Continuous System Modeling

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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
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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 wasnt 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
  higher-level 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 pre-defined 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.

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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.

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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 wont
  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.

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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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Method of Lines II

• Programmed in PDELib

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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 doesnt offer nifty 3D graphics
  capabilities yet.
• The results could have been exported to MATLAB
  for graphing, but we chose not to do so.

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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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Finite Volume Method II

• We integrate
• from t to t?t, and divide by ?t and by ?x

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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.

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Advection Equation
advection equation
initial condition
boundary condition
Method of Lines
Finite Volume Method
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Simulation Results
Finite Volume Method
Method of Lines
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Burgers Equation
Burgers equation
initial condition
boundary conditions
Method of Lines
Finite Volume Method
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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.

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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.

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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 havent 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.

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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 user-friendly fashion.

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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.