PDE-based regularization of shape optimization is inspired by the Helmholtz filter introduced for topology optimization; see Ref. 1 and Ref. 2. The shape deformation of the boundary is thus typically given by

where the maximum displacement, dmax, and filter length, Rmin, effectively define a maximum slope of deformation. Boundary conditions can be added to fix the boundary deformation or allow sliding along planar boundaries. If the initial normal vector is orthogonal to the boundary, this property can be preserved by imposing d · n = 0 on the edge. If the orthogonality should not be preserved, one can solve a Helmholtz filter on the edge and impose the result as a Dirichlet boundary condition.

If different values for dmax are present for boundaries that share an edge, the lower value of dmax will be used. The same logic is used when one boundary is regularized with a PDE while another is regularized with a polynomial, except in 3D where the polynomial boundary will always control the edge.