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We present an extension of the projection method proposed by Challis et al.(Int J Solids Struct 45(14$\unicode{x2013}$15):4130$\unicode{x2013}$4146, 2008)for constrained level set-based topology optimisation that harnesses theHilbertian velocity extension-regularisation framework. Our Hilbertianprojection method chooses a normal velocity for the level set function as alinear combination of (1) an orthogonal projection operator applied to theextended optimisation objective shape sensitivity and (2) a weighted sum oforthogonal basis functions for the extended constraint shape sensitivities.This combination aims for the best possible first-order improvement of theoptimisation objective in addition to first-order improvement of theconstraints. Our formulation utilising basis orthogonalisation naturallyhandles linearly dependent constraint shape sensitivities. Furthermore, use ofthe Hilbertian extension-regularisation framework ensures that the resultingnormal velocity is extended away from the boundary and enriched with additionalregularity. Our approach is generally applicable to any topology optimisationproblem to be solved in the level set framework. We consider several benchmarkconstrained microstructure optimisation problems and demonstrate that ourmethod is effective with little-to-no parameter tuning. We also find that ourmethod performs well when compared to a Hilbertian sequential linearprogramming method.

A Hilbertian projection method for constrained level set-based topology optimisation