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
1987671130 2092680586 2097230877
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. Isogeometric analysis has been proposed as a methodology for bridging the gap between computer aided design (CAD) and finite element analysis (FEA). Although both the traditional and isogeometric pipelines rely upon the same conceptualization to solid model steps, they drastically differ in how they bring the solid model both to and through the analysis process. The isogeometric analysis process circumvents many of the meshing pitfalls experienced by the traditional pipeline by working directly within the approximation spaces used by the model representation. In this paper, we demonstrate that in a similar way as how mesh quality is used in traditional FEA to help characterize the impact of the mesh on analysis, an analogous concept of model quality exists within isogeometric analysis. The consequence of these observations is the need for a new area within modeling – analysis-aware modeling – in which model properties and parameters are selected to facilitate isogeometric analysis. 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.