Analysis of semi-discretized differential algebraic equation from coupled circuit device simulation

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• 作者：Sascha Baumanns ; Lennart Jansen ; Monica Selva-Soto…
• 关键词：Circuit device simulation ; DAEs index ; Finite elements ; Mixed finite elements ; Monolithic simulation ; Unique solvability ; 34K28 ; 65L05 ; 65M20 ; 65M22 ; 65M60 ; 94C05
• 刊名：Computational and Applied Mathematics
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：34
• 期：3
• 页码：933-955
• 全文大小：639 KB
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• 作者单位：Sascha Baumanns (1)
  Lennart Jansen (2)
  Monica Selva-Soto (3)
  Caren Tischendorf (2) (4)
  
  1. Generali Deutschland Holding AG, Cologne, Germany
  2. Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099?, Berlin, Germany
  3. Department of Mathematical Engineering, University of Concepción, Casilla 160-C, Concepción, Chile
  4. DFG Research Center Matheon, Berlin, Germany
  
• 刊物主题：Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematical Applications in the Physical Sciences; Mathematical Applications in Computer Science;
• 出版者：Springer Basel
• ISSN：1807-0302

文摘

In the last years, several mathematical models coupling differential algebraic equations and partial differential equations for describing the behavior of electrical circuits have been proposed in the literature. Most of them investigate the properties of coupled systems that include one-dimensional drift–diffusion equations for describing the highly sensitive semi-conducting elements in the circuit. Here, we extend the results to coupled systems with higher dimensional drift–diffusion models that allow a proper modeling of semiconductor devices constituted of regions with different material properties. For stability reasons, we investigate a monolithic simulation approach and consider two common variants of PDE discretizations for semiconductors in the system: besides a finite element method that may be combined with the Scharfetter–Gummel approach, a mixed finite element discretization. The resulting differential algebraic equations share important properties that allow us to show that their index is always less or equal to two and depends only on the circuit’s topology. The information about the index enables us to conclude unique solvability of the associated initial value problems as well as to choose appropriate time-stepping methods for them. Keywords Circuit device simulation DAEs index Finite elements Mixed finite elements Monolithic simulation Unique solvability