On a cellular developmental method for layout optimization via the two-point topological derivative

This paper introduces a two-point topological derivative for elasticity and a cellular developmental method for layout optimization of structures. The two-point derivative measures the change on a functional due to the addition of a thin ligament between parts of the domain. It contrasts with the existing single-point topological derivatives that measure the change of a functional due to the subtraction of material. Based on the two-point topological derivative, a cellular developmental method is presented, which builds the layout of a structure in a sequence of cellular divisions and cellular dynamics developmental cycles. The methodology introduced in this work can easily account for member sizes and angle constraints among members of the frame, and the optimization can be performed in the free space or in a bounded domain. Benchmark problems are solved to demonstrate the capability of the cellular developmental method to solve optimization problems in structural mechanics.