3.5. Interface Conditions

In Fig. 3.1 an example of an interface between two subdomains of a problem was demonstrated. When the properties of these two subdomains are different, Contact or an interface boundary condition can be used to describe the abrupt change in conditions. At the interface between two subdomains $\symDomain_i$ and $\symDomain_j$ , we assume a zero jump in the flux of a quantity in the direction normal to the interface,

(3.97) $\ljump q_n \rjump = 0$

The follow section provides some examples of treatment of the conserved quantity at the interface, following this assumption.

It is worth noting that interface conditions can be applied in one of two ways, depending on the interface. If the mesh is contiguous at an interface (i.e. nodes are shared by the blocks connected to the interface), an interface boundary condition is sufficient. However if the nodes are not shared (as is demonstrated in Fig. 3.8), a contact definition must be made at the interface (see Contact). For example in the sketch below, one would need contact for a jump between block_2 and block_3, but only an interface condition for a jump between block_1 and block_2

            |
            o
  block_1   |
            |
--o----o----o o----o----o---
            | |
  block_2   | |  block_3
            o o
            | |

Note that this section is not a comprehensive list of all contact enforcement conditions available in Aria, but is rather a subset of representative contact enforcement conditions that are commonly used. Details on what a specific boundary condition applies can be found in its command summary. Similarly, refer to Contact for information on how to setup a contact definition.

3.5.1. Perfect or Tied Contact

The first and most common condition is to enforce continuity of a conserved quantity $b$ at the interface,

(3.98) $b\left|_{\partial\Omega_{i}} - b\right|_{\partial\Omega_{j}} = 0,$

where the notation $b\left.\right|_{\partial\Omega_i}$ indicates that the temperature is to be evaluated on the surface $\partial\Omega_i$ , which is associated with the subdomain $\Omega_{i}$ .

3.5.2. Contact Resistance

The second condition allows for a model of imperfect contact between two surfaces, which can take account of surface roughness. The physical meaning of this contact resistance depends based on the equation it is applied to. In general, though, this model accounts for an imperfect mating of the two contact surfaces causing effective flux of the conserved quantity to occur only on a subset of the contacting surface.

We treat this contact by setting the “gap” flux across the interface proportional to the drop in the conserved quantity,

(3.99) $\left. q_n \right|_{\partial\Omega_{i}} - \left. q_n \right|_{\partial\Omega_{j}} = h_c \left( b\left|_{\partial\Omega_{i}} - b\right|_{\partial\Omega_{j}} \right),$

where the constant $h_c$ is the contact conductance, (analogous to a heat transfer coefficient, for a thermal problem). The value of contact conductance is dependent upon the following 1 :

values of $b$ of the two materials at the contact surface;
the materials in contact;
surface finish and cleanliness;
pressure at which the surfaces are forced together;
the substance, or lack of it, in the interstitial spaces.

Footnotes

1: More on the theory of contact conductance, and the values of conductances for typical surface finishes and moderate contact pressures can be found in [15, 16, 17]