Abstract In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries. Instead of the computational domain bounded by four B-spline curves, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including Bezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior Bezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C 1 ∕ G 1 -continuity constraints on the interface of neighboring Bezier patches with respect to each quad in the quadrangulation, the high-quality Bezier patch parameterization is obtained by a local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach.
: A variational approach for constructing NURBS parameterizations of swept volumes is proposed by M. @cite_21 . proposed a constrained optimization framework to construct analysis-suitable volume parameterizations @cite_46 . Spline volume faring is proposed by Pettersen and Skytt to obtain high-quality volume parameterization @cite_4 . The construction of conformal solid T-splines from boundary T-spline representations is studied by using octree structure and boundary offset @cite_2 . In @cite_29 , a variational harmonic method is proposed to construct analysis-suitable parameterizations of computational domains from given CAD boundary information. Wang and Qian proposed an efficient method by combining divide-and-conquer, constraint aggregation and the hierarchical optimization technique to obtain valid trivariate B-spline solids from six boundary B-spline surfaces @cite_12 . Analysis-suitable trivariate NURBS representations of composite panels is constructed with a new curve surface offset algorithm @cite_15 . proposed a two-stage scheme to construct the analysis-suitable NURBS volumetric parameterization by a uniformity-improved boundary reparameterization method @cite_34 . Recently, given a template domain, B-spline based consistent volumetric parameterization is proposed for a set of models with similar semantic features @cite_22 .
@cite_4 @cite_22 @cite_15 @cite_29 @cite_21 @cite_2 @cite_46 @cite_34 @cite_12
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High-quality volumetric parameterization of computational domain plays an important role in three-dimensional isogeometric analysis. Reparameterization technique can improve the distribution of isoparametric curves surfaces without changing the geometry. In this paper, using the reparameterization method, we investigate the high-quality construction of analysis-suitable NURBS volumetric parameterization. Firstly, we introduce the concept of volumetric reparameterization, and propose an optimal Mobius transformation to improve the quality of the isoparametric structure based on a new uniformity metric. Secondly, from given boundary NURBS surfaces, we present a two-stage scheme to construct the analysis-suitable volumetric parameterization: in the first step, uniformity-improved reparameterization is performed on the boundary surfaces to achieve high-quality isoparametric structure without changing the shape; in the second step, from a new variational harmonic metric and the reparameterized boundary surfaces, we construct the optimal inner control points and weights to achieve an analysis-suitable NURBS solid. Several examples with complicated geometry are presented to illustrate the effectiveness of proposed methods. We present a methodology based on discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate B-spline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric analysis [Hughes, T.J., Cottrell, J.A., B., Y., 2005. Isogeometric analysis: Cad, finite elements, nurbs, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, 4135-4195]. Input data consists of both a closed triangle mesh representing the exterior geometric shape of the object and interior triangle meshes that can represent material attributes or other interior features. The trivariate B-spline geometric and attribute representations are generated from the resulting parameterization, creating trivariate B-spline material property representations over the same parameterization in a way that is related to [Martin, W., Cohen, E., 2001. Representation and extraction of volumetric attributes using trivariate splines. In: Symposium on Solid and Physical Modeling, pp. 234-240] but is suitable for application to a much larger family of shapes and attributes. The technique constructs a B-spline representation with guaranteed quality of approximation to the original data. Then we focus attention on a model of simulation interest, a femur, consisting of hard outer cortical bone and inner trabecular bone. The femur is a reasonably complex object to model with a single trivariate B-spline since the shape overhangs make it impossible to model by sweeping planar slices. The representation is used in an elastostatic isogeometric analysis, demonstrating its ability to suitably represent objects for isogeometric analysis. In this paper, we present a novel algorithm for constructing a volumetric T-spline from B-reps inspired by constructive solid geometry Boolean operations. By solving a harmonic field with proper boundary conditions, the input surface is automatically decomposed into regions that are classified into two groups represented, topologically, by either a cube or a torus. We perform two Boolean operations (union and difference) with the primitives and convert them into polycubes through parametric mapping. With these polycubes, octree subdivision is carried out to obtain a volumetric T-mesh, and sharp features detected from the input model are also preserved. An optimization is then performed to improve the quality of the volumetric T-spline. The obtained T-spline surface is C 2 everywhere except the local region surrounding irregular nodes, where the surface continuity is elevated from C 0 to G 1. Finally, we extract trivariate Bezier elements from the volumetric T-spline and use them directly in isogeometric analysis. In this paper, we present an approach that automatically constructs a trivariate tensor-product B-spline solid via a gradient-based optimization approach. Given six boundary B-spline surfaces for a solid, this approach finds the internal control points so that the resulting trivariate B-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 B-spline solid even with moderate shape complexity, direct optimization of the Jacobian of the B-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 B-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.