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. (C) 2013 Elsevier B.V. All rights reserved.