Analytic solutions Graphics Important modifications Installation and running MMP solver New features PFD solver System requirements
OpenMaXwell provides several tools for computing analytic solutions for a given problem. Although OpenMaXwell is designed for computational electromagnetics, you can also solve problems in other areas of computational physics.
Since analytic solutions are known for only relatively simple problems, it is unreasonable to expect OpenMaXwell to compute analytic solutions for complicated problems. Even if you know the analytic solution to a problem, it may be impossible to tell OpenMaXwell how to solve the problem. Instead you can write your own program to solve the problem analytically and save the results in an ASCII file format that is compatible with OpenMaXwell. Then you can use OpenMaXwell to display the results graphically taking advantage of OpenMaXwell’s animation tools. Note that you can also run your analytic code within OpenMaXwell with the movie directive RUN.
For obtaining analytic solutions of simple problems, OpenMaXwell provides three tools:
1) The field formula definition with the OpenMaXwell formula interpreter,
2) Grid transformations, especially conformal mapping, and
3) MMP expansions.
Select Project… from the Tools menu to open the Project dialog. Specify the main properties of the problem to be solved. The main question here is: Is the field complex or real, i.e., is it time-harmonic or not? Note that static problems can be considered as non-harmonic or as harmonic with frequency = 0.
Select Field… from the Tools menu to open the Field dialog. Press the Field Formula… button. The Field formula dialog pops up. In this dialog you can enter the analytic formula for all original field components.
When you have specified the field formula, close the dialog and go back to the Field dialog. You now must specify other field data that are required for the visualization: The grid, where the original field has to be evaluated, the components of the original field that are included in your solution, etc.
When you have specified all field data, press the Clear all !! button. OpenMaXwell will close the Field dialog, allocate memory for the original and derived field, apply the field formula to all grid points for computing the original field, and compute the derived field according to the specifications in the Field dialog.
Make sure that the options of the graphic window are appropriately set and select Field… from the Draw menu for drawing the resulting field.
When your problem has several domains with different material properties, it is inconvenient to specify the material properties as constants in the field formula. Instead, you can specify these data in the Domain dialog. In the field formula you can access the material properties with the corresponding functions. Select Domains… from the Tools menu for obtaining the Domain dialog and for specifying the material properties.
When you have several domains, it is usually not convenient to specify the domain numbers of each grid point manually. It is much easier in most cases to specify the boundaries between the domains. You can define boundaries graphically (select Boundaries… in the Modify menu) and manually (select Boundaries… in the Tools menu). OpenMaXwell can evaluate the domain numbers of all grid points from the boundary data.
Sometimes, you know the analytic solution only of a part of the original field. For example, you know it for the scalar potential of an electrostatic field, but you are too lazy to derive the electric field from the potential. In such cases, you can use the GFD feature for obtaining a GFD approximation of the part of the original field that is not defined analytically.
Conformal mapping is a well-known method for obtaining analytic solutions to 2D static problems. Here, the 2D xy plane is considered as a complex z plane with x=Real(z), y=Imag(z). A conformal mapping solution consists of the following steps:
Transform the original z plane into a transformed w plane. For OpenMaXwell this means that the coordinates of all original grid points are transformed into coordinates of the transformed grid points.
Compute the field on the transformed grid. When you want to obtain an analytic solution, this must be the analytic solution of the transformed problem in the w plane.
Apply the inverse grid transformation to obtain the field on the original grid in the original z plane.
Conformal transformations can be specified in the Grid transformation dialog. This dialog pops up when you press the Grid trans..!! button in the Field dialog. Select Field… from the Tools menu to open the Field dialog.
Since you will have to apply the inverse transformation later, make sure that you know both the transformation and its inverse. Moreover, it is reasonable to check and save the corresponding formula on files as long as the Grid transformation dialog is open. When you have specified the transformation, press the Apply !! button. OpenMaXwell will close the dialogs and compute the coordinates of the transformed grid.
On the transformed grid, you have to specify the analytic solution as indicated in the Field formula section.
Note that the boundaries are mapped together with the grid. OpenMaXwell does not provide any feature for conformal boundary transformation. Since you need to know the analytic solution on the transformed grid only, you specify the boundaries after the grid transformation. When you have computed the field, the boundaries are no longer required.
As soon as the field has been computed on the transformed grid, you can apply the inverse grid transformation exactly as you have applied the grid transformation.
Make sure that the options of the graphic window are appropriately set and select Field… from the Draw menu to draw the resulting field.
If you do not know the analytic solution in the transformed w plane, you can use either the MMP or the GFD tool for obtaining an approximate solution. This solution can also be transformed back to the original z plane.
MMP is a semi-analytic method. This means that all MMP expansions are analytic solutions of the field equations within a domain. In order to obtain analytic solutions of a given problem, all parameters in the MMP expansions must be specified in such a way that all boundary conditions on the boundaries between the domains are exactly met. This can only be achieved when the geometry is sufficiently simple.
Select Project… from the Tools menu to open the Project dialog. Specify the main properties of the problem to be solved. For details see Project dialog.
Select Domains… from the Tools menu to open the Domain dialog. Specify the material properties of the problem to be solved. For details see Domain dialog.
Select Boundaries… from the Tools menu to open the Boundary dialog. Specify the boundaries of the problem to be solved. In addition to this, you can also graphically modify the boundaries. For details see Boundary dialog.
Select Expansions… from the Tools menu to open the Expansion dialog. Specify the MMP expansions of the problem to be solved. In addition to this, you can also graphically modify the expansions. For details see Expansion dialog. For obtaining an analytic solution, you must specify all parameters of all MMP expansions. Note that this is not necessary when you let the MMP tool compute the parameters. When the set of expansions is complete for obtaining an analytic solution, you can either specify the analytic values of the parameters or you can let MMP compute them. You will see that MMP can very accurately compute the parameters.
Select Objects… from the Tools menu to open the 3D objects dialog. Specify how 3D objects shall be generated from 2D boundaries and 2D expansions.
When you have specified the MMP expansions and parameters, open the Field dialog. You now must specify further field data that is required for display purposes: the grid, where the original field is to be evaluated, the components of the original field that are involved in your solution, etc.
Press the Field formula… button, to open the Field formula dialog. In this dialog, you must check the Ignore formula, use expansion box. When this box is checked, OpenMaXwell computes the original field from the MMP expansions. If the expansion number is zero, the sum over all MMP expansions is computed. If you are curious about the field of a certain expansion, you can specify the number of this expansion.
When you have closed the Field formula dialog, press the Clear all !! button. OpenMaXwell will close the Field dialog, allocate memory for the original and derived field, apply the field formula to all grid points for computing the original field, and compute the derived field according to the specifications in the Field dialog.
Make sure that graphics window options are appropriately set and select Field… from the Draw menu for drawing the resulting field.
Responsible for this web page: Ch. Hafner, Computational Optics Group, IEF, ETH, 8092 Zurich, Switzerland
Last update 17.02.2014