Products: ABAQUS/Standard ABAQUS/Explicit
ABAQUS/Standard allows heat flow across an interface via conduction or radiation. Generally, both modes of heat transfer are present to some degree. Regardless of the mode, heat transfer across the interface is assumed to occur only in the normal direction.
Heat conduction across the interface is assumed to be defined by
is the heat flux per unit area crossing the interface from point 
on one surface to point 
on the other, 
and 
are the temperatures of the points on the surfaces, and 
is the gap conductance.The derivatives of are
The heat flow per unit area between corresponding points is assumed to be given by
is the value of absolute zero temperature on the temperature scale being used; is the heat flux per unit surface area crossing the gap at this point, from surface to surface ; and are the temperatures of the two surfaces; and 
is the gap radiation constant derived from the emissivities of the two surfaces.The derivatives of are
The contribution to the variational statement of thermal equilibrium is
is the area. The contribution to the Jacobian matrix for the Newton solution is 
For “tied” thermal contact the temperature at point is constrained to have the same temperature as point . The Lagrange multiplier method is used to impose the constraint by augmenting the thermal equilibrium statement as follows:
is the Lagrange multiplier. The contribution to the Jacobian matrix for the Newton solution is 