We will present two approaches to the decomposition of a large matrix inequality into several smaller ones with the goal to efficiently use existing SDP solvers. The approaches will be demonstrated on an SDP problem arising in topology optimization of mechanical structures. The first, well-known, technique is based on the decomposition of chordal graphs. The second one, motivated by the structure of the underlying PDE and its discretization, is applicable to arrow matrices satisfying certain conditions. We will show that the second approach can be considered a sparse low-rank version of the first one and that it leads to a significantly more efficient algorithm.