An optimization approach for constructing trivariate BB-spline solids

• Automatically construct a trivariate tensor-product BB-spline solid.
• The minimal Jacobian of the resulting solid is positive.
• A volumetric functional is minimized to improve parametrization quality.
• Deformation, constraint aggregation, and divide-and-conquer techniques combined.

In this paper, we present an approach that automatically constructs a trivariate tensor-product BB-spline solid via a gradient-based optimization approach. Given six boundary BB-spline surfaces for a solid, this approach finds the internal control points so that the resulting trivariate BB-spline solid is valid in the sense the minimal Jacobian of the solid is positive. It further minimizes a volumetric functional to improve resulting parametrization quality.For a trivariate BB-spline solid even with moderate shape complexity, direct optimization of the Jacobian of the BB-spline solid is computationally prohibitive since it would involve thousands of design variables and hundreds of thousands of constraints. We developed several techniques to address this challenge. First, we develop initialization methods that can rapidly generate initial parametrization that are valid or near-valid. We then use a divide-and-conquer approach to partition the large optimization problem into a set of separable sub-problems. For each sub-problem, we group the BB-spline coefficients of the Jacobian determinant into different blocks and make one constraint for each block of coefficients. This is achieved by taking an aggregate function, the Kreisselmeier–Steinhauser function value of the elements in each block. With block aggregation, it reduces the dimension of the problem dramatically. In order to further reduce the computing time at each iteration, a hierarchical optimization approach is used where the input boundary surfaces are coarsened to difference levels. We optimize the distribution of internal control points for the coarse representation first, then use the result as initial parametrization for optimization at the next level. The resulting parametrization can then be further optimized to improve the mesh quality.Optimized trivariate parametrization from various boundary surfaces and the corresponding parametrization metric are given to illustrate the effectiveness of the approach.