Conference 2008
Support for Dymola in the Modeling
and Simulation of Physical Systems
with Distributed Parameters
Farid Dshabarow
ABB Turbo Systems, Switzerland
François E. Cellier
ETH Zürich, Switzerland
Dirk Zimmer
ETH Zürich, Switzerland
March 4, 2008
© Prof. Dr. François E. Cellier
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Conference 2008
History of General Purpose PDE Solvers
• In the 1970s, a number of general-purpose PDE
solvers for parabolic and hyperbolic PDEs in one
and two space dimensions were developed.
• The most versatile among those was FORSIM VI, a
FORTRAN library for simulating such systems.
• FORSIM was based on the Method of Lines, a
technique that discretizes the spatial axes of a PDE,
while leaving the time axis continuous, thereby
converting a PDE into a set of ODEs.
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History of General Purpose PDE Solvers II
• FORSIM VI offered a set of FORTRAN subroutines that the
user could call for computing approximations of spatial
derivatives and for integrating the resulting ODE system over
time.
• The tool was versatile but not user-friendly. It was versatile,
because it offered the user all of the tools that s/he needed for
solving a PDE problem, but it wasn’t user-friendly, because it
forced the user to manually decompose the problem down to
a fairly low level.
• Other tools, such as PDEL and LEANS III offered higherlevel interfaces, but this came at the price of these tools being
more specialized tools that could only solve problems
belonging to predefined sets of PDEs.
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History of General Purpose PDE Solvers III
• In the long run, none of these tools survived, because
engineers wanted to solve more and more complex problems,
and neither of these tools were versatile and/or efficient
enough to do so.
• Consequently, most engineers and physicists created their
own spaghetti codes written for solving one particular PDE
problem only in as efficient a manner as possible.
• Evidently, this caused a lot of duplication of very similar
code segments that were rewritten by many programmers
over and over again, and ultimately, it must be recognized
that this approach is wasteful in terms of human resource
utilization.
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History of General Purpose PDE Solvers IV
• Only quite recently, a renaissance of general-purpose PDE
codes could be observed. The best and most advanced
among these second-generation codes is FEMLAB
(COMSOL).
• FEMLAB essentially copied the approach taken earlier by
PDEL and LEANS III, i.e., it offers solutions to a predefined set of PDE problems that are being parameterized to
allow a user access to this tool who knows very little about
the properties of numerical PDE solvers.
• Compared to the earlier tools, FEMLAB offers a much
improved user interface with neat graphing capabilities. The
code has been rather successful and is being widely accepted
as the state of the art in numerical PDE technology.
March 4, 2008
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Conference 2008
PDELib
• We chose to take the other route and re-implement
facets of FORSIM VI in Dymola/Modelica, making
use of the built-in matrix manipulation capabilities
of Modelica and the graphical user interface of
Dymola.
• In this way, simple PDE problems can be coded
graphically without forcing the user to write any
programming code directly.
• Of course, we pay a heavy price in that users of
PDELib will have to be somewhat knowledgeable
about the algorithms underlying numerical PDE
solvers.
March 4, 2008
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Conference 2008
PDELib II
• Yet, the object-oriented nature of Modelica allows to wrap
PDELib codes, offering a “mathematical” user interface, into
higher-level abstractions, promoting a “physical” user
interface.
• At that higher level of abstraction, users won’t need to
understand the underlying numerical algorithms any longer.
• Yet, as these higher-level codes will be mapped onto PDELib
first, maintaining these higher-level codes will be much
easier than if these higher-level codes were programmed in
C++ directly, for example.
• Due to the symbolic formulae manipulation techniques
offered by tools such as Dymola, the resulting codes can be
made to run as efficiently as any hand-written spaghetti codes
of the past.
March 4, 2008
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Conference 2008
Method of Lines
• Let us explain by means of a simple example how the method
of lines operates.
• Given the 1D diffusion equation:
• The method of lines discretizes the space dimension:
leading to:
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Conference 2008
Method of Lines II
• Programmed in PDELib:
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Conference 2008
Simulation Results
• The spatial axis had been
discretized into 40 segments.
• The numerical solution is
compared to the analytical
solution.
• With 40 segments, the results are
not yet highly accurate.
• Dymola doesn’t offer nifty 3D
graphics capabilities yet.
• The results could have been
exported to MATLAB for
graphing, but we chose not to do
so.
March 4, 2008
© Prof. Dr. François E. Cellier
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Conference 2008
Finite Volume Method
• A second method implemented in PDELib is the Finite Volume method.
It works as follows. The space gets subdivided into a finite number of
cells. The ith cell can be written as:
• The average value of the solution u(t) for this cell can be written as:
• Mass conservation can be written as:
• where f(t) is the flux function.
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Conference 2008
Finite Volume Method II
• We integrate:
• from t to t+t, and divide by t and by x:
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Conference 2008
Finite Volume Method III
• With the abbreviation:
• we can rewrite as follows:
• which can be reinterpreted as:
• In this way, we have once again converted a PDE to a set of ODEs. The
only remaining question is how we approximate the average flux values.
March 4, 2008
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Conference 2008
Advection Equation
advection equation
initial condition
Method of Lines
March 4, 2008
boundary condition
Finite Volume Method
© Prof. Dr. François E. Cellier
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Conference 2008
Simulation Results
Finite Volume Method
Method of Lines
March 4, 2008
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Conference 2008
Burger’s Equation
Burger’s equation
initial condition
Method of Lines
March 4, 2008
boundary conditions
Finite Volume Method
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Conference 2008
Simulation Results
•
•
The FVM solution is less accurate, but
remains numerically stable for an
arbitrarily large number of segments.
Hence a numerically accurate solution can
be obtained using sufficiently many
segments.
Finite Volume Method
•
•
•
•
Method of Lines
The MoL solution is more accurate, but
less stable.
With 10 segments, the solution remains
stable, but is very inaccurate.
With 20 segments, it is more accurate, but
turns unstable after 0.6 sec.
With 40 segments, it is always unstable.
March 4, 2008
© Prof. Dr. François E. Cellier
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Conference 2008
MoL vs. FVM
• The method of lines works well for parabolic PDEs. It has
a tendency to lead to numerically unstable solutions in the
case of hyperbolic PDEs, unless a geometric integration
algorithm is being used (not currently available in
Dymola).
• The MoL solution of hyperbolic PDEs can frequently be
stabilized also by use of an upwind discretization scheme.
• The finite volume method is not as straightforward.
However, it can be stabilized more easily.
• The FVM was designed for hyperbolic PDEs primarily.
Whereas the method can sometimes be abused to solve
parabolic PDEs as well, the user has no reason to do so.
March 4, 2008
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Conference 2008
Conclusions
• The first release of PDELib only considers parabolic and
hyperbolic PDEs in a single space dimension.
• Later versions will extend the tool to 2D and 3D problems
also.
• However, this requires finite element approximations in
space, which haven’t been implemented yet (similar to
what FEMLAB does).
• Whereas 1D problems lead to ODE sets in a few dozen
ODEs, 2D problems quickly lead to thousands of ODEs,
and 3D problems almost immediately lead to problems
with several hundreds of thousands of ODEs. Hence
efficiency matters.
• Boundary conditions are much more difficult to satisfy in
2D and 3D problems.
March 4, 2008
© Prof. Dr. François E. Cellier
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Conference 2008
Conclusions II
• At the current time, PDELib is only a test bed
implementation.
• It can be used for solving toy problems, and also
for teaching purposes, but not yet for solving real
engineering problems as they need to be solved in
practice.
• Future versions shall enhance the tool, as well as
offer higher-level (physical layer) abstractions for
dealing with complex problems in a more userfriendly fashion.
March 4, 2008
© Prof. Dr. François E. Cellier
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Conference 2008
References
• Dshabarow, F. (2007), Support for Dymola in the
Modeling and Simulation of Physical Systems
with Distributed Parameters, MS Thesis,
Department of Computer Science, ETH Zurich,
Switzerland.
March 4, 2008
© Prof. Dr. François E. Cellier
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