PDEModelica1

PDEModelica1 is nonstandardised experimental Modelica language extension for 1-dimensional partial differential extensions (PDE).

It is enabled using compiler flag --grammar=PDEModelica. Compiler flags may be set e.g. in OMEdit (Tools->Options->Simulation->OMC Flags) or in the OpenModelica script using command

PDEModelica1 language elements

Let us introduce new PDEModelica1 language elements by an advection equation example model:

model Advection "advection equation"
  parameter Real pi = Modelica.Constants.pi;
  parameter DomainLineSegment1D omega(L = 1, N = 100)  "domain";
  field Real u(domain = omega)                         "field";
initial equation
  u = sin(2*pi*omega.x)                                "IC";
equation
  der(u) + pder(u,x) = 0   indomain omega              "PDE";
  u = 0                    indomain omega.left         "BC";
  u = extrapolateField(u)  indomain omega.right        "extrapolation";
end Advection;

Error

[<interactive>:4:14-4:14:writable] Error: Missing token: SEMICOLON

The domain omega represents the geometrical domain where the PDE holds. The domain is defined using the built-in record DomainLineSegment1D. This record contains among others L – the length of the domain, N – the number of grid points, x – the coordinate variable and the regions left, right and interior, representing the left and right boundaries and the interior of the domain.

The field variable u is defined using a new keyword field. The domain is a mandatory attribute to specify the domain of the field.

The indomain operator specifies where the equation containing the field variable holds. It is utilised in the initial conditions (IC) of the fields, in the PDE and in the boundary conditions (BC). The syntax is

anEquation indomain aDomain.aRegion;

If the region is omitted, interior is the default (e.g. the PDE in the example above).

The IC of the field variable u is written using an expression containing the coordinate variable omega.x.

The PDE contains a partial space derivative written using the pder operator. Also the second derivative is allowed (not in this example), the syntax is e.g. pder(u,x,x). It is not necessary to specify the domain of coordinate in pder (to write e.g. pder(u,omega.x), even though x is a member of omega.

Limitations

BCs may be written only in terms of variables that are spatially differentiated currently.

All fields that are spatially differentiated must have either BC or extrapolation at each boundary. This extrapolation should be done automatically by the compiler, but this has not been implemented yet. The current workaround is the usage of the extrapolateField() operator directly in the model.

If-equations are not spported yet, if-expressions must be used instead.

Viewing results

During translation field variables are replaced with arrays. These arrays may be plotted using Array Plot or even better using Array Parametric Plot (to plot x-coordinate versus a field).