In isogeometric analysis, it is frequently required to handle the geometric models enclosed by four-sided or non-four-sided boundary patches, such as trimmed surfaces. In this paper, we develop a Gregory solid based method to parameterize those models. First, we extend the Gregory patch representation to the trivariate Gregory solid representation. Second, the trivariate Gregory solid representation is employed to interpolate the boundary patches of a geometric model, thus generating the polyhedral volume parametrization. To improve the regularity of the polyhedral volume parametrization, we formulate the construction of the trivariate Gregory solid as a sparse optimization problem, where the optimization objective function is a linear combination of some terms, including a sparse term aiming to reduce the negative Jacobian area of the Gregory solid. Then, the alternating direction method of multipliers (ADMM) is used to solve the sparse optimization problem. Lots of experimental examples illustrated in this paper demonstrate the effectiveness and efficiency of the developed method.
In this paper, we developed the representation of the trivariate Gregory solid, and employed it to fill the models enclosed by boundary patches, thus generating the polyhedral volume parametrization of the input models. The Gregory patch @cite_8 @cite_18 arose from the Gregory's method @cite_4 , which produces the 8 inner control points from four boundary edges and four corner points, one pair per corner. And then, the four pairs of inner control points are blended so that the generated patch interpolates the boundary straight line segments. Similarly, a triangular Gregory patch can be constructed using the method proposed in @cite_12 . Moreover, @cite_28 defined the Gregory patch as a mapping from an @math -sided parametric domain with straight line boundaries to an @math -sided parametric domain of a trimmed surface, and non-self-overlapping structured grids can be generated on it, as well as the trimmed surface.
@cite_18 @cite_4 @cite_8 @cite_28 @cite_12
1975803044 2079132777 2013052821 2498162759
Abstract We present a new method to construct a trivariate T-spline representation of complex genus-zero solids for the application of isogeometric analysis. The proposed technique only demands a surface triangulation of the solid as input data. The key of this method lies in obtaining a volumetric parameterization between the solid and the parametric domain, the unitary cube. To do that, an adaptive tetrahedral mesh of the parametric domain is isomorphically transformed onto the solid by applying a mesh untangling and smoothing procedure. The control points of the trivariate T-spline are calculated by imposing the interpolation conditions on points sited both on the inner and on the surface of the solid. The distribution of the interpolating points is adapted to the singularities of the domain in order to preserve the features of the surface triangulation. 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. A comprehensive scheme is described to construct rational trivariate solid T-splines from boundary triangulations with arbitrary topology. To extract the topology of the input geometry, we first compute a smooth harmonic scalar field defined over the mesh, and saddle points are extracted to determine the topology. By dealing with the saddle points, a polycube whose topology is equivalent to the input geometry is built, and it serves as the parametric domain for the trivariate T-spline. A polycube mapping is then used to build a one-to-one correspondence between the input triangulation and the polycube boundary. After that, we choose the deformed octree subdivision of the polycube as the initial T-mesh, and make it valid through pillowing, quality improvement and applying templates to handle extraordinary nodes and partial extraordinary nodes. The T-spline that is obtained is C^2-continuous everywhere over the boundary surface except for the local region surrounding polycube corner nodes. The efficiency and robustness of the presented technique are demonstrated with several applications in isogeometric analysis. In this paper we present a methodology to create higher order parametric trivariate representations such as B-splines or T-splines, from closed triangle meshes with higher genus or bifurcations. The input can consist of multiple interior boundaries which represent inner object material attributes. Fundamental to our approach is the use of a midsurface in combination with harmonic functions to decompose the object into a small number of trivariate tensor-product patches that respect material attributes. The methodology is applicable to thin solid models which we extend using the flexibility of harmonic functions and demonstrate our technique, among other objects, on a genus-1 pelvis data set containing an interior triangle mesh separating the cortical part of the bone from the trabecular part. Finally, a B-spline representation is generated from the parameterization.