Publications
Scientific publication on Concepts:
2002

Frauenfelder, P. and Lage, C. Concepts  An objectoriented software package for partial differential equations. ESAIM: M2AN, 36(5), 937951, 2002. [DOI] Abstract: Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arena that follow this design methodology providing extensible, reusable, and flexible software while staying competitive to traditionally designed point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such that the level of abstraction is essentially not raised. In this paper we discuss an alternative way to approach the design challenge which we call “concept oriented design”. We apply this design methodology to PetrovGalerkin methods leading to a class library for both, boundary element methods (BEM) and finite element methods (FEM). We show as a particular example the implementation of hpFEM using the library with special attention to the handling of inconsistent meshes. 
1998

Lage, C. Concept oriented design of numerical software. SAM Report 199807, Seminar for Applied Mathematics, ETH Zürich, 1998. [PDF] Abstract: The continuously growing computing power of modern computers admits to tackle numerical problems of extreme complexity. This complexity carries over to the numerical methods applied to solve the problems. Whereas the mathematical formulation of these methods does not raise any difficulties, their implementation turns out to be the bottleneck in the realization of numerical applications. In the last years, in order to afford relief, object oriented methods were applied to promote reusable and extensible numerical software, since this kind of flexibility is the key to manage complexity. It became evident that a carefully chosen modularization of the considered methods is a necessary requirement to provide flexible software components. In this paper we give a brief review of object oriented methods to identify the key issues that support a flexible software design and discuss a modularization technique based on mathematical concepts. Finally, the application of this concept oriented approach to boundary element methods is presented. 
Scientific publications in which Concepts is applied:
Engström, C. Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations. Num. Math., 126(3): 413440, 2014. [DOI] 2014

Klindworth, D. and Schmidt, K. An efficient calculation of photonic crystal band structures using Taylor expansions. Accepted for publication in Commun. Comput. Phys., available as Matheon Preprint #1068, 2014. [PDF] In this paper we used the same matrices as produced for the paper with Sonia Fliss on DirichlettoNeumann transparent boundary conditions for photonic crystal waveguides (see below). We exported these matrices to Matlab's binary format using Concepts's concepts::MatfileIO class. All postprocessing is then done with Matlab. 

Schmidt, K. and ThönsZueva, A. Impedance boundary conditions for acoustic time harmonic wave propagation in viscous gases. Preprint series of the Institute of Mathematics 62014, Technische Universität Berlin, 2014. [PDF]


Schmidt, K. and Hiptmair, R. Asymptotic boundary element methods for thin conducting sheets in two dimensions. IEEE Trans. Magn., 50: 469472, 2014. [DOI], [PDF]


Schmidt, K. and Chernov, A. Robust transmission conditions of high order for thin conducting sheets in two dimensions. IEEE Trans. Magn., 50(2): 4144, 2014. [DOI], [PDF]


Klindworth, D. and Schmidt, K. DirichlettoNeumann transparent boundary conditions for photonic crystal waveguides. IEEE Trans. Magn., 50: 217220, 2014. [DOI], [PDF]


Schmidt, K. and Heier, C. An analysis of Feng's and other symmetric local absorbing boundary conditions. Accepted for publication in ESAIM: M2AN, 2014. In this article we use a continuous FEM discretisation for local absorbing boundary conditions which involve derivatives of order 4 and higher. For this we use an interior penalty discretisation which additional terms on the nodes on FE mesh on the boundary. 

Schmidt, K., ThönsZueva, A. and Joly, P. Asymptotic analysis for acoustics in viscous gases close to rigid walls. Math. Models Meth. Appl. Sci., 24(9): 18231855, 2014. [DOI]


Fliss, S., Klindworth, D. and Schmidt, K. RobintoRobin transparent boundary conditions for the computation of guided modes in photonic crystal waveguides. Submitted to BIT, 2014. As for the project with DirichlettoNeumann operators (see below) we used Concepts to construct finite element spaces on coarse meshes that perfectly resolve the circular holes in the photonic crystal unit cells. In addition to the project with DirichlettoNeumann we have to implement mixed variational formulations for which we need finite element subspaces the trace spaces and, if the material cofficient jumps at the boundaries, we additionally need their corresponding dual spaces to cope for the lower regularity of the Neumann and Robin traces at the boundaries. To this end, we employed Concepts' hp2D::TraceSpace and hp1D::DualSpace classes. 

Klindworth, D., Schmidt, K. and Fliss, S. Numerical realization of DirichlettoNeumann transparent boundary conditions for photonic crystal waveguides. Comput. Math. Appl., 67(4): 918943, 2014. [DOI], [PDF] For the exact computation of guided modes in photonic crystal waveguides we employ DirichlettoNeumann operators. These operators are computed with the help of Dirichlet problems in a unit cell of the photonic crystal. We used Concepts to construct finite element spaces on coarse meshes with curved cells and large polynomial degrees (greater or equal to five). Concepts is ideal for this purpose as it offers the construction of periodic spaces and it allows for curved cell boundaries, which is important to perfectly resolve the holes/rods (grey circles in the image on the right hand side). Moreover, it is straightforward with Concepts to construct the space in a way that the degrees of freedom are sorted for your needs. In this context it is for example beneficial if one can easily access the degrees of freedom on the boundaries of the unit cell. To this end, the space is constructed such that the first degrees of freedom lie on the boundaries. 

Engström, C. Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations. Num. Math., 126(3): 413440, 2014. [DOI] For the band structure calculation for frequency dependent material the quadratic eigenvalue problem in the wave vector k as well as its finite element approximation is analysed. Numerical experiments has been performed with Concepts in 2D using pFEM with curved quadrilateral cells, where exponential convergence of the wave vectors has been achieved. 

2013

Schmidt, K. and Hiptmair, R. Asymptotic boundary element methods for thin conducting sheets. Preprint series of the Institute of Mathematics 152013, Technische Universität Berlin, 2013. [PDF]


Wang, M., Schmidt, K., Alparslan, A. and Hafner, C. hpFEM and PML analysis of plasmonic particles in layered media. Prog. Electromagn. Res., 142: 523544, 2013. [PDF]


Schmidt, K. and Chernov, A. A unified analysis of transmission conditions for thin conducting sheets in the timeharmonic eddy current model. SIAM J. Appl. Math, 73(6): 19802003, 2013. [DOI], [PDF]


Claeys X., Hiptmair, R., and Spindler, E. A secondkind Galerkin boundary element method for scattering at composite objects. SAM Report 201313 (revised), Seminar for Applied Mathematics, ETH Zürich, 2013. [PDF] Concepts was used to obtain reliable reference solutions by hpadaptive FEM discretisation for the scattering problem on curved multimaterial bodies with corner singularities. The solution computed on the adaptive meshes is evaluated on a uniformly distributed set of points on the material interfaces. 

2011

Schmidt, K. and Tordeux, S. High order transmission conditions for thin conductive sheets in magnetoquasistatics. ESAIM: M2AN, 45(6): 11151140, 2011. [PDF]


Wang, M., Engström, C., Schmidt, K. and Hafner, C. On highorder FEM applied to canonical scattering problems in plasmonics. J. Comput. Theor. Nanosci., 8(8): 19, 2011. [PDF]


Brandsmeier, H., Schmidt, K. and Schwab, C. A multiscale hpFEM for 2D photonic crystal bands. J. Comput. Phys., 230(2): 349374, 2011. [DOI], [PDF]


2010

Schmidt, K. and Kappeler, R. Efficient computation of photonic crystal waveguide modes with dispersive material. Optics Express, 18(7): 73077322, 2010. [PDF]


Schmidt, K. and Tordeux, S. Asymptotic modelling of conductive thin sheets. Z. Angew. Math. Phys., 61(4): 603626, 2010. [DOI], [PDF]


2009

Schmidt, K. and Kauf, P. Computation of the band structure of twodimensional photonic crystals with hp finite elements. Comp. Meth. App. Mech. Engr., 198: 12491259, 2009. [PDF]


Engström, C., Hafner, C. and Schmidt, K. Computations of lossy Bloch waves in twodimensional photonic crystals. J. Comput. Theor. Nanosci., 6: 775783, 2009. [PDF]


2008

Schmidt, K., Sterz, O. and Hiptmair, R. Estimating the eddycurrent modelling error. IEEE Trans. Magn., 44: 686689, 2008. [PDF]
