**3D Modeling in Gui_plop**

Gui_plop now supports 3D finite element modeling to a limited extent. Starting with release 3.0.0, gui_plop can generate 3D modeling files for the FEM program Z88. You can download Z88 separately from their home page here or at sourceforge.net, but the gui_plop distribution includes slightly modified versions of the Z88 programs you will need. These are z88i1plop and z88i2plop which do the preprocessing and solution of the 3D FEM problem. The only reason that they are modified is to look in the windows temp directory for the data, and to run without requring you to click on any buttons.

The original version of gui_plop used the plate program written by Toshimi Taki. To use this in Plop, I rewrote parts of it to dramatically speed it up so that it could be used for optimization, which can require hundreds of FEM runs. However, plate, as its name implies, uses a plate model of the mirror. That is, it models the mirror as a collection of 2 dimensional plates and can calculate the deflection in the third dimension. This has a couple of limitations in that (1) it can’t model the spreading out of the support force from the back to the front, and (2) it can’t model the sideways forces and support force imbalance when the mirror is tilted.

To see the effect of the force spreading, compare a 2D plate model with a 3D. On the left side we see the result from a force exterted in a plate using 2D modeling. Since it is a 2D modelt the deflection at the bottom of the mirror is propagated through to the top. This is unrealistic as obviously you won’t have a perfect point deflection at the top.

A 3D modeling program, as the name implies, divides the object into 3D pieces and can model the distortion of the object in all 3 dimensions. So as represented below, it can calculate the compression of the mirror along the Z axis and show how the mirror bends and the deflection is spread across the surface of the mirror.

Here is a visual comparison of a 3 point cell using plate and z88. First, the 2D plate model.

Here is the z88 model

You can see that the error is reduced about 10%. You can also see the effect of smearing out the peaks as desribed above.

With an 18 point cell and the supports so close together that the peaks dominate the error, the effect is more pronounced. Here is a 2D model of the 18 point cell and the 3D model.

2D Plate Model:

3D model:

The error is halved due to modeling the spreading effect better.

The other important effect that cannot be modeled using 2D models is off axis.
When the telescope is tilted from zenith, the forces on the cell become
unequal. The exact way that they are redistributed depends on the method used
to keep the mirror on the cell. The two popular methods are a sling, and to
simply glue the mirror to the cell using a compliant adhesive such as silicone.

Gui_plop supports modeling the mirror using both of these methods. However there are certain assumptions and approximations that must be made and we describe them here. We’ll just use a simple 2D cross section of a mirror to explain.

First, when a mirror is aimed at zenith, all the force is along the z axis and the mirror cell acts with the force balance as decribed in your cell design. Here gravity exerts a downward force through the center of gravity of the mirror and the cell exerts a compensating upwards force.

When the mirror is tilted, the gravitational force is no longer on-axis and some compensating forces are applied. In the case of gluing the mirror to the supports, we assume that each support has a contribution proportional to its Z axis load when aimed at zenith. Then the Z forces are adjusted so that the mirror is balanced. In the example below, the vertical force on the left support is larger to compensate for the torque induced by the off-axis gravitational support.

In the case of a sling, force is assumed to be applied along the edge of the mirror. There is still some very small imbalance in the support force, only due to the fact that we assume that the sling applies equal force along the height of the mirror, while the sagitta of the mirror means that the center of gravity along Z is not exactly in the middle. Therefore there is still a small imbalance in the mirror cell support forces.

The sling is defined by the angle that it spans around the edge of the mirror. The sling is assumed to have no friction against the edge of the mirror so it applies an equal force along the part of the edge that it supports.

Now lets compare the effects of the two approaches on the 18 point cell shown above. First the glued cell:

The error is now more than 30 times larger, and is now about 1/7 wave P-V. With a sling and a 180 degree included angle the error is dramatically reduced.