In this subproject we develop a MOR method that admits the analysis of the electromagnetic influence on semiconductors. Furthermore, certain variations of the feature structure size caused, e.g., by inaccuracies of the resolution during the lithography, are considered. These variations effect the dynamical model as uncertain parameters with a distribution function given by the production process. For that purpose, it is necessary to develop MOR methods for dynamical systems arising from the discretization of Maxwell's equations. As an alternative to the methods considered in the subprojects 1 and 2, a version of balanced truncation based on the method described in [6] will be implemented for a discretization via the boundary element method (BEM) in conjunction with hierarchical matrices provided by SP1. The resulting method is tested on realistic problems by use of the EM solver of SP2.
Fast simulation of the coupled semiconductor electromagnetic models with uncertain parameters requires to further develop the parametric MOR methods introduced in [5, 7, 42, 40, 39]. This leads to variational MOR where not only the parameters as symbolic quantities but also their stochastic properties are preserved by the reduced model. In case that the theoretical analysis turns out to be infeasible, the preservation of the distribution function has to be proven in an empirical way. Variational MOR methods are based on balanced truncation for stochastic systems as proposed in [10].
The computation of the required Gramians is done via bilinear Lyapunov equations. Therefore, numerical methods for solving large scale systems have to be developed, which might require the evaluation of stochastic integrals, similar to the case of non-linear systems [68]. Sparse grid techniques [23, 48, 91] which are already applied in [7] will be used for interpolation based quadrature rules [4], as was already proposed for stochastic applications in [77]. In cooperation with SP4, subspace recycling techniques [42], which where already applied to parametric MOR, will be enhanced to optimize the computing time.
The MOR methods for EM systems and the systems with uncertain parameters are coupled to realize the desired fast variational analysis of semiconductors with EM effect. The resulting methods are implemented in the software of the industrial partners during the last stage of the project. Variational MOR methods seem to be applicable at Infineon Technologies AG while the holistic approach for coupled electromagnetic semiconductor models is integrated at the other industrial partners in CAE or EDA software, respectively.
Milestones:
[Months 1-10] Development and implementation of a MOR method for systems with parametric uncertainties based on balanced truncation (computation via bilinear Lyapunov equations, alternatively by means of sparse grid quadrature/ collocation).
[Months 11-14] Implementation of a MOR method for EM systems based on BEM discretization with hierarchical matrices.
[Months 15-24] Development and implementation of a MOR method to analyze the influence of the surrounding electromagnetic field on semiconductors with parametric uncertainties based on the coupling of the methods developed in the first two milestones.
[Months 20-24] Analysis of the statistic properties of the simulation results of the reduced model. Analytic proof of the preservation of the distribution function, alternatively empirical confirmation.
[Months 25-36] Verification of the new MOR methods in the industrial environment and implementation of a prototype in a standalone version of TITAN, provided by Infineon Technologies AG. Comparison with complete EM simulation with CST Software as well as in EDA applications at MunEDA.
Printable Version
Top