Digital Twins
Physical and data driven models

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.