Exact analytical theory of topology optimization with some pre-existing members or elements

This note deals with topological optimization of structures in which some members or elements of given cross-section exist prior to design and new members are to be added to the system. Existing members are costless, but new members and additions to the cross-section of existing members have a non-zero cost. The added weight is minimized for given behavioural constraints. The proposed analytical theory is illustrated with examples of least-weight (Michell) trusses having (a) stress or compliance constraints, (b) one loading condition and (c) some pre-existing members. Different permissible stresses in tension and compression are also considered. The proposed theory is also confirmed by finite element (FE)-based numerical solutions.