The advent of isogeometric analysis has prompted a need for methods to generate Trivariate B-spline Solids (TBS) with positive Jacobian. However, it is difficult to guarantee a positive Jacobian of a TBS since the geometric pre-condition for ensuring the positive Jacobian is very complicated. In this paper, we propose a method for generating TBSs with guaranteed positive Jacobian. For the study, we used a tetrahedral (tet) mesh model and segmented it into sub-volumes using the pillow operation. Then, to reduce the difficulty in ensuring a positive Jacobian, we separately fitted the boundary curves and surfaces and the sub-volumes using a geometric iterative fitting algorithm. Finally, the smoothness between adjacent TBSs is improved. The experimental examples presented in this paper demonstrate the effectiveness and efficiency of the developed algorithm.
To analyze the arterial blood flow by IGA, a trivariate NURBS-solid modeling the artery was constructed using a skeleton-based method . Following volume parameterization by a harmonic function, a cylinder-like trivariate B-spline solid was generated with a singular centric curve @cite_5 . Moreover, trivariate B-spline solids with positive Jacobian values were produced from boundary representations using optimization-based approaches . However, the optimization method may fail when the objective function is highly nonlinear. Based on the given boundary conditions and guiding curves, a NURBS solid was constructed to model a swept volume by the variational approach .
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2079132777 1418831932 2084654821 2094528257
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. The use of trivariate NURBS in isogeometric analysis has put the quality of parametrization of NURBS volumes on the agenda. Sometimes a NURBS volume needs a better parametrization to meet requirements regarding smoothness, approximation or periodicity. In this paper we generalize various smoothing methods that already exist for bivariate parametric spline surfaces to trivariate parametric spline volumes. We will also address how rational and polynomial spline volumes create different challenges and solutions in the algorithms. Trivariate NURBS (non-uniform rational B-splines) representation of composite panels which is suitable for three-dimensional isogeometric analysis (IGA) is constructed with a new curve surface offset algorithm. The proposed offset algorithm, which is required by IGA, is non-existent in the CAD literature. Using the presented approach, finite element analysis of composite panels can be performed with the only input being the geometry representation of the composite surface. The method proposed provides a bi-directional system in which one can go forward from CAD to analysis and backwards from analysis to CAD. This is believed to facilitate the design of composite structures. Different parts (patches) can be parametrized independently of each other and glued together, in the finite element solver, by a discontinuous Galerkin method. A stress analysis of curved composite panel with stiffeners is provided to demonstrate the proposed framework. 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.