In engineering, the use of mathematical or physical models is as old as the discipline itself, and the idea of a twin has also been used at NASA since the early Apollo missions, see Rosen 2015. However, more efficient algorithms and the enormous growth in data and computing power have recently made it possible to create a new concept, the digital twin, see Shafto 2010. Digital twins integrate multiphysical models and data-based approaches and contain all knowledge about a product or system as executable code. They enable new types of support, e.g. for design optimization, process control or life cycle management. Digital twins are so important to the economy today that Gartner has named them one of the top 10 strategic technology trends in 2019.

Digital twins are often based on Differential Algebraic Equations (DAEs) which are a combination of differential equations and algebraic constraints. Originally, this was particularly relevant in mechanics and robotics, as well as in electrical circuit simulation. They occur often in multiphysics problems due to coupling conditions. The algebraic equations create severe numerical difficulties because the computation necessitates not only integration but also differentiation.

Recalling from calculus that differentiation is an unbounded operation, it is much more difficult to handle than the integration used for solving ordinary differential equations. Let us consider a DAE with a sinusoidal excitation of small amplitude but at high frequency, e.g.

\begin{aligned}\dot{x}(t) & = y(t)\\0 & = x(t)\;-\; \varepsilon\sin(\omega t).\end{aligned}

This sine wave may be numerical noise as small as machine precision but nonetheless its magnitude is amplified by the frequency

\begin{aligned}x(t) & = \varepsilon\sin(\omega t)\\y(t) & = \varepsilon \;\omega \cos(\omega t).\end{aligned}

The more derivatives involved in the exact solution of a DAE, the more problems can occur in the numerical computations. The DAE-index is a measure for this difficultly. That is why it is relevant to know the index before simulation.

Idoia Cortes Garcia ; Herbert De Gersem ; Sebastian Schöps (2020):
A Structural Analysis of Field/Circuit Coupled Problems Based on a Generalised Circuit Element.
In: Numerical Algorithms, 83, (1), pp. 373–394, ISSN: 1017-1398, DOI: 10.1007/s11075-019-00686-x, arXiv: 1801.07081. [Article]

Idoia Cortes Garcia ; Jonas Pade ; Sebastian Schöps ; Christian Strohm ; Caren Tischendorf (2019):
Waveform relaxation for field/circuit coupled problems with cutsets of inductances and current sources.
In: Proceedings of the 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA), 1286–1286. IEEE. ISBN: 978-1-7281-0562-8, DOI: 10.1109/ICEAA.2019.8878955, URL: https://ieeexplore.ieee.org/xpl/conhome/8868012/proceeding. [In Proceedings]

Sebastian Schöps ; Andreas Bartel ; Michael Günther ; E. Jan W. ter Maten ; Peter C. Müller (editors) (2014):
Progress in Differential-Algebraic Equations, Differential-Algebraic Equations Forum, Springer. ISBN: 978-3662449257 DOI: 10.1007/978-3-662-44926-4 [Proceedings]

Andreas Bartel ; Markus Brunk ; Sebastian Schöps (2014):
On the Convergence Rate of Dynamic Iteration for Coupled Problems With Multiple Subsystems.
In: Journal of Computational and Applied Mathematics, 262, pp. 14–24, ISSN: 0377-0427, DOI: 10.1016/j.cam.2013.07.031. [Article]

Sebastian Schöps ; Andreas Bartel ; Michael Günther (2013):
An Optimal p-Refinement Strategy for Dynamic Iteration of Ordinary and Differential Algebraic Equations.
In: Proceedings in Applied Mathematics and Mechanics, volume 13, 549–552. DOI: 10.1002/pamm.201310262. [In Proceedings]

Andreas Bartel ; Markus Brunk ; Michael Günther ; Sebastian Schöps (2013):
Dynamic Iteration for Coupled Problems of Electric Circuits and Distributed Devices.
In: SIAM Journal on Scientific Computing, 35, (2), pp. B315–B335, ISSN: 1064-8275, DOI: 10.1137/120867111. [Article]

Sebastian Schöps ; Andreas Bartel ; Herbert De Gersem (2012):
Multirate Time-Integration of Field/Circuit Coupled Problems by Schur Complements.
In: Scientific Computing in Electrical Engineering SCEE 2010, volume 16 of Mathematics in Industry, 243–251. Springer. ISBN: 978-3-540-71979-3, DOI: 10.1007/978-3-642-22453-9_26. [In Proceedings]

Markus Clemens ; Sebastian Schöps ; Herbert De Gersem ; Andreas Bartel (2011):
Decomposition and Regularization of Nonlinear Anisotropic Curl-Curl DAEs.
In: COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 30, (6), pp. 1701–1714, DOI: 10.1108/03321641111168039. [Article]

Andreas Bartel ; Sascha Baumanns ; Sebastian Schöps (2011):
Structural Analysis of Electrical Circuits Including Magnetoquasistatic Devices.
In: Applied Numerical Mathematics, 61, pp. 1257–1270, ISSN: 0168-9274, DOI: 10.1016/j.apnum.2011.08.004. [Article]

Sebastian Schöps ; Andreas Bartel ; Herbert De Gersem ; Michael Günther (2010):
DAE-Index and Convergence Analysis of Lumped Electric Circuits Refined by 3-D MQS Conductor Models.
In: Scientific Computing in Electrical Engineering SCEE 2008, volume 14 of Mathematics in Industry, 341–350. Springer. ISBN: 978-3-642-12293-4, DOI: 10.1007/978-3-642-12294-1_43. [In Proceedings]