Automatic model reduction of differential algebraic systems by proper orthogonal decomposition
Introduction
Many modern mathematical models of real-life processes impose difficulties when it comes to their numerical solution. This holds especially for models represented by nonlinear distributed parameter systems, which are frequent in engineering. Usually, for the numerical solution of distributed parameter systems the original system of infinite order is approximated by one with a finite system order by a semi-discretization, which results in a system of differential algebraic equations. The resulting number of degrees of freedom is usually very high and makes the use of the discretized model inconvenient for model-based process design, process control and optimization (Shi et al., 2006). Thus there is a need for reduced models. Through model reduction, a small system with reduced number of equations is derived. The numerical solution of reduced models should be much easier and faster than the solution of the original problem. On the other hand, the reduced model should be able to reproduce the system behavior with sufficient accuracy in the relevant window of operation conditions and in the relevant range of system parameters.
Various methods for nonlinear and linear model reduction have been proposed, particularly in the areas of electrical and mechanical engineering, control design and computational fluid dynamics. Some of them are based on physical simplifications like assumption of perfect mixing, introduction of compartments, equilibrium assumptions, etc. This approach requires physical insight of the modeler and hence is hard to automatize. Another successful approach, which may also be considered as a physical model reduction method, is based on nonlinear wave propagation theory (Marquardt, 1990, Kienle, 2000). It produces reduced model by approximation of the spatially distributed solution by profile with a given shape. As in the previous case, this method requires physical process understanding from the user and can be applied only for special systems. The generalized method of moments (Marchisio and Fox, 2005, Lebaz et al., 2016) is a widely used mathematical reduction technique for population balance equations. In this case, the reduced model does not preserve full information on spatial profile. Another mathematical possibility to obtain reduced models is to separate fast and slow subsystems. Slow manifold approximation (Christofides and Daoutidis, 1997) requires complicated symbolic operations, which impose difficulties on the automatization of this method. To sum up, widely used methods for nonlinear model reduction require experienced user; automatic application and integration in a simulation tool is a difficult and challenging task, which has hardly been attempted to our knowledge. On the other hand, there are linear model reduction techniques like balanced truncation (Benner et al., 2000, Heinkenschloss et al., 2011), which are applicable to high order systems and can be automatized quite easily. However, the resulting linear reduced models are only valid locally and not able to capture nonlinear properties of the original system.
In this work proper orthogonal decomposition (POD) (Kunisch and Volkwein, 2002, Park and Cho, 1996, Sirovich, 1987, Antoulas, 2005) is used for the development of an automatic procedure for model reduction. This method has been successfully applied for numerous problems in the fields of fluid dynamics, optimal control, and for population balance systems like crystallizers (Krasnyk and Mangold, 2010, Mangold et al., 2015), and granulators (Mangold, 2012). To put it in other words, the model reduction by POD is a proven approach. Nevertheless, applying model reduction by POD manually to complex engineering models is a challenging and tedious task. The idea of this work is to provide a software environment that performs the model reduction by POD automatically with minimal additional input from the user.
The work is structured as follows. Section 2 discusses the model reduction method. Technical details of the developed software tool for automatic model reduction are described in Section 3. Section 4 shows the developed software tool in action by applying it to two test models: a nonlinear heat conductor and a continuous fluidized bed crystallizer.
Section snippets
Reference model representation
Before applying a reduction procedure to the reference model, it has to be transformed into a spatially discretized form by applying the method of lines (Schiesser, 1991). Discretization results in a system of differential algebraic equations, which may be written aswhere x(t) is the discretized state vector, B and A are the system matrices, where B may be singular, c is a constant vector, and g(x(t)) is a function that comprises the nonlinearities of the system.
Software implementation
The automatic procedure for the model reduction is implemented in the modeling and simulation environment ProMoT/Diana (Mangold et al., 2014). ProMoT is a modeling tool written in Common Lisp with a graphical user interface written in Java (Ginkel et al., 2003). ProMoT supports the structured implementation of dynamic models described by systems of nonlinear implicit differential algebraic equations. ProMoT itself is a purely symbolic modeling tool and hence has no restriction with respect to
Heat conductor
One of the first spatially distributed chemical engineering models to which POD was applied is a nonlinear heat conduction system defined on a two-dimensional plane (Park and Cho, 1996). In Park and Cho (1996), the model reduction was done manually, separating the system into a part with homogeneous boundary conditions and another one with inhomogeneous boundary conditions. This separation is quite tedious. Therefore, the model is a nice test example for the developed automatic model reduction
Conclusions
The automatic tool for the model reduction has been developed by using proper orthogonal decomposition combined with empirical interpolation. For demonstration purposes a virtual machine has been prepared with all the needed software installed. It is freely available for download from http://promottrac.mpi-magdeburg.mpg.de/dist/pod/promot-pod-reducer-32bit.ova.
Although the basis functions from snapshots of the reference model give some hints on the accuracy to be expected from the reduced
Latin symbols:
Symbol Description Unit Value A Right system matrix of reference model Ared Right system matrix of reduced model A(x) Cross-sectional area of crystallizer m2 Aeff(x) Effective area of crystallizer m2 B Left system matrix of reference model Bred Left system matrix of reduced model c Constant vector of reference model cred Constant vector of reduced model csat Saturated solution concentration 1 0.0051 c(x) Liquid phase concentration 1 D Dispersion coefficient in particle phase m2 s−1 10−4 Df Dispersion coefficient in liquid phase m2
Acknowledgements
This work has been supported financially by DFG in the framework of SPP 1679, contract nr. MA 2051/2-1.
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