- General information
- Subprojects
- SP 2 (Darmstadt)
- Partners
- Activities

Mathematics for Innovations in Industry and Services

--MoreSim4Nano--

- Subproject leader:
**Prof. Dr.-Ing. Thomas Weiland, TU Darmstadt**

In this part of the project, the efficient simulation of electromagnetic wave propagation in complex semiconductor structures based on the numerical solution of the Maxwell's equations is investigated. Special focus is put on the extension of existing solvers as well as on the construction of appropriate systems which can be subsequently used to apply different MOR techniques.

Within the aspired nanotechnology and micro-systems modeling, the direct solution of the Maxwell's equations leads to prohibitively large matrices, as for realistic problems a vast amount of degrees of freedom is required to properly describe the field variations in geometrically complex structures. The necessary discretization of the semiconductor devices can be achieved with various existing numerical methods but due to the expected large scale model size the very efficient finite integration technique (FIT) is particularly suited.

Following this approach, the description of material properties can be formulated efficiently because of the favorable advantage that FIT renders material matrices in block diagonal form with very small block sizes even if fully populated material tensors have to be applied. Specifically, for either isotropic material or diagonal material tensors with possibly different weighting of the individual components, this kind of modeling results in pure diagonal material matrices. With a successive model order reduction in mind this trivially invertible form is of particular interest.

When dealing with practical problems with possibly more than 1010 degrees of freedom special care has to be taken to guarantee an efficient setup of the large system of equations. In such cases, the specifications of the layered structures can be immediately utilized with the help of a specialized problem-adapted formulation, e.g. through physically motivated reduced bases. This enables to reduce the necessary parameters already on the modeling level and supplementary extends the available algebraic reduction procedures.

Using domain decomposition and parallel processing of geometry and material data during the setup of the system matrices, one can further improve the run-time efficiency of the developed programs. The mutual coupling can then be applied directly via the exchange of the field components located at the boundaries of the various domains or can be alternatively related based on the scattering parameters of each functional block. Fortunately, in those cases the model reduction can be applied independently of each other and all individual pieces are put together afterwards to properly represent the total system. Depending on the specific application, one can take advantage of a mixed coupling considering clusters of neighboring subdivisions. By using an improved matrix description even higher memory efficiency is conceivable, if one already employs a suitable compression method at this stage. In collaboration with the SP3, the construction of the system matrices and the application of subsequent MOR methods can be tailored appropriately.

Milestones:

- [Months 1-6] Determination of the transmission behavior of simple structures and comparisons with existing classical MOR methods based on Krylov-spaces. Consideration of more realistic structures in which the behavior can be approved directly at least in parts.
- [Months 7-24] Incorporation of problem-adapted formulations with reduced bases to generate physically-motivated small models before performing the algebraic reduction. Since the automatic detection of profitable sections is difficult, manual labeling is used for reduction-feasible regions.
- [Months 25-36] Implementation of parallelized algorithms on shared-memory and distributed-memory architectures to handle realistic problems.