Class for computing the scalar Green's function for homogeneous, linear, and isotropic materials, with the singular term(s) implicitly subtracted using the Taylor series expansion of the kernel. More...
#include <hgf.hpp>
Inheritance diagram for bem::SingularitySubtractedTaylorHGF:
Detailed Description
Class for computing the scalar Green's function for homogeneous, linear, and isotropic materials, with the singular term(s) implicitly subtracted using the Taylor series expansion of the kernel.
Member Function Documentation
◆ kernel() [1/2]
| Complex bem::SingularitySubtractedTaylorHGF::kernel | ( | ConstEigRef< EigColVecN< Float, 3 > > | r_obs, |
| ConstEigRef< EigColVecN< Float, 3 > > | r_src, | ||
| const Complex | k | ||
| ) | const |
Computes the kernel for given observation and source points.
- Parameters
-
[in] r_obs - Observer position vector. [in] r_src - Source position vector. [in] k - Complex wavenumber.
- Returns
- Kernel value.
Computes
\[ \frac{e^{-jk|\vec{r} - \vec{r}\,'|}}{4\pi|\vec{r} - \vec{r}\,'|} - \frac{1}{4\pi|\vec{r} - \vec{r}\,'|} \]
where \( \vec{r} \) is the observer position vector, \( \vec{r}\,' \) is the source position vector, and \( k \) is the complex wavenumber. The singular term is never actually computed; it is implicitly subtracted out using the Taylor series expansion
\[ \frac{e^{-jk|\vec{r} - \vec{r}\,'|}}{4\pi|\vec{r} - \vec{r}\,'|} = \frac{1}{4\pi|\vec{r} - \vec{r}\,'|} + \frac{(-jk)}{4\pi} + \frac{(-jk)^2}{4\pi\cdot 2!}|\vec{r} - \vec{r}\,'| + \ldots \]
truncated when the next term is smaller than the tolerance KERNEL_DEFAULT_TOL relative to the sum so far.
◆ grad_kernel() [1/2]
| EigColVecN< Complex, 3 > bem::SingularitySubtractedTaylorHGF::grad_kernel | ( | ConstEigRef< EigColVecN< Float, 3 > > | r_obs, |
| ConstEigRef< EigColVecN< Float, 3 > > | r_src, | ||
| const Complex | k | ||
| ) | const |
Computes the gradient of the kernel for given observation and source points.
- Parameters
-
[in] r_obs - Observer position vector. [in] r_src - Source position vector. [in] k - Complex wavenumber.
- Returns
- Components of the gradient of the kernel.
Computes
\begin{align*} \nabla\left(\frac{e^{-jk|\vec{r} - \vec{r}\,'|}}{4\pi|\vec{r} - \vec{r}\,'|} - \frac{1}{4\pi|\vec{r} - \vec{r}\,'|}\right) &=\\ -(\vec{r} - \vec{r}\,')&\left[\left(1 + jk|\vec{r} - \vec{r}\,'|\right) \frac{e^{-jk|\vec{r} - \vec{r}\,'|}}{4\pi|\vec{r} - \vec{r}\,'|^3} - \frac{1}{4\pi|\vec{r} - \vec{r}\,'|^3} - \left(\frac{1}{2}\right) \frac{k^2}{4\pi|\vec{r} - \vec{r}\,'|}\right] \end{align*}
where \( \vec{r} \) is the observer position vector, \( \vec{r}\,' \) is the source position vector, and \( k \) is the complex wavenumber. The singular term is never actually computed; it is implicitly subtracted out using the Taylor series expansion
\[ \nabla\left(\frac{e^{-jk|\vec{r} - \vec{r}\,'|}}{4\pi|\vec{r} - \vec{r}\,'|}\right) = -\frac{\vec{r} - \vec{r}\,'}{4\pi|\vec{r} - \vec{r}\,'|}\left[ \frac{1}{|\vec{r} - \vec{r}\,'|^2} - \frac{(-jk)^2}{2!} - 2\frac{(-jk)^3}{3!}|\vec{r} - \vec{r}\,'| + \ldots\right] \]
truncated when the next term is smaller than the tolerance KERNEL_DEFAULT_TOL relative to the sum so far.
◆ kernel() [2/2]
|
pure virtualinherited |
Computes the scalar kernel for given observation and source points.
- Parameters
-
[in] r_obs - Observer position vector. [in] r_src - Source position vector. [in] k - Complex wavenumber.
- Returns
- Scalar kernel value.
◆ grad_kernel() [2/2]
|
pure virtualinherited |
Computes the gradient of the scalar kernel for given observation and source points.
- Parameters
-
[in] r_obs - Observer position vector. [in] r_src - Source position vector. [in] k - Complex wavenumber.
- Returns
- Components of the gradient of the scalar kernel.
The documentation for this class was generated from the following files:
