Solving Implicit Differential Equations within a Modeling Environment
by Jacques de Swart
Many industrial applications can be modeled by sets of Implicit Differential Equations (IDEs). In joint work at CWI and Paragon Decision Technology (PDT), sponsored by the Technology Foundation STW, it is shown how the integration of an IDE solver in an algebraic modeling system can help to overcome many of the difficulties that a user currently encounters in trying to solve such systems of IDEs.
In industrial design processes, testing designs is of major importance. Before products are manufactured based on some design, one wants to know how a product based on this design would behave under several circumstances. For this purpose, one models the product in terms of its design. By solving the model, one simulates the working of the product. This procedure is much cheaper than building and testing prototypes. Often the modeling of a product results in IDEs equations in which derivatives of the unknowns with respect to one independent variable, typically time, appear implicitly. The modeling of time-dependent processes often results in IDEs. To solve IDEs, numerical methods are indispensable.
Examples of applications where simulation processes involve IDEs are testing the design of an electrical circuit, and the formulation of safety requirements for trains. Other areas include the modeling of turbulent flows in water tube systems, the description of demand-supply curves in liberalized markets, and the simulation of chemical reactions.
On the one hand, the complexity and size of the applications require a user friendly modeling environment, which speaks the language of the modeler, and offers the possibility to test and compare several scenarios and instances of the model data efficiently. Moreover, the modeler does not want to be involved in the often cumbersome interfacing with solvers. On the other hand, modern numerical techniques are required to solve ill-conditioned IDE systems of high dimension. How to meet these requirements is studied by integrating the novel IDE solver PSIDE, developed at CWI, in the advanced modeling environment AIMMS (Advanced Interactive Multi-dimensional Modeling System), which is a product of PDT.
If the time scales of the various solution components vary greatly, and if the rapidly changing components are physically irrelevant, then we call a problem stiff. For example, if both high and low frequency signals are present in an electrical circuit, but the high-frequent signals are small in magnitude, then the modeling of such a circuit gives rise to a stiff system of IDEs. To solve such IDEs, an implicit method is required, which means that the numerical approximations are not directly available, but have to be computed from nonlinear systems. This computation requires the evaluation of Jacobians of the IDEs with respect to the unknowns. In AIMMS these Jacobians are available in analytical form.
If some solution components of an IDE are more sensitive to perturbations than others, then the IDE is said to be of higher index. In order to integrate IDEs numerically with variable stepsize, one usually estimates a local error , in which the index has to be taken into account. Existing IDE solvers do not have a facility to compute the index. An automatic index determination facility for PSIDE, which uses the analytical Jacobian available in AIMMS, is currently under development.
The IDE solver has to know not only the values of all variables at the start of the integration interval, but also their derivatives. Especially the latter are in practice often unknown to the modeler and have to be computed from an - often nonlinear - system of algebraic equations. A powerful commercial nonlinear solver available in AIMMS is CONOPT. We used CONOPT successfully to compute the missing initial values. For higher index problems the problem of finding initial values is even more complicated, because the initial values have to satisfy differentiated equations as well. Based on the index determination facility and the capability of AIMMS to differentiate equations automatically, we are working on an automatic procedure for finding initial values for higher index problems.
One problem in the CWI Test Set is the Transistor Amplifier, whose circuit diagram is shown in the Figure. The task is to compute the behaviour of the voltages in the nodes and the currents in the wires over time. There are several symbolic equations, such as Kirchoff&Mac226;s Law, whereas for every type of electrical component there is an equation describing its working. These equations are independent of the specific electrical circuit. To simulate another circuit one only has to adapt model data. The application shown in the figure is clickable. One can play the movie of the circuit&Mac226;s working and select the wires and nodes of which one wants to see more information. As additional information, the maximum of the currents over all wires is displayed. The <<, <, > and >> buttons serve to step through the movie.
Jacques de Swart - CWI/PDT
Tel: +31 20 592 4176
E-mail: email@example.com, Jacques.de.Swart@paragon.nl