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/***********************************************/
/**
* @file kernelStokes.cpp
*
* @brief Stokes kernel (gravity anomalies).
* @see Kernel
*
* @author Torsten Mayer-Guerr
* @date 2003-09-20
*
*/
/***********************************************/
#include "base/import.h"
#include "classes/gravityfield/gravityfield.h"
#include "classes/kernel/kernel.h"
#include "classes/kernel/kernelStokes.h"
/***********************************************/
Vector KernelStokes::coefficients(Vector3d const &q, UInt degree) const
{
if(degree==INFINITYDEGREE)
throw(Exception("In KernelStokes::coefficients: INFINITYDEGREE requested"));
Double R = q.r();
Vector k(degree+1);
Double *kn = k.field();
for(UInt n=0; n<=degree; n++)
*kn++ = R * ((n==1) ? (0.0) : (1.0/(n-1.0)) );
return k;
}
/***********************************************/
Vector KernelStokes::inverseCoefficients(Vector3d const &p, UInt degree, Bool interior) const
{
if(degree==INFINITYDEGREE)
throw(Exception("In KernelStokes::inverseCoefficients: INFINITYDEGREE requested"));
Double r = p.r();
Vector k(degree+1);
Double *kn = k.field();
if(interior)
for(UInt n=0; n<=degree; n++)
*kn++ = -(n+2.)/r;
else
for(UInt n=0; n<=degree; n++)
*kn++ = (n-1.)/r;
return k;
}
/***********************************************/
Double KernelStokes::kernel(Vector3d const &p, Vector3d const &q) const
{
Vector3d diff = p-q;
Double r = p.r();
Double R = q.r();
Double l = diff.r();
Double cos_psi = (R*R+r*r-l*l)/(2*R*r);
return R*(2*R/l-3*R*l/(r*r)-R*R/(r*r)*cos_psi*(5+3*log((l+r-R*cos_psi)/(2*r))));
}
/***********************************************/
Double KernelStokes::radialDerivative(Vector3d const &p, Vector3d const &q) const
{
Vector3d diff = p-q;
Double r = p.r();
Double R = q.r();
Double l = diff.r();
Double cos_psi = (R*R+r*r-l*l)/(2*R*r);
Double term0 = (R*R+r*r-l*l);
Double term = (r+l-R)*(r+l+R);
Double ln_term = log(term/(4*r*r));
Double r2 = r*r;
Double r3 = r2*r;
Double r4 = r3*r;
// Ableitungen Stokes nach r,l
// Kettenregel: ds/dr = dS/du*du/dr + dS/dl*dl/dr mit u = r
Double dS_du = R*(6*R*l/r3+1.5*R*term0*(5+3*ln_term)/r4
-R*(5+3*ln_term)/r2
-3*R*term0*(0.25*term0/r3-0.5*(r-0.5*term0/r+l)/r2)/(r2*(r-0.5*term0/r+l)));
Double dS_dl = R*(R/r3*(-2*r3/(l*l)-3*r+5*l+3*l*ln_term
-3*term0*(l+r)/term));
Double dl_dr = (r-R*cos_psi)/l;
Double dS_dr = dS_du+dS_dl*dl_dr;
return dS_dr;
}
/***********************************************/
Vector3d KernelStokes::gradient(Vector3d const &p, Vector3d const &q) const
{
// Vorausberechnungen
Vector3d diff = p-q;
Double r = p.r();
Double R = q.r();
Double l = diff.r();
Double term0 = (R*R+r*r-l*l);
Double term = (r+l-R)*(r+l+R);
Double ln_term = log(term/(4*r*r));
Double r2 = r*r;
Double r3 = r2*r;
Double r4 = r3*r;
// Ableitungen r,l nach x,y,z
Vector3d dr = 1/r * p;
Vector3d dl = 1/l * diff;
// Ableitungen Stokes nach r,l
Double dS_dr = -R/r2+6*R*l/r3+1.5*R/r4*term0*(5+3*ln_term)
-R/r2*(5+3*ln_term)
-3*R/r3*term0*((r+l)/term-1/r);
Double dS_dl = R/r3*(-2*r3/(l*l)-3*r+5*l+3*l*ln_term
-3*term0*(l+r)/term);
// Kettenregel
return R*(dS_dr*dr + dS_dl*dl);
}
/***********************************************/
Tensor3d KernelStokes::gradientGradient(Vector3d const &p, Vector3d const &q) const
{
Tensor3d tns;
// Vorausberechnungen
Double r = p.r();
Double R = q.r();
Double l = (p-q).r();
Double term0 = (R*R+r*r-l*l);
Double term = (r+l-R)*(r+l+R);
Double ln_term = log(term/(4*r*r));
Double r2 = r*r;
Double r3 = r2*r;
Double r4 = r3*r;
Double r5 = r4*r;
// Ableitungen r,l nach x,y,z
Double dr_dx = p.x()/r; Double dl_dx = (p.x()-q.x())/l;
Double dr_dy = p.y()/r; Double dl_dy = (p.y()-q.y())/l;
Double dr_dz = p.z()/r; Double dl_dz = (p.z()-q.z())/l;
// Zweite Ableitungen r,l nach x,y,z
Double d2r_dx2 = 1/r-p.x()*p.x()/r3;
Double d2r_dy2 = 1/r-p.y()*p.y()/r3;
Double d2r_dz2 = 1/r-p.z()*p.z()/r3;
Double d2r_dxdy = -p.x()*p.y()/r3;
Double d2r_dxdz = -p.x()*p.z()/r3;
Double d2r_dydz = -p.y()*p.z()/r3;
Double d2l_dx2 = 1/l-(p.x()-q.x())*(p.x()-q.x())/(l*l*l);
Double d2l_dy2 = 1/l-(p.y()-q.y())*(p.y()-q.y())/(l*l*l);
Double d2l_dz2 = 1/l-(p.z()-q.z())*(p.z()-q.z())/(l*l*l);
Double d2l_dxdy = -(p.x()-q.x())*(p.y()-q.y())/(l*l*l);
Double d2l_dxdz = -(p.x()-q.x())*(p.z()-q.z())/(l*l*l);
Double d2l_dydz = -(p.y()-q.y())*(p.z()-q.z())/(l*l*l);
// Ableitungen Stokes nach r,l
Double dS_dr = -R/r2+6*R*l/r3+1.5*R*term0*(5+3*ln_term)/r4
-R*(5+3*ln_term)/r2
-3*R*term0*(0.25*term0/r3
-0.5*(r-0.5*term0/r+l)/r2)/(r2*(r-0.5*term0/r+l));
Double dS_dl = R/r3*(-2*r3/(l*l)-3*r+5*l+3*l*ln_term
-3*term0*(l+r)/term);
// Zweite Ableitungen Stokes nach r,l
Double d2S_dr2 = R*(-3/r3-18*l/r4-30*(R*R-l*l)/r5
+(15*r2-18*term0)*ln_term/r5
+12*(R*R-r*l-l*l)*(R*R-l*l)/(r5*term)
+6*term0*(R*R-r*l-l*l)*(r+l)/(r4*term*term)
-3*term0*(3*l*l+2*r*l-3*R*R)/(r5*term));
Double d2S_drdl= R*(6.0/r3 - 15*l/r4 - 9*l*ln_term/r4
+ 6*(l*(R*R-l*r-l*l)-r2*(l+r))/(r4*term)
+ 3*term0*(3*l+2*r)/(r4*term)
+ 6*(r+l)*(r+l)*term0/(r3*term*term));
Double d2S_dl2 = R/(r3)*(4*r3/(l*l*l)+5+3*ln_term
+ (12*l*(r+l)-3*term0)/term
+ 6*(r+l)*(r+l)*term0/(term*term));
// Kettenregel
tns.xx() = dS_dr*d2r_dx2 + dS_dl*d2l_dx2
+ d2S_dr2*dr_dx*dr_dx + 2*d2S_drdl*dr_dx*dl_dx + d2S_dl2*dl_dx*dl_dx;
tns.yy() = dS_dr*d2r_dy2 + dS_dl*d2l_dy2
+ d2S_dr2*dr_dy*dr_dy + 2*d2S_drdl*dr_dy*dl_dy + d2S_dl2*dl_dy*dl_dy;
tns.zz() = dS_dr*d2r_dz2 + dS_dl*d2l_dz2
+ d2S_dr2*dr_dz*dr_dz + 2*d2S_drdl*dr_dz*dl_dz + d2S_dl2*dl_dz*dl_dz;
tns.xy() = dS_dr*d2r_dxdy + dS_dl*d2l_dxdy
+ d2S_dr2*dr_dx*dr_dy + d2S_drdl*dr_dx*dl_dy + d2S_drdl*dr_dy*dl_dx + d2S_dl2*dl_dx*dl_dy;
tns.xz() = dS_dr*d2r_dxdz + dS_dl*d2l_dxdz
+ d2S_dr2*dr_dx*dr_dz + d2S_drdl*dr_dx*dl_dz + d2S_drdl*dr_dz*dl_dx + d2S_dl2*dl_dx*dl_dz;
tns.yz() = dS_dr*d2r_dydz + dS_dl*d2l_dydz
+ d2S_dr2*dr_dy*dr_dz + d2S_drdl*dr_dy*dl_dz + d2S_drdl*dr_dz*dl_dy + d2S_dl2*dl_dy*dl_dz;
return R*tns;
}
/***********************************************/
Double KernelStokes::inverseKernel(Vector3d const &p, Vector3d const &q, const Kernel &kernel) const
{
// anomalies = -dK/dr - 2K/r
return -kernel.radialDerivative(p,q) - 2*kernel.kernel(p,q)/p.r();;
}
/***********************************************/
Double KernelStokes::inverseKernel(const Time &time, const Vector3d &p, const GravityfieldBase &field) const
{
// anomalies = -dK/dr - 2K/r
return -field.radialGradient(time, p) - 2*field.potential(time, p)/p.r();
}
/***********************************************/
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