XGC1
dbc_mod Module Reference

"(D)irichlet (B)oundary (C)onditions" A module for evaluating Dirichlet boundary conditions for the Fourier decomposed Ampere and Poisson equations. More...

## Public Member Functions

subroutine, public dbc_weights (weights, R, Z, n, grid)
Returns weights $$w_i(R,Z)$$ for approximating the integral $$\Phi(R,Z)=\int_{\Omega'}\frac{R' f(R',Z')}{\sqrt{R'^2+R^2+(Z-Z')^2}}g(k,n)dR'dZ'$$ as $$\Phi(R,Z)\approx\sum_{i=1}^{n_{\textrm{nodes}}} w_i f_i$$. Here $$g(k,n)=\int_0^{2\pi}\frac{\cos(nx)}{\sqrt{1-k\cos(x)}}dx$$ and $$k(R,Z,R',Z')=\frac{2RR'}{R^2+R'^2+(R-R')^2}$$. The data $$f(R',Z')$$ is assumed piecewise linear over the triangulation of the domain $$\Omega$$ and Gauss quadrature is used for the $$dR'dZ'$$ integral. More...

real(kind=8) function, public dbc_triangle (R, Z, RS, ZS, FS, n)
Evaluates the integral $$Phi(R',Z')=\int_{\Delta}\frac{R' f(R',Z')}{\sqrt{R'^2+R^2+(Z-Z')^2}}g(k,n)dR'dZ'$$ where $$g(k,n)=\int_0^{2\pi}\frac{\cos(nx)}{\sqrt{1-k\cos(x)}}dx$$ and $$k(R,Z,R',Z')=\frac{2RR'}{R^2+R'^2+(R-R')^2}$$ using linear interpolation for the data $$f(R',Z')$$ over the triangle $$\Delta$$ and Gauss quadrature for the $$dR'dZ'$$ integral. More...

## Private Member Functions

real(kind=8) function, dimension(3) dbc_triangle2dbarycoords (xs, ys, x, y)

## Private Attributes

integer, parameter ngauss =6
number of points for the Gauss-quadrature over a unit triangle More...

real(kind=8), parameter pi = 4*atan(1.0D0)
value for pi used inside the module. More...

real(kind=8), dimension(6),
parameter
xi =(/ 0.816847572980459D0, 0.091576213509771D0, 0.091576213509771D0, 0.108103018168070D0, 0.445948490915965D0, 0.445948490915965D0 /)
x coordinates for the Gauss-quadrature over a unit triangle More...

real(kind=8), dimension(6),
parameter
yi =(/ 0.091576213509771D0, 0.816847572980459D0, 0.091576213509771D0, 0.445948490915965D0, 0.108103018168070D0, 0.445948490915965D0 /)
y coordinates for the Gauss-quadrature over a unit triangle More...

real(kind=8), dimension(6),
parameter
wi =(/ 0.109951743655322D0, 0.109951743655322D0, 0.109951743655322D0, 0.223381589678011D0, 0.223381589678011D0, 0.223381589678011D0 /)
weight for the Gauss point in Gauss-quadrature over a unit triangle More...

## Detailed Description

"(D)irichlet (B)oundary (C)onditions" A module for evaluating Dirichlet boundary conditions for the Fourier decomposed Ampere and Poisson equations.

## Member Function/Subroutine Documentation

 real(kind=8) function, public dbc_mod::dbc_triangle ( real(kind=8), intent(in) R, real(kind=8), intent(in) Z, real(kind=8), dimension(3), intent(in) RS, real(kind=8), dimension(3), intent(in) ZS, real(kind=8), dimension(3), intent(in) FS, integer, intent(in) n )

Evaluates the integral $$Phi(R',Z')=\int_{\Delta}\frac{R' f(R',Z')}{\sqrt{R'^2+R^2+(Z-Z')^2}}g(k,n)dR'dZ'$$ where $$g(k,n)=\int_0^{2\pi}\frac{\cos(nx)}{\sqrt{1-k\cos(x)}}dx$$ and $$k(R,Z,R',Z')=\frac{2RR'}{R^2+R'^2+(R-R')^2}$$ using linear interpolation for the data $$f(R',Z')$$ over the triangle $$\Delta$$ and Gauss quadrature for the $$dR'dZ'$$ integral.

Parameters
 [in] R cylindrical R for the position where to evaluate the integral [in] Z cylindrical Z for the position where to evaluate the integral [in] RS cylindrical R for the triangle vertices [in] ZS cylindrical Z for the triangle vertices [in] FS the function $$f(R,Z)$$ values at the triangle vertices [in] n the mode number for the toroidal angular integrals.
Todo:
Right now I'm not aware of anything that should be improved.

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 real(kind=8) function, dimension(3) dbc_mod::dbc_triangle2dbarycoords ( real(kind=8), dimension(3), intent(in) xs, real(kind=8), dimension(3), intent(in) ys, real(kind=8), intent(in) x, real(kind=8), intent(in) y )
private

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 subroutine, public dbc_mod::dbc_weights ( real(kind=8), dimension(grid%nnode), intent(out) weights, real(kind=8), intent(in) R, real(kind=8), intent(in) Z, integer, intent(in) n, type(grid_type), intent(in) grid )

Returns weights $$w_i(R,Z)$$ for approximating the integral $$\Phi(R,Z)=\int_{\Omega'}\frac{R' f(R',Z')}{\sqrt{R'^2+R^2+(Z-Z')^2}}g(k,n)dR'dZ'$$ as $$\Phi(R,Z)\approx\sum_{i=1}^{n_{\textrm{nodes}}} w_i f_i$$. Here $$g(k,n)=\int_0^{2\pi}\frac{\cos(nx)}{\sqrt{1-k\cos(x)}}dx$$ and $$k(R,Z,R',Z')=\frac{2RR'}{R^2+R'^2+(R-R')^2}$$. The data $$f(R',Z')$$ is assumed piecewise linear over the triangulation of the domain $$\Omega$$ and Gauss quadrature is used for the $$dR'dZ'$$ integral.

Parameters
 [out] weights the precomputed weights for evaluating the integral [in] R cylindrical R of the position for which the weights are precomputed [in] Z cylindrical Z of the position for which the weights are precomputed [in] n the mode number for the angular integrals. [in] grid the structure containing the mesh and connectivity
Todo:
Right now I'm not aware of anything that should be improved.

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## Member Data Documentation

 integer, parameter dbc_mod::ngauss =6
private

number of points for the Gauss-quadrature over a unit triangle

 real(kind=8), parameter dbc_mod::pi = 4*atan(1.0D0)
private

value for pi used inside the module.

 real(kind=8), dimension(6), parameter dbc_mod::wi =(/ 0.109951743655322D0, 0.109951743655322D0, 0.109951743655322D0, 0.223381589678011D0, 0.223381589678011D0, 0.223381589678011D0 /)
private

weight for the Gauss point in Gauss-quadrature over a unit triangle

 real(kind=8), dimension(6), parameter dbc_mod::xi =(/ 0.816847572980459D0, 0.091576213509771D0, 0.091576213509771D0, 0.108103018168070D0, 0.445948490915965D0, 0.445948490915965D0 /)
private

x coordinates for the Gauss-quadrature over a unit triangle

 real(kind=8), dimension(6), parameter dbc_mod::yi =(/ 0.091576213509771D0, 0.816847572980459D0, 0.091576213509771D0, 0.445948490915965D0, 0.108103018168070D0, 0.445948490915965D0 /)
private

y coordinates for the Gauss-quadrature over a unit triangle

The documentation for this module was generated from the following file:
• /u/gitlab-xgc/builds/YGMz2TJ8/0/xgc/XGC-Devel/XGC_core/dbc.F90