Heat conduction in two-dimensional parquets and optimization of spine shape

By the methods of complex analysis media composed of double- periodic phases with different (arbitrary) conductivity are considered. Within each phase temperature is a harmonic function and along interfaces two conjugation conditions are rigorously satisfied (continuity of temperature and normal flux component) as well as periodicity conditions along the boundaries of elementary cells. For specific examples of composites (lensed structure and chequer-board) explicit analytic solutions are derived in terms of the thermogradients. The effective conductivities are calculated. For the chequer-board composite the hodograph of effective conductivity absolute value is shown to coincide exactly with an ellipse. As a limiting case of parquets developed surfaces are studied. In the class of semi-ellipsoidal spines two non-trivial local extrema of the total flux exist alongside two global ones. Both for two-dimensional and three-dimensional protrusions these extrema appear if the conductivity ratio exceeds some critical value.