  This paper explores the electrical and thermal conductivity of complex
contact spots on the surface of a half-space. Employing an in-house Fast
Boundary Element Method implementation, various complex geometries were
studied. Our investigation begins with annulus contact spots to assess the
impact of connectedness. We then study shape effects on "multi-petal" spots
exhibiting dihedral symmetry, resembling flowers, stars, and gears. The
analysis culminates with self-affine shapes, representing a multi-scale
generalization of the multi-petal forms. In each case, we introduce appropriate
normalizations and develop phenomenological models. For multi-petal shapes, our
model relies on a single geometric parameter: the normalized number of
"petals". This approach inspired the form of the phenomenological model for
self-affine spots, which maintains physical consistency and relies on four
geometric characteristics: standard deviation, second spectral moment, Nayak
parameter, and Hurst exponent. As a by product, these models enabled us to
suggest flux estimations for an infinite number of petals and the fractal
limit. This study represents an initial step into understanding the
conductivity of complex contact interfaces, which commonly occur in the contact
of rough surfaces.
