An exact volume constraint method for topology optimization via reaction–diffusion equation

Abstract

A novel approach is proposed to determine the exact Lagrangian multiplier Λ for the volume constraint of topology optimization via the reaction–diffusion equation (RDE). Such Λ enables an exact volume constraint. The mathematical approach involves splitting the density function (or level-set function) and their RDEs into two parts. Both become independent of Λ, and the exact value of Λ can then be determined by superposition to satisfy the constraint at the current step. Because only a few iterations are required on average, our proposed volume constraint method is computationally inexpensive. In the existing volume constraint method (the augmented Lagrangian method for constrained problems), its Λ is determined from the previous optimization step, and hence the volume constraint at the current step is inexact. Such inexactness leads to a large fluctuation in the as-constrained volume fraction, which can jeopardize the appearance of minor geometrical features and hence alter the topology of the optimized structure. The exactness of our proposed volume constraint method can ensure convergence for the minimum compliance problem by only reconstructing and remeshing the material domain, whereas the existing volume constraint method fails to do so. The proposed exact volume constraint during the entire optimization process differs from merely adjusting the volume by choosing a level set other than zero. The latter is merely an engineering trick that neither affects the convergence nor corresponds to the obtained objective functional.