Scalable solving of boundary element methods utilizing the Green cross approximation method and GPUs
This project is maintained by Bennet Carstensen
Set of kernel functions and their partial derivative usable in the Green cross approximation method. More...
Functions | |
HEADER_PREFIX real | laplace_kernel (const field *x, const field *y, const uint dim) |
Implementation of the fundamental solution of the negative Laplace operator \(- \Delta\) in 2D and 3D. More... | |
HEADER_PREFIX real | pdx_laplace_kernel (const field *x, const field *y, const uint dim, const uint i) |
Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i th component of the first operand in 2D and 3D. More... | |
HEADER_PREFIX real | pdy_laplace_kernel (const field *x, const field *y, const uint dim, const uint i) |
Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i th component of the second operand in 2D and 3D. More... | |
Variables | |
static const real | r_minus_two_pi = -0.159154943091895336 |
\(- \frac{1}{2 \pi}\) More... | |
static const real | r_four_pi = 0.0795774715459476679 |
\(\frac{1}{4 \pi}\) More... | |
static const real | r_minus_pi = -0.318309886183790671 |
\(- \frac{1}{\pi}\) More... | |
Set of kernel functions and their partial derivative usable in the Green cross approximation method.
HEADER_PREFIX real laplace_kernel | ( | const field * | x, |
const field * | y, | ||
const uint | dim | ||
) |
Implementation of the fundamental solution of the negative Laplace operator \(- \Delta\) in 2D and 3D.
Calculates the fundamental solution of the negative Laplace operator:
\[ g(x, y) := \begin{cases} -\frac{1}{2 \pi} \log{\left \Vert x - y \right \Vert_2} & \text{if } x \neq y, \ dim = 2 \\ \\ \frac{1}{4 \pi \left \Vert x - y \right \Vert_2} & \text{if } x \neq y, \ dim = 3 \\ \\ 0 & \text{otherwise} \end{cases} \]
[in] | x | First operand as an array of dim coefficients. |
[in] | y | Second operand as an array of dim coefficients. |
[in] | dim | Dimension in which to evaluate the negative Laplace operator. |
dim
dimensional negative Laplace operator in the parameters x
and y
. HEADER_PREFIX real pdx_laplace_kernel | ( | const field * | x, |
const field * | y, | ||
const uint | dim, | ||
const uint | i | ||
) |
Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i
th component of the first operand in 2D and 3D.
Calculates the partial derivative of the fundamental solution of the negative Laplace operatorin in the i
th component of the first operand:
\[ \frac{\delta g}{\delta x_{i}}(x, y) := \begin{cases} - \frac{x_i - y_i}{\pi \left \Vert x - y \right \Vert_{2}^{2}} & \text{if } x \neq y, \ dim = 2 \\ \\ - \frac{x_i - y_i}{4 \pi \left \Vert x - y \right \Vert_{2}^{3}} & \text{if } x \neq y, \ dim = 3 \\ \\ 0 & \text{otherwise} \end{cases} \]
[in] | x | First operand as an array of dim coefficients. Parital derivating the i th component of this operand. |
[in] | y | Second operand as an array of dim coefficients. |
[in] | dim | Dimension in which to evaluate the partial derivative of the negative Laplace operator |
[in] | i | Index of the component of x in which to parital derive. |
dim
dimensional partial derivative of the fundamental solution of the negative Laplace operator. HEADER_PREFIX real pdy_laplace_kernel | ( | const field * | x, |
const field * | y, | ||
const uint | dim, | ||
const uint | i | ||
) |
Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i
th component of the second operand in 2D and 3D.
Calculates the partial derivative of the fundamental solution of the negative Laplace operatorin in the i
th component of the second operand:
\[ \frac{\delta g}{\delta y_{i}}(x, y) := \begin{cases} \frac{x_i - y_i}{\pi \left \Vert x - y \right \Vert_{2}^{2}} & \text{if } x \neq y, \ dim = 2 \\ \\ \frac{x_i - y_i}{4 \pi \left \Vert x - y \right \Vert_{2}^{3}} & \text{if } x \neq y, \ dim = 3 \\ \\ 0 & \text{otherwise} \end{cases} \]
[in] | x | First operand as an array of dim coefficients. |
[in] | y | Second operand as an array of dim coefficients. Parital derivating the i th component of this operand. |
[in] | dim | Dimension in which to evaluate the partial derivative of the negative Laplace operator |
[in] | i | Index of the component of y in which to parital derive. |
dim
dimensional partial derivative of the fundamental solution of the negative Laplace operator.
|
static |
\(\frac{1}{4 \pi}\)
|
static |
\(- \frac{1}{\pi}\)
|
static |
\(- \frac{1}{2 \pi}\)