The optimum anisotropy in problems of heat conduction in solids

Some problems of optimizing the internal structure of solids, made of a material which is locally orthotropic with respect to the heat-conducting properties, are formulated. The state variable (the inverse temperature) is determined from the solution of the boundary value problem of heat conduction. The orthogonal rotation tensor, which defines the optimum orientation of the orthotropy axes of the material that delivers an extremum to the dissipation functional, is used as the control variable. The necessary conditions for an extremum are derived and some properties of the equations defining the optimal structures are investigated. Examples are given of the solution of problems of the optimum arrangement of the orthotropic material, and the possibility of effectively using the membrane analogy for this purpose is pointed out.