We present a numerical approach able to capture the dependence of the optimal shape profiles of thermal fins on the conductivity parameters. We consider the two-dimensional cross-section of a periodic array of fins and involve the third dimension via the thermal boundary layer. The highly conductive fins converge to "sharp-pointed", narrow base shapes while the low conductivity ones prefer blunted, wide base fins. The optimal shapes we obtain are similar to the shapes of intestinal villi and stegosaurus plates. A meshfree method, coupled with a gradient-based optimization algorithm, is used to handle the significant shape changes from a simple, generic initial guess to the final, optimal shape. We reach the optimal shapes without remeshing. (C) 2004 Elsevier Ltd. All rights reserved.