Arbeitsgruppe Angewandte Mathematik / Numerische Analysis
Bergische Universität Wuppertal
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Dynamic Iteration Schemes
Standard time-integration methods solve transient problems all at once. This may become very inefficient or impossible for large systems of equations. Imaging that such large systems often stem from a coupled problem formulation, where different physical phenomena interact and need to be coupled in order to produce a precise mathematical model.
E.g. highly integrated electric circuits (as in memory chips or CPUs) produce heat, which effects in turn their behavior as electrical system; thus one needs to couple electric and thermal subproblem descriptions. On the one hand, this creates multiple time scales due to different physical phenomena, which demands an efficient treatment, see multirate. On the other hand, in a professional environment one usually has dedicated solvers for the subproblems, which need to be used, and an overall problem formulation is not feasible for any of the involved tools.
For those partitioned problems a dynamic iteration method becomes beneficial or even the sole way-out: it keeps the subproblems separate, solves subproblems sequentially (or in parallel) and iterates until convergence (fixed-point interation). Thus the subproblem's structure can be exploited in the respective integration.
To guarantee or to speed up convergence the time interval of interest is split into a series of windows. Then the time-integration of the windows is applied sequentially and in each window the subproblems are solved iteratively by your favoured method.
- Herbert De Gersem, Katholieke Universiteit Leuven
On the convergence rate of dynamic iteration for coupled problems with multiple subsystemsJCAM, :accepted
Dynamic iteration for coupled problems of electric circuits and distributed devicesSIAM J. Sci. Comput., 35(2):B315-B335
An Optimal p-Refinement Strategy for Dynamic Iteration of Ordinary and Differential Algebraic Equations
Proc. Appl. Math. Mech Band 13 , Seite 549.
Multiscale Modeling and Multirate Time-Integration of Field/Circuit Coupled Problems
Herausgeber: VDI Verlag, Düsseldorf
Zusammenfassung: This treatise is intended for mathematicians and computational engineers that work on modeling, coupling and simulation of electromagnetic problems. This includes lumped electric networks, magnetoquasistatic field and semiconductor devices. Their coupling yields a multiscale system of partial differential algebraic equations containing device models of any dimension interconnected by the electric network. It is solved in time domain by multirate techniques that efficiently exploit the structure. The central idea is the usage of lumped surrogate models that describe latent model parts sufficiently accurate (e.g. the field model by an inductance) even if other model parts (e.g. the circuit) exhibit highly dynamic behavior. We propose dynamic iteration and a bypassing technique using surrogate Schur complements. A mathematical convergence analysis is given and numerical examples are discussed. They show a clear reduction in the computational costs compared to single rate approaches.
Preconditioned splitting in dynamic iteration schemes for coupled DAE systems in RC network design
In Buikis, A. and Ciegis, R. and Fitt, A. D., Editor, Progress in Industrial Mathematics at ECMI 2002 , Seite 173 -- 177.
Herausgeber: Springer, Berlin
Dynamic iteration may fail for partitioned network simulationProceedings in Applied Mathematics and Mechanics, 2(1):499 -- 502
Preconditioned dynamic iteration for coupled differential-algebraic systemsZeitschrift für Bibliothek, Information und Technologie, 41(1):1 -- 25