The BEM-based engineering library
Bembel is the Boundary Element Method Based Engineering Library written in C++ to solve boundary value problems governed by the Laplace, Helmholtz or electric wave equation within the isogeometric framework [3,4,5,6]. It was developed as part of a cooperation between the TU Darmstadt and the University of Basel, coordinated by H. Harbrecht, S. Kurz and S. Schöps. The code is based on the Laplace BEM of J. Dölz, H. Harbrecht, and M. Multerer, [2,7] as well as the spline and geometry framework of F. Wolf. The code was extended by J. Dölz and F. Wolf in early 2018 to cover electromagnetic applications [5,6]. A short introduction to the code has been written [3]. If you utilise our code as part of a publication, we would appreciate it if you cite it.
A traditional German ceramic, as depicted in our logo. Quoting Wikipedia:
Most establishments also serve Apfelwein by the Bembel (a specific Apfelwein jug), much like how beer can be purchased by the pitcher in many countries. The paunchy Bembel (made from salt-glazed stoneware) usually has a basic grey colour with blue-painted detailing.
Current key features include
You need to install the Eigen3 library, see Eigen’s Documentation for help.
We do not rely on any other external libraries, except for the standard template
library. Thus, having installed Eigen, Bembel should run out of the box. If you want to use
Bembel as part of your application, simply add the Bembel/
directory to your includes.
If you want to run the provided examples and tests, you can utilize the procided CMakeLists.txt
.
We recommend setting up a build/
directory
in the root directory of the Bembel, to switch to it and then to call cmake ..
and make debug
or make release
, respectively.
The CMake-File checks for installations of Eigen via the
corresponding Eigen3Config.cmake
. On Unix you may
run apt install libeigen3-dev
, or on Mac with
Homebrew brew install eigen
, and everything should work.
Alternatively, you can delete the line
find_package (Eigen3 3.3 REQUIRED NO_MODULE)
and all lines of the type
target_link_librarires(... Eigen3::Eigen)
from CMakeLists.txt
, and give a path to the eigen headers as an include directory manually.
Then, the examples and tests should compile without any issues. You may run all tests by calling
make test
from the build/
directory after a successful compilation.
The geometry files required to run the examples can be found in the geo/
folder.
The general structure of the repository looks as follows.
assets/
only contains things relevant for GitHub pages.geo/
contains geometry files in the format of the octave
NURBS package. They can be utilized for computations. Note that
the normal vector must be outward directed on all patches!
geo/octave_example/
includes .m
files that showcase how geometries can be
constructed using the NURBS package
of octave.Bembel/
contains the library. In Bembel/
, module files have been created analogously to the
design of the Eigen library. If Bembel is used, only these module files should be included. These modules are the following:
AnsatzSpace
contains the files whose routines manage the discrete space on the surface of the geometry. Specifically, this is realised through the four classes Superspace
, Projector
, Glue
, and AnsatzSpace
. Therein, the Superspace
manages local polynomial bases on every element. Through a transformation matrix generated by the template class Projector
which depends on the LinearOperator
and its defined traits, the Superspace
can be related to B-Spline bases on every patch. To build conforming spaces (in the case of DifferentialForm::DivergenceConforming
through continuity of the normal component across patch interfaces, and in the case of DifferentialForm::Continuous
through global C0-continuity), the template class Glue
assembles another transformation matrix to identify degrees of freedom across edges. Then, a coefficient vector in the Superspace
can be related to one of the smooth B-Spline basis, as explained in [4,5].ClusterTree
implements an element structure for refinement. Currently, only uniform refinement has been implemented. A ClusterTree
object itself is comprised of a Geometry
object and an ElementTree
. The ElementTree
provides routines for the extraction of topological information, refinement, as well as iterators to effortlessly loop over elements.DuffyTrick
provides quadrature routines for (nearly) singular integrals. Therein, quadrature routines for 4 cases are implemented: Separated elements, elements sharing a corner, elements sharing an edge, and identical elements.DummyOperator
provides a LinearOperator
for testing.Geometry
implements the handling of the geometry.H2Matrix
handles the compression of the large, densely populated matrices. The algorithm was introduced in [2] and was generalized to the Maxwell case in [5].Helmholtz
provides the necessary specializations to solve Helmholtz problems.IO
provides input-output functionality, including routines for VTK file export, timing, and writing log files.Laplace
provides the necessary specializations to solve Laplace problems.LinearForm
implements trace operators, i.e., routines to generate the right hand side of the linear systems. Currently, only a Dirichlet trace and a rotated tangential trace are provided.LinearOperator
provides a framework to implement linear operators that can be used to solve PDEs. Therein, the kernel function, a routine for the evaluation, and certain traits (like the scalar type, the operator order, and the DifferentialForm
, i.e., weather the basis must be discontinuous, vector-valued and divergence conforming, or globally continuous) required for discretization must be provided.Maxwell
provides the necessary specializations to solve Maxwell problems.Potential
provides routines for the evaluation of the Potential
, i.e., the solution to the PDE via the solution of (non-singular) integrals based on the LinearOperator
s.Quadrature
Spline
provided basic routines related to spline function and local polynomials.The Bembel/src/
directory leads to folders corresponding to these modules, which include the actual implementation.
The folder examples/
contains short and well documented examples for Bembels functionality and tests/
contains the tests for Bembel.
A good place to start are the examples in the examples/
folder, together with publication [3]. Apart from that, a Doxygen documentation is available.
For a list of known bugs and upcoming features, please have a look at the issue tracker on github.
The following articles and preprints influenced the development of Bembel. We appreciate a citation of [6] if you use it in one of your articles.
[1] A. Buffa, J. Dölz, S. Kurz, S. Schöps, R. Vázques, and F. Wolf. Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis. In: Numer. Math., 144, 201-236, 2020. To the paper. To the preprint.
[2] J. Dölz. A Higher Order Perturbation Approach for Electromagnetic Scattering Problems on Random Domains. In: SIAM/ASA J. UQ, 2020, 8(2). To the paper.
[3] J. Dölz. Data sparse multilevel covariance estimation in optimal complexity. Preprint available, 2023. To the preprint.
[4] J. Dölz, H. Harbrecht, C. Jerez-Hanckes and M. Multerer. Isogeometric multilevel quadrature for forward and inverse random acoustic scattering. In: Comput. Methods Appl. Mech. Engrg., 388, 114242, 2022. To the paper.
[5] J. Dölz, H. Harbrecht, and M. Peters. An interpolation-based fast multipole method for higher-order boundary elements on parametric surfaces. In: Int. J. Numer. Meth. Eng., 108(13):1705-1728, 2016. To the paper.
[6] J. Dölz, H. Harbrecht, S. Kurz, M. Multerer, S. Schöps, and F. Wolf. Bembel: The Fast Isogeometric Boundary Element C++ Library for Laplace, Helmholtz, and Electric Wave Equation. In: SoftwareX, 11, 10476. To the paper.
[7] J. Dölz, H. Harbrecht, S. Kurz, S. Schöps, and F. Wolf. A fast isogeometric BEM for the three dimensional Laplace- and Helmholtz problems. In: Comput. Methods Appl. Mech. Engrg., 330:83-101, 2018. To the paper. To the preprint.
[8] J. Dölz, S. Kurz, S. Schöps, and F. Wolf. Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples. In: SIAM J. Sci. Comput., 41(5), B983-B1010, 2019. To the paper. To the preprint.
[9] J. Dölz, S. Kurz, S. Schöps, and F. Wolf. A Numerical Comparison of an Isogeometric and a Parametric Higher Order Raviart–Thomas Approach to the Electric Field Integral Equation. In: IEEE Trans. Antenn. Prop., 68(1), 593–597, 2020. To the paper. To the preprint.
[10] J. Dölz, O. Palii and M. Schlottbom. On robustly convergent and efficient iterative methods for anisotropic radiative transfer. In: J. Sci. Comput., 2022, 90(3). To the paper.
[11] M. Elasmi, C. Erath and S. Kurz. Non-symmetric isogeometric FEM-BEM couplings. In: Adv. Appl. Math. Mech., 2021, 47(61). To the paper.
[12] H. Harbrecht, M. Multerer and R. von Rickenbach. Isogeometric shape optimization of periodic structures in three dimensions. In: Comput. Methods Appl. Mech. Engrg., 391, 114552, 2022. To the paper.
[13] H. Harbrecht and M. Peters. Comparison of fast boundary element methods on parametric surfaces. In: Comput. Methods Appl. Mech. Engrg., 261-262:39-55, 2013. To the paper.
[14] W. Huang and M. Multerer. Isogeometric analysis of diffusion problems on random surfaces. In: APNUM, 179, 50-65, 2022. To the paper.
[15] S. Kurz, S. Schöps, G. Unger and F. Wolf. Solving Maxwell’s Eigenvalue Problem via Isogeometric Boundary Elements and a Contour Integral Method. In: Math. Meth. Appl. Sci., 2021, 44(13). To the paper.
[16] M. Liebsch, S. Russenschuck and S. Kurz. BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering. In: Comput. Methods Appl. Mech. Engrg., 2022, forthcoming. To the paper.
[17] R. Torchio, M. Nolte, S. Schöps and A. E. Ruehli. A spline-based partial element equivalent circuit method for electrostatics. In: IEEE Trans. Dielectr. Electr. Insul., 2022, forthcoming. To the paper.
Contributors include (alphabetically):
Jürgen Dölz currently holds a professorship at the University of Bonn. He may also be found on GitHub.
Helmut Harbrecht currently holds a professorship at the Departement Mathematik und Informatik at the University of Basel.
Stefan Kurz currently holds a professorship at the University of Jyväskylä.
Michael Multerer currently holds a professorship at the Institute of Computational Science at the Università della Svizzera italiana in Lugano. He may also be found on GitHub.
Maximilian Nolte is currently a PhD student at the Institute TEMF at TU Darmstadt. He may also be found on GitHub.
Sebastian Schöps currently holds a professorship at the Institute TEMF at TU Darmstadt. He may also be found on GitHub.
Felix Wolf is currently a lecturer at the Institute TEMF at TU Darmstadt. He may also be found on GitHub.
Work on Bembel was conducted with the following financial support (alphabetically):