] = (m[0][1] * m[1][2] - m[0][2] * m[1][1]) * det_inv;
inv.m[1][0] = (m[1][2] * m[2][0] - m[1][0] * m[2][2]) * det_inv;
inv.m[1][1] = (m[0][0] * m[2][2] - m[0][2] * m[2][0]) * det_inv;
inv.m[1][2] = (m[0][2] * m[1][0] - m[0][0] * m[1][2]) * det_inv;
inv.m[2][0] = (m[1][0] * m[2][1] - m[1][1] * m[2][0]) * det_inv;
inv.m[2][1] = (m[0][1] * m[2][0] - m[0][0] * m[2][1]) * det_inv;
inv.m[2][2] = (m[0][0] * m[1][1] - m[0][1] * m[1][0]) * det_inv;
return inv;
}
// conversion
//note: quaternion conversion math based on notes from
// http://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles
void matrix3::from_quaternion(const quaternion &q) {
//TODO: optimize
m[0][0] = 1.0f - 2.0f * q.y * q.y - 2.0f * q.z*q.z; m[0][1] = 2.0f * q.x * q.y - 2.0f * q.z * q.w; m[0][2] = 2.0f * q.x * q.z + 2.0f * q.y * q.w;
m[1][0] = 2.0f * q.x * q.y + 2.0f * q.z * q.w; m[1][1] = 1.0f - 2.0f * q.x * q.x - 2.0f * q.z * q.z; m[1][2] = 2.0f * q.y * q.z - 2.0f * q.x * q.w;
m[2][0] = 2.0f * q.x * q.z - 2.0f * q.y * q.w; m[2][1] = 2.0f * q.y * q.z + 2.0f * q.x * q.w; m[2][2] = 1.0f - 2.0f * q.x * q.x - 2.0f * q.y * q.y;
}
void matrix3::to_quaternion(quaternion &out_q) const {
out_q.from_matrix((*this));
}
//note: euler conversion math based on notes from
// http://www.euclideanspace.com/maths/geometry/rotations/euler/index.htm
// rotation matrix is constructed according to:
// R = ([Ry(heading)][Rz(attitude)])[Rx(bank)]
void matrix3::from_euler(const float heading, const float attitude, const float bank) {
float sin_heading, cos_heading, sin_attitude, cos_attitude, sin_bank, cos_bank;
fsincos(heading, sin_heading, cos_heading);
fsincos(attitude, sin_attitude, cos_attitude);
fsincos(bank, sin_bank, cos_bank);
m[0][0] = cos_heading * cos_attitude; m[0][1] = sin_heading * sin_bank - cos_heading * sin_attitude * cos_bank; m[0][2] = cos_heading * sin_attitude * sin_bank + sin_heading * cos_bank;
m[1][0] = sin_attitude; m[1][1] = cos_attitude * cos_bank; m[1][2] =-cos_attitude * sin_bank;
m[2][0] = -sin_heading * cos_attitude; m[2][1] = sin_heading * sin_attitude * cos_bank + cos_heading * sin_bank; m[2][2] =-sin_heading * sin_attitude * sin_bank + cos_heading * cos_bank;
}
void matrix3::to_euler(float &out_heading, float &out_attitude, float &out_bank) const {
// m20 in [~-1,~1]
if ((m[1][0] > mth::upb(-1.0f)) && (m[1][0] < mth::lwb(1.0f))) {
out_heading = atan2f(-m[2][0], m[0][0]);
// force m10 to [-1,1]
out_attitude = asinf(mth::clamp(m[1][0], -1.0f, 1.0f));
out_bank = atan2f(-m[1][2], m[1][1]);
// singularity at south or north pole
} else {
out_heading = atan2f(m[0][2], m[2][2]);
out_attitude = (m[1][0] < 0.0f) ? -PI / 2.0f : PI / 2.0f;
out_bank = 0;
}
}
void matrix3::from_rotation(const float angle, const vector3& axis) {
float sin_angle, cos_angle;
fsincos(angle, sin_angle, cos_angle);
const vector3 axis2 = axis * axis;
const float l2 = (axis2.x * axis2.x) + (axis2.y * axis2.y) + (axis2.z * axis2.z);
const float l2_sqrt = sqrtf(l2);
if (approx_ne(l2, 0.f)) {
m[0][0] = (axis2.x + (axis2.y + axis2.z) * cos_angle) / l2;
m[0][1] = (axis.x * axis.y * (1 - cos_angle) - axis.z * l2_sqrt * sin_angle) / l2;
m[0][2] = (axis.x * axis.z * (1 - cos_angle) + axis.y * l2_sqrt * sin_angle) / l2;
m[1][0] = (axis.x * axis.y * (1 - cos_angle) + axis.z * l2_sqrt * sin_angle) / l2;
m[1][1] = (axis2.y + (axis2.x + axis2.z) * cos_angle) / l2;
m[1][2] = (axis.y * axis.z * (1 - cos_angle) - axis.x * l2_sqrt * sin_angle) / l2;
m[2][0] = (axis.x * axis.z * (1 - cos_angle) - axis.y * l2_sqrt * sin_angle) / l2;
m[2][1] = (axis.y * axis.z * (1 - cos_angle) + axis.x * l2_sqrt * sin_angle) / l2;
m[2][2] = (axis2.z + (axis2.x + axis2.y) * cos_angle) / l2;
}
else {
(*this) = matrix3(mth::IDENTITY_INITIALIZATION);
}
}
} // namespace mth //
void StaticMatrixTest::determinant()
{
// 3x3 matrix filled with 2's
chemkit::StaticMatrix<chemkit::Float, 3, 3> matrix3;
matrix3.fill(2);
QCOMPARE(qRound(matrix3.determinant()), 0);
// another matrix
matrix3.fill(0);
matrix3(0, 0) = 6;
matrix3(0, 1) = 3;
matrix3(0, 2) = 2;
matrix3(1, 0) = 4;
matrix3(1, 1) = -3;
matrix3(1, 2) = 2;
matrix3(2, 0) = -1;
matrix3(2, 1) = 9;
matrix3(2, 2) = -2;
QCOMPARE(qRound(matrix3.determinant()), 12);
// change last row
matrix3(2, 0) = 0;
matrix3(2, 1) = 4;
matrix3(2, 2) = 0;
QCOMPARE(qRound(matrix3.determinant()), -16);
// change first row
matrix3(0, 0) = 0;
matrix3(0, 1) = 4;
matrix3(0, 2) = 0;
QCOMPARE(qRound(matrix3.determinant()), 0);
}
int main(int argc, char *argv[])
{
if (argc != 2)
{
puts("\a\n usage: math_tst matrix_file_name\n");
exit( 1 );
}
::printf("\n\n------------- MACHINE (CPU) DEPENDENT THINGS --------------\n\n");
// defining machine epsilon for different built in data types supported by C++
float float_eps = f_macheps(); // or use float.h values FLT_EPSILON
double double_eps = d_macheps(); // or use float.h values DBL_EPSILON
//long double long_double_eps = ld_macheps(); // or use float.h values LDBL_EPSILON
::printf("\n float macheps = %.20e \n",float_eps);
::printf(" double macheps = %.20e \n",double_eps);
//::printf(" long double macheps = %.20Le \n\n\n",long_double_eps);
// Usual tolerance is defined as square root of machine epsilon
float sqrt_f_eps = sqrt(f_macheps());
double sqrt_d_eps = sqrt(d_macheps());
//long double sqrt_ld_eps = sqrt(ld_macheps());
::printf("\n sqrt float macheps = %.20e \n",sqrt_f_eps);
::printf(" sqrt double macheps = %.20e \n",sqrt_d_eps);
//::printf(" sqrt long double macheps = %.20Le \n",sqrt_ld_eps);
::printf("\n\n---------------- MATRIX -----------------\n\n");
// New data type MATRIX is defined in such a way that it can be regarded
// as concrete data type.
// default constructor:
BJmatrix matrix1;
matrix1.print("m1","matrix1");
static double initvalue[] = {1.0, 2.0,
3.0, 4.0};
// constructor with matrix initialization
BJmatrix matrix2(2,2,initvalue);
matrix2.print("m2","matrix2");
// unit matrix initialization ( third order )
BJmatrix matrix3("I",3);
matrix3.print("m3","matrix3");
// matrix initialization from the file that is in predefined format!
BJmatrix m(argv[1]);
m.print("m","m is");
//====
// matrix assignment
BJmatrix first_matrix = m;
first_matrix.print("fm","first_matrix is");
// ---------------------------------
::printf("\n\n--------EIGENVECTORS and EIGENVALUES-----------\n");
static double symm_initvalue[] = {1.0, 0.0,
0.0, 0.0};
// constructor with matrix initialization
BJmatrix sym_matrix(2,2,symm_initvalue);
sym_matrix.print("m","Sym_matrix");
// Eigenvalues
BJvector eval = sym_matrix.eigenvalues();
eval.print("ev","Eigenvalues of sym_matrix");
// Eigenvectors
BJmatrix evect = sym_matrix.eigenvectors();
evect.print("ev","Eigenvectors of sym_matrix");
// matrix to file "test.$$$"
m.write_standard("test.$$$", "this is a test");
// equivalence of two matrices ( within tolerance sqrt(macheps) )
if ( first_matrix == m ) ::printf("\n first_matrix == m TRUE \n");
else ::printf("\a\n first_matrix == m NOTTRUE \n");
// matrix addition
BJmatrix m2 =m+m;
m2.print("m2","\nm+m = \n");
// equivalence of two matrices ( within tolerance sqrt(macheps) )
if ( m2 == m ) ::printf("\a\n m2 == m TRUE \n");
else ::printf("\n m2 == m NOTTRUE \n");
// matrix substraction
BJmatrix m3 =m-m;
m3.print("m3","\nm-m = \n");
// matrix multiplication
BJmatrix m4 = m*m;
m4.print("m4","\n m*m = \n");
// matrix transposition
BJmatrix mtran = m.transpose();
mtran.print("mt","transpose of m is");
// determinant of matrix
printf("determinant = %6.6f \n",m.determinant());
// matrix inversion
BJmatrix minv = m.inverse();
minv.print("mi","inverse of m is");
// check matrix inversion
( m * minv ).print("1"," m * m.inverse() ");
// minima, maxima, mean and variance values for matrix:
printf("min = %6.6f \n", m.mmin() );
printf("max = %6.6f \n", m.mmax() );
printf("mean = %6.6f \n", m.mean() );
printf("variance = %6.6f \n", m.variance() );
//====//===printf("\n\n----------------- SKY MATRIX ----------------------\n\n");
//====//===
//====//===// skymatrix as defined by K.J. Bathe in "FE procedures in Engineering Analysis"
//====//===// The example is from that book too!
//====//===
//====//===int maxa[] = { 1, 2, 4, 6, 8, 13};
//====//===double a[] = {2.0, 3.0, -2.0, 5.0, -2.0, 10.0, -3.0, 10.0, 4.0, 0.0, 0.0, -1.0};
//====//===double rhs[] = { 0.0, 1.0, 0.0, 0.0, 0.0 };
//====//===
//====//===// skymatrix constructor
//====//=== skymatrix first( 5, maxa, a);
//====//===
//====//===// simple print functions:
//====//=== first.full_print("\nfirst Sparse Skyline Matrix as FULL :\n ");
//====//=== first.lower_print("\nfirst Sparse Skyline Matrix :\n ");
//====//=== first.upper_print("\nfirst Sparse Skyline Matrix :\n ");
//====//===
//====//===// skymatrix assignment
//====//=== skymatrix second = first;
//====//=== second.full_print("\nsecond Sparse Skyline Matrix as FULL :\n ");
//====//===
//====//===// skymatrix third; // not defined !
//====//===// assignments:
//====//=== skymatrix third = second = first;
//====//===
//====//===// LDL factorization ( see K.J. Bathe's book! )
//====//=== first = first.v_ldl_factorize();
//====//===
//====//===// simple print functions:
//====//=== first.lower_print("\n lower first but factorized :\n ");
//====//=== first.full_print("\nfirst but factorized Sparse Skyline Matrix as FULL :\n ");
//====//=== first.upper_print("\n upper first but factorized :\n ");
//====//===
//====//===// right hand side reduction ( see K.J. Bathe's book! ) ( vetor of unknowns is rhs )
//====//=== first.d_reduce_r_h_s_l_v( rhs );
//====//=== int i=0;
//====//=== for( i=0 ; i<5 ; i++ )
//====//=== {
//====//=== printf("intermediate_rhs[%d] = %8.4f\n",i, rhs[i]);
//====//=== }
//====//===
//====//===// backsubstitution
//====//=== first.d_back_substitute ( rhs );
//====//=== for( i=0 ; i<5 ; i++ )
//====//=== {
//====//=== printf("solution_rhs[%d] = %8.4f\n",i, rhs[i]);
//====//=== }
//====//===
//====//===// minima, maxima and mean values for skymatrix:
//====//=== printf("min = %6.6f \n", first.mmin() );
//====//=== printf("max = %6.6f \n", first.mmax() );
//====//=== printf("mean = %6.6f \n", first.mean() );
//====//===//
//====//===//
//====//===
//====//===printf("\n\n----------------- PROFILE MATRIX ----------------------\n");
//====//===printf("----------------- EXAMPLE 1 ----------------------\n\n");
//====//===
//====//===// profilematrix as defined by O.C. Zienkiewicz and R.L. Taylor
//====//===// in "The FEM - fourth edition"
//====//===// The example is from: K.J. Bathe: "FE procedures in Engineering Analysis"
//====//===
//====//===int jp[] = { 0, 1, 2, 3, 7};
//====//===double au[] = { -2.0, -2.0, -3.0, -1.0, 0.0, 0.0, 4.0};
//====//===double ad[] = { 2.0, 3.0, 5.0, 10.0, 10.0 };
//====//===double* al = au;
//====//===double b[] = { 0.0, 1.0, 0.0, 0.0, 0.0 };
//====//===
//====//===// profile matrix constructor
//====//=== profilematrix first_profile( 5, jp, 'S', au, al, ad);
//====//===
//====//===// simple print function
//====//=== first_profile.full_print("\nfirst_profile Sparse Profile Matrix as FULL :\n ");
//====//===
//====//===// asignment operetor
//====//=== profilematrix second_profile = first_profile;
//====//=== second_profile.full_print("\nsecond_profile Sparse Profile Matrix as FULL :\n ");
//====//===
//====//===// triangularization ( Crout's method ); see: O.C. Zienkiewicz and R.L. Taylor:
//====//===// "The FEM - fourth edition"
//====//=== first_profile = first_profile.datri();
//====//=== first_profile.full_print("\nfirst but factorized Sparse Profile Matrix as FULL :\n ");
//====//=== ::printf("\n\n");
//====//===
//====//===// right hand side reduction and backsubstitution ( vector of unknowns is b )
//====//=== first_profile.dasol(b);
//====//=== for( i=0 ; i<5 ; i++ )
//====//=== {
//====//=== printf("solution_b[%d] = %8.4f\n",i, b[i]);
//====//=== }
//====//===
//====//=== printf("\n\nmin = %6.6f \n", first_profile.mmin() );
//====//=== printf("max = %6.6f \n", first_profile.mmax() );
//====//=== printf("mean = %6.6f \n", first_profile.mean() );
//====//===
//====//===
//====//===printf("\n----------------- EXAMPLE 2 ----------------------\n\n");
//====//===
//====//===// The example is from: O.C. Zienkiewicz and R.L. Taylor:
//====//===// "The FEM - fourth edition"
//====//===int jp2[] = { 0, 1, 3};
//====//===double au2[] = { 2.0, 1.0, 2.0};
//====//===double ad2[] = { 4.0, 4.0, 4.0};
//====//===double* al2 = au2;
//====//===double b2[] = { 0.0, 0.0, 1.0 };
//====//===
//====//===// profile matrix constructor
//====//=== profilematrix first_profile2( 3, jp2, 'S', au2, al2, ad2);
//====//===
//====//===// simple print function
//====//=== first_profile2.full_print("\nfirst_profile Sparse Profile Matrix as FULL :\n ");
//====//===
//====//===// asignment operetor
//====//=== profilematrix second_profile2 = first_profile2;
//====//=== second_profile2.full_print("\nsecond_profile Sparse Profile Matrix as FULL :\n ");
//====//===
//====//===// triangularization ( Crout's method ); see: O.C. Zienkiewicz and R.L. Taylor:
//====//===// "The FEM - fourth edition"
//====//=== profilematrix first_profile2TRI = first_profile2.datri();
//====//=== first_profile2TRI.full_print("\nfirst but factorized Sparse Profile Matrix as FULL :\n ");
//====//=== ::printf("\n\n");
//====//===
//====//===// right hand side reduction and backsubstitution ( vector of unknowns is b2 )
//====//=== first_profile2TRI.dasol( b2 );
//====//=== for( i=0 ; i<3 ; i++ )
//====//=== {
//====//=== printf("solution_b[%d] = %8.4f\n",i, b2[i]);
//====//=== }
//====//===
//====//=== printf("\n\nmin = %6.6f \n", first_profile2.mmin() );
//====//=== printf("max = %6.6f \n", first_profile2.mmax() );
//====//=== printf("mean = %6.6f \n", first_profile2.mean() );
//====//===
//====//===
//====//===printf("\n----------------- EXAMPLE 3 ----------------------\n\n");
//====//===
//====//===// The main advantage of profile solver as described in
//====//===// O.C. Zienkiewicz and R.L. Taylor: "The FEM - fourth edition"
//====//===// is that it supports nonsymetric matrices stored in profile format.
//====//===
//====//===int jp3[] = { 0, 1, 3};
//====//===double au3[] = { 2.0, 3.0, 2.0};
//====//===double ad3[] = { 4.0, 4.0, 4.0};
//====//===double al3[] = { 2.0, 1.0, 2.0};
//====//===double b3[] = { 1.0, 1.0, 1.0 };
//====//===
//====//===// nonsymetric profile matrix constructor
//====//=== profilematrix first_profile3( 3, jp3, 'N', au3, al3, ad3);
//====//===
//====//===// print out
//====//=== first_profile3.full_print("\nfirst_profile Sparse Profile Matrix as FULL :\n ");
//====//===
//====//===// triangularization ( Crout's method ); see: O.C. Zienkiewicz and R.L. Taylor:
//====//===// "The FEM - fourth edition"
//====//=== profilematrix first_profile3TRI = first_profile3.datri();
//====//=== first_profile3TRI.full_print("\nfirst but factorized \
//====//=== Sparse Profile Matrix as FULL :\n ");
//====//=== ::printf("\n\n");
//====//===
//====//===// right hand side reduction and backsubstitution ( vector of unknowns is b3 )
//====//=== first_profile3TRI.dasol( b3 );
//====//=== for( i=0 ; i<3 ; i++ )
//====//=== {
//====//=== printf("solution_b[%d] = %8.4f\n",i, b3
//====//=== [i]);
//====//=== }
//====//===
//====//=== printf("\n\nmin = %6.6f \n", first_profile3.mmin() );
//====//=== printf("max = %6.6f \n", first_profile3.mmax() );
//====//=== printf("mean = %6.6f \n", first_profile3.mean() );
//====//===
//====//===
//====printf("\n\n------------ VECTOR -------------------\n\n");
static double vect1[] = {2.0, 3.0, -2.0, 5.0, -2.0};
static double vect2[] = {1.0, 1.0, 1.0, 1.0, 1.0};
// vector constructor
BJvector first_vector( 5, vect1);
first_vector.print("f","\n\nfirst_vector vector\n");
// vector constructor
BJvector second_vector ( 5, vect2);
second_vector.print("s","\n\nsecond_vector vector\n");
// default constructor for vector
BJvector third_vector;
// vector assignment
third_vector = first_vector + second_vector;
third_vector.print("t","\nfirst_vector + second_vector = \n");
BJvector dot_product = first_vector.transpose() * second_vector;
// vector dot product;
// dot_product = first_vector.transpose() * second_vector;
//==== dot_product.print("d","\nfirst_vector * second_vector \n");
//====
printf("\n\n------------VECTOR AND MATRIX -----------------\n\n");
static double vect3[] = {1.0, 2.0, 3.0, 4.0, 5.0};
//default construcotrs
matrix res1;
matrix res2;
// matrix constructor
matrix vect_as_matrix(5, 1, vect3);
BJvector vect(5, vect3);
first_matrix.print("f","\n first matrix");
// vector and matrix are quite similar !
vect_as_matrix.print("vm","\n vector as matrix");
vect.print("v","\n vector");
// this should give the same answer! ( vector multiplication with matrix
res1 = vect_as_matrix.transpose() * first_matrix;
res2 = vect.transpose() * first_matrix;
res1.print("r1","\n result");
res2.print("r2","\n result");
::printf("\n\n----------------- TENSOR --------------------\n\n");
// tensor default construcotr
tensor t1;
t1.print("t1","\n tensor t1");
static double t2values[] = { 1,2,3,
4,5,6,
7,8,9 };
// tensor constructor with values assignments ( order 2 ; like matrix )
tensor t2( 2, def_dim_2, t2values);
t2.print("t2","\ntensor t2 (2nd-order with values assignement)");
tensor tst( 2, def_dim_2, t2values);
// t2 = t2*2.0;
// t2.print("t2x2","\ntensor t2 x 2.0");
t2.print("t2","\noriginal tensor t2 ");
tst.print("tst","\ntst ");
tensor t2xt2 = t2("ij")*t2("jk");
t2xt2.print("t2xt2","\ntensor t2 x t2");
tensor t2xtst = t2("ij")*tst("jk");
t2xtst.print("t2xtst","\ntensor t2 x tst");
t2.print("t2","\noriginal tensor t2 ");
static double Tvalues[] = { 1,2,3,
4,5,6,
7,8,9 };
tensor multTest( 2, def_dim_2, t2values);
double a = 5.0;
double b = a * multTest("ab"):
// tensor constructor with 0.0 assignment ( order 4 ; 4D array )
tensor ZERO(4,def_dim_4,0.0);
ZERO.print("z","\ntensor ZERO (4-th order with value assignment)");
static double t4val[] =
{ 11.100, 12.300, 13.100, 15.230, 16.450, 14.450, 17.450, 18.657, 19.340,
110.020, 111.000, 112.030, 114.034, 115.043, 113.450, 116.670, 117.009, 118.060,
128.400, 129.500, 130.060, 132.560, 133.000, 131.680, 134.100, 135.030, 136.700,
119.003, 120.400, 121.004, 123.045, 124.023, 122.546, 125.023, 126.043, 127.000,
137.500, 138.600, 139.000, 141.067, 142.230, 140.120, 143.250, 144.030, 145.900,
146.060, 147.700, 148.990, 150.870, 151.000, 149.780, 152.310, 153.200, 154.700,
164.050, 165.900, 166.003, 168.450, 169.023, 167.067, 170.500, 171.567, 172.500,
155.070, 156.800, 157.067, 159.000, 160.230, 158.234, 161.470, 162.234, 163.800,
173.090, 174.030, 175.340, 177.060, 178.500, 176.000, 179.070, 180.088, 181.200};
// tensor constructor with values assignment ( order 4 ; 4D array )
tensor t4(4, def_dim_4, t4val);
t4.print("t4","\ntensor t4 (4th-order with values assignment)");
// Some relevant notes from the Dec. 96 Draft (as pointed out by KAI folks):
//
// "A binary operator shall be implemented either by a non-static member
// function (_class.mfct_) with one parameter or by a non-member function
// with two parameters. Thus, for any binary operator @, x@y can be
// interpreted as either x.operator@(y) or operator@(x,y)." [13.5.2]
//
// "The order of evaluation of arguments [of a function call] is
// unspecified." [5.2.2 paragraph 9]
//
//
// SO THAT IS WHY I NEED THIS temp
// It is not evident which operator will be called up first: operator() or operator* .
//
//% ////////////////////////////////////////////////////////////////////////
// testing the multi products using indicial notation:
tensor tst1;
tensor temp = t2("ij") * t4("ijkl");
tst1 = temp("kl") * t4("klpq") * t2("pq");
tst1.print("tst1","\ntensor tst1 = t2(\"ij\")*t4(\"ijkl\")*t4(\"klpq\")*t2(\"pq\");");
// testing inverse function for 4. order tensor. Inverse for 4. order tensor
// is done by converting that tensor to matrix, inverting that matrix
// and finally converting matrix back to tensor
tensor t4inv_2 = t4.inverse_2();
t4inv_2.print("t4i","\ntensor t4 inverted_2");
tensor unity_2 = t4("ijkl")*t4inv_2("klpq");
unity_2.print("uno2","\ntensor unity_2");
//....................................................................
// There are several built in tensor types. One of them is:
// Levi-Civita permutation tensor
tensor e("e",3,def_dim_3);
e.print("e","\nLevi-Civita permutation tensor e");
// The other one is:
// Kronecker delta tensor
tensor I2("I", 2, def_dim_2);
I2.print("I2","\ntensor I2 (2nd order unit tensor: Kronecker Delta -> d_ij)");
// tensor multiplications
tensor I2I2 = I2("ij")*I2("ij");
I2I2.print("I2I2","\ntensor I2 * I2");
// tensor multiplications ( from right)
tensor I2PI = I2*PI;
I2PI.print("I2PI","\ntensor I2 * PI");
// tensor multiplications ( from left)
tensor PII2 = PI*I2;
PII2.print("PII2","\ntensor PI * I2");
// Unary minus
tensor mPII2 = -PII2;
mPII2.print("mPII2","\ntensor -PI * I2");
// trace function
double deltatrace = I2.trace();
::printf("\ntrace of Kronecker delta is %f \n",deltatrace);
// determinant of 2. order tensor only!
double deltadet = I2.determinant();
::printf("\ndeterminant of Kronecker delta is %f \n",deltadet);
// comparison of tensor ( again within sqrt(macheps) tolerance )
tensor I2again = I2;
if ( I2again == I2 ) ::printf("\n I2again == I2 TRUE (OK)\n");
else ::printf("\a\n\n\n I2again == I2 NOTTRUE \n\n\n");
//....................................................................
// some of the 4. order tensor used in conituum mechanics:
// the most general representation of the fourth order isotropic tensor
// includes the following fourth orther unit isotropic tensors:
tensor I_ijkl( 4, def_dim_4, 0.0 );
I_ijkl = I2("ij")*I2("kl");
// I_ijkl.print("ijkl","\ntensor I_ijkl = d_ij d_kl)");
tensor I_ikjl( 4, def_dim_4, 0.0 );
// I_ikjl = I2("ij")*I2("kl");
I_ikjl = I_ijkl.transpose0110();
// I_ikjl.print("ikjl","\ntensor I_ikjl) ");
tensor I_ikjl_inv_2 = I_ikjl.inverse_2();
// I_ikjl_inv_2.print("I_ikjl","\ntensor I_ikjl_inverse_2)");
if ( I_ikjl == I_ikjl_inv_2 ) ::printf("\n\n I_ikjl == I_ikjl_inv_2 !\n");
else ::printf("\a\n\n I_ikjl != I_ikjl_inv_2 !\n");
tensor I_iljk( 4, def_dim_4, 0.0 );
// I_ikjl = I2("ij")*I2("kl");
//.. I_iljk = I_ikjl.transpose0101();
I_iljk = I_ijkl.transpose0111();
// I_iljk.print("iljk","\ntensor I_iljk) ");
// To check out this three fourth-order isotropic tensor please see
// W.Michael Lai, David Rubin, Erhard Krempl
// " Introduction to Continuum Mechanics"
// QA808.2
// ISBN 0-08-022699-X
//....................................................................
// Multiplications between 4. order tensors
::printf("\n\n intermultiplications \n");
// tensor I_ijkl_I_ikjl = I_ijkl("ijkl")*I_ikjl("ikjl");
tensor I_ijkl_I_ikjl = I_ijkl("ijkl")*I_ikjl("ijkl");
I_ijkl_I_ikjl.print("i","\n\n I_ijkl*I_ikjl \n");
// tensor I_ijkl_I_iljk = I_ijkl("ijkl")*I_iljk("iljk");
tensor I_ijkl_I_iljk = I_ijkl("ijkl")*I_iljk("ijkl");
I_ijkl_I_iljk.print("i","\n\n I_ijkl*I_iljk \n");
// tensor I_ikjl_I_iljk = I_ikjl("ikjl")*I_iljk("iljk");
tensor I_ikjl_I_iljk = I_ikjl("ijkl")*I_iljk("ijkl");
I_ikjl_I_iljk.print("i","\n\n I_ikjl*I_iljk \n");
//....................................................................
// More multiplications between 4. order tensors
::printf("\n\n intermultiplications same with same \n");
tensor I_ijkl_I_ijkl = I_ijkl("ijkl")*I_ijkl("ijkl");
I_ijkl_I_ijkl.print("i","\n\n I_ijkl*I_ijkl \n");
tensor I_ikjl_I_ikjl = I_ikjl("ikjl")*I_ikjl("ikjl");
I_ikjl_I_ikjl.print("i","\n\n I_ikjl*I_ikjl \n");
tensor I_iljk_I_iljk = I_iljk("iljk")*I_iljk("iljk");
I_iljk_I_iljk.print("i","\n\n I_iljk*I_iljk \n");
// tensor additions ans scalar multiplications
// symmetric part of fourth oder unit isotropic tensor
tensor I4s = (I_ikjl+I_iljk)*0.5;
// tensor I4s = 0.5*(I_ikjl+I_iljk);
I4s.print("I4s","\n symmetric tensor I4s = (I_ikjl+I_iljk)*0.5 ");
// skew-symmetric part of fourth oder unit isotropic tensor
tensor I4sk = (I_ikjl-I_iljk)*0.5;
I4sk.print("I4sk","\n skew-symmetric tensor I4sk = (I_ikjl-I_iljk)*0.5 ");
// equvalence check ( they should be different )
if ( I4s == I4sk ) ::printf("\n\n I4s == I4sk !\n");
else ::printf("\a\n\n I4s != I4sk !\n");
//....................................................................
// e-delta identity ( see Lubliner ( QA931 .L939 1990) page 2.
// and Lai ( QA808.2 L3 1978) page 12 )
tensor ee = e("ijm")*e("klm");
ee.print("ee","\ntensor e_ijm*e_klm");
tensor id = ee - (I_ikjl - I_iljk);
if ( id == ZERO ) ::printf("\n\n e-delta identity HOLDS !! \n\n");
tensor ee1 = e("ilm")*e("jlm");
ee1.print("ee1","\n\n e(\"ilm\")*e(\"jlm\") \n");
tensor ee2 = e("ijk")*e("ijk");
ee2.print("ee2","\n\n e(\"ijk\")*e(\"ijk\") \n");
//....................................................................
// Linear Isotropic Elasticity Tensor
// building elasticity tensor
double Ey = 20000;
double nu = 0.2;
// building stiffness tensor in one command line
tensor E1 = (I_ijkl*((Ey*nu*2.0)/(2.0*(1.0+nu)*(1-2.0*nu)))) +
( (I_ikjl+I_iljk)*(Ey/(2.0*(1.0+nu))));
// building compliance tensor in one command line
tensor D1= (I_ijkl * (-nu/Ey)) + ( (I_ikjl+I_iljk) * ((1.0+nu)/(2.0*Ey)));
// multiplication between them
tensor test = E1("ijkl")*D1("klpq");
test.print("t","\n\n\n test = E1(\"ijkl\")*D1(\"klpq\") \n");
// and this should be equal to the symmetric part of 4. order unit tensor
if ( test == I4s ) ::printf("\n test == I4s TRUE ( up to sqrt(macheps())) \n");
else ::printf("\a\n\n\n test == I4s NOTTRUE ( up to sqrt(macheps())) \n\n\n");
//............Different Way...........................................
// Linear Isotropic Elasticity Tensor
double lambda = nu * Ey / (1. + nu) / (1. - 2. * nu);
double mu = Ey / (2. * (1. + nu));
::printf("\n lambda = %.12e\n",lambda);
::printf(" mu = %.12e\n",mu);
::printf("\nYoung Modulus = %.4e",Ey);
::printf("\nPoisson ratio = %.4e\n",nu);
::printf( " lambda + 2 mu = %.4e\n", (lambda + 2 * mu));
// building elasticity tensor
tensor E2 = (I_ijkl * lambda) + (I4s * (2. * mu));
E2.print("E2","tensor E2: elastic moduli Tensor = lambda*I2*I2+2*mu*I4s");
tensor E2inv = E2.inverse();
// building compliance tensor
tensor D2= (I_ijkl * (-nu/Ey)) + (I4s * (1./(2.*mu)));
D2.print("D2","tensor D2: compliance Tensor (I_ijkl * (-nu/Ey)) + (I4s * (1./2./mu))");
if ( E2inv == D2 ) ::printf("\n E2inv == D2 TRUE \n");
else ::printf("\a\n\n\n E2inv == D2 NOTTRUE \n\n\n");
if ( E1 == E2 ) ::printf("\n E1 == E2 TRUE \n");
else ::printf("\a\n\n\n E1 == E2 NOTTRUE \n\n\n");
if ( D1 == D2 ) ::printf("\n D1 == D2 TRUE \n");
else ::printf("\a\n\n\n D1 == D2 NOTTRUE \n\n\n");
// --------------
::printf("\n\n\n --------------------------TENSOR SCALAR MULTIPLICATION ----\n\n");
tensor EEEE1 = E2;
tensor EEEE2 = EEEE1*2.0;
EEEE1.print("E1","tensor EEEE1");
EEEE2.print("E2","tensor EEEE2");
//###printf("\n\n------------VECTOR AND SKYMATRIX -----------------\n\n");
//###// 1) use of member function full_val is of crucial importance here
//###// and it enable us to use skymatrix as if it is full matrix.
//###// There is however small time overhead but that can be optimized . . .
//###double vect4[] = {1.0, 2.0, 3.0, 4.0, 5.0};
//###
//###matrix res2;
//###vector vector2(5, vect4);
//###second.full_print();
//###vector2.print();
//###res2 = vector2.transpose() * second;
//###res2.print();
//###
//###
//###printf("\n\n------------ MATRIX AND SKYMATRIX -----------------\n\n");
//###// 1) use of member function full_val is of crucial importance here
//###// and it enable us to use skymatrix as if it is full matrix.
//###// There is however small time overhead but that can be optimized . . .
//###
//###matrix big_result;
//###big_result = first_matrix * second;
//###big_result.print();
exit(1);
// return 1;
}