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infcoeff.c
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/*==================================================
SET TABSTOPS AT EVERY FOUR SPACES FOR PROPER DISPLAY
====================================================*/
/*****************************************************************************
* FILE: infcoeff.c
* DATE: June, 1993
* BY: Andrew L. Thomas
*
* The following functions calculate displacement and strain influence
* coefficients for the angular dislocation as described by Comninou &
* Dunders (1975). The contributions from the dislocation and its image
* have been split into separate equations, so that whole- and half-space
* problems may be addressed with a single set of functions.
*****************************************************************************/
/***************************** Includes/Defines *****************************/
#include <math.h>
#include "infcoeff.h"
#include "safetan.h"
#include "pi.h"
/*************************** Function Declarations *************************/
#if defined(__STDC__) || defined(ANSI) /* ANSI */
double reduce_angle(double angle);
#else
double reduce_angle();
#endif
/*********************** Function: comninou_displ_ics ***********************
* Calculates displacement influence coefficients for the angular dislocation.
*
* In: y - coords of point at which to calc inf coeffs
* a - depth to angular dislocation vertex
* beta - angle subtended by dislocation
* pr - poisson's ratio
* half_space - half/whole-space flag
* Out: displ_ic[i][j] - jth component of displ at y due to unit ith Burgers
* vector component
*****************************************************************************/
#if defined(__STDC__) || defined(ANSI) /* ANSI */
void comninou_displ_ics(double y[3], double a, double beta, double pr,int half_space, double displ_ic[3][3])
#else
void comninou_displ_ics(y, a, beta, pr, half_space, displ_ic)
double y[3];
double a;
double beta;
double pr;
int half_space;
double displ_ic[3][3];
#endif
{
/*double trash1[3][3];*/
/*double trash2[3][3];*/
double dic2[3][3];
double term1, term2, term3;
int i, j;
double displ_ic_ft[3][3];
double y1 = y[0];
double y2 = y[1];
double y3 = y[2];
double sinb = sin(beta);
double cosb = cos(beta);
double cotb = 1/tan(beta);
double y3b = y3 + 2*a;
double z1 = y1*cosb - y3*sinb;
double z3 = y1*sinb + y3*cosb;
double r = sqrt(y1*y1 + y2*y2 + y3*y3);
double z1b = y1*cosb + y3b*sinb;
double z3b = -y1*sinb + y3b*cosb;
double rb = sqrt(y1*y1 + y2*y2 + y3b*y3b);
/* For debugging only
---------------------*/
/*
double f = reduce_angle(-safe_atan2(y2,y1) + safe_atan2(y2,z1) +
safe_atan2(y2*r*sinb,(y1*z1+(y2*y2)*cosb)));
double fb = reduce_angle(-safe_atan2(y2,y1) + safe_atan2(y2,z1b) +
safe_atan2(y2*rb*sinb,(y1*z1b+(y2*y2)*cosb)));
*/
double f = -safe_atan2(y2,y1) + safe_atan2(y2,z1) +
safe_atan2(y2*r*sinb,(y1*z1+(y2*y2)*cosb));
double fb = -safe_atan2(y2,y1) + safe_atan2(y2,z1b) +
safe_atan2(y2*rb*sinb,(y1*z1b+(y2*y2)*cosb));
/* Dislocation-Induced Displacements
------------------------------------*/
displ_ic[0][0] =
-y1*y2*(1/(r*(r-y3))) -
y2*cosb*((r*sinb-y1)/(r*(r-z3)));
displ_ic_ft[0][0] = 2*(1-pr)*(f);
displ_ic[0][1] =
(1-2*pr)*(log(r-y3) - cosb * (log(r-z3))) -
(y2*y2)*(1/(r*(r-y3)) - cosb*(1/(r*(r-z3))));
displ_ic_ft[0][1] = 0.0;
displ_ic[0][2] =
y2 * (1/r - cosb * ((r*cosb - y3)/(r*(r-z3))));
displ_ic_ft[0][2] = 0.0;
displ_ic[1][0] =
(-(1-2*pr))*(log(r-y3)-cosb*(log(r-z3))) +
(y1*y1)*(1/(r*(r-y3))) + z1*(r*sinb-y1)/(r*(r-z3));
displ_ic_ft[1][0] = 0.0;
displ_ic[1][1] =
y1*y2*(1/(r*(r-y3))) - y2*(z1/(r*(r-z3)));
displ_ic_ft[1][1] = 2*(1-pr)*(f);
displ_ic[1][2] =
-(1-2*pr)*sinb*(log(r-z3)) -
y1*(1/r) + z1*(r*cosb-y3)/(r*(r-z3));
displ_ic_ft[1][2] = 0.0;
displ_ic[2][0] =
y2*sinb*((r*sinb-y1)/(r*(r-z3)));
displ_ic_ft[2][0] = 0.0;
displ_ic[2][1] =
(1-2*pr)*sinb*(log(r-z3)) -
(y2*y2)*sinb*(1/(r*(r-z3)));
displ_ic_ft[2][1] = 0.0;
displ_ic[2][2] =
y2*sinb*((r*cosb-y3)/(r*(r-z3)));
displ_ic_ft[2][2] = 2*(1-pr)*(f);
if (half_space) {
/* Image-Induced Displacements
------------------------------*/
displ_ic[0][0] +=
-y1*y2*(1/(rb*(rb+y3b))) -
y2*cosb*((rb*sinb-y1)/(rb*(rb+z3b)));
displ_ic_ft[0][0] += 2*(1-pr)*(fb);
displ_ic[0][1] +=
(1-2*pr)*(log(rb+y3b) - cosb * (log(rb+z3b))) -
(y2*y2)*(1/(rb*(rb+y3b)) - cosb*(1/(rb*(rb+z3b))));
displ_ic_ft[0][1] += 0.0;
displ_ic[0][2] +=
y2 * (-1/rb - cosb * (-(rb*cosb + y3b)/(rb*(rb+z3b))));
displ_ic_ft[0][2] += 0.0;
displ_ic[1][0] +=
(-(1-2*pr))*(log(rb+y3b)-cosb*(log(rb+z3b))) +
(y1*y1)*(1/(rb*(rb+y3b))) + z1b*(rb*sinb-y1)/(rb*(rb+z3b));
displ_ic_ft[1][0] += 0.0;
displ_ic[1][1] +=
y1*y2*(1/(rb*(rb+y3b))) - y2*(z1b/(rb*(rb+z3b)));
displ_ic_ft[1][1] += 2*(1-pr)*(fb);
displ_ic[1][2] +=
-(1-2*pr)*sinb*(-log(rb+z3b)) -
y1*(-1/rb) - z1b*(rb*cosb+y3b)/(rb*(rb+z3b));
displ_ic_ft[1][2] += 0.0;
displ_ic[2][0] +=
y2*sinb*((rb*sinb - y1)/(rb*(rb+z3b)));
displ_ic_ft[2][0] += 0.0;
displ_ic[2][1] +=
(1-2*pr)*sinb*(log(rb+z3b)) -
(y2*y2)*sinb*(1/(rb*(rb+z3b)));
displ_ic_ft[2][1] += 0.0;
displ_ic[2][2] +=
y2*sinb*(-(rb*cosb+y3b)/(rb*(rb+z3b)));
displ_ic_ft[2][2] += 2*(1-pr)*(-fb);
/* Corrective Displacements
---------------------------*/
displ_ic[0][0] += 2.0*(
/* term1 */
(1-2*pr)*y2/(rb+y3b) *
((1-2*pr-a/rb)*cotb - y1/(rb+y3b)*(pr+a/rb)) +
/* term2 */
(1-2*pr)*y2*cosb*cotb/(rb+z3b)*(cosb+a/rb) +
/* term3 */
a*y2*(y3b-a)*cotb/pow(rb,3.0) +
/* term4 */
y2*(y3b-a)/(rb*(rb+y3b))*(-(1-2*pr)*cotb + y1/(rb+y3b) *
(2*pr+a/rb) + a*y1/(rb*rb)) +
/* term5 */
y2*(y3b-a)/(rb*(rb+z3b))*(cosb/(rb+z3b)*((rb*cosb+y3b) *
((1-2*pr)*cosb-a/rb)*cotb +
2*(1-pr)*(rb*sinb-y1)*cosb) -
a*y3b*cosb*cotb/(rb*rb))
);
displ_ic_ft[0][0] += 2* -2*(1-pr)*(1-2*pr)*fb*(cotb*cotb);
displ_ic[0][1] += 2.0*(
/* term1 */
(1-2*pr)*((2*(1-pr)*(cotb*cotb)-pr)*log(rb+y3b) -
(2*(1-pr)*(cotb*cotb)+1-2*pr)*cosb*log(rb+z3b)) -
/* term2 */
(1-2*pr)/(rb+y3b)*(y1*cotb*(1-2*pr-a/rb) + pr*y3b - a +
(y2*y2)/(rb+y3b)*(pr+a/rb)) -
/* term3 */
(1-2*pr)*z1b*cotb/(rb+z3b)*(cosb+a/rb) -
a*y1*(y3b-a)*cotb/(pow(rb,3.0)) +
/* term4 */
(y3b-a)/(rb+y3b)*(-2*pr + 1/rb*((1-2*pr)*y1*cotb-a) +
(y2*y2)/(rb*(rb+y3b))*(2*pr+a/rb)+a*(y2*y2)/pow(rb,3.0)) +
/* term5 */
(y3b-a)/(rb+z3b)*((cosb*cosb) -
1/rb*((1-2*pr)*z1b*cotb+a*cosb) +
a*y3b*z1b*cotb/pow(rb,3.0) - 1/(rb*(rb+z3b)) *
((y2*y2)*(cosb*cosb) - a*z1b*cotb/rb*(rb*cosb+y3b)))
);
displ_ic_ft[0][1] += 0.0;
displ_ic[0][2] += 2.0*(
/* term1 */
2*(1-pr)*(y2/(rb+y3b)*(2*pr+a/rb) -
y2*cosb/(rb+z3b)*(cosb+a/rb)) +
/* term2 */
y2*(y3b-a)/rb*(2*pr/(rb+y3b)+a/(rb*rb)) +
/* term3 */
y2*(y3b-a)*cosb/(rb*(rb+z3b))*
(1-2*pr-(rb*cosb+y3b)/(rb+z3b) *
(cosb + a/rb) - a*y3b/(rb*rb))
);
displ_ic_ft[0][2] += 2* 2*(1-pr)*((1-2*pr)*fb*cotb);
displ_ic[1][0] += 2.0*(
/* term1 */
(1-2*pr)*((2*(1-pr)*(cotb*cotb)+pr)*log(rb+y3b) -
(2*(1-pr)*(cotb*cotb)+1)*cosb*log(rb+z3b)) +
/* term2 */
(1-2*pr)/(rb+y3b)*
(-(1-2*pr)*y1*cotb+pr*y3b-a+a*y1*cotb/rb +
(y1*y1)/(rb+y3b)*(pr+a/rb)) -
/* term3 */
(1-2*pr)*cotb/(rb+z3b)*(z1b*cosb -
a*(rb*sinb-y1)/(rb*cosb)) -
/* term4 */
a*y1*(y3b-a)*cotb/pow(rb,3.0) +
/* term5 */
(y3b-a)/(rb+y3b)*(2*pr + 1/rb*((1-2*pr)*y1*cotb+a) -
(y1*y1)/(rb*(rb+y3b))*(2*pr+a/rb) - a*(y1*y1)/pow(rb,3.0)) +
/* term6 */
(y3b-a)*cotb/(rb+z3b)*
(-cosb*sinb+a*y1*y3b/(pow(rb,3.0)*cosb) +
(rb*sinb-y1)/rb*(2*(1-pr)*cosb -
(rb*cosb+y3b)/(rb+z3b) *
(1+a/(rb*cosb))))
);
displ_ic_ft[1][0] += 0.0;
displ_ic[1][1] += 2.0*(
/* term1 */
(1-2*pr)*y2/(rb+y3b)*(-(1-2*pr-a/rb)*cotb +
y1/(rb+y3b)*(pr+a/rb)) -
/* term2 */
(1-2*pr)*y2*cotb/(rb+z3b)*(1+a/(rb*cosb)) -
/* term3 */
a*y2*(y3b-a)*cotb/pow(rb,3.0) +
/* term4 */
y2*(y3b-a)/(rb*(rb+y3b))*((1-2*pr)*cotb - 2*pr*y1/(rb+y3b) -
a*y1/rb*(1/rb+1/(rb+y3b))) +
/* term5 */
y2*(y3b-a)*cotb/(rb*(rb+z3b))*(-2*(1-pr)*cosb +
(rb*cosb+y3b)/(rb+z3b)*(1+a/(rb*cosb)) +
a*y3b/((rb*rb)*cosb))
);
displ_ic_ft[1][1] += 2* 2*(1-pr)*(1-2*pr)*fb*(cotb*cotb);
displ_ic[1][2] += 2.0*(
/* term1 */
-2*(1-pr)*(1-2*pr)*cotb *
(log(rb+y3b)-cosb*log(rb+z3b)) -
/* term2 */
2*(1-pr)*y1/(rb+y3b)*(2*pr+a/rb) +
/* term3 */
2*(1-pr)*z1b/(rb+z3b)*(cosb+a/rb) +
/* term4 */
(y3b-a)/rb*((1-2*pr)*cotb-2*pr*y1/(rb+y3b)-a*y1/(rb*rb)) -
/* term5 */
(y3b-a)/(rb+z3b)*(cosb*sinb +
(rb*cosb+y3b)*cotb/rb*(2*(1-pr)*cosb -
(rb*cosb+y3b)/(rb+z3b)) +
a/rb*(sinb - y3b*z1b/(rb*rb) -
z1b*(rb*cosb+y3b)/(rb*(rb+z3b))))
);
displ_ic_ft[1][2] += 0.0;
term1 =
(1-2*pr)*(y2/(rb+y3b)*(1+a/rb) -
y2*cosb/(rb+z3b)*(cosb+a/rb));
term2 = y2*(y3b-a)/rb*(a/(rb*rb) + 1/(rb+y3b));
term3 = y2*(y3b-a)*cosb/(rb*(rb+z3b))*
((rb*cosb+y3b)/(rb+z3b)*(cosb+a/rb) +
a*y3b/(rb*rb));
displ_ic[2][0] += 2.0*(
/* term1 */
(1-2*pr)*(y2/(rb+y3b)*(1+a/rb) -
y2*cosb/(rb+z3b)*(cosb+a/rb)) -
/* term2 */
y2*(y3b-a)/rb*(a/(rb*rb) + 1/(rb+y3b)) +
/* term3 */
y2*(y3b-a)*cosb/(rb*(rb+z3b))*
((rb*cosb+y3b)/(rb+z3b)*(cosb+a/rb) +
a*y3b/(rb*rb))
);
displ_ic_ft[2][0] += 0.0;
displ_ic[2][1] += 2.0*(
/* term1 */
(1-2*pr)*(-sinb*log(rb+z3b) - y1/(rb+y3b)*(1+a/rb) +
z1b/(rb+z3b)*(cosb+a/rb)) +
/* term2 */
y1*(y3b-a)/rb*(a/(rb*rb) + 1/(rb+y3b)) -
/* term3 */
(y3b-a)/(rb+z3b)*(sinb*(cosb-a/rb) +
z1b/rb*(1+a*y3b/(rb*rb)) -
1/(rb*(rb+z3b))*((y2*y2)*cosb*sinb -
a*z1b/rb*(rb*cosb+y3b)))
);
displ_ic_ft[2][1] += 0.0;
displ_ic[2][2] += 2.0*(
/* term1 */
2*(1-pr) * (y2*sinb/(rb+z3b)*(cosb + a/rb)) +
/* term2 */
y2*(y3b-a)*sinb/(rb*(rb+z3b))*
(1 + (rb*cosb+y3b)/(rb+z3b) *
(cosb+a/rb) + a*y3b/(rb*rb))
);
displ_ic_ft[2][2] += 2* 2*(1-pr) * (fb);
}
/* For debugging only -- Delete me */
for (i=0; i < 3; i++) {
for (j=0; j < 3; j++) {
dic2[i][j] = 2.0*displ_ic[i][j];
}
}
for (i=0; i < 3; i++) {
for (j=0; j < 3; j++) {
/* For debugging only
----------------------
displ_ic_ft[i][j] /= 2.*2.*(1-pr);
trash1[i][j] = displ_ic_ft[i][j];
displ_ic_ft[i][j] = reduce_angle(displ_ic_ft[i][j]);
trash2[i][j] = displ_ic_ft[i][j];
displ_ic_ft[i][j] *= 2.*2.*(1-pr);
*/
/* Add _ft terms to inf coeff eqns
----------------------------------*/
displ_ic[i][j] += displ_ic_ft[i][j];
/* Divide through by consts on l.h.s.
-------------------------------------*/
displ_ic[i][j] /= 8.0*PI*(1-pr);
}
}
}
/***************** Function: comninou_strain_ics **********************
* Calculates the influence coefficients for strain for an angular dislocation.
*
* In: y - coords of point at which to calc inf coeffs
* a - depth to angular dislocation vertex
* beta - angle subtended by dislocation
* pr - poisson's ratio
* half_space - half/whole-space flag
* Out: strain_ic[i][j][k] - jk component of strain at y due to unit ith
* Burgers vector component
*****************************************************************************/
#if defined(__STDC__) || defined(ANSI) /* ANSI */
void comninou_strain_ics(double y[3], double a, double beta, double pr,int half_space, double strain_ic[3][3][3])
#else
void comninou_strain_ics(y, a, beta, pr, half_space, strain_ic)
double y[3];
double a;
double beta;
double pr;
int half_space;
double strain_ic[3][3][3];
#endif
{
double dummy[3][3][3];
double dg_ic[3][3][3];
int i, j, k;
double y1 = y[0];
double y2 = y[1];
double y3 = y[2];
double y3b = y3 + 2.*a;
double sinb = sin(beta);
double cosb = cos(beta);
double cotb = 1./tan(beta);
double z1 = y1*cosb - y3*sinb;
double z3 = y1*sinb + y3*cosb;
double r = sqrt(y1*y1 + y2*y2 + y3*y3);
double z1b = y1*cosb + y3b*sinb;
double z3b = -y1*sinb + y3b*cosb;
double rb = sqrt(y1*y1 + y2*y2 + y3b*y3b);
double rMy3P2 = (r-y3)*(r-y3);
double rP2 = r*r;
double y1P2 = y1*y1;
double y2P2 = y2*y2;
double y2P3 = y2P2*y2;
double y2P4 = y2P2*y2P2;
double rMz3P2 = (r-z3)*(r-z3);
double cosbP2 = cosb*cosb;
double sinbP2 = sinb*sinb;
double cotbP2 = cotb*cotb;
double rbPy3bP2 = (rb+y3b)*(rb+y3b);
double rbP2 = rb*rb;
double rbPz3bP2 = (rb+z3b)*(rb+z3b);
double rM1 = 1./r;
double rbM1 = 1./rb;
double rbP3 = rbP2*rb;
double rbP4 = rbP2*rbP2;
double rbPy3bM1 = 1./(rb+y3b);
double rbPz3b = rb+z3b;
double rbPy3b = rb+y3b;
double rMz3 = r-z3;
double rTsinbMy1 = r*sinb - y1;
double rTcosbMy3 = r*cosb - y3;
double rbTcosbPy3b = rb*cosb + y3b;
double rbTsinbMy1 = rb*sinb - y1;
double rMy3 = r-y3;
double y3bMa = y3b-a;
double pr2 = 2.*(1.-pr);
double pr3 = (1.-2.*pr);
double aQrb = a/rb;
double z1P2 = z1*z1;
double z1bP2 = z1b*z1b;
double Dy3bDy3 = 1.0;
double Dz1Dy1 = cosb;
double Dz1Dy3 = -sinb;
double Dz3Dy1 = sinb;
double Dz3Dy3 = cosb;
double DrDy1 = y1/r;
double DrDy2 = y2/r;
double DrDy3 = y3/r;
double Dz1bDy1 = cosb;
double Dz1bDy3b = sinb;
double Dz3bDy1 = -sinb;
double Dz3bDy3b = cosb;
double DrbDy1 = y1/rb;
double DrbDy2 = y2/rb;
double DrbDy3b = y3b/rb;
double Df_denom =
rP2*sinbP2*y2P2 + cosbP2*y2P4 + 2*cosb*y1*y2P2*z1 + y1P2*z1P2;
double Dfb_denom =
rbP2*sinbP2*y2P2 + cosbP2*y2P4 + 2*cosb*y1*y2P2*z1b + y1P2*z1bP2;
double DfDy1 =
y2/(y1P2 + y2P2) - (r*sinb*y2*z1)/ Df_denom;
double DfDy2 =
-(y1/(y1P2 + y2P2)) + z1/(y2P2 + z1P2) +
(-(cosb*r*sinb*y2P2) + r*sinb*y1*z1)/ Df_denom;
double DfDz1 =
-(y2/(y2P2 + z1P2)) - (r*sinb*y1*y2)/ Df_denom;
double DfDr =
(sinb*y2*(cosb*y2P2 + y1*z1))/ Df_denom;
double DfbDy1 =
y2/(y1P2 + y2P2) - (rb*sinb*y2*z1b)/ Dfb_denom;
double DfbDy2 =
-(y1/(y1P2 + y2P2)) + z1b/(y2P2 + z1bP2) +
(-(cosb*rb*sinb*y2P2) + rb*sinb*y1*z1b)/ Dfb_denom;
double DfbDz1b =
-(y2/(y2P2 + z1bP2)) - (rb*sinb*y1*y2)/ Dfb_denom;
double DfbDrb =
(sinb*y2*(cosb*y2P2 + y1*z1b))/ Dfb_denom;
/* Dislocation-Indiced Displacement Gradients */
dg_ic[0][0][0] =
pr2*(DrDy1* DfDr + Dz1Dy1* DfDz1 + DfDy1) -
y2/(rMy3*r) - (cosb*y2*(-1. + sinb*DrDy1))/ (rMz3*r) +
(cosb*y2*rTsinbMy1* (-Dz3Dy1 + DrDy1))/ (rMz3P2*r) +
(y1*y2*DrDy1)/ (rMy3*rP2) + (cosb*y2*rTsinbMy1*DrDy1)/
(rMz3*rP2) + (y1*y2*DrDy1)/ (rMy3P2*r);
dg_ic[0][0][1] =
pr2*(DrDy2* DfDr + DfDy2) - y1/(rMy3*r) -
(cosb*rTsinbMy1)/ (rMz3*r) + (y1*y2*DrDy2)/ (rMy3*rP2) +
(cosb*y2*rTsinbMy1*DrDy2)/ (rMz3*rP2) + (y1*y2*DrDy2)/ (rMy3P2*r) +
(cosb*y2*rTsinbMy1*DrDy2)/ (rMz3P2*r) -
(cosb*sinb*y2*DrDy2)/ (rMz3*r);
dg_ic[0][0][2] =
pr2*(DrDy3* DfDr + Dz1Dy3* DfDz1) +
(y1*y2*(-1. + DrDy3))/ (rMy3P2*r) +
(cosb*y2*rTsinbMy1* (-Dz3Dy3 + DrDy3))/ (rMz3P2*r) +
(y1*y2*DrDy3)/ (rMy3*rP2) +
(cosb*y2*rTsinbMy1*DrDy3)/ (rMz3*rP2) -
(cosb*sinb*y2*DrDy3)/ (rMz3*r);
dg_ic[0][1][0] =
pr3*(-((cosb*(-Dz3Dy1 + DrDy1))/ rMz3) +
DrDy1/rMy3) - y2P2*((cosb*(-Dz3Dy1 + DrDy1))/ (rMz3P2*r) -
DrDy1/ (rMy3*rP2) + (cosb*DrDy1)/ (rMz3*rP2) -
DrDy1/ (rMy3P2*r));
dg_ic[0][1][1] =
-2.*y2*(1/(rMy3*r) - cosb/(rMz3*r)) + pr3*(DrDy2/rMy3 -
(cosb*DrDy2)/ rMz3) - y2P2*(-(DrDy2/ (rMy3*rP2)) +
(cosb*DrDy2)/ (rMz3*rP2) - DrDy2/ (rMy3P2*r) +
(cosb*DrDy2)/ (rMz3P2*r));
dg_ic[0][1][2] =
pr3*((-1. + DrDy3)/rMy3 - (cosb*(-Dz3Dy3 + DrDy3))/ rMz3) -
y2P2*(-((-1. + DrDy3)/ (rMy3P2*r)) + (cosb*(-Dz3Dy3 +
DrDy3))/ (rMz3P2*r) - DrDy3/ (rMy3*rP2) + (cosb*DrDy3)/ (rMz3*rP2));
dg_ic[0][2][0] =
y2*((cosb*rTcosbMy3* (-Dz3Dy1 + DrDy1))/ (rMz3P2*r) -
DrDy1/rP2 + (cosb*rTcosbMy3*DrDy1)/ (rMz3*rP2) -
(cosbP2*DrDy1)/ (rMz3*r));
dg_ic[0][2][1] =
y2*(-(DrDy2/rP2) + (cosb*rTcosbMy3*DrDy2) /(rMz3*rP2) +
(cosb*rTcosbMy3*DrDy2) /(rMz3P2*r) - (cosbP2*DrDy2)/ (rMz3*r)) +
rM1 - (cosb*rTcosbMy3)/ (rMz3*r);
dg_ic[0][2][2] =
y2*(-((cosb*(-1. + cosb*DrDy3))/ (rMz3*r)) +
(cosb*rTcosbMy3* (-Dz3Dy3 + DrDy3))/ (rMz3P2*r) -
DrDy3/rP2 + (cosb*rTcosbMy3*DrDy3)/ (rMz3*rP2));
dg_ic[1][0][0] =
-(pr3*(-((cosb*(-Dz3Dy1 + DrDy1))/ rMz3) + DrDy1/rMy3)) +
(2.*y1)/(rMy3*r) + ((-1. + sinb*DrDy1)*z1)/ (rMz3*r) -
(rTsinbMy1* (-Dz3Dy1 + DrDy1)* z1)/(rMz3P2*r) +
(rTsinbMy1*Dz1Dy1)/ (rMz3*r) - (y1P2*DrDy1)/ (rMy3*rP2) -
(y1P2*DrDy1)/ (rMy3P2*r) - (rTsinbMy1*z1*DrDy1)/ (rMz3*rP2);
dg_ic[1][0][1] =
-(pr3*(DrDy2/rMy3 - (cosb*DrDy2)/ rMz3)) - (y1P2*DrDy2)/ (rMy3*rP2) -
(y1P2*DrDy2)/ (rMy3P2*r) - (rTsinbMy1*z1*DrDy2)/ (rMz3*rP2) -
(rTsinbMy1*z1*DrDy2)/ (rMz3P2*r) + (sinb*z1*DrDy2)/ (rMz3*r);
dg_ic[1][0][2] =
-(pr3*((-1. + DrDy3)/rMy3 - (cosb*(-Dz3Dy3 + DrDy3))/ rMz3)) -
(y1P2*(-1. + DrDy3))/ (rMy3P2*r) -
(rTsinbMy1* (-Dz3Dy3 + DrDy3)* z1)/(rMz3P2*r) +
(rTsinbMy1*Dz1Dy3)/ (rMz3*r) - (y1P2*DrDy3)/ (rMy3*rP2) -
(rTsinbMy1*z1*DrDy3)/ (rMz3*rP2) + (sinb*z1*DrDy3)/ (rMz3*r);
dg_ic[1][1][0] =
pr2*(DrDy1* DfDr + Dz1Dy1* DfDz1 + DfDy1) + y2/(rMy3*r) +
(y2*(-Dz3Dy1 + DrDy1)* z1)/(rMz3P2*r) - (y2*Dz1Dy1)/ (rMz3*r) -
(y1*y2*DrDy1)/ (rMy3*rP2) - (y1*y2*DrDy1)/ (rMy3P2*r) +
(y2*z1*DrDy1)/ (rMz3*rP2);
dg_ic[1][1][1] =
pr2*(DrDy2* DfDr + DfDy2) + y1/(rMy3*r) - z1/(rMz3*r) -
(y1*y2*DrDy2)/ (rMy3*rP2) - (y1*y2*DrDy2)/ (rMy3P2*r) +
(y2*z1*DrDy2)/ (rMz3*rP2) + (y2*z1*DrDy2)/ (rMz3P2*r);
dg_ic[1][1][2] =
pr2*(DrDy3* DfDr + Dz1Dy3* DfDz1) -
(y1*y2*(-1. + DrDy3))/ (rMy3P2*r) +
(y2*(-Dz3Dy3 + DrDy3)* z1)/(rMz3P2*r) - (y2*Dz1Dy3)/ (rMz3*r) -
(y1*y2*DrDy3)/ (rMy3*rP2) + (y2*z1*DrDy3)/ (rMz3*rP2);
dg_ic[1][2][0] =
-((sinb*pr3*(-Dz3Dy1 + DrDy1))/ rMz3) - rM1 -
(rTcosbMy3* (-Dz3Dy1 + DrDy1)* z1)/(rMz3P2*r) +
(rTcosbMy3*Dz1Dy1)/ (rMz3*r) + (y1*DrDy1)/rP2 -
(rTcosbMy3*z1*DrDy1)/ (rMz3*rP2) + (cosb*z1*DrDy1)/ (rMz3*r);
dg_ic[1][2][1] =
-((sinb*pr3*DrDy2)/ rMz3) + (y1*DrDy2)/rP2 -
(rTcosbMy3*z1*DrDy2)/ (rMz3*rP2) -
(rTcosbMy3*z1*DrDy2)/ (rMz3P2*r) +
(cosb*z1*DrDy2)/ (rMz3*r);
dg_ic[1][2][2] =
-((sinb*pr3*(-Dz3Dy3 + DrDy3))/ rMz3) +
((-1. + cosb*DrDy3)*z1)/ (rMz3*r) -
(rTcosbMy3* (-Dz3Dy3 + DrDy3)* z1)/(rMz3P2*r) +
(rTcosbMy3*Dz1Dy3)/ (rMz3*r) + (y1*DrDy3)/rP2 -
(rTcosbMy3*z1*DrDy3)/ (rMz3*rP2);
dg_ic[2][0][0] =
(sinb*y2*(-1. + sinb*DrDy1))/ (rMz3*r) -
(sinb*y2*rTsinbMy1* (-Dz3Dy1 + DrDy1))/ (rMz3P2*r) -
(sinb*y2*rTsinbMy1*DrDy1)/ (rMz3*rP2);
dg_ic[2][0][1] =
(sinb*rTsinbMy1)/(rMz3*r) - (sinb*y2*rTsinbMy1*DrDy2)/
(rMz3*rP2) - (sinb*y2*rTsinbMy1*DrDy2)/ (rMz3P2*r) +
(sinbP2*y2*DrDy2)/ (rMz3*r);
dg_ic[2][0][2] =
-((sinb*y2*rTsinbMy1* (-Dz3Dy3 + DrDy3))/ (rMz3P2*r)) -
(sinb*y2*rTsinbMy1*DrDy3)/ (rMz3*rP2) + (sinbP2*y2*DrDy3)/ (rMz3*r);
dg_ic[2][1][0] =
(sinb*pr3*(-Dz3Dy1 + DrDy1))/ rMz3 + (sinb*y2P2*(-Dz3Dy1 +
DrDy1))/ (rMz3P2*r) + (sinb*y2P2*DrDy1)/ (rMz3*rP2);
dg_ic[2][1][1] =
(-2.*sinb*y2)/(rMz3*r) + (sinb*pr3*DrDy2)/ rMz3 +
(sinb*y2P2*DrDy2)/ (rMz3*rP2) + (sinb*y2P2*DrDy2)/ (rMz3P2*r);
dg_ic[2][1][2] =
(sinb*pr3*(-Dz3Dy3 + DrDy3))/ rMz3 + (sinb*y2P2*(-Dz3Dy3 +
DrDy3))/ (rMz3P2*r) + (sinb*y2P2*DrDy3)/ (rMz3*rP2);
dg_ic[2][2][0] =
pr2*(DrDy1* DfDr + Dz1Dy1* DfDz1 + DfDy1) -
(sinb*y2*rTcosbMy3* (-Dz3Dy1 + DrDy1))/ (rMz3P2*r) -
(sinb*y2*rTcosbMy3*DrDy1)/ (rMz3*rP2) +
(cosb*sinb*y2*DrDy1)/ (rMz3*r);
dg_ic[2][2][1] =
pr2*(DrDy2* DfDr + DfDy2) +
(sinb*rTcosbMy3)/ (rMz3*r) - (sinb*y2*rTcosbMy3*DrDy2)/ (rMz3*rP2) -
(sinb*y2*rTcosbMy3*DrDy2)/ (rMz3P2*r) + (cosb*sinb*y2*DrDy2)/ (rMz3*r);
dg_ic[2][2][2] =
pr2*(DrDy3* DfDr + Dz1Dy3* DfDz1) +
(sinb*y2*(-1. + cosb*DrDy3))/ (rMz3*r) -
(sinb*y2*rTcosbMy3* (-Dz3Dy3 + DrDy3))/ (rMz3P2*r) -
(sinb*y2*rTcosbMy3*DrDy3)/ (rMz3*rP2);
if (half_space) {
/* Image-Induced Displacement Gradients */
dg_ic[0][0][0] +=
pr2*(DrbDy1* DfbDrb + Dz1bDy1* DfbDz1b + DfbDy1) -
y2/(rbPy3b*rb) - (cosb*y2*(-1. + sinb*DrbDy1))/
(rbPz3b*rb) + (cosb*y2*rbTsinbMy1* (Dz3bDy1 + DrbDy1))/
(rbPz3bP2* rb) + (y1*y2*DrbDy1)/ (rbPy3b*rbP2) +
(cosb*y2*rbTsinbMy1* DrbDy1)/ (rbPz3b* rbP2) +
(y1*y2*DrbDy1)/ (rbPy3bP2*rb);
dg_ic[0][0][1] +=
pr2*(DrbDy2* DfbDrb + DfbDy2) -
y1/(rbPy3b*rb) - (cosb*rbTsinbMy1)/ (rbPz3b*rb) + (y1*y2*DrbDy2)/
(rbPy3b*rbP2) + (cosb*y2*rbTsinbMy1* DrbDy2)/ (rbPz3b* rbP2) +
(y1*y2*DrbDy2)/ (rbPy3bP2*rb) + (cosb*y2*rbTsinbMy1* DrbDy2)/
(rbPz3bP2* rb) - (cosb*sinb*y2*DrbDy2)/ (rbPz3b*rb);
dg_ic[0][0][2] +=
pr2*(Dy3bDy3* DrbDy3b* DfbDrb + Dz1bDy3b*Dy3bDy3* DfbDz1b) +
(y1*y2*(Dy3bDy3 + Dy3bDy3* DrbDy3b))/ (rbPy3bP2*rb) +
(cosb*y2*rbTsinbMy1* (Dz3bDy3b*Dy3bDy3 + Dy3bDy3* DrbDy3b))/
(rbPz3bP2* rb) + (y1*y2*Dy3bDy3* DrbDy3b)/ (rbPy3b*rbP2) +
(cosb*y2*rbTsinbMy1* Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2) -
(cosb*sinb*y2*Dy3bDy3* DrbDy3b)/ (rbPz3b*rb);
dg_ic[0][1][0] +=
pr3*(-((cosb*(Dz3bDy1 + DrbDy1))/ rbPz3b) +
DrbDy1/ rbPy3b) - y2P2*((cosb*(Dz3bDy1 + DrbDy1))/
(rbPz3bP2* rb) - DrbDy1/ (rbPy3b*rbP2) + (cosb*DrbDy1)/
(rbPz3b* rbP2) - DrbDy1/ (rbPy3bP2*rb));
dg_ic[0][1][1] +=
-2.*y2*(1/(rbPy3b*rb) - cosb/ (rbPz3b* rb)) +
pr3*(DrbDy2/ rbPy3b - (cosb*DrbDy2)/ rbPz3b) -
y2P2*(-(DrbDy2/ (rbPy3b* rbP2)) + (cosb*DrbDy2)/ (rbPz3b* rbP2) -
DrbDy2/ (rbPy3bP2*rb) + (cosb*DrbDy2)/ (rbPz3bP2* rb));
dg_ic[0][1][2] +=
pr3*((Dy3bDy3 + Dy3bDy3* DrbDy3b)/ rbPy3b -
(cosb*(Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/ rbPz3b) -
y2P2*(-((Dy3bDy3 + Dy3bDy3* DrbDy3b)/ (rbPy3bP2* rb)) +
(cosb*(Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/ (rbPz3bP2* rb) -
(Dy3bDy3* DrbDy3b)/ (rbPy3b*rbP2) + (cosb*Dy3bDy3*
DrbDy3b)/ (rbPz3b* rbP2));
dg_ic[0][2][0] +=
y2*(-((cosb*rbTcosbPy3b* (Dz3bDy1 + DrbDy1))/ (rbPz3bP2* rb)) +
DrbDy1/rbP2 - (cosb*rbTcosbPy3b* DrbDy1)/ (rbPz3b* rbP2) +
(cosbP2*DrbDy1)/ (rbPz3b* rb));
dg_ic[0][2][1] +=
y2*(DrbDy2/rbP2 - (cosb*rbTcosbPy3b* DrbDy2)/ (rbPz3b* rbP2) +
(cosbP2*DrbDy2)/ (rbPz3b* rb) -
(cosb*rbTcosbPy3b* DrbDy2)/ (rbPz3bP2* rb)) - rbM1 +
(cosb*rbTcosbPy3b)/ (rbPz3b*rb);
dg_ic[0][2][2] +=
y2*((cosb*(Dy3bDy3 + cosb*Dy3bDy3* DrbDy3b))/ (rbPz3b* rb) -
(cosb*rbTcosbPy3b* (Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/
(rbPz3bP2* rb) + (Dy3bDy3* DrbDy3b)/ rbP2 -
(cosb*rbTcosbPy3b* Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2));
dg_ic[1][0][0] +=
-(pr3*(-((cosb*(Dz3bDy1 + DrbDy1))/ rbPz3b) + DrbDy1/ rbPy3b)) +
(2.*y1)/(rbPy3b*rb) + ((-1. + sinb*DrbDy1)* z1b)/
(rbPz3b*rb) - (rbTsinbMy1* (Dz3bDy1 + DrbDy1)* z1b)/
(rbPz3bP2* rb) + (rbTsinbMy1*Dz1bDy1) /(rbPz3b* rb) -
(y1P2*DrbDy1)/ (rbPy3b*rbP2) - (y1P2*DrbDy1)/ (rbPy3bP2*rb) -
(rbTsinbMy1*z1b* DrbDy1)/ (rbPz3b* rbP2);
dg_ic[1][0][1] +=
-(pr3*(DrbDy2/ rbPy3b - (cosb*DrbDy2)/ rbPz3b)) -
(y1P2*DrbDy2)/ (rbPy3b*rbP2) - (y1P2*DrbDy2)/ (rbPy3bP2*rb) -
(rbTsinbMy1*z1b* DrbDy2)/ (rbPz3b* rbP2) -
(rbTsinbMy1*z1b* DrbDy2)/ (rbPz3bP2* rb) +
(sinb*z1b*DrbDy2)/ (rbPz3b*rb);
dg_ic[1][0][2] +=
-(pr3*((Dy3bDy3 + Dy3bDy3* DrbDy3b)/ rbPy3b -
(cosb*(Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/ rbPz3b)) -
(y1P2*(Dy3bDy3 + Dy3bDy3* DrbDy3b))/ (rbPy3bP2*rb) -
(rbTsinbMy1* (Dz3bDy3b*Dy3bDy3 + Dy3bDy3* DrbDy3b)*
z1b)/ (rbPz3bP2* rb) + (rbTsinbMy1* Dz1bDy3b*Dy3bDy3)/
(rbPz3b*rb) - (y1P2*Dy3bDy3* DrbDy3b)/ (rbPy3b*rbP2) -
(rbTsinbMy1*z1b* Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2) +
(sinb*z1b*Dy3bDy3* DrbDy3b)/ (rbPz3b*rb);
dg_ic[1][1][0] +=
pr2*(DrbDy1* DfbDrb + Dz1bDy1* DfbDz1b + DfbDy1) +
y2/(rbPy3b*rb) + (y2*(Dz3bDy1 + DrbDy1)* z1b)/ (rbPz3bP2* rb) -
(y2*Dz1bDy1)/ (rbPz3b*rb) - (y1*y2*DrbDy1)/ (rbPy3b*rbP2) -
(y1*y2*DrbDy1)/ (rbPy3bP2*rb) + (y2*z1b*DrbDy1)/ (rbPz3b* rbP2);
dg_ic[1][1][1] +=
pr2*(DrbDy2* DfbDrb + DfbDy2) + y1/(rbPy3b*rb) -
z1b/ (rbPz3b*rb) - (y1*y2*DrbDy2)/ (rbPy3b*rbP2) -
(y1*y2*DrbDy2)/ (rbPy3bP2*rb) + (y2*z1b*DrbDy2)/ (rbPz3b* rbP2) +
(y2*z1b*DrbDy2)/ (rbPz3bP2* rb);
dg_ic[1][1][2] +=
pr2*(Dy3bDy3* DrbDy3b* DfbDrb + Dz1bDy3b*Dy3bDy3* DfbDz1b) -
(y1*y2*(Dy3bDy3 + Dy3bDy3* DrbDy3b))/ (rbPy3bP2*rb) +
(y2*(Dz3bDy3b*Dy3bDy3 + Dy3bDy3* DrbDy3b)* z1b)/ (rbPz3bP2* rb) -
(y2*Dz1bDy3b*Dy3bDy3)/ (rbPz3b*rb) - (y1*y2*Dy3bDy3* DrbDy3b)/
(rbPy3b*rbP2) + (y2*z1b*Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2);
dg_ic[1][2][0] +=
(sinb*pr3*(Dz3bDy1 + DrbDy1))/ rbPz3b + rbM1 +
(rbTcosbPy3b* (Dz3bDy1 + DrbDy1)* z1b)/ (rbPz3bP2* rb) -
(rbTcosbPy3b* Dz1bDy1)/
(rbPz3b*rb) - (y1*DrbDy1)/rbP2 + (rbTcosbPy3b*z1b* DrbDy1)/
(rbPz3b* rbP2) - (cosb*z1b*DrbDy1)/ (rbPz3b*rb);
dg_ic[1][2][1] +=
(sinb*pr3*DrbDy2)/ rbPz3b - (y1*DrbDy2)/rbP2 +
(rbTcosbPy3b*z1b* DrbDy2)/ (rbPz3b* rbP2) -
(cosb*z1b*DrbDy2)/ (rbPz3b*rb) + (rbTcosbPy3b*z1b* DrbDy2)/
(rbPz3bP2* rb);
dg_ic[1][2][2] +=
(sinb*pr3*(Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/
rbPz3b - ((Dy3bDy3 + cosb*Dy3bDy3* DrbDy3b)*
z1b)/ (rbPz3b*rb) + (rbTcosbPy3b* (Dz3bDy3b*Dy3bDy3 + Dy3bDy3*
DrbDy3b)* z1b)/ (rbPz3bP2* rb) - (rbTcosbPy3b*
Dz1bDy3b*Dy3bDy3)/ (rbPz3b*rb) - (y1*Dy3bDy3* DrbDy3b)/ rbP2 +
(rbTcosbPy3b*z1b* Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2);
dg_ic[2][0][0] +=
(sinb*y2*(-1. + sinb*DrbDy1))/
(rbPz3b*rb) - (sinb*y2*rbTsinbMy1* (Dz3bDy1 + DrbDy1))/
(rbPz3bP2* rb) - (sinb*y2*rbTsinbMy1* DrbDy1)/ (rbPz3b* rbP2);
dg_ic[2][0][1] +=
(sinb*rbTsinbMy1)/ (rbPz3b*rb) - (sinb*y2*rbTsinbMy1* DrbDy2)/
(rbPz3b* rbP2) - (sinb*y2*rbTsinbMy1* DrbDy2)/ (rbPz3bP2* rb) +
(sinbP2*y2*DrbDy2)/ (rbPz3b*rb);
dg_ic[2][0][2] +=
-((sinb*y2*rbTsinbMy1* (Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/
(rbPz3bP2* rb)) - (sinb*y2*rbTsinbMy1* Dy3bDy3* DrbDy3b)/
(rbPz3b* rbP2) + (sinbP2*y2*Dy3bDy3* DrbDy3b)/ (rbPz3b*rb);
dg_ic[2][1][0] +=
(sinb*pr3*(Dz3bDy1 + DrbDy1))/ rbPz3b +
(sinb*y2P2*(Dz3bDy1 + DrbDy1))/ (rbPz3bP2* rb) +
(sinb*y2P2*DrbDy1)/ (rbPz3b* rbP2);
dg_ic[2][1][1] +=
(-2.*sinb*y2)/ (rbPz3b*rb) + (sinb*pr3*DrbDy2)/ rbPz3b +
(sinb*y2P2*DrbDy2)/ (rbPz3b* rbP2) + (sinb*y2P2*DrbDy2)/
(rbPz3bP2* rb);
dg_ic[2][1][2] +=
(sinb*pr3*(Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/
rbPz3b + (sinb*y2P2*(Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/
(rbPz3bP2* rb) + (sinb*y2P2*Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2);
dg_ic[2][2][0] +=
-pr2*(DrbDy1* DfbDrb + Dz1bDy1* DfbDz1b + DfbDy1) +
(sinb*y2*rbTcosbPy3b* (Dz3bDy1 + DrbDy1))/ (rbPz3bP2* rb) +
(sinb*y2*rbTcosbPy3b* DrbDy1)/ (rbPz3b* rbP2) -
(cosb*sinb*y2*DrbDy1)/ (rbPz3b*rb);
dg_ic[2][2][1] +=
-pr2*(DrbDy2* DfbDrb + DfbDy2) - (sinb*rbTcosbPy3b)/
(rbPz3b*rb) + (sinb*y2*rbTcosbPy3b* DrbDy2)/ (rbPz3b*
rbP2) - (cosb*sinb*y2*DrbDy2)/ (rbPz3b*rb) + (sinb*y2*rbTcosbPy3b*
DrbDy2)/ (rbPz3bP2* rb);
dg_ic[2][2][2] +=
-pr2*(Dy3bDy3* DrbDy3b* DfbDrb + Dz1bDy3b*Dy3bDy3* DfbDz1b) -
(sinb*y2*(Dy3bDy3 + cosb*Dy3bDy3* DrbDy3b))/
(rbPz3b*rb) + (sinb*y2*rbTcosbPy3b* (Dz3bDy3b*Dy3bDy3 + Dy3bDy3*
DrbDy3b))/ (rbPz3bP2* rb) + (sinb*y2*rbTcosbPy3b*
Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2);
/* Corrective Displacement Gradients */
dg_ic[0][0][0] +=
2.*(-((cosb*cotb*y2*pr3*(cosb + aQrb)* (Dz3bDy1 + DrbDy1))/
rbPz3bP2) - 2.*cotbP2*pr3*(1. - pr)* (DrbDy1* DfbDrb +
Dz1bDy1* DfbDz1b + DfbDy1) + (y2*pr3*(-((pr + aQrb)/
rbPy3b) + (y1*(pr + aQrb)* DrbDy1)/ rbPy3bP2 + (a*cotb*DrbDy1)/
rbP2 + (a*y1*DrbDy1)/ (rbPy3b* rbP2)))/ rbPy3b - (y2*y3bMa*
((cosb*(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b))/ rbPz3b - (a*cosb*cotb*y3b)/rbP2)* (Dz3bDy1 +
DrbDy1))/ (rbPz3bP2* rb) + (y2*y3bMa*
(-((cosb*(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b)* (Dz3bDy1 + DrbDy1))/ rbPz3bP2) +
(cosb* (2.*cosb* (-1. + sinb*DrbDy1)* (1. - pr) + cosb*cotb*
(cosb*pr3 - aQrb)* DrbDy1 + (a*cotb*rbTcosbPy3b* DrbDy1)/
rbP2))/ rbPz3b + (2.*a*cosb*cotb*y3b* DrbDy1)/ rbP3))/
(rbPz3b* rb) + (y2*y3bMa* ((2.*pr + aQrb)/ rbPy3b + a/rbP2 -
(y1*(2.*pr + aQrb)* DrbDy1)/ rbPy3bP2 - (2.*a*y1*DrbDy1)/
rbP3 - (a*y1*DrbDy1)/ (rbPy3b* rbP2)))/ (rbPy3b*rb) -
(y2*pr3*(-((y1*(pr + aQrb))/ rbPy3b) + cotb*(1. - 2.*pr - aQrb))*
DrbDy1)/ rbPy3bP2 - (3*a*cotb*y2*y3bMa* DrbDy1)/ rbP4 -
(a*cosb*cotb*y2*pr3*DrbDy1) /(rbPz3b* rbP2) - (y2*y3bMa*
((cosb*(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b))/ rbPz3b - (a*cosb*cotb*y3b)/rbP2)* DrbDy1)/
(rbPz3b* rbP2) - (y2*y3bMa* (-(cotb*pr3) + (y1*(2.*pr + aQrb))/
rbPy3b + (a*y1)/rbP2)* DrbDy1)/ (rbPy3b*rbP2) - (y2*y3bMa*
(-(cotb*pr3) + (y1*(2.*pr + aQrb))/ rbPy3b + (a*y1)/rbP2)*
DrbDy1)/ (rbPy3bP2*rb));
dg_ic[0][0][1] +=
2.*((pr3*(-((y1*(pr + aQrb))/ rbPy3b) +
cotb*(1. - 2.*pr - aQrb)))/ rbPy3b +
(cosb*cotb*pr3*(cosb + aQrb))/ rbPz3b -
2.*cotbP2*pr3*(1. - pr)* (DrbDy2* DfbDrb + DfbDy2) +
(y2*pr3*((y1*(pr + aQrb)* DrbDy2)/ rbPy3bP2 + (a*cotb*DrbDy2)/
rbP2 + (a*y1*DrbDy2)/ (rbPy3b* rbP2)))/ rbPy3b +
(a*cotb*y3bMa)/rbP3 + (y3bMa*((cosb*
(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b))/ rbPz3b - (a*cosb*cotb*y3b)/rbP2))/ (rbPz3b* rb) +
(y3bMa*(-(cotb*pr3) + (y1*(2.*pr + aQrb))/ rbPy3b +
(a*y1)/rbP2))/ (rbPy3b*rb) + (y2*y3bMa*
((cosb*(2.*cosb*sinb*(1. - pr)* DrbDy2 + cosb*cotb*
(cosb*pr3 - aQrb)* DrbDy2 + (a*cotb*rbTcosbPy3b* DrbDy2)/
rbP2))/ rbPz3b - (cosb*
(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b)* DrbDy2)/ rbPz3bP2 + (2.*a*cosb*cotb*y3b*
DrbDy2)/ rbP3))/ (rbPz3b* rb) + (y2*y3bMa* (-((y1*(2.*pr + aQrb)*
DrbDy2)/ rbPy3bP2) - (2.*a*y1*DrbDy2)/ rbP3 - (a*y1*DrbDy2)/
(rbPy3b* rbP2)))/ (rbPy3b*rb) - (y2*pr3*(-((y1*(pr + aQrb))/
rbPy3b) + cotb*(1. - 2.*pr - aQrb))* DrbDy2)/
rbPy3bP2 - (cosb*cotb*y2*pr3*(cosb + aQrb)* DrbDy2)/
rbPz3bP2 - (3*a*cotb*y2*y3bMa* DrbDy2)/ rbP4 -
(a*cosb*cotb*y2*pr3*DrbDy2) /(rbPz3b* rbP2) - (y2*y3bMa*
((cosb*(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b))/ rbPz3b - (a*cosb*cotb*y3b)/rbP2)* DrbDy2)/
(rbPz3b* rbP2) - (y2*y3bMa* (-(cotb*pr3) + (y1*(2.*pr + aQrb))/
rbPy3b + (a*y1)/rbP2)* DrbDy2)/ (rbPy3b*rbP2) - (y2*y3bMa*
((cosb*(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b))/ rbPz3b - (a*cosb*cotb*y3b)/rbP2)* DrbDy2)/
(rbPz3bP2* rb) - (y2*y3bMa* (-(cotb*pr3) + (y1*(2.*pr + aQrb))/
rbPy3b + (a*y1)/rbP2)* DrbDy2)/ (rbPy3bP2*rb));
dg_ic[0][0][2] +=
2.*(-((y2*pr3*(-((y1*(pr + aQrb))/ rbPy3b) +
cotb*(1. - 2.*pr - aQrb))* (Dy3bDy3 + Dy3bDy3* DrbDy3b))/
rbPy3bP2) - (cosb*cotb*y2*pr3*(cosb + aQrb)* (Dz3bDy3b*
Dy3bDy3 + Dy3bDy3* DrbDy3b))/ rbPz3bP2 -
2.*cotbP2*pr3*(1. - pr)* (Dy3bDy3* DrbDy3b* DfbDrb +
Dz1bDy3b* Dy3bDy3* DfbDz1b) + (y2*pr3*((y1*(pr + aQrb)*
(Dy3bDy3 + Dy3bDy3* DrbDy3b))/ rbPy3bP2 + (a*cotb*Dy3bDy3*
DrbDy3b)/ rbP2 + (a*y1*Dy3bDy3* DrbDy3b)/ (rbPy3b* rbP2)))/
rbPy3b - (y2*y3bMa* ((cosb*(pr2*cosb*rbTsinbMy1 +
cotb*(cosb*pr3 - aQrb)* rbTcosbPy3b))/ rbPz3b -
(a*cosb*cotb*y3b)/rbP2)* (Dz3bDy3b* Dy3bDy3 + Dy3bDy3*
DrbDy3b))/ (rbPz3bP2* rb) - (y2*y3bMa* (Dy3bDy3 + Dy3bDy3*
DrbDy3b)* (-(cotb*pr3) + (y1*(2.*pr + aQrb))/ rbPy3b +
(a*y1)/rbP2))/ (rbPy3bP2*rb) + (y2*y3bMa* (-((y1*(2.*pr + aQrb)*
(Dy3bDy3 + Dy3bDy3* DrbDy3b))/ rbPy3bP2) - (2.*a*y1*Dy3bDy3*
DrbDy3b)/ rbP3 - (a*y1*Dy3bDy3* DrbDy3b)/ (rbPy3b* rbP2)))/
(rbPy3b*rb) + (y2*y3bMa*
(-((cosb*(pr2*cosb*rbTsinbMy1 + cotb*(cosb*pr3 - aQrb)*
rbTcosbPy3b)* (Dz3bDy3b* Dy3bDy3 + Dy3bDy3* DrbDy3b))/
rbPz3bP2) + (cosb* (cotb*(cosb*pr3 - aQrb)* (Dy3bDy3 +
cosb*Dy3bDy3* DrbDy3b) + 2.*cosb*sinb*(1. - pr)*Dy3bDy3*
DrbDy3b + (a*cotb*rbTcosbPy3b* Dy3bDy3* DrbDy3b)/ rbP2))/
rbPz3b - (a*cosb*cotb*Dy3bDy3)/ rbP2 + (2.*a*cosb*cotb*y3b*
Dy3bDy3* DrbDy3b)/ rbP3))/ (rbPz3b* rb) +
(a*cotb*y2*Dy3bDy3)/rbP3 + (y2*((cosb*(pr2*cosb*rbTsinbMy1 +
cotb*(cosb*pr3 - aQrb)* rbTcosbPy3b))/ rbPz3b -
(a*cosb*cotb*y3b)/rbP2)* Dy3bDy3)/ (rbPz3b* rb) +
(y2*(-(cotb*pr3) + (y1*(2.*pr + aQrb))/ rbPy3b + (a*y1)/rbP2)*
Dy3bDy3)/ (rbPy3b*rb) - (3*a*cotb*y2*y3bMa*Dy3bDy3*
DrbDy3b)/ rbP4 - (a*cosb*cotb*y2*pr3*Dy3bDy3* DrbDy3b)/
(rbPz3b* rbP2) - (y2*y3bMa* ((cosb*(pr2*cosb*rbTsinbMy1 +
cotb*(cosb*pr3 - aQrb)* rbTcosbPy3b))/ rbPz3b -
(a*cosb*cotb*y3b)/rbP2)* Dy3bDy3* DrbDy3b)/ (rbPz3b* rbP2) -
(y2*y3bMa* (-(cotb*pr3) + (y1*(2.*pr + aQrb))/ rbPy3b +
(a*y1)/rbP2)* Dy3bDy3* DrbDy3b)/ (rbPy3b*rbP2));
dg_ic[0][1][0] +=
2.*(pr3*(-((cosb*(Dz3bDy1 + DrbDy1)*
(1. - 2.*pr + 2.*cotbP2*(1. - pr)))/ rbPz3b) +
((-pr + 2.*cotbP2*(1. - pr))* DrbDy1)/ rbPy3b) -
(y3bMa*(Dz3bDy1 + DrbDy1)* (cosbP2 - (a*cosb + cotb*pr3*z1b)/
rb - (cosbP2*y2P2 - (a*cotb*rbTcosbPy3b* z1b)/rb)/ (rbPz3b*
rb) + (a*cotb*y3b*z1b)/ rbP3))/ rbPz3bP2 -
(pr3*(cotb*(1. - 2.*pr - aQrb) - (y2P2*(pr + aQrb)*
DrbDy1)/ rbPy3bP2 + (a*cotb*y1*DrbDy1)/ rbP2 -
(a*y2P2*DrbDy1)/ (rbPy3b* rbP2)))/ rbPy3b +
(y3bMa*((cotb*pr3)/rb - (3*a*y2P2*DrbDy1)/
rbP4 - (a*y2P2*DrbDy1)/ (rbPy3b* rbP3) -
((-a + cotb*y1*pr3)* DrbDy1)/ rbP2 - (y2P2*(2.*pr + aQrb)*
DrbDy1)/ (rbPy3b* rbP2) - (y2P2*(2.*pr + aQrb)*
DrbDy1)/ (rbPy3bP2* rb)))/ rbPy3b + (y3bMa*(((cosbP2*y2P2 -
(a*cotb*rbTcosbPy3b* z1b)/rb)* (Dz3bDy1 + DrbDy1))/
(rbPz3bP2* rb) - (-((a*cotb* rbTcosbPy3b* Dz1bDy1)/
rb) + (a*cotb*rbTcosbPy3b* z1b* DrbDy1)/ rbP2 -
(a*cosb*cotb*z1b* DrbDy1)/ rb)/ (rbPz3b* rb) -
(cotb*pr3*Dz1bDy1)/ rb + (a*cotb*y3b* Dz1bDy1)/ rbP3 +
((a*cosb + cotb*pr3*z1b)* DrbDy1)/ rbP2 + ((cosbP2*y2P2 -
(a*cotb*rbTcosbPy3b* z1b)/rb)* DrbDy1)/ (rbPz3b*
rbP2) - (3*a*cotb*y3b*z1b* DrbDy1)/ rbP4))/ rbPz3b -
(a*cotb*y3bMa)/rbP3 + (cotb*pr3*(cosb + aQrb)* (Dz3bDy1 +
DrbDy1)* z1b)/ rbPz3bP2 - (cotb*pr3*(cosb + aQrb)*
Dz1bDy1)/ rbPz3b + (pr3*(-a + (y2P2*(pr + aQrb))/ rbPy3b +
cotb*y1*(1. - 2.*pr - aQrb) + pr*y3b)* DrbDy1)/ rbPy3bP2 -
(y3bMa*(-2.*pr + (a*y2P2)/rbP3 + (-a + cotb*y1*pr3)/rb +
(y2P2*(2.*pr + aQrb))/ (rbPy3b* rb))* DrbDy1)/ rbPy3bP2 +