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/**
* @file model_manager.cc
*
* @author Alejandro M. Aragón <alejandro.aragon@epfl.ch>
*
* @date creation: Mon Jan 07 2013
* @date last modification: Tue May 07 2013
*
* @brief higher order object that deals with collections of models
*
* @section LICENSE
*
* Copyright (©) 2014 EPFL (Ecole Polytechnique Fédérale de Lausanne)
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* Akantu is free software: you can redistribute it and/or modify it under the
* terms of the GNU Lesser General Public License as published by the Free
* Software Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
* details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
*
*/
// akantu header files
#include "model_manager.hh"
__BEGIN_AKANTU__
#define SWAPD(a,b) do { double tmp = b ; b = a ; a = tmp ; } while(0)
uint32_t solve_quadratic(Real a, Real b, Real c,
Real *x0, Real *x1 )
{
#ifdef DEBUG_POLYSOLVER
std::cout << "solve_quadratic( "
<< a << ", "
<< b << ", "
<< c << " )\n";
#endif
// Handle linear case
if( a == 0.0 ) {
if( b == 0.0 ) {
#ifdef DEBUG_POLYSOLVER
std::cout << "linear with no roots\n";
#endif
return( 0 );
} else {
*x0 = -c / b;
#ifdef DEBUG_POLYSOLVER
std::cout << "linear with one root at x = " << *x0 << "\n";
#endif
return( 1 );
}
}
Real disc = b*b - 4.0*a*c;
if( disc > 0 ) {
if( b == 0 ) {
Real r = fabs( 0.5 * sqrt (disc) / a );
*x0 = -r;
*x1 = r;
} else {
Real sgnb = (b > 0 ? 1 : -1);
Real temp = -0.5 * (b + sgnb * sqrt (disc));
Real r1 = temp / a ;
Real r2 = c / temp ;
if( r1 < r2 ) {
*x0 = r1 ;
*x1 = r2 ;
} else {
*x0 = r2 ;
*x1 = r1 ;
}
}
return 2;
} else if( disc == 0 ) {
*x0 = -0.5 * b / a ;
*x1 = -0.5 * b / a ;
return 2;
} else
return 0;
}
/* Finds the real roots of x^4 + a x^3 + b x^2 + c x + d = 0
*/
//template<>
uint32_t Kinematic_traits<Consider_t>::solve_quartic(Real a, Real b, Real c, Real d,
Real *x0, Real *x1, Real *x2, Real *x3) {
Real u[3];
Real aa, pp, qq, rr, rc, sc, tc, mt;
Real w1r, w1i, w2r, w2i, w3r;
Real v[3], v1, v2, arg, theta;
Real disc, h;
int k1 = 0, k2 = 0;
Real zarr[4];
/* Deal easily with the cases where the quartic is degenerate. The
* ordering of solutions is done explicitly. */
if (0 == b && 0 == c)
{
if (0 == d)
{
if (a > 0)
{
*x0 = -a;
*x1 = 0.0;
*x2 = 0.0;
*x3 = 0.0;
}
else
{
*x0 = 0.0;
*x1 = 0.0;
*x2 = 0.0;
*x3 = -a;
}
return 4;
}
else if (0 == a)
{
if (d > 0)
{
return 0;
}
else
{
*x1 = sqrt (sqrt (-d));
*x0 = -(*x1);
return 2;
}
}
}
if (0.0 == c && 0.0 == d)
{
*x0=0.0;
*x1=0.0;
if( solve_quadratic( 1.0, a, b, x2, x3 ) == 0 ) {
mt=3;
} else {
mt=1;
}
}
else
{
/* For non-degenerate solutions, proceed by constructing and
* solving the resolvent cubic */
aa = a * a;
pp = b - (3.0/8.0) * aa;
qq = c - (1.0/2.0) * a * (b - (1.0/4.0) * aa);
rr = d - (1.0/4.0) * (a * c - (1.0/4.0) * aa * (b - (3.0/16.0) * aa));
rc = (1.0/2.0) * pp;
sc = (1.0/4.0) * ((1.0/4.0) * pp * pp - rr);
tc = -((1.0/8.0) * qq * (1.0/8.0) * qq);
/* This code solves the resolvent cubic in a convenient fashion
* for this implementation of the quartic. If there are three real
* roots, then they are placed directly into u[]. If two are
* complex, then the real root is put into u[0] and the real
* and imaginary part of the complex roots are placed into
* u[1] and u[2], respectively. Additionally, this
* calculates the discriminant of the cubic and puts it into the
* variable disc. */
{
Real qcub = (rc * rc - 3 * sc);
Real rcub = (2 * rc * rc * rc - 9 * rc * sc + 27 * tc);
Real Q = qcub / 9;
Real R = rcub / 54;
Real Q3 = Q * Q * Q;
Real R2 = R * R;
Real CR2 = 729 * rcub * rcub;
Real CQ3 = 2916 * qcub * qcub * qcub;
disc = (CR2 - CQ3) / 2125764.0;
if (0 == R && 0 == Q)
{
u[0] = -rc / 3;
u[1] = -rc / 3;
u[2] = -rc / 3;
}
else if (CR2 == CQ3)
{
Real sqrtQ = sqrt (Q);
if (R > 0)
{
u[0] = -2 * sqrtQ - rc / 3;
u[1] = sqrtQ - rc / 3;
u[2] = sqrtQ - rc / 3;
}
else
{
u[0] = -sqrtQ - rc / 3;
u[1] = -sqrtQ - rc / 3;
u[2] = 2 * sqrtQ - rc / 3;
}
}
else if (CR2 < CQ3)
{
Real sqrtQ = sqrt (Q);
Real sqrtQ3 = sqrtQ * sqrtQ * sqrtQ;
Real theta = acos (R / sqrtQ3);
if (R / sqrtQ3 >= 1.0) theta = 0.0;
{
Real norm = -2 * sqrtQ;
u[0] = norm * cos (theta / 3) - rc / 3;
u[1] = norm * cos ((theta + 2.0 * M_PI) / 3) - rc / 3;
u[2] = norm * cos ((theta - 2.0 * M_PI) / 3) - rc / 3;
}
}
else
{
Real sgnR = (R >= 0 ? 1 : -1);
Real modR = fabs (R);
Real sqrt_disc = sqrt (R2 - Q3);
Real A = -sgnR * pow (modR + sqrt_disc, 1.0 / 3.0);
Real B = Q / A;
Real mod_diffAB = fabs (A - B);
u[0] = A + B - rc / 3;
u[1] = -0.5 * (A + B) - rc / 3;
u[2] = -(sqrt (3.0) / 2.0) * mod_diffAB;
}
}
/* End of solution to resolvent cubic */
/* Combine the square roots of the roots of the cubic
* resolvent appropriately. Also, calculate 'mt' which
* designates the nature of the roots:
* mt=1 : 4 real roots (disc == 0)
* mt=2 : 0 real roots (disc < 0)
* mt=3 : 2 real roots (disc > 0)
*/
if (0.0 == disc)
u[2] = u[1];
if (0 >= disc)
{
mt = 2;
/* One would think that we could return 0 here and exit,
* since mt=2. However, this assignment is temporary and
* changes to mt=1 under certain conditions below.
*/
v[0] = fabs (u[0]);
v[1] = fabs (u[1]);
v[2] = fabs (u[2]);
v1 = std::max(std::max(v[0], v[1] ), v[2] );
/* Work out which two roots have the largest moduli */
k1 = 0, k2 = 0;
if (v1 == v[0])
{
k1 = 0;
v2 = std::max( v[1], v[2] );
}
else if (v1 == v[1])
{
k1 = 1;
v2 = std::max( v[0], v[2] );
}
else
{
k1 = 2;
v2 = std::max( v[0], v[1] );
}
if (v2 == v[0])
{
k2 = 0;
}
else if (v2 == v[1])
{
k2 = 1;
}
else
{
k2 = 2;
}
if (0.0 <= u[k1])
{
w1r=sqrt(u[k1]);
w1i=0.0;
}
else
{
w1r=0.0;
w1i=sqrt(-u[k1]);
}
if (0.0 <= u[k2])
{
w2r=sqrt(u[k2]);
w2i=0.0;
}
else
{
w2r=0.0;
w2i=sqrt(-u[k2]);
}
}
else
{
mt = 3;
if (0.0 == u[1] && 0.0 == u[2])
{
arg = 0.0;
}
else
{
arg = sqrt(sqrt(u[1] * u[1] + u[2] * u[2]));
}
theta = atan2(u[2], u[1]);
w1r = arg * cos(theta / 2.0);
w1i = arg * sin(theta / 2.0);
w2r = w1r;
w2i = -w1i;
}
/* Solve the quadratic to obtain the roots to the quartic */
w3r = qq / 8.0 * (w1i * w2i - w1r * w2r) /
(w1i * w1i + w1r * w1r) / (w2i * w2i + w2r * w2r);
h = a / 4.0;
zarr[0] = w1r + w2r + w3r - h;
zarr[1] = -w1r - w2r + w3r - h;
zarr[2] = -w1r + w2r - w3r - h;
zarr[3] = w1r - w2r - w3r - h;
/* Arrange the roots into the variables z0, z1, z2, z3 */
if (2 == mt)
{
if (u[k1] >= 0 && u[k2] >= 0)
{
mt = 1;
*x0 = zarr[0];
*x1 = zarr[1];
*x2 = zarr[2];
*x3 = zarr[3];
}
else
{
return 0;
}
}
else
{
*x0 = zarr[0];
*x1 = zarr[1];
}
}
/* Sort the roots as usual */
if (1 == mt)
{
/* Roots are all real, sort them by the real part */
if (*x0 > *x1)
SWAPD (*x0, *x1);
if (*x0 > *x2)
SWAPD (*x0, *x2);
if (*x0 > *x3)
SWAPD (*x0, *x3);
if (*x1 > *x2)
SWAPD (*x1, *x2);
if (*x2 > *x3) {
SWAPD (*x2, *x3);
if (*x1 > *x2)
SWAPD (*x1, *x2);
}
return 4;
}
else
{
/* 2 real roots */
if (*x0 > *x1)
SWAPD (*x0, *x1);
}
return 2;
}
__END_AKANTU__
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