Design and Implementation of Object-Oriented ... - Automatic Control

Chapter 2. Modeling Techniques
• Send the modeler back to the step 1 to resolve the problem with
pen and paper, reformulating the model into an equivalent index 1
problem.
• Build the capacity of recognizing and resolving high index problems
into the modeling tool.
Because Modelica is designed as a multi-domain modeling tool, it has to
support automatic index handling. This does not mean that automatic
index reduction is the silver bullet that solves all high index problems in
the best possible way. There are a few exceptions when manual derivation
of an index 1 problem is preferable to the automatic procedure. This
is the case when the automatic procedure gives results with numerical
disadvantages, at least with the current version of the Modelica language
and the index reduction algorithms in the Dymola tool. Examples of these
rare cases are presented in Chapter 4. But independent of an automatic
handling, model users have to understand the implications of high index
problems in order to provide the right initial conditions.
Partial Differential Equations
Partial differential equations (PDE) are the mathematical formulations
that permits the highest level of detail for system level models. They
provide the tool corresponding to the magnifying glass with the highest
magnification power. PDEs form an essential part of the mathematical
description of physical systems depending on space and time. They arise
in a large number of modeling applications, but are not that common in
systems level modeling. The reason for this is that PDE can have widely
differing properties and that they are much more difficult to deal with
numerically than ODEs. Using PDE for dynamical systems means that
we have at least one more independent variable apart from time. Usually
these are spatial coordinates, but variables can also be distributed in other
dimensions like particle sizes in crystallization processes.
Numerical solutions to PDE for design calculations are used in almost
all engineering disciplines and are established as independent research
areas, e. g., Computational Fluid Dynamics (CFD) and Finite Element
Methods (FEM).
Consider a set of PDEs expressed over the solution domain Ω which is
a compact region in IR m+1 and its boundary δ Ω ∈ IR m , compare Figure 2.3
for a two-dimensional example. If the independent variables are separated
into time t and a vector of spatial variables x, then the dependent variables
u can be written as
u = u(x, t)
where u ∈ IR n , x ∈ IR m and t ∈ IR ≥ 0. In mathematical texts the independent
space and time variables are usually treated alike, here Ω is used
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