Boundary layer in shape optimization of convective fins using a meshfree approach

We show how shape optimization of fin arrays for increased heat flux through the base under area constraint leads to non-existence of optimal solutions. An additional constraint in terms of the boundary layer eliminates the apparent paradox. We consider a variable heat transfer coefficient and we use a fixed-point iteration scheme to solve the resulting non-linear boundary value problem for the steadystate heat operator with temperature, flux, and convection boundary conditions. We propose a simple yet effective algorithm for evaluating the boundary layer constraint and eliminating the constraint violation. There are large shape changes between the initial and final design but no remeshing is required because we use a meshfree method that is not sensitive to shape distortion of integration cells as long as they remain convex. The resulting optimal unit cell is repeated by periodicity to produce the optimal fin array. The obtained shapes display similarities to shapes seen in natural systems governed by diffusion/convection and conduction processes. A length-scale for the unit cell is naturally introduced by the non-overlap condition imposed on the thermal boundary layer in the cooling ambient fluid