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.
: E. @cite_32 proposed the concept of , in which the parameters of CAD models are selected to facilitate isogeometric analysis. @cite_11 showed that the quality of parameterization has a great impact on the analysis results and the efficiency. Pilgerstorfer and J "uttler @cite_44 showed that the condition number of the stiffness matrix, which is a key factor for the stability of the linear system, depends strongly on the quality of the domain parameterization.
@cite_44 @cite_32 @cite_11
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In the isogeometric analysis framework, a computational domain is exactly described using the same representation as the one employed in the CAD process. For a CAD object, various computational domains can be constructed with the same shape but with different parameterizations; however one basic requirement is that the resulting parameterization should have no self-intersections. Moreover we will show, with an example of a 3D thermal conduction problem, that different parameterizations of a computational domain have different impacts on the simulation results and efficiency in isogeometric analysis. In this paper, a linear and easy-to-check sufficient condition for the injectivity of a trivariate B-spline parameterization is proposed. For problems with exact solutions, we will describe a shape optimization method to obtain an optimal parameterization of a computational domain. The proposed injective condition is used to check the injectivity of the initial trivariate B-spline parameterization constructed by discrete Coons volume method, which is a generalization of the discrete Coons patch method. Several examples and comparisons are presented to show the effectiveness of the proposed method. During the refinement step, the optimal parameterization can achieve the same accuracy as the initial parameterization but with less degrees of freedom. Abstract Isogeometric Analysis (IGA) was introduced by (2005) [1] as a new method to bridge the gap between the geometry description and numerical analysis. Similar to the finite element approach, the IGA concept to solve a partial differential equation leads to a (linear) system of equations. The condition number of the coefficient matrix is a crucial factor for the stability of the system. It depends strongly on the domain parameterization, which provides the isogeometric discretization. In this paper we derive a bound for the condition number of the stiffness matrix of the Poisson equation. In particular, we investigate the influence of the domain parameterization and the knot spacing on the stability of the numerical system. The factors appearing in our bound reflect the stability properties of a given domain parameterization. Parameterization of the computational domain is a key step in isogeometric analysis just as mesh generation is in finite element analysis. In this paper, we study the volume parameterization problem of the multi-block computational domain in an isogeometric version, i.e., how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Firstly, we show how to find good volume parameterization of the single-block computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parameterization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. By using this method, the resulting volume parameterization has no self-intersections, and the isoparametric structure has good uniformity and orthogonality. Then we extend this method to the multi-block case, in which the continuity condition between the neighbor B-spline volumes should be added to the constraint term. The effectiveness of the proposed method is illustrated by several examples based on the three-dimensional heat conduction problem. Volumetric spline parameterization and computational efficiency are two main challenges in isogeometric analysis (IGA). To tackle this problem, we propose a framework of computation reuse in IGA on a set of three-dimensional models with similar semantic features. Given a template domain, B-spline based consistent volumetric parameterization is first constructed for a set of models with similar semantic features. An efficient quadrature-free method is investigated in our framework to compute the entries of stiffness matrix by Bezier extraction and polynomial approximation. In our approach, evaluation on the stiffness matrix and imposition of the boundary conditions can be pre-computed and reused during IGA on a set of CAD models. Examples with complex geometry are presented to show the effectiveness of our methods, and efficiency similar to the computation in linear finite element analysis can be achieved for IGA taken on a set of models.