Substructure containing callback functions to different types of kernel evaluations. More...

#include <bem3d.h>

Data Fields
void(*	fundamental )(pcbem3d bem, const real(X)[3], const real(Y)[3], pamatrix V)
	Evaluate the fundamental solution at given point sets `X` and `Y`. More...

void(*	fundamental_wave )(pcbem3d bem, const real(X)[3], const real(Y)[3], pcreal dir, pamatrix V)
	Evaluate the modified fundamental solution in direction $c_\iota$ at given point sets `X` and `Y`. More...

void(*	dny_fundamental )(pcbem3d bem, const real(X)[3], const real(Y)[3], const real(*NY)[3], pamatrix V)
	Evaluate the normal derivative of the fundamental solution in respect to the 2nd component at given point sets `X` and `Y` More...

void(*	dnx_dny_fundamental )(pcbem3d bem, const real(X)[3], const real(NX)[3], const real(Y)[3], const real(NY)[3], pamatrix V)
	Evaluate the normal derivatives of the fundamental solution in respect to the 1st and 2nd component at given point sets `X` and `Y` More...

void(*	kernel_row )(const uint idx, const real(Z)[3], pcbem3d bem, pamatrix A)
	Integrate the kernel function within the 1st component. More...

void(*	kernel_col )(const uint idx, const real(Z)[3], pcbem3d bem, pamatrix B)
	Integrate the kernel function within the 2nd component. More...

void(*	dnz_kernel_row )(const uint idx, const real(Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix A)
	Integrate the normal derivative of the kernel function with respect to the 2nd component within the 1st component. More...

void(*	dnz_kernel_col )(const uint idx, const real(Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix B)
	Integrate the normal derivative of the kernel function with respect to the 1st component within the 2nd component. More...

void(*	fundamental_row )(const uint idx, const real(Z)[3], pcbem3d bem, pamatrix A)
	Integrate the fundamental solution within the 1st component. More...

void(*	fundamental_col )(const uint idx, const real(Z)[3], pcbem3d bem, pamatrix B)
	Integrate the fundamental solution within the 2nd component. More...

void(*	dnz_fundamental_row )(const uint idx, const real(Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix A)
	Integrate the normal derivative of the fundamental solution with respect to the 2nd component within the 1st component. More...

void(*	dnz_fundamental_col )(const uint idx, const real(Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix B)
	Integrate the normal derivative of the fundamental solution with respect to the 1st component within the 2nd component. More...

void(*	lagrange_row )(const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcbem3d bem, pamatrix V)
	Integrate the Lagrange polynomials or their derivatives within the 1st component. More...

void(*	lagrange_wave_row )(const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcreal dir, pcbem3d bem, pamatrix V)
	Integrate the modified Lagrange polynomials in direction $c_\iota$ or their derivatives within the 1st component. More...

void(*	lagrange_col )(const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcbem3d bem, pamatrix W)
	Integrate the Lagrange polynomials or their derivatives within the 2nd component. More...

void(*	lagrange_wave_col )(const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcreal dir, pcbem3d bem, pamatrix W)
	Integrate the modified Lagrange polynomials in direction $c_\iota$ or their derivatives within the 2nd component. More...

Detailed Description

Substructure containing callback functions to different types of kernel evaluations.

This struct consists of a set of callback function, that will evaluate the kernel, its derivatives or integrals over the kernel and / or its derivatives. Of course they depend on an appropriate problem to be solved. The kernel function will be a function of the following form:

$g : \mathbb R^3 \times \mathbb R^3 \to \mathbb K$

In many application the field $\mathbb K$ will just be the field of the real number $\mathbb R$ , as in the laplace-problem. But it can also be the complex field $\mathbb C$ as in the helmholtz-equation or a vector valued function as in the lame-problem.

Field Documentation

void(* dnx_dny_fundamental) (pcbem3d bem, const real(*X)[3], const real(*NX)[3], const real(*Y)[3], const real(*NY)[3], pamatrix V)

Evaluate the normal derivatives of the fundamental solution in respect to the 1st and 2nd component at given point sets X and Y

This callback will evaluate the normal derivatives of the fundamental solution $ g $ corresponding to $ x $ and $ y $ at points $x_i, y_j \in \mathbb R^3$ and store the result into matrix V at position $V_{ij}$ .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
X	An array of 3D-vectors. `X[i][0]` will be the first component of the i-th vector. Analogously `X[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->rows` .
NX	An array of normal vectors correspoding to the vectors defined by X and are stored in the same way as X and Y.
Y	An array of 3D-vectors. `Y[i][0]` will be the first component of the i-th vector. Analogously `Y[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->cols` .
NY	An array of normal vectors correspoding to the vectors defined by Y and are stored in the same way as X and Y.
V	V will contain the results of the kernel evaluations. It applies $V_{ij} = \frac{\partial^2 g}{\partial n_x \partial n_y}(\vec x_i, \vec y_j) .$

void(* dny_fundamental) (pcbem3d bem, const real(*X)[3], const real(*Y)[3], const real(*NY)[3], pamatrix V)

Evaluate the normal derivative of the fundamental solution in respect to the 2nd component at given point sets X and Y

This callback will evaluate the normal derivatives of the fundamental solution $ g $ corresponding to $ y $ at points $x_i, y_j \in \mathbb R^3$ and store the result into matrix V at position $V_{ij}$ .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
X	An array of 3D-vectors. `X[i][0]` will be the first component of the i-th vector. Analogously `X[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->rows` .
Y	An array of 3D-vectors. `Y[i][0]` will be the first component of the i-th vector. Analogously `Y[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->cols` .
NY	An array of normal vectors correspoding to the vectors defined by Y and are stored in the same way as X and Y.
V	V will contain the results of the kernel evaluations. It applies $V_{ij} = \frac{\partial g}{\partial n_y}(\vec x_i, \vec y_j) .$

void(* dnz_fundamental_col) (const uint *idx, const real(*Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix B)

Integrate the normal derivative of the fundamental solution with respect to the 1st component within the 2nd component.

This callback will evaluate the integral

$\int_\Gamma \frac{\partial g}{\partial n_z} (\vec z, \vec y) \, \psi(\vec y) \, \mathrm d \vec y$

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `B->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{B->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `B->cols` .
NZ	An array Normal vectors corresponding to the vectors Z.
bem	BEM-object containing additional information for computation of the matrix entries.
B	B will contain the results of the integral evaluations. It applies $B_{ij} = \int_\Gamma \frac{\partial g}{\partial n_z} (\vec z_j, \vec y) \, \psi_i(\vec y) \, \mathrm d \vec y .$

void(* dnz_fundamental_row) (const uint *idx, const real(*Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix A)

Integrate the normal derivative of the fundamental solution with respect to the 2nd component within the 1st component.

This callback will evaluate the integral

$\int_\Gamma \varphi(\vec x) \, \frac{\partial g}{\partial n_z} (\vec x, \vec z) \, \mathrm d \vec x$

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `A->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{A->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `A->cols` .
NZ	An array Normal vectors corresponding to the vectors Z.
bem	BEM-object containing additional information for computation of the matrix entries.
A	A will contain the results of the integral evaluations. It applies $A_{ij} = \int_\Gamma \varphi_i(\vec x) \, \frac{\partial g}{\partial n_z} (\vec x, \vec z_j) \, \mathrm d \vec x .$

void(* dnz_kernel_col) (const uint *idx, const real(*Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix B)

Integrate the normal derivative of the kernel function with respect to the 1st component within the 2nd component.

This callback will evaluate the integral

$\int_\Gamma \frac{\partial \gamma}{\partial n_z} (\vec z, \vec y) \, \psi(\vec y) \, \mathrm d \vec y$

with $\gamma$ being a kernel function depending on a specific integral operator. In case of the single layer potential it applies $\gamma = g$ . But in case of the double layer potential e.g. it applies $\gamma = \frac{\partial g}{\partial n_y}$ .

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `B->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{B->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `B->cols` .
NZ	An array Normal vectors corresponding to the vectors Z.
bem	BEM-object containing additional information for computation of the matrix entries.
B	B will contain the results of the integral evaluations. It applies $B_{ij} = \int_\Gamma \frac{\partial \gamma}{\partial n_z} (\vec z_j, \vec y) \, \psi_i(\vec y) \, \mathrm d \vec y .$

void(* dnz_kernel_row) (const uint *idx, const real(*Z)[3], const real(*NZ)[3], pcbem3d bem, pamatrix A)

Integrate the normal derivative of the kernel function with respect to the 2nd component within the 1st component.

This callback will evaluate the integral

$\int_\Gamma \varphi(\vec x) \, \frac{\partial \gamma}{\partial n_z} (\vec x, \vec z) \, \mathrm d \vec x$

with $\gamma$ being a kernel function depending on a specific integral operator. In case of the single layer potential it applies $\gamma = g$ . But in case of the double layer potential e.g. it applies $\gamma = \frac{\partial g}{\partial n_y}$ .

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `A->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{A->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `A->cols` .
NZ	An array Normal vectors corresponding to the vectors Z.
bem	BEM-object containing additional information for computation of the matrix entries.
A	A will contain the results of the integral evaluations. It applies $A_{ij} = \int_\Gamma \varphi_i(\vec x) \, \frac{\partial \gamma}{\partial n_z} (\vec x, \vec z_j) \, \mathrm d \vec x .$

void(* fundamental) (pcbem3d bem, const real(*X)[3], const real(*Y)[3], pamatrix V)

Evaluate the fundamental solution at given point sets X and Y.

This callback will evaluate the fundamental solution $ g $ at points $x_i, y_j \in \mathbb R^3$ and store the result into matrix V at position $V_{ij}$ .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
X	An array of 3D-vectors. `X[i][0]` will be the first component of the i-th vector. Analogous `X[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->rows` .
Y	An array of 3D-vectors. `Y[i][0]` will be the first component of the i-th vector. Analogous `Y[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->cols` .
V	V will contain the results of the kernel evaluations. It applies $V_{ij} = g(\vec x_i, \vec y_j) .$

void(* fundamental_col) (const uint *idx, const real(*Z)[3], pcbem3d bem, pamatrix B)

Integrate the fundamental solution within the 2nd component.

This callback will evaluate the integral

$\int_\Gamma g(\vec z, \vec y) \, \psi(\vec y) \, \mathrm d \vec y$

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `B->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{B->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `B->cols` .
bem	BEM-object containing additional information for computation of the matrix entries.
B	B will contain the results of the integral evaluations. It applies $B_{ij} = \int_\Gamma g(\vec z_j, \vec y) \, \psi_i(\vec y) \, \mathrm d \vec y .$

void(* fundamental_row) (const uint *idx, const real(*Z)[3], pcbem3d bem, pamatrix A)

Integrate the fundamental solution within the 1st component.

This callback will evaluate the integral

$\int_\Gamma \varphi(\vec x) \, g(\vec x, \vec z) \, \mathrm d \vec x$

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `A->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{A->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `A->cols` .
bem	BEM-object containing additional information for computation of the matrix entries.
A	A will contain the results of the integral evaluations. It applies $A_{ij} = \int_\Gamma \varphi_i(\vec x) \, g(\vec x, \vec z_j) \, \mathrm d \vec x .$

void(* fundamental_wave) (pcbem3d bem, const real(*X)[3], const real(*Y)[3], pcreal dir, pamatrix V)

Evaluate the modified fundamental solution in direction $c_\iota$ at given point sets X and Y.

This callback will evaluate the modified fundamental solution $g_{c_\iota}$ with direction $c_\iota$ at points $x_i, y_j \in \mathbb R^3$ and store the result into matrix V at position $V_{ij}$ .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
X	An array of 3D-vectors. `X[i][0]` will be the first component of the i-th vector. Analogous `X[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->rows` .
Y	An array of 3D-vectors. `Y[i][0]` will be the first component of the i-th vector. Analogous `Y[i][1]` will be the second component of the i-th vector. The length of this array is determined by `V->cols` .
dir	Direction $c_\iota$ in which the modified fundamental solution should be evaluated.
V	V will contain the results of the kernel evaluations. It applies $V_{ij} = g_{c_\iota}(\vec x_i, \vec y_j) = g(\vec x_i, \vec y_j) e^{-\langle c_\iota, \vec x_i - \vec y_j \rangle}.$

void(* kernel_col) (const uint *idx, const real(*Z)[3], pcbem3d bem, pamatrix B)

Integrate the kernel function within the 2nd component.

This callback will evaluate the integral

$\int_\Gamma \gamma(\vec z, \vec y) \, \psi(\vec y) \, \mathrm d \vec y$

with $\gamma$ being a kernel function depending on a specific integral operator. In case of the single layer potential it applies $\gamma = g$ . But in case of the double layer potential e.g. it applies $\gamma = \frac{\partial g}{\partial n_y}$ .

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `B->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{B->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `B->cols` .
bem	BEM-object containing additional information for computation of the matrix entries.
B	B will contain the results of the integral evaluations. It applies $B_{ij} = \int_\Gamma \gamma(\vec z_j, \vec y) \, \psi_i(\vec y) \, \mathrm d \vec y .$

void(* kernel_row) (const uint *idx, const real(*Z)[3], pcbem3d bem, pamatrix A)

Integrate the kernel function within the 1st component.

This callback will evaluate the integral

$\int_\Gamma \varphi(\vec x) \, \gamma(\vec x, \vec z) \, \mathrm d \vec x$

with $\gamma$ being a kernel function depending on a specific integral operator. In case of the single layer potential it applies $\gamma = g$ . But in case of the double layer potential e.g. it applies $\gamma = \frac{\partial g}{\partial n_y}$ .

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `A->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{A->rows} -1$ instead.
Z	An array of 3D-vectors. `Z[i][0]` will be the first component of the i-th vector. Analogously `Z[i][1]` will be the second component of the i-th vector. The length of this array is determined by `A->cols` .
bem	BEM-object containing additional information for computation of the matrix entries.
A	A will contain the results of the integral evaluations. It applies $A_{ij} = \int_\Gamma \varphi_i(\vec x) \, \gamma(\vec x, \vec z_j) \, \mathrm d \vec x .$

void(* lagrange_col) (const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcbem3d bem, pamatrix W)

Integrate the Lagrange polynomials or their derivatives within the 2nd component.

This callback will evaluate the integral

$\int_\Gamma \mathcal L(\vec y) \, \psi(\vec y) \, \mathrm d \vec y$

with $\mathcal L$ being a Lagrange polynomial or its derivative.

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `W->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{W->rows} -1$ instead.
px	A Vector of type avector that contains interpolation points in x-direction.
py	A Vector of type avector that contains interpolation points in y-direction.
pz	A Vector of type avector that contains interpolation points in z-direction.
bem	BEM-object containing additional information for computation of the matrix entries.
W	W will contain the results of the integral evaluations. It applies $W_{ij} = \int_\Gamma \mathcal L_j(\vec y) \, \psi_i(\vec y) \, \mathrm d \vec y .$ The index is computed in a tensor way. Having Points in x-direction, Points in y-direction and Points in z-direction, then using the -th, the -th and the -th point will result in matrix index $j = j_x + j_y \cdot m_x + j_z \cdot m_x \cdot m_y$ .

void(* lagrange_row) (const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcbem3d bem, pamatrix V)

Integrate the Lagrange polynomials or their derivatives within the 1st component.

This callback will evaluate the integral

$\int_\Gamma \varphi(\vec x) \, \mathcal L(\vec x) \, \mathrm d \vec x$

with $\mathcal L$ being a Lagrange polynomial or its derivative.

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `V->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{V->rows} -1$ instead.
px	A Vector of type avector that contains interpolation points in x-direction.
py	A Vector of type avector that contains interpolation points in y-direction.
pz	A Vector of type avector that contains interpolation points in z-direction.
bem	BEM-object containing additional information for computation of the matrix entries.
V	V will contain the results of the integral evaluations. It applies $V_{ij} = \int_\Gamma \varphi_i(\vec x) \, \mathcal L_j(\vec x) \, \mathrm d \vec x .$ The index is computed in a tensor way. Having Points in x-direction, Points in y-direction and Points in z-direction, then using the -th, the -th and the -th point will result in matrix index $j = j_x + j_y \cdot m_x + j_z \cdot m_x \cdot m_y$ .

void(* lagrange_wave_col) (const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcreal dir, pcbem3d bem, pamatrix W)

Integrate the modified Lagrange polynomials in direction $c_\iota$ or their derivatives within the 2nd component.

This callback will evaluate the integral

$\int_\Gamma \mathcal L_{c_\iota}(\vec y) \, \psi(\vec y) \, \mathrm d \vec y = \int_\Gamma \mathcal L(\vec y) \, e^{\langle c_\iota, \vec y \rangle} \,\psi(\vec y) \, \mathrm d \vec y$

with $\mathcal L$ being a Lagrange polynomial or its derivative.

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `W->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{W->rows} -1$ instead.
px	A Vector of type avector that contains interpolation points in x-direction.
py	A Vector of type avector that contains interpolation points in y-direction.
pz	A Vector of type avector that contains interpolation points in z-direction.
dir	Direction $c_\iota$ in which the modified Lagrange polynomials should be evaluated.
bem	BEM-object containing additional information for computation of the matrix entries.
W	W will contain the results of the integral evaluations. It applies $W_{ij} = \int_\Gamma \mathcal L_{c_\iota,\, j}(\vec y) \, \psi_i(\vec y) \, \mathrm d \vec y .$ The index is computed in a tensor way. Having Points in x-direction, Points in y-direction and Points in z-direction, then using the -th, the -th and the -th point will result in matrix index $j = j_x + j_y \cdot m_x + j_z \cdot m_x \cdot m_y$ .

void(* lagrange_wave_row) (const uint *idx, pcrealavector px, pcrealavector py, pcrealavector pz, pcreal dir, pcbem3d bem, pamatrix V)

Integrate the modified Lagrange polynomials in direction $c_\iota$ or their derivatives within the 1st component.

This callback will evaluate the integral

$\int_\Gamma \varphi(\vec x) \, \mathcal L_{c_\iota}(\vec x) \, \mathrm d \vec x = \int_\Gamma \varphi(\vec x) \, \mathcal L(\vec x) \, e^{\langle c_\iota, \vec x \rangle} \, \mathrm d \vec x$

with $\mathcal L$ being a Lagrange polynomial or its derivative.

Parameters

idx	This array describes the permutation of the degrees of freedom. Its length is determined by `V->rows` . In case `idx == NULL` it is assumed the degrees of freedom are $0, 1, \ldots , \texttt{V->rows} -1$ instead.
px	A Vector of type avector that contains interpolation points in x-direction.
py	A Vector of type avector that contains interpolation points in y-direction.
pz	A Vector of type avector that contains interpolation points in z-direction.
dir	Direction $c_\iota$ in which the modified Lagrange polynomials should be evaluated.
bem	BEM-object containing additional information for computation of the matrix entries.
V	V will contain the results of the integral evaluations. It applies $V_{ij} = \int_\Gamma \varphi_i(\vec x) \, \mathcal L_{c_\iota, \, j}(\vec x) \, \mathrm d \vec x .$ The index is computed in a tensor way. Having Points in x-direction, Points in y-direction and Points in z-direction, then using the -th, the -th and the -th point will result in matrix index $j = j_x + j_y \cdot m_x + j_z \cdot m_x \cdot m_y$ .

The documentation for this struct was generated from the following file:

Library/bem3d.h

Data Fields

Detailed Description

Field Documentation