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Inspired by the r-refinement method in isogeometric analysis, in this paper, the authors propose a curvature-based r-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surface patches. The authors construct three absolute curvature metrics of isogeometric solution surface to characterize its gradient information, which is more straightforward and effective. The proposed method takes the internal weights as optimization variables and the resulting parameterization is analysis-suitable and injectivity-preserving with a theoretical guarantee. Several PDEs are solved over multi-sided computational domains parameterized by toric surface patches to demonstrate the effectiveness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

De Casteljau algorithm and degree elevation of Bézier and NURBS curves/surfaces are two important techniques in computer aided geometric design. This paper presents the de Casteljau algorithm and degree elevation of toric surface patches, which include tensor product and triangular rational Bézier surfaces as special cases. Some representative examples of toric surface patches with common shapes are illustrated to verify these two algorithms. Moreover, the authors also apply the degree elevation of toric surface patches to isogeometric analysis. And two more examples show the effectiveness of proposed method. [ABSTRACT FROM AUTHOR]

Toric surface patches are a multi-sided generalization of classical rational Bézier surface patches which are widely used in free-form surface modeling. In this paper, we present the first derivatives of toric surface patches along the boundary and study the G 1 continuity between adjacent toric surface patches by the toric degenerations. Furthermore, some practical G 1 sufficient conditions of toric surface patches are developed and the representative examples are given. [ABSTRACT FROM AUTHOR]

• An improved subdivision algorithm for toric surface patches is proposed. • An N-sided toric surface patch is subdivided into N rational tensor product Bézier surface patches. • A novel h-refinement scheme for planar toric parameterization in IGA is developed. Toric surface patches are a class of multi-sided surface patches that can represent multi-sided domains without mesh degeneration. In this paper, we propose an improved subdivision algorithm for toric surface patches, which subdivides an N -sided toric surface patch into N rational tensor product Bézier surface patches. By the proposed subdivision algorithm, a C k -continuous spline surface composed of piecewise toric surface patches is subdivided into a set of rational tensor product Bézier surface patches with G k -continuity. Additionally, after subdivision, toric surface patches are compatible with CAD systems. Combining the subdivision algorithm with the classical knot insertion algorithm of non-uniform rational B-splines, we develop a novel h -refinement scheme for isogeometric analysis with planar toric parameterizations. Several numerical examples are given to demonstrate the effectiveness and numerical stability of the presented method. [ABSTRACT FROM AUTHOR]

We propose a new isogeometric method using Toric surface patches for trimmed CAD planar surfaces. This method converts each trimmed spline element into a Toric surface patch with conforming boundary representation and converts each non-trimmed spline element into a Bézier element. Because the Toric surface patches are a multi-sided generalization of classical Bézier surface patches, all trimmed and non-trimmed elements of a trimmed CAD surface have a unified geometric representation using Toric surface patches. Toric surface patches share the advantages of isogeometric continuum elements in that they can exactly model the geometry and can be easily implemented in standard finite-element code architectures. Several numerical examples are used to demonstrate the reliability of the proposed method. • We develop a new isogeometric method for dealing with trimmed CAD surfaces. • The presented method can convert each trimmed element into a toric surface patch. • The proposed method can reach the optimized convergence rate. • The presented method integrates IGA with standard finite-element code architectures. [ABSTRACT FROM AUTHOR]