The SRV constraint for 0/1 topological design

In density-based topological design, 0/1 solutions are often sought, that is, one expects that the final design includes either elements with full material or no material, excluding grey areas. The accepted technique for achieving binary values for the densities is to use a solid isotropic microstructure with penalization (SIMP) material for which Young's modulus is a polynomial function of the otherwise continuous relative densities. This approach indeed enhances 0/1 solutions in a significant manner and as such it has achieved prominent status in topological design. Nevertheless, this paper proposes a possible alternative to the SIMP methodology for generating 0/1 structures. The design variables are still the densities of the finite elements but Young's modulus is a linear function of these densities (in some sense, a SIMP material without penalty). In order to drive the solution to a 0/1 layout a new constraint, labeled the sum of the reciprocal variables (SRV), is introduced. The constraint stipulates that the SRV must be larger or equal to its value at a discrete design for a specified amount of material. It is understood that this implies a minimum gage on the design variables, a provision which is also present in the standard fixed-grid formulation to avoid singular stiffness matrices. The technique turned out to be very effective in conjunction with the method of moving asymptotes (MMA) when using topological design methods for finding optimal layouts of patches of piezo-electric (PZT) material in order to minimize the mechanical noise emanating from vibrating surfaces. It also performed satisfactorily in classical structural topological design instances, as can be seen in the numerical examples that illustrate this work.