Main container object for computation of BEM related matrices and vectors. More...

#include <bem2d.h>

Data Fields
pccurve2d	gr
	Polygonal two dimensional mesh. More...

psingquad1d	sq
	Quadrature rules used within BEM computation. More...

uint	N_neumann
	Number of degrees of freedom for neumann data.

basisfunctionbem2d	basis_neumann
	Type of basis function for neumann-data.

uint	N_dirichlet
	Number of degrees of freedom for dirichlet data.

basisfunctionbem2d	basis_dirichlet
	Type of basis function for dirichlet-data.

real *	mass
	Mass-matrix of reference basis functions. More...

field	alpha
	Boundary integral operator + $\alpha$ mass matrix.

uint *	v2t
	vertex to triangle map. More...

uint *	v2ts
	Vertex to triangle map stating indices. More...

void(*	nearfield )(const uint ridx, const uint cidx, pcbem2d bem, bool ntrans, pamatrix N)
	Computes nearfield entries of Galerkin matrices. More...

void(*	farfield_rk )(pccluster rc, uint rname, pccluster cc, uint cname, pcbem2d bem, prkmatrix R)
	Computes rank-k-approximations of a given block. More...

void(*	farfield_u )(uint rname, uint cname, pcbem2d bem, puniform U)
	Computes coupling matrix for uniform matrices. More...

void(*	leaf_row )(pcbem2d bem, pclusterbasis rb, uint rname)
	Computes the the matrix for a leaf cluster $t \in \mathcal L_{\mathcal I}$ . More...

void(*	leaf_col )(pcbem2d bem, pclusterbasis cb, uint cname)
	Computes the the matrix for a leaf cluster $s \in \mathcal L_{\mathcal J}$ . More...

void(*	transfer_row )(pcbem2d bem, pclusterbasis rb, uint rname)
	Computes the transfermatrices $E_{t'}$ for a cluster $t \in \mathcal T_{\mathcal I} \, , t' \in \operatorname{sons}(t)$ . More...

void(*	transfer_col )(pcbem2d bem, pclusterbasis cb, uint cname)
	Computes the transfermatrices $F_{s'}$ for a cluster $s \in \mathcal T_{\mathcal J} \, , s' \in \operatorname{sons}(s)$ . More...

paprxbem2d	aprx
	A collection of necessary data structures for approximating matrices with various techniques. More...

pparbem2d	par
	Some helpers to make parallel computations possible. More...

pkernelbem2d	kernels
	A collection of callback functions for different types of kernel evaluations. More...

Detailed Description

Main container object for computation of BEM related matrices and vectors.

This struct defines the essential part of all calculations concerning BEM. Just creating a simple bem2d object is not intended, though possible. Therefore call a problem specific constructor such as new_slp_laplace_bem2d or new_dlp_laplace_bem2d . This will initialize all needed callback functions to build up at least dense matrices. To be able to create h- or h2matrix approximations one needs to call one of the set_hmatrix_aprx* or set_h2matrix_aprx* depending on the type of your matrix, to fully initialize the bem2d object. Afterwards you can build up your desired matrix approximation with a few simple functions calls, delivered by the bem2d object, using your favored approximation technique. Calling another of these " set " functions will reinitialize the bem2d object and you can build another approximation with a different technique.

Field Documentation

paprxbem2d aprx

A collection of necessary data structures for approximating matrices with various techniques.

Necessary Objects for approximation matrices with different approximation schemes are stored within aprx.

void(* farfield_rk) (pccluster rc, uint rname, pccluster cc, uint cname, pcbem2d bem, prkmatrix R)

Computes rank-k-approximations of a given block.

This callback function computes a rank-k-approximation of a matrix block given by row cluster rc and column cluster cc. The Matrices A and B are stored in a rank-k-matrix R.

Parameters

rc	Current row cluster .
rname	Enumerated number of current row cluster `rc`.
cc	Current column cluster
cname	Enumerated number of current column cluster `cc`.
bem	BEM-object containing additional information for computation of the matrix entries.
R	Rank-k-matrix, which takes up the resulting matrices `A` and `B` . The rank k is defined by the underlying approximation technique.

void(* farfield_u) (uint rname, uint cname, pcbem2d bem, puniform U)

Computes coupling matrix $ S_b $ for uniform matrices.

While using h2-matrices the coupling matrices $ S_b $ are computed by this callback function. Coupling matrices are stored in the struct uniform defining a certain block of the whole h2-matrix .

Parameters

rname	Enumerated number of current row cluster clusterbasis `U->rb`.
cname	Enumerated number of current column clusterbasis `U->cb`.
bem	BEM-object containing additional information for computation of the matrix entries.
U	An Object of type uniform containing the coupling matrix , the row clusterbasis `U->rb` and the column clusterbasis `U->cb` .

pccurve2d gr

Polygonal two dimensional mesh.

Polygonal boundary mesh used for the BEM-problem.

See also: curve2d.

pkernelbem2d kernels

A collection of callback functions for different types of kernel evaluations.

This struct defines a set of callback functions that are based on the fundamental solution of the underlying problem. They are used to build up approximations that will be used within farfield_rk , farfield_u , leaf_row , leaf_col , transfer_row and transfer_col. In Order to receive correct result, the callback functions defined in kernels have to be set correctly to the given problem by a constructor. These callback functions also depend on the utilized basis functions for either neumann- and dirichlet-data. As an example the constructors new_slp_laplace_bem2d and new_dlp_laplace_bem2d should serve.

See also: kernelbem2d

void(* leaf_col) (pcbem2d bem, pclusterbasis cb, uint cname)

Computes the the matrix $ W_s $ for a leaf cluster $s \in \mathcal L_{\mathcal J}$ .

leaf_col is used for building up a clusterbasis for the set $\mathcal L_{\mathcal J}$ for h2matrices. For each $s \in \mathcal L_{\mathcal J}$ leaf_col will construct the matrix $ W_s $ . In case of nested clusterbasis transfer_col has also to be set to a valid function constructing suitable transfer matrices $ F_s $ .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
cb	current column clusterbasis. The Matrix will be saved into `cb->V` .
cname	Enumerated number of current column clusterbasis `cb`.

void(* leaf_row) (pcbem2d bem, pclusterbasis rb, uint rname)

Computes the the matrix $ V_t $ for a leaf cluster $t \in \mathcal L_{\mathcal I}$ .

leaf_row is used for building up a clusterbasis for the set $\mathcal L_{\mathcal I}$ for h2matrices. For each $t \in \mathcal L_{\mathcal I}$ leaf_row will construct the matrix $ V_t $ . In case of nested clusterbasis transfer_row has also to be set to a valid function constructing suitable transfer matrices $ E_t $ .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
rb	current row clusterbasis. The Matrix will be saved into `rb->V` .
rname	Enumerated number of current row clusterbasis `rb`.

real* mass

Mass-matrix of reference basis functions.

Mass-matrix for all combinations of reference basis functions $\widehat\varphi_i$ and $\widehat\psi_j$ . Entries are computed as:

$M_{ij} = \int_\Gamma \widehat\varphi_i(x) \cdot \widehat\psi_j(x) \, \mathrm d x$

void(* nearfield) (const uint *ridx, const uint *cidx, pcbem2d bem, bool ntrans, pamatrix N)

Computes nearfield entries of Galerkin matrices.

This callback function computes 'nearfield' entries of the underlying problem and integral-operator.

Parameters

ridx	Defines the indices of row boundary elements used. If `ridx` equals NULL, then Elements `0, ..., N->rows-1` are used.
cidx	Defines the indices of column boundary elements used. If `cidx` equals NULL, then Elements `0, ..., N->cols-1` are used.
bem	Contains additional Information for the computation of the matrix entries.
ntrans	Is a boolean flag to indicates the way of storing the entries in matrix `N` . If `ntrans == true` the matrix entries will be stored in a transposed way.
N	For `ntrans == false` the matrix entries are computed as: $N_{ij} = \int_\Gamma \int_\Gamma \varphi_i(x) \, g(x,y) \, \psi_j(y) \, \mathrm d y \, \mathrm d x \, ,$ otherwise they are stored as $N_{ji} = \int_\Gamma \int_\Gamma \varphi_i(x) \, g(x,y) \, \psi_j(y) \, \mathrm d y \, \mathrm d x \, .$

pparbem2d par

Some helpers to make parallel computations possible.

For the sake of parallelism we need some auxiliary data structure, which are all collected within the struct par. Most commonly used is this struct for enumerated non linear data structures such als cluster- or blocktrees.

psingquad1d sq

Quadrature rules used within BEM computation.

Singular quadrature formulas used within boundary-integral-Operator computation.

See also: _singquad1d.

void(* transfer_col) (pcbem2d bem, pclusterbasis cb, uint cname)

Computes the transfermatrices $F_{s'}$ for a cluster $s \in \mathcal T_{\mathcal J} \, , s' \in \operatorname{sons}(s)$ .

transfer_col will build up the transfer matrices $F_{s'}$ , for all $s \in \mathcal T_{\mathcal J}$ and $s' \in \operatorname{sons}(s)$ . For a cluster $s \in \mathcal T_{\mathcal J}$ transfer_col will compute the matrices $F_{s_1}, \ldots, F_{t_\sigma}$ , $\# \operatorname{sons}(s) = \sigma$ . The matrix $F_{s_1}$ is stored in cb->son[0]->E, $F_{s_2}$ is stored in cb->son[1]->E, $\ldots F_{t_\sigma}$ is stored in cb->son[ ]->E .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
cb	Current column clusterbasis. The Matrices $F_{s'}, s' \in \operatorname{sons}(s)$ will be saved into `cb->son[s'-1]->E` .
cname	Enumerated number of current column clusterbasis `cb`.

void(* transfer_row) (pcbem2d bem, pclusterbasis rb, uint rname)

Computes the transfermatrices $E_{t'}$ for a cluster $t \in \mathcal T_{\mathcal I} \, , t' \in \operatorname{sons}(t)$ .

transfer_row will build up the transfer matrices $E_{t'}$ , for all $t \in \mathcal T_{\mathcal I}$ and $t' \in \operatorname{sons}(t)$ . For a cluster $t \in \mathcal T_{\mathcal I}$ transfer_row will compute the matrices $E_{t_1}, \ldots, E_{t_\tau}$ , $\# \operatorname{sons}(t) = \tau$ . The matrix $E_{t_1}$ is stored in rb->son[0]->E, $E_{t_2}$ is stored in rb->son[1]->E, $\ldots E_{t_\tau}$ is stored in rb->son[ ]->E .

Parameters

bem	BEM-object containing additional information for computation of the matrix entries.
rb	Current row clusterbasis. The Matrices $E_{t'}, t' \in \operatorname{sons}(t)$ will be saved into `rb->son[t'-1]->E` .
rname	Enumerated number of current row clusterbasis `rb`.

uint* v2t

vertex to triangle map.

A list of triangle indices corresponding to each vertex index. A vertex $ v $ is part of all triangles ranging from $\texttt{v2t[v2ts[v]]} \ldots \texttt{v2t[v2ts[v+1]-1]}$

uint* v2ts

Vertex to triangle map stating indices.

Array of starting indices for each vertex needed by v2t. v2ts has length vertices + 1.

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

Library/bem2d.h

Data Fields

Detailed Description

Field Documentation