Using the Thermal Module of Pro/MECHANICA to Optimize a 3D Electrostatic Problem

by Johannes Tredoux, ABB Trasmissione & Distribuzione

We can take advantage of the fact the governing equations for heat conduction and electrostatics are the same, using the Thermal module of Pro/MECHANICA to solve basic electrostatic di-electric problems (see “Solving 2D Electrostatic Problems Using the Thermal Module of Pro/MECHANICA,” Winter 2003). Even 3D problems can be solved quite easily in a simple and straightforward way. Furthermore, we can use the powerful functionality in Pro/MECHANICA to optimize the geometry, thus improving from the normal “merely acceptable” to a “near optimum” result.

Brief Explanation of the Theory

Heat conduction can be described by the following differential equation:

where:

 is the conductivity,

 is the Laplacian operator,

 is the temperature, and

 is the internal heat generated (source or sink).

For basic electrostatics, we have the same governing equation, where:

 is the relative permittivity,

 is the field potential (voltage), and

This differential equation can be solved by the finite element method and implemented in the Pro/MECHANICA Thermal module.

Getting Started

When designing parts in a di-electric environment, we usually have to find a geometry that ensures the voltage gradient is below some acceptable limit. Quite a large range of geometric variations would satisfy these limits, and most are not close to the optimum. If we could use the optimum, however, the overall dimensions would likely be smaller, saving space and cost. Even if the dimensions were the same size, the safety margin could be improved.

The following example shows how to use Pro/MECHANICA Wildfire 2.0 to find the optimum geometry in a three-dimensional di-electric problem. We will find the ideal dimensions (inner diameter and ring diameter) of a torus, with a rod (of given diameter) passing through its center, enclosed in a metal duct (of given inner diameter) as shown below. In this case, “ideal” means the geometry that gives the lowest maximum voltage gradient over the whole model.

Building the Geometry in Pro/ENGINEER

For the analysis in Pro/MECHANICA, we need 3D geometry. When working in 3D, it is wise to keep the model as simple as possible to save computation time and resources. Symmetry should be exploited whenever possible. At the same time, the model must be robust enough so that the parameters can vary within the specified ranges without causing regeneration failures. Careful planning during this stage can save a lot of time later.

In this example, everything is enclosed inside the duct, so we need to model only the metal parts that are inside, plus the volume of the gas enclosed by the duct. And since there are three symmetry planes, we have to model only one-eighth of the full geometry.

1. Create the parts.

The gas volume is modeled as a quarter of a cylinder, with an outer diameter equal to the inner diameter of the metal duct. The length is less important in this case, but it must be long enough so that the free end is far enough to have no influence on the field around the torus.

The torus should be modeled in such a way that we have dimensions for the inner diameter and the ring diameter, since these values will be varied to find the optimum geometry.

The rod is modeled as a cylinder, long enough to pass through the duct.

2. Create an assembly.

The gas volume, torus, and rod are assembled as shown here.

Because we need to vary the Pro/ENGINEER parameters during the optimization analysis, we must create two parameters in the assembly and assign them to the required part dimensions, using relations.

Whlie not essential, we can assign the current values of these dimensions to the parameters.

We need to write relations so that the part dimensions are driven by the two parameters as follows.

In this case, we must also make sure that the outer diameter of the torus is always smaller than the duct inner diameter, so we need a relation to define the outer diameter.

To cut out all the parts from the gas volume, choose Edit, Component Operations.

Select the part (gas volume) from which it will be cut and click OK. Then select all parts (torus and rod) that will be used to cut the first part. Choose Reference as the type and confirm for each part.

3. Create a simplified representation.

We need to create a simplified representation with only the non-conducting parts (the gas volume in this case), since this is where the voltage field is distributed. All conducting parts should be excluded because they will be represented with assigned boundary conditions. (An exception to this is when there is a conducting part with a “floating” or unknown voltage.)

Create a new simplified representation (DE in this case). Exclude all metal parts. This representation can also be redefined using Edit, Redefine.

Make sure it is active by right-clicking to get the pop-up menu.

The geometry looks like this, and we are now ready to enter Pro/MECHANICA.

Analysis in Pro/MECHANICA

Enter Pro/MECHANICA under the Applications menu.

Make sure the units are correct. I always work in [N,mm,s,C], setting one degree Celsius to one kilovolt and getting the results in kV/mm.

Now enter the Thermal module of Pro/MECHANICA.

1. Define the model.

We need to apply a constant temperature (voltage) to all the boundary surfaces where the metal parts have been used to cut the gas volume.

In this case, we assign a value of 100°C (100 kV) to the surface that represents the torus.

To the surfaces representing the duct inner surface and the rod outer surface, we assign a value of 0 since they are at ground potential.

We have to define some design controls that will be used to change the geometry during the optimization study. Under the Analysis menu, choose Mechanica Design Controls…

Now define two parameters “D_RING” and “D_INNER” and assign the two previously defined Pro/ENGINEER parameters to them, setting their minimum and maximum values as follows.

To be able to check that the outer diameter of the torus stays inside the duct during the optimization study, we define a measure “OD” as follows.

Next we define a new di-electric material (the gas volume) for the non-conducting material in the model.

Since the relative permittivity is equivalent to the thermal conductivity, we only need to fill this value into the “Thermal conductivity” box on the Thermal tab. For example, for gases and vacuum, this is 1.  Leave the units of the conductivity as [N/(sec) C].

Pro/MECHANICA also requires numerical values in the “Specific Heat Capacity” box, the “Density” box, and in the “Young's Modulus” box (on the Structure tab). For purposes here, just set them equal to 1.

Assign the material to the gas volume.

Keep in mind that we are mainly interested in the voltage (temperature) gradient distribution, which is a derivative of the voltage (temperature) distribution. To ensure a smooth and accurate distribution, it is therefore important to have a very good mesh in the important areas and that the analysis converges properly.

So, before creating the mesh, set more desirable settings by selecting AutoGEM, Settings…

Under the Settings tab, it also helps to select “Modify or Delete Existing Elements.”

On the Limits tab, we need to change the default settings so that the created elements have better proportions, but without setting the limits too tight and causing other problems (such as extremely long meshing and solving times). This is always a compromise. When working in 3D, it is usually a good idea to start with values close to the default settings and then adjust them as necessary. The simpler the model, the better these values can be without running into problems. Good starting values are shown below.

2. Define the analysis.

First we define a new steady state thermal analysis, which will be used by the optimization study.

To ensure the analysis converges well, we set it up as follows. The convergence method should be “Multi-Pass Adaptive,”  starting with a polynomial order of 3 and setting the maximum to 9.  Normally it should not go above 6 or so. (If goes up to 9, this is usually a sign that something is wrong with the model.)  We also need to converge on the measures that are important—the energy norm and the maximum gradient measures. The percentage convergence should be set to a very small value like 0,25% to get repeatable results with different meshes.

3. Define the design study.

Now we create a new design study.

Select “Optimization” as the type and define the goal to find the minimum of the “max_grad_mag” (maximum gradient magnitude). We assign a limit of 480mm to the outer diameter of the torus, and set the parameters “D_INNER” and “D_RING” to vary between the minimums and maximums defined earlier. Starting values for the parameters can be anything between the minimum and maximum. Normally, it is best to run a few studies with different starting values to make sure the absolute (not just a local) minimum is found. Make sure “Repeat P-Loop Convergence” is selected.

4. Run the design study.

Make sure the correct study is selected and click on the green flag to start the run.

While it is running, you can watch the progress by selecting the “Display study status” button.

5. Review results.

Looking at the “Run Status” report, we see that the optimum has been found.

Keep in mind that there are small inherent inaccuracies in the finite element method. Even so, we can come very close to the real optimum. It is good practice to confirm any results with other means if possible. In this example, a few global sensitivity studies could be run, varying the parameters around the optimum values.

Note: It may be necessary to set the following environment variable if your optimization studies keep failing for no logical reason.

Johannes Tredoux is an analysis engineer with ABB Trasmissione & Distribuzione S.p.A. in Lodi, Italy. He can be reached via e-mail at Johannes.Tredoux@it.abb.com.