Topology optimization in B-spline space

In this paper, we present a new form of density based topology optimization where the design space is restricted to the B-spline space. An arbitrarily shaped design domain is embedded into a rectangular domain in which tensor-product B-splines are used to represent the density field. We show that, with proper choice of B-spline degrees and knot spans, the B-spline design space is free from checkerboards without extraneous filtering or penalty. We further reveal that the B-spline representation provides an intrinsic filter for topology optimization where the filter size is controlled by B-spline degrees and knot spans. This B-spline filter is effective in removing numerical artifacts and controlling minimal feature length in optimized structures when the B-spline basis functions span multiple analysis elements. We demonstrate that the B-spline filter is linear in storage cost and does not require neighboring element information. Further, this B-spline based density representation decouples the design representation of density distribution from the finite element mesh thus multi-resolution designs can be obtained without re-meshing the design domain. In particular, successive optimization with respect to design resolutions leads to topologically simple features obtainable in either coarse or fine design resolutions, thus achieving a form of mesh independency with respect to design representation. This approach is versatile in the sense a variety of finite element and isogeometric analysis techniques can be used for solution of equilibrium equations and a variety of projection methods can be used to approximate B-spline density in analysis. Numerical studies have been conducted over several representative topology optimization problems, including minimal compliance of MBB beams, compliant mechanism inverters, and heat conductions.