Reference solutions
Within MaX-1, analytic solutions are mainly required as a reference for testing the accuracy of numeric solutions obtained using MMP or GFD.
Initial solutions for FD iterations
Another purpose of analytic solutions is to set up an initial solution for an iterative FD solution. In this case, the analytic solution can be the solution of a simplified, 'unperturbed' problem and GFD is applied for solving a more complex, 'perturbed' problem for which no analytic solution is known.
Analytic MaX-1 solutions
MaX-1 provides a formula interpreter for defining analytic solutions. Since this interpreter contains a much smaller set of functions and tools than well-known programs such as Mathematica and Maple, you can solve only very simple problems analytically. Open the Field dialog and press the Field formula… button. The Field formula dialog will pop up, allowing you to specify the formula defining the field.
Imported analytic solutions
For obtaining analytic solutions to more sophisticated problems, you are better off using Mathematica, Maple, or a similar package. Note however you can import analytic solutions obtained from other programs into MaX-1 via ASCII files. As the ACII files generated by Mathematica or Maple have a format are incompatible to MaX-1 file formats, you will need to write a conversion routine.
Geometry and symmetry
Generally, finding analytic solutions is a very demanding task that requires extremely good mathematical knowledge. Relatively simple analytic solutions can only be found for problems with simple, symmetries and a high symmetry. The proper definition of the term 'high symmetry' requires knowledge of the theory of symmetry groups. A typical example of a highly symmetric object is an idealized coaxial cable consisting of concentric, circular cylinders. When such a structure is moved parallel to its axis by any displacement vector, or rotated around its axis with any angle, it is left unchanged. Therefore, one has infinitely many symmetry operations and the dimension of the corresponding symmetry group is infinite. Moreover, the structure exhibits some reflective symmetries that play a minor role. We will consider only problems with high symmetry in the following sections.
Material properties
Open the Domain dialog for defining the material properties of all domains of the problem to be solved. In electrostatics, only the permittivity is required, whereas the permeability is required in magnetostatics. For any obsolete material properties you should set some dummy constant value.
Boundaries
When the analytic formula describing the resulting field takes the boundary conditions into account, the explicit definition of the boundaries between domains is not required. In many cases, the analytic formula is restricted to certain domains only. In this case, you should provide the boundaries of the corresponding domain. This allows MaX-1 to detect the domain numbers of the grid points.
Grid points
Although analytic formulas can be evaluated anywhere in space, MaX-1 solves the field on a grid for display purposes. Therefore you must specify a grid in the Grid group of the Field dialog.
