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Advanced Topics in Model Order Reduction

  • Chapter
Coupled Multiscale Simulation and Optimization in Nanoelectronics

Part of the book series: Mathematics in Industry ((TECMI,volume 21))

Abstract

This chapter contains three advanced topics in model order reduction (MOR): nonlinear MOR, MOR for multi-terminals (or multi-ports) and finally an application in deriving a nonlinear macromodel covering phase shift when coupling oscillators. The sections are offered in a preferred order for reading, but can be read independently.

Section 6.1, written by Michael Striebel and E. Jan W. ter Maten, deals with MOR for nonlinear problems. Well-known methods like TPWL (Trajectory PieceWise Linear) and POD (Proper Orthogonal Decomposition) are presented. Development for POD led to some extensions: Missing Point Estimation, Adapted POD, DEIM (Discrete Empirical Interpolation Method).

Section 6.2, written by Roxana Ionutiu and Joost Rommes, deals with the multi-terminal (or multi-port) problem. A crucial outcome of the research is that one should detect “important” internal unknowns, which one should not eliminate in order to keep a sparse reduced model. Such circuits come from verification problems, in which lots of parasitic elements are added to the original design. Analysis of effects due to parasitics is of vital importance during the design of large-scale integrated circuits, since it gives insight into how circuit performance is affected by undesired parasitic effects. Due to the increasing amount of interconnect and metal layers, parasitic extraction and simulation may become very time consuming or even unfeasible. Developments are presented, for reducing systems describing R and \(\mathit{RC}\) netlists resulting from parasitic extraction. The methods exploit tools from graph theory to improve sparsity preservation especially for circuits with multi-terminals. Circuit synthesis is applied after model reduction, and the resulting reduced netlists are tested with industrial circuit simulators. With the novel RC reduction method SparseMA, experiments show reduction of 95 % in the number of elements and 46x speed-up in simulation time.

Section 6.3, written by Davit Harutyunyan, Joost Rommes, E. Jan W. ter Maten and Wil H.A. Schilders, addresses the determination of phase shift when perturbing or coupling oscillators. It appears that for each oscillator the phase shift can be approximated by solving an additional scalar ordinary differential equation coupled to the main system of equations. This introduces a nonlinear coupling effect to the phase shift. That just one scalar evolution equation can describe this is a great outcome of Model Order Reduction. The motivation behind this example is described as follows. Design of integrated RF circuits requires detailed insight in the behavior of the used components. Unintended coupling and perturbation effects need to be accounted for before production, but full simulation of these effects can be expensive or infeasible. In this section we present a method to build nonlinear phase macromodels of voltage controlled oscillators. These models can be used to accurately predict the behavior of individual and mutually coupled oscillators under perturbation at a lower cost than full circuit simulations. The approach is illustrated by numerical experiments with realistic designs.

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Notes

  1. 1.

    Section 6.1 has been written by Michael Striebel and E. Jan W. ter Maten.

  2. 2.

    Most frequently V is constructed to be orthogonal, such that W = V can be chosen.

  3. 3.

    This means, the matrix has exactly one non-zero entry per row at most one non-zero per column.

  4. 4.

    Section 6.2 has been written by Roxana Ionutiu and Joost Rommes. For an extended treatment on the topics of this section see also the Ph.D. Thesis of the first author [68].

  5. 5.

    Section 6.3 has been written by Davit Harutyunyan, Joost Rommes, E. Jan W. ter Maten and Wil H.A. Schilders.

  6. 6.

    Similar to singular values of matrices, the Hankel singular values and corresponding vectors can be used to identify the dominant subspaces of the system’s statespace: the larger the Hankel singular value, the more dominant.

  7. 7.

    Matlab code for plotting the PSD is given in [107].

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Authors and Affiliations

  1. ASML, De Run 6501, 5504, DR Veldhoven, The Netherlands

    Davit Harutyunyan

  2. ATTE, ABB Switzerland Ltd, Austraße, 5300, Turgi, Switzerland

    Roxana Ionutiu

  3. Applied Mathematics/Numerical Analysis, Bergische Universität Wuppertal, Gaußstraße 20, D-42119, Wuppertal, Germany

    E. Jan W. ter Maten (Chair)

  4. Department of Mathematics and Computer Science, CASA, Eindhoven University of Technology, 513, 5600, Eindhoven, The Netherlands

    E. Jan W. ter Maten (Chair) & Wil H. A. Schilders

  5. Mentor Graphics, DSM/AMS, Le Viseo – Bâtiment B, 110 rue Blaise Pacal, Inovalee, 38330, Montbonnot, France

    Joost Rommes

  6. ZF Lenksysteme GmbH, Richard-Bullinger-Straße 77, D-73527, Schwäbisch Gmünd, Germany

    Michael Striebel

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  1. Davit Harutyunyan
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    Michael Günther

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Harutyunyan, D., Ionutiu, R., ter Maten, E.J.W., Rommes, J., Schilders, W.H.A., Striebel, M. (2015). Advanced Topics in Model Order Reduction. In: Günther, M. (eds) Coupled Multiscale Simulation and Optimization in Nanoelectronics. Mathematics in Industry(), vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46672-8_6

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