With the release of version 6.3 of the COMSOL Multiphysics® software, there is a new Interior Contact boundary condition available in the Solid Mechanics interface. This boundary condition simplifies the workflow when setting up a structural mechanics problem, and it has advantages in multiphysics modeling, especially when solving for electromagnetic fields. Let’s take a look at the new workflow and the advantages of using this boundary condition in the context of computing the electrical contact at a bolted connection between AC busbars.
Contact Conditions of a Bolted Busbar Model
The situation that we will consider is a bolted connection between two copper busbars carrying 1 kA at 60 Hz. The bolt is made of steel and is tightened so that there is high contact pressure at the mating surface between the busbars. This contact pressure reduces the electrical contact resistance between the two pieces of copper, so the current will tend to flow primarily through the contact area. However, when AC current flows through a conductor, it is forced toward the outside boundaries of the conductor due to the skin effect. These two phenomena act in opposition, and it is this behavior that we want to capture with our model.
A bolted connection between two copper busbars. The electrical resistance to current flow (blue arrows) depends upon how much the bolt is tightened. The symmetry plane is highlighted in gray.
The Structural Modeling Problem
The approach that we will take begins with the assumption that the relative displacement between the two busbars and the bolt is not significant. That is, we assume that the structural or electrical behavior is only affected by the contact pressure and not by the relative motion of the contacting surfaces. Under this assumption, it is possible to treat this as a geometrically linear problem, and, hence, we do not need to consider the change in shape or orientation of the domains. This assumption means that we are within the regime where the Interior Contact boundary condition can be used.
The usability advantage of the Interior Contact condition lies in that it can be used in combination with the Form Union geometry finalization step method, rather than the Form Assembly method. As a consequence of using the Form Union method, the mesh itself is always contiguous across all boundaries, even though the computed displacement field can be discontinuous across the boundaries. The advantage of this approach is that the computational expense of the contact search algorithm is avoided. It is still desirable to refine the mesh in this contact area since we do want to get a good resolution of the stresses at the interface.
Image of the mesh used in the contact area between the two busbars.
Note: If you instead use the Form Assembly geometry finalization method, it will automatically recognize all mating faces and create so-called contact pairs. However, this workflow does have additional setup and solution costs. The benefit of the Form Assembly approach is that it allows for arbitrary sliding and large relative deformations.
Learn more in our Learning Center article “Structural Contact Modeling Guidelines”.
In addition to the Interior Contact boundary condition, the Solid Mechanics interface includes a Bolt Pretension feature, which we will apply to a simplified through bolt geometry in our model. There are a number of different ways in which such bolted connections can be modeled, and the approach used here assumes continuity of the fields between the bolt head, the nut, and the busbars. The model also includes a Thermal Expansion feature (a subnode of the Linear Elastic Material node) that accounts for stresses that arise due to the mismatch between the thermal expansion coefficients of the steel bolt and copper busbars. The assumption in this case is that the assembly is isothermal, a reasonable assumption under many operating conditions since copper is a very good thermal conductor.
The case under consideration can be further simplified by exploiting symmetry about the center plane. Solving first for the bolt pretension and then the resultant deformations and stresses lets us visualize the contact pressure. As expected, this is centered around the bolt and drops off in magnitude. It is this contact pressure that will affect the electrical resistance between the busbars, and we will next incorporate this phenomenon into the electromagnetic model.
Visualization of the magnitude of the contact pressure around the bolt.
The Electromagnetic Modeling Problem
We are interested here specifically in the skin effect, or induced currents, around the contact area that arise due to the AC excitation. This type of analysis requires using the Magnetic and Electric Fields interface, which includes an Electrical Contact boundary condition that can model the resistive loss at a boundary between conductors. This boundary condition is applied as a subnode of the Magnetic Continuity boundary condition, which enforces continuity of the magnetic field. Both the magnetic field and the current will be continuous across the boundary, but there will be an electric field across the boundary due to the contact resistance. This resistance can be computed from either the Cooper–Mikic–Yovanovich correlation or the Mikic elastic correlation, both of which take as inputs the contact pressure computations from the Internal Contact feature of the Solid Mechanics interface.
When using the Magnetic Continuity boundary condition, it is necessary that any adjacent boundaries within the modeling domain also have the Magnetic Continuity boundary condition applied. That is, there can be no free edges of the Magnetic Continuity condition within the modeled space. For the situation being modeled here, this means that all boundaries between the conductors and the air have the Magnetic Continuity condition applied using its Electric Insulation subnode. This condition enforces that there can be no current flow — neither conduction current nor displacement (capacitive) current — across the boundaries from the conductor into the air.
Illustration highlighting the faces where the Contact Impedance (magenta) and Electric Insulation (cyan) boundary conditions are applied. There are no free edges of these boundaries within the modeling space.
The exterior boundaries of the modeling space are modeled via combination of a Perfect Magnetic Conductor to enforce the symmetry condition and the Magnetic Insulation condition, with Ground, Electric Insulation, and Terminal subnodes added to excite current flow through the busbar assembly. The electromagnetic problem is solved in the frequency domain in a subsequent study step after solving the structural problem, thus exploiting the assumption of one-way coupling from the structural–thermal model to the electrical model.
Screenshot of the model setup. The Contact pressure in the settings of the Electrical Contact feature subnode of the Magnetic Continuity boundary condition is computed using the output of the Interior Contact in the Solid Mechanics interface.
The losses at the surface can be plotted to show the competing effects of the contact resistance being lower close to the bolt but the current wanting to flow near the exterior boundaries of the busbars.
Plot of the electromagnetic interface losses, highlighting the competing effects of the lowered contact resistance near the center and the skin effect driving the current away from the center.
A streamline plot of the current flow through the assembly also highlights this skin effect, along with the pinching of the current flow, in the contact area.
Streamline plot of the electric currents highlighting the pinching of the current flow.
Faster, Easier Modeling of Bolted Connections
The new Contact boundary condition in the Solid Mechanics interface allows you to quickly and easily model situations where there is no significant relative movement between the contacting faces, as is commonly the case around bolted connections. This condition can be used with the Form Union geometry finalization method and thus allows for a contiguous mesh at boundaries between parts. This both speeds up convergence and lets us easily add in other physics — using, for example, the Magnetic and Electric Fields interface — that require contiguous meshes. This combination is useful in the modeling of electrical contact at bolted connections and can be used in many other situations as well.
To gain hands-on experience with the new boundary condition and the model discussed in this blog post, click the button below.
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