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 . 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
We recommend setting up a
in the root directory of the Bembel, to switch to it and then to call
cmake .. and
make release, respectively.
The CMake-File checks for installations of Eigen via the
Eigen3Config.cmake. On Unix you may
apt install libeigen3-dev, or on Mac with
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
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
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!
.mfiles 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:
AnsatzSpacecontains the files whose routines manage the discrete space on the surface of the geometry. Specifically, this is realised through the four classes
AnsatzSpace. Therein, the
Superspacemanages local polynomial bases on every element. Through a transformation matrix generated by the template class
Projectorwhich depends on the
LinearOperatorand its defined traits, the
Superspacecan be related to B-Spline bases on every patch. To build conforming spaces (in the case of
DifferentialForm::DivergenceConformingthrough continuity of the normal component across patch interfaces, and in the case of
DifferentialForm::Continuousthrough global C0-continuity), the template class
Glueassembles another transformation matrix to identify degrees of freedom across edges. Then, a coefficient vector in the
Superspacecan be related to one of the smooth B-Spline basis, as explained in [4,5].
ClusterTreeimplements an element structure for refinement. Currently, only uniform refinement has been implemented. A
ClusterTreeobject itself is comprised of a
Geometryobject and an
ElementTreeprovides routines for the extraction of topological information, refinement, as well as iterators to effortlessly loop over elements.
DuffyTrickprovides 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.
Geometryimplements the handling of the geometry.
H2Matrixhandles the compression of the large, densely populated matrices. The algorithm was introduced in  and was generalized to the Maxwell case in .
Helmholtzprovides the necessary specializations to solve Helmholtz problems.
IOprovides input-output functionality, including routines for VTK file export, timing, and writing log files.
Laplaceprovides the necessary specializations to solve Laplace problems.
LinearFormimplements 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.
LinearOperatorprovides 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.
Maxwellprovides the necessary specializations to solve Maxwell problems.
Potentialprovides routines for the evaluation of the
Potential, i.e., the solution to the PDE via the solution of (non-singular) integrals based on the
Splineprovided basic routines related to spline function and local polynomials.
Bembel/src/ directory leads to folders corresponding to these modules, which include the actual implementation.
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 . 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  if you use it in one of your articles.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
Contributors include (alphabetically):
Work on Bembel was conducted with the following financial support (alphabetically):