HLIBpro
2.7

𝖧𝖫𝖨𝖡𝗉𝗋𝗈 implements various tools for discretization and solving boundary element methods, namely grid generation, function spaces and bilinear forms.
The implementation is based on solving the integral equation
\begin{equation*} \int_{\Gamma} k(x,y) \mathbf{u}(y) dy = \mathbf{f}(x), \quad x \in \Gamma \end{equation*}
with \(\Gamma = \partial \Omega \subset \mathbf{R}^3\) and \(\mathbf{u} : \Gamma \to \mathbf{R}\) being sought for a given right hand side \(\mathbf{f} : \Gamma \to \mathbf{R}\).
The solution should be computed using Galerkin discretisation with ansatz functions \(V = \{\phi_i, 0 \le i < n\} \) and test functions \(W = \{\psi_i, 0 \le i < m\} \), which results in a linear equation system
\[A u = f\]
where \(u\) contains the coefficients of the discretised \(\mathbf{u}\) and \(A\) is defined by
\[ a_{ij} = \int_\Gamma \int_\Gamma \phi_i(x) k(x,y) \psi_j(y) dy dx \]
The right hand side \(f\) is given by
\[ f_i = \int_{\Gamma} \psi_i(x) f(x) dx. \]
Please see Boundary Element Methods for a full example (Helmholtz).
Grids can be loaded from files. Here different file formats are supported, e.g., the HLIB, PLY, GMSH and SurfaceMesh format, with the restriction of triangular grids. One can either use file format auto detection and use the general function
or directly the read
function from the classes TAutoGridIO
, THLibGridIO
, TPlyGridIO
, TSurfMeshGridIO
and TGMSHGridIO
.
Furthermore, basic grids can be constructed with one of
which will construct a sphere (diameter 2, centered at \((0,0,0)^T\)), cube ( \((0,0,0)^T \ldots (1,1,1)^T\)) or square ( \((0,0,0)^T \ldots (1,1,0)^T\)) grid.
As the return type TRefinableGrid
suggest, these grids can be refined using the function
This can be used to refine the grid until the number of triangles is sufficiently large:
Based on these grids functions spaces are constructed, defining the degrees of freedom (DoF). 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 implements TConstFnSpace
for piecewise constant functions, TLinearFnSpace
for piecewise linear functions and TConstEdgeFnSpace
for constant normal linear tangential (CN/LT) edge elements (for Maxwell applications).
During construction, the single argument is the grid:
As function spaces define the DoFs, their number is available with the n_indices
function
Furthermore, the coordinates of all DoFs are provided by the function spaces using the build_coord
function:
which can be used for clustering.
𝖧𝖫𝖨𝖡𝗉𝗋𝗈 implements the bilinear forms for the following BEM kernels:
TLaplaceSLPBF
and TLaplaceDLPBF
)THelmholtzSLPBF
and THelmholtzDLPBF
)TMaxwellEFIEBF
and TMaxwellMFIEBF
)TMassBF
, TMaxwellEFIEMassBF
and TMaxwellMFIEMassBF
)TExpBF
)The implementation of all these is based on quadrature (SauterSchwab triangular quadrature), the order of which is am optional parameter for the corresponding constructors. Beside this, the functions spaces for the ansatz and test function spaces are the other parameters:
The computation
\[ f_i = \int_{\Gamma} \psi_i(x) f(x) dx. \]
of the right hand side is also implemented using quadrature and implemented for the scalar valued types in TQuadBEMRHS
.
For this, the user needs to implement the function \(f\) by deriving from TBEMFunction
, which defines an eval
function for a point \(x\) and normal direction \(n\):
This function is abstract in TBEMFunction
and needs to be implemented in derived classes:
With the help of TQuadBEMRHS
, the discrete version of \(f\) can be computed:
For Maxwell computations, the RHS is vector valued. A special type complex3_t
is available for holding three complex values. With this, the user defined RHS function looks like
The computation of the RHS vector is performed by TMaxwellEFIERHS
and TMaxwellMFIERHS
.