In this paper we consider the geometrical optimization of a complex cavity, namely a T-Y-shaped cavity with two additional lateral intrusions into a solid conducting wall. The assemblies of cavities were cooled by a steady convective stream while the solid generates heat uniformly and it is insulated on the external perimeter. Two sets of geometrical configurations, having different displacement of the lateral cavities, named "Design 1" and "Design 2", have been numerically investigated. The objective is to minimize the global thermal resistance between the solid and the complex assembly of cavities. The basic idea is to give the assembly more freedom to morph by increasing its degrees of freedom: it is expected, according to the constructal design, that an increase of the geometrical complexity will lead to an augmentation of the thermal performance. The analysis demonstrated that not all degrees of freedom play the same role in the performance of the complex cavity design: an improvement of approximately 45% has been observed after the optimization of the ratio H-3/L-3 while the optimization of the parameters H-1/L-1 and H-2/L-2 led to an additional gain of only 1%. Design 2's performance proofed to be superior to Design l's performance, because its lateral cavities are able to penetrate more efficiently into the solid body. Furthermore, Design 2's maximal excess temperature is approximately half of the corresponding dimensionless maximal excess temperature of the basic T-Y assembly, i.e., a cavity shaped as T and Y without lateral intrusions. (C) 2011 Elsevier Ltd. All rights reserved.