With respect to the real magnet, having hexagonal structure, the main magnetic analysis was carried out for the plane intersecting the hexagon at the center of two opposite sides. A further analysis for the plane rotated of 30 (intersecting two opposite corners) was also performed.

The single cables of the winding were not modeled; the two layers are thin cylinders, with 1 cm radial thickness, in order to reproduce the thickness of the Rutherford superconducting cable. The solenoid radius and axial length are considered at 4.5 K.

The flux bars and the gap plates are not continuous along the azimuthal direction. In order to carry out an axi-symmetric analysis, they are modeled as continuous over the 360, with suitable (averaged) magnetic properties.

FINITE ELEMENT CODES

Two finite element codes have been used to perform the magnetic analysis, OPERA 2D and ANSYS, implemented on a Digital Alpha-VAX. For both the mesh was defined so that an acceptable CPU time was required for calculations (see Fig. 1-2 and Fig. 1-3 ).

Fig. 1-2 OPERA 2D meshed model (~40000 nodes)
Fig. 1-3 ANSYS meshed model (~ 60000 nodes)

Both the codes have very powerful post-processors, allowing every kind of analysis (field maps, uniformity maps, calculations on magnetic forces, ...). Ansys has the further possibility of performing sub-modelisation, which is useful to analyse in more detail restricted regions of the meshed model.

MAGNETIC FIELD CALCULATIONS

The first results of a magnetic analysis are the flux line and magnetic field maps on the overall model (see Fig. 1-4 Fig. 1-5 ); secondly it is possible to better analyse the regions of particular interest, such as the zones between the forward, backward and barrel plates. In these regions a particularly fine mesh has been performed (at least three elements between two steel plates), in order to obtain a more accurate field map and to calculate the magnetic force on the single plate (see Fig. 1-6 ). Furthermore a list of the nodes and their corresponding field values of those regions can be obtained, in order to perform simulations with the Montecarlo method.

Fig. 1-4 Potential vector lines
Fig. 1-5 Overall field map
Fig. 1-6 Particular of the forward door field map

It is important to say that the results of ANSYS and OPERA 2D, in terms of central magnetic field, force on the winding, uniformity in the drift chamber, etc., are the same within 2%; this result is generally true and is very slightly dependent on the applied mesh.

DRIFT CHAMBER UNIFORMITY

The aim of the magnetic design was to obtain a magnetic field of 1.5 T with a uniformity of 2% in the drift chamber; this region covers the range from -1015 mm backward to 1749 mm forward axially, and from 239 mm to 805 mm radially.

With both the codes the resultant central field is 1.50 T; the field uniformity in the drift chamber is shown in Fig. 1-7.

Fig. 1-7 Field uniformity in the drift chamber

FUTURE DEVELOPMENTS

Up to now only 2D calculations were performed; this is because in the past only a limited version of Ansys was available. Now the complete Opera 3D is available, so that, in the future, when the geometry of the iron will be completely defined, 2D and 3D calculations will be carried out.

MAGNETIC FIELD MAP IMPLEMENTATION FOR BBSIM

GENERALITIES

The previously described field model has been implemented in terms of a FORTRAN routine intended to be used within BBSIM, i.e. conforming to the GEANT GUFLD requirements (one 3-d vector input argument, containing the Cartesian coordinates of the current point expressed in cm, and a 3-d vector output argument returning the Cartesian coordinates of the field in that point, expressed in kgauss). In order to allow the test of this new field map without loosing the default one, the routine has been named GUFLDNEW and is called by the default GUFLD when the corresponding option is selected.

The main problems concerning this implementations are the following:

a) the original input data (the ANSYS results) are given as axial and radial components of the field in the nodes of an irregular mesh, which has been defined according the needs of ANSYS but is not practical for determining the field in an arbitrary point

b) the field in the laminated flux return regions has strong variations when passing from iron to air.

GENERATION OF THE INPUT FILE

These problems required different methods to be applied in different regions. The region defined by 0 < r < 182 cm and -155 cm < z < 231 cm has no iron, and in the following will be called "coil". All the other regions contain iron slabs, and they will be called respectively "barrel" (182 cm < r< 400 cm, -155 < z < 231 cm), "backward" (0 < r< 400 cm, -357 < z < -155 cm) and "forward" (0 < r< 400 cm, 231 cm < z < 434 cm). Outside these regions the field is assumed to be vanishing, with the exception of the beam pipe that has a special treatment, with a call to a specific routine.

A preparatory job creates a data file ( gnbase/dat/bfield.dat ) containing the axial and radial fields components in a regular grid, separately for the different regions. In the case of the "coil" region the fields components are given in grid of 5 cm x 5 cm. For each point of this grid the 3 nearest points in the ANSYS mesh are looked for, and the field value is determined by a 2-d linear interpolation.

In the other regions it is necessary to distiguish the case of points in the iron or in the air, and the interpolations are difficult because of the irregularity of the field. For this reason the ANSYS nodes in iron and in air are kept separate, and for each point of the regular grid two values of each field components are given, one taken from the nearest ANSYS node in air and the other from the nearest ANSYS node in iron. The grid spacing are choosen in such a way to have points on each slab and inside each gap (for "barrel" 2 cm (r) x 5 cm (z), for backward and forward 10 cm (r) x 2 cm (z)). There is no detailed description of all support and stuffening pieces, and a crude treatment of the forward and backward end plugs.

GUFLDNEW OPERATION

At the first entry in GUFLDNEW at run time the data file (bfield.dat) is read. After, and for all the subsequent calls, the region of the current point is determined. When it happens not to be the "coil", it is also determined whether the current point is in iron or not. In the "coil" the field components are found with a linear 2-d interpolation (subroutine FINT from CERNLIB); in the other regions the field components are found using as a look-up table the field components in the grid concerning the same material as the current one. This method is easy to implement and fast at execution time, but is somehow crude and could be eventually improved once a more final computation of the field will be available.

RESULTS

The dependence of the field components on z for a fixed r value or on r for a fixed z value is shown in Fig. 2-1.

Fig. 2-1 Axial and radial components as function of Z and R

In order to test the consistency of the procedure the GUFLDNEW field has been computed in the points of the original ANSYS mesh, and compared to the ANSYS values.

The comparison is shown in Fig. 2-2 for nodes in "air" and in Fig. 2-3 for nodes in "iron".

Fig. 2-2 ANSYS nodes in air
Fig. 2-3 ANSYS nodes in the iron

In each figure three plots are presented: in the first the dots represent the position of all the ANSYS nodes, in the second one only the nodes where the absolute value of the difference between the axial components as given by ANSYS and found by GUFLDNEW is more than 0.3 kgauss (about 2% of the field in the central region of the coil) and the third one the same but for the radial components.

It is clear that the nodes with a high difference are relatively frequent, but they are essentially localized outside of the coil and tipically in regions where the field is highly nonuniform.

In order to get a quantitative estimate of the differences between ANSYS and GUFLDNEW, the respective histograms are presented in Fig. 2-4 for the coil region (fig. 2-4-a concerns the radial component, 2-4-c the axial one, while 2-4-b and 2-4-d are restricted to the drift volume). Figs. 2-5, Fig. 2-6 and Fig. 2-7 show similar distributions, separately for the nodes in iron and in air, respectively for nodes in the "barrel", in the "backward" and "forward" regions. It appears evident that, although in a limited number of points, there are difference much higher than desirable. As a consequence we assume that the next generation of a field routine, that will follow the final field calculation, will require a more refined treatment than the present one, at least in the critical zones. As possible solution to this problem we can foresee a more fine spacing of the grid, at least where it is necessary, and some controlled interpolation procedure in the zones where iron and air are intermixed.

Fig. 2-4 [Delta]B_R and [Delta]B_Z in the coil
Fig. 2-5 [Delta]B_R and [Delta]B_Z in the barrel
Fig. 2-6 [Delta]B_R and [Delta]B_Z in the backward region
Fig. 2-7 [Delta]B_R and [Delta]B_Z in the forward region