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Isogeometric analysis is an innovative numerical paradigm with the potential to bridge the gap between Computer-Aided Design and Computer-Aided Engineering. However, constructing analysis-suitable parameterizations from a given boundary representation remains a critical challenge in the isogeometric design-through-analysis pipeline, particularly for computational domains with complex geometries, such as high-genus cases. To tackle this issue, we propose a multi-patch parameterization method for computational domains grounded in the singular structure of cross-fields. Initially, the vector field functions over the computational domain are solved using the boundary element method. The cross-field is then obtained through the one-to-one mapping between the vector field and the cross-field. Subsequently, we acquire the position information and topological connection relations of singularities and streamlines by analyzing the singular structure of the cross-field. Moreover, we introduce a simple and effective method for computing streamlines. We propose a novel segmentation strategy to divide the computational domain into several quadrilateral NURBS sub-patches. Once the multi-patch structure is established, we develop two methods to construct analysis-suitable multi-patch parameterizations. The first method is a direct generalization of the barrier function-based approach, while the second method yields smoother parameterizations by incorporating the interface control points of sub-patches into the optimization model. Numerical experiments demonstrate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

For potential applications in geometric design and manufacturing of material, the G 1 connection of many pieces of developable surfaces is an important issue. In this paper, by using de Casteljau algorithm we study the G 1 connection of four pieces of developable surfaces with Bézier boundary curves. We convert these surfaces to tensor form firstly, then characterize the constrains of the control points of the surfaces need to satisfy when G 1 connecting them. This method can also be extended to the case when the developable surfaces possess Bézier boundary curves with different degrees. [ABSTRACT FROM AUTHOR]

Through a rational map, a toric patch is defined associated to a lattice polygon, which is the convex of a given finite integer lattice points set A . The classical rational Bézier curves, rational triangular and tensor-product patches are special cases of toric patches. One of the geometric meanings of toric patch is that the limiting of the patch is its regular control surface, when all weights tend to infinity. In this paper, we study the number of regular decompositions of A , and the relationship between regular decompositions and the corresponding secondary polytope. What is more, we indicate that the number of regular control surfaces of toric patch associated with A is equal to the number of regular decompositions of A . [ABSTRACT FROM AUTHOR]

NURBS surface is very useful in geometric modeling, animation, image morphing and deformation. Constructing non-self-intersecting (injective) NURBS surfaces is an important process in surface and solid modeling. In this paper, the injective conditions of tensor product NURBS surface are studied, based on the geometric positions of control points, which are equivalent to the surface to be non-self-intersecting for all positive weights. Finally, some representative examples are provided. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) becomes an effective tool for solving partial differential equations (PDEs). Compared with isogeometric Galerkin method, isogeometric collocation (IGC) method has higher computational efficiency. In this paper, we aim to optimize the domain parameterization in IGC for better numerical accuracy of solving PDEs. Firstly, we present a new parameterization method of planar physical domain, called residual parameterization, which is obtained by minimizing the objective functions consisting of geometry-related functionals and the analysis-related residual norms in an unconstrained optimization problem. Secondly, the reduced quadrature rules are applied to residual norms due to high computational cost in evaluating the integration of residual terms. Finally, based on the residual parameterization, we solve boundary value problems (BVPs) by IGC with Greville points and superconvergent points to verify the strength of our proposed residual parameterization of planar physical domain. Several numerical examples show that the numerical accuracy of the proposed method is improved nearly half order of magnitude compared with standard IGC method. [ABSTRACT FROM AUTHOR]

The injectivity of NURBS curve implies the curve has no self-intersection. In this paper, we propose a geometric condition on the control polygon which guarantees the NURBS curve to be injective for all possible choices of positive weights. The proof is based on the degree elevation algorithm and toric degeneration theory of NURBS curve. [ABSTRACT FROM AUTHOR]

Rational Bézier surface is a common fitting tool in Computer Aided Geometric Design. The injectivity of curve/surface implies the one-to-one property and there is no self-intersection of curve/surface. In this paper, we propose a geometric method for checking the injectivity of rational Bézier surface based on its control points. [ABSTRACT FROM AUTHOR]

Rational Bernstein operators are widely used in approximation theory and geometric modeling but in general they do not reproduce linear polynomials. Based on the work of P. Piţul and P. Sablonnière, we construct a new family of triangular and tensor product bivariate rational Bernstein operators, which are positive and reproduce the linear polynomials. The main result is a proof of convergence of the bivariate rational Bernstein operators defined on the square or triangle. [ABSTRACT FROM AUTHOR]

Abstract: As a piecewise polynomial with a certain smoothness, the spline plays an important role in computational geometry. The algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a generalization of the algebraic variety. In this paper, the correspondence between piecewise algebraic varieties and spline ideals is discussed. Furthermore, Hilbert’s Nullstellensatz for the piecewise algebraic variety is also studied. [Copyright &y& Elsevier]

Abstract: In this paper, numerical solution of Burgers–Fisher equation is presented based on the cubic B-spline quasi-interpolation. At first, the generalized Burgers–Fisher equation and the cubic B-spline quasi-interpolation are introduced. Moreover, the numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Moreover, the stability of this method is studied. At last, the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. [Copyright &y& Elsevier]

Abstract: In this paper, numerical solution of the Burgers’ equation is presented based on the cubic B-spline quasi-interpolation. At first the cubic B-spline quasi-interpolation is introduced. Moreover, the numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. The accuracy of the proposed method is demonstrated by some test problems. The numerical results are found in good agreement with exact solutions. The advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. [Copyright &y& Elsevier]

Abstract: Implicit curves and surfaces are extensively used in interpolation, approximation and blending. [Li J, Hoschek J, Hartmann E. -functional splines for interpolation and approximation of curves, surfaces and solids. Computer Aided Geometric Design 1990;7:209–20] presented a functional method for constructing curves and surfaces which are called functional splines. In this paper, functional splines with different degrees of smoothness are presented and applied to some typical problems. [Copyright &y& Elsevier]

Abstract: The piecewise algebraic curve is a generalization of the classical algebraic curve. In this paper, we give the Nöther-type theorem of piecewise algebraic curves on cross-cut partitions. By using interpolation along a piecewise algebraic curve, we also present a method to construct Lagrange interpolation sets for bivariate spline spaces on cross-cut partitions. [Copyright &y& Elsevier]

The piecewise algebraic curve, determined by a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, by using Bezout's theorem and No¨ther-type theorem of piecewise algebraic curves, the Cayley–Bacharach theorem and Hilbert function of C0 piecewise algebraic curves are presented. [Copyright &y& Elsevier]

The surface of partial differential equation (PDE surface) is a surface that satisfies the PDE with boundary conditions, which can be applied in surface modeling and construction. In this paper, the construction of tensor product Bézier surfaces of triharmonic equation from different boundary conditions is presented. The internal control points of the resulting triharmonic Bézier surface can be obtained uniquely by the given boundary condition. Some representative examples show the effectiveness of the presented method. [ABSTRACT FROM AUTHOR]

Injective parametrizations are widely used both in theory and in applications. The injectivity of parameteric curves and surfaces means that there are no self-intersections. Toric surface patch is defined by a set of integer lattice points A and corresponding control points and weights, which includes rational tensor product and triangle Bézier patches as special cases. In 2011, Sottile and Zhu presented a geometric method to check the injectivity of 2D toric surface patches. In this paper, we present an improved algorithm of their method. The complexity of the improved algorithm is reduced from O (n 3) to O (n 2) , where n = # (A). Some examples are shown to illustrate the effectiveness of our algorithm. Moreover, the algorithm is also applied to check the injectivity of parameterization in isogeometric analysis. [ABSTRACT FROM AUTHOR]

The normalized totally positive bases are widely used in many fields. Based on the generalized Vandermonde determinant, the normalized total positivity of a kind of generalized toric-Bernstein basis is proved, which is defined on a set of real points. By this result, then the progressive iterative approximation property of the generalized toric-Bézier curve is obtained and some examples are provided to illustrate this property. [ABSTRACT FROM AUTHOR]

NURBS method is the standard mathematical method for describing the shapes of curves/surfaces/volumes, and it is extensively used in computer-aided design, computer-aided manufacturing, and computer graphics. The injectivity of NURBS volumes means that they do not have self-intersections. Since the injectivity of parameterizations depend on the signs of their Jacobian functions, and the Jacobian of a NURBS volume is determined by the determinant of its tangent vectors in three directions, we first propose a method to compute the bounding vectors of the tangent cones of NURBS volume in this paper. Then the sufficient condition for the injectivity of NURBS volume is proposed. A checking algorithm is also presented. Some examples are given to verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Traditional end conditions for cubic spline interpolation consist of values, the first or the second derivatives of interpolated functions on the boundary interpolation knots. The not-a-knot end condition proposed by de Boor (1985) is a kind of end condition of cubic spline interpolation for the practical application without the requirements of the derivatives at the end knots. However, a significant disadvantage of such end condition is that there is a sharp decrease in the accuracy of the interpolation at boundary intervals. In this paper, by changing the locations of two spline knots in not-a-knot end condition, we propose the optimal arrangement of shifted spline knots for cubic spline interpolation. The proposed scheme leads to an approximately 3.4 times increasing in the accuracy of the interpolation compared to the de Boor's not-a-knot end condition. Furthermore, we also present the optimal end conditions of cubic spline interpolation to approximate the first and the second derivatives of interpolated functions. The representative examples show the effectiveness and the superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: The signed distance function can effectively support many geometry processing tasks such as decimates, smoothing and shape reconstruction since it provides efficient access to distance estimates. In this paper, we present an adaptive method to approximate the signed distance function of a smooth curve by using polynomial splines over type-2 triangulation. The trimmed offsets are also studied. [Copyright &y& Elsevier]

The Geometry-Independent Field approximaTion (GIFT) technique, an extension of isogeometric analysis (IGA), allows for separate spaces to parameterize the computational domain and approximate solution field. Based on the GIFT approach, this paper proposes a novel IGA methodology that incorporates toric surface patches for multi-sided geometry representation, while utilizing B-spline or truncated hierarchical B-spline (THB-spline) basis for analysis. By creating an appropriate bijection between the parametric domains of distinct bases for modeling and approximation, our method ensures smoothness within the computational domain and combines the compact support of B-splines or the local refinement potential of THB-splines, resulting in more efficient and precise solutions. To enhance the quality of parameterization and consequently boost the accuracy of downstream analysis, we suggest optimizing the composite toric parameterization. Numerical examples validate the effectiveness and superiority of our suggested approach. [Display omitted] • MS-GIFT adopts multi-sided surface with global smoothness for IGA. • MS-GIFT introduces the locality of B-spline or THB-spline to multi-sided surface. • We develop a bijection between polygonal and square, with two methods to compute its inverse. • We propose the concept of composite parameterization optimization. [ABSTRACT FROM AUTHOR]

Highlights: [•] We deduce the condition for the given curve to be the line of curvature on surface when the marching-scale functions are in more general expression. [•] Two functions (s) and (s) control the shape of the surface. [•] We classify the conditions by the expression of (s). [ABSTRACT FROM AUTHOR]

Abstract: Geodesic is an important curve in practical application, especially in shoe design and garment design. In practical applications, we not only hope the shoe and garment surfaces possess characteristic curves, but also we hope minimal cost of material to build surfaces. In this paper, we combine the geodesic and minimal surface. We study the approximation minimal surface with geodesics by using Dirichlet function. The extremal of such a function can be easily computed as the solutions of linear systems, which avoid the high nonlinearity of the area function. They are not extremal of the area function but they are a fine approximation in some cases. [Copyright &y& Elsevier]

Abstract: Developable surface and line of curvature play an important role in geometric design and surface analysis. This paper proposes a new method to construct a developable surface possessing a given curve as the line of curvature of it. We analyze the necessary and sufficient conditions when the resulting developable surface is a cylinder, cone or tangent surface. Finally, we illustrate the convenience and efficiency of this method by some representative examples. [Copyright &y& Elsevier]

Abstract: For potential application in shoemaking and garment manufacture industries, the G 1 connection of (1, k) developable surfaces with abutting geodesic is important. In this paper, we discuss the developable surface which contains a given 3D Bézier curve as geodesic and prove the corresponding conclusions in detail. Primarily we study G 1 connection of developable surfaces through abutting cubic Bézier geodesics and give some examples. [Copyright &y& Elsevier]

Abstract: In this paper, we use the so-called conformality method of smoothing cofactor (abbr. CSC) and hyperplane arrangements to study truncated powers and box splines in . By the relation between hyperplane arrangements and truncated powers, we give the expressions of the truncated powers. Moreover, by means of the CSC method, the least smoothness degrees of the truncated powers and the box splines on each direction of partition edges are studied. [Copyright &y& Elsevier]

Abstract: Constructing multi-step difference schemes is important for solving ODE numerically. In this paper, by using an algebraic function approximation to the exponent function, linear multi-step schemes for solving ODE are deduced. The approximation order of this linear multi-step scheme equals that of the algebraic function approximation to . Moreover, we show that the linear -step difference scheme of order is unstable, which is proved in a novel way. [Copyright &y& Elsevier]

Abstract: Line of curvature on a surface plays an important role in practical applications. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. By utilizing the Frenet frame, the surface pencil can be expressed as a linear combination of the components of the local frame. With this parametric representation, we derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface. Moreover, the necessary and sufficient condition for the given curve to satisfy the line of curvature and the geodesic requirements is also analyzed. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new multilevel univariate multiquadric (MQ) quasi-interpolation approach with higher approximation order compared with the existing MQ quasi-interpolations. The proposed approach includes two schemes, which are based on inverse multiquadric radial basis function (IMQ-RBF) interpolation, and Wu & Schaback’s MQ quasi-interpolation operator . Error analysis shows that our operators produce higher accuracy. Numerical examples demonstrate that the proposed MQ quasi-interpolation schemes are valid. [ABSTRACT FROM AUTHOR]

• A sufficient and necessary condition for the injectivity of toric volumes is proposed. • The sign of mixed product of three vectors is used to check the compatibility between the sets of lattice points and control points. • An algorithm is proposed to check the injectivity of toric volumes for arbitrary positive weights. • The algorithm is improved based on the property of clean and empty tetrahedron. [Display omitted] Parameterizations, which map parametric domains into certain domains, are widely used in computer aided design, computer aided geometric design, computer graphics, isogeometric analysis, and related fields. The parameterizations of curves, surfaces, and volumes are injective means that they do not have self-intersections. A 3D toric volume is defined via a set of 3D control points with weights that correspond to a set of finite 3D lattice points. Rational tensor product or tetrahedral Bézier volumes are special cases of toric volumes. In this paper, we proved that a toric volume is injective for any positive weights if and only if the lattice points set and control points set are compatible. An algorithm is also presented for checking the compatibility of the two sets by the mixed product of three vectors. Some examples illustrate the effectiveness of the proposed method. Moreover, we improve the algorithm based on the properties and results of clean and empty tetrahedrons in combinatorics. [ABSTRACT FROM AUTHOR]

The Least-Squares Progressive-Iterative Approximation (LSPIA) method offers a powerful and intuitive approach for data fitting. Non-Uniform Rational B-splines (NURBS) are a popular choice for approximation functions in data fitting, due to their robust capabilities in shape representation. However, a restriction of the traditional LSPIA application to NURBS is that it only iteratively adjusts control points to approximate the provided data, with weights and knots remaining static. To enhance fitting precision and overcome this constraint, we present Full-LSPIA, an innovative LSPIA method that jointly optimizes weights and knots alongside control points adjustments for superior NURBS curves and surfaces creation. We achieve this by constructing an objective function that incorporates control points, weights, and knots as variables, and solving the resultant optimization problem. Specifically, control points are adjusted using LSPIA, while weights and knots are optimized through the LBFGS method based on the analytical gradients of the objective function with respect to weights and knots. Additionally, we present a knot removal strategy known as Decremental Full-LSPIA. This strategy reduces the number of knots within a specified error tolerance, and determines optimal knot locations. The proposed Full-LSPIA and Decremental Full-LSPIA maximize the strengths of LSPIA, with numerical examples further highlighting the superior performance and effectiveness of these methods. Compared to the classical LSPIA, Full-LSPIA offers greater fitting accuracy for NURBS curves and surfaces while maintaining the same number of control points, and automatically determines suitable weights and knots. Moreover, Decremental Full-LSPIA yields fitting results with fewer knots while maintaining the same error tolerance. • We propose Full-LSPIA, a flexible and automatic fitting framework for LSPIA. • We develop analytical gradient formulations for the weights and knots of NURBS. • Full-LSPIA enhances the precision in NURBS approximation maintaining fixed DOFs. • Decremental Full-LSPIA yields lightweight NURBS curves with given error tolerance. [ABSTRACT FROM AUTHOR]

Localized and anisotropic features extensively exist in various physical phenomena. The present work focuses on the r -adaptive parameterization technique for isogeometric analysis (IGA), which aims to acquire higher numerical accuracy while keeping the degrees of freedom constant. The principal feature is utilizing the so-called absolute principal curvature of the IGA solution surfaces to characterize numerical errors instead of posteriori error estimations, which establishes the relation between analysis results and geometric quantity. The bijectivity is a fundamental requirement for analysis-suitable parameterization. With the cooperation of a minor regularization and common line search criteria, the proposed method guarantees the bijectivity of the resulting parameterizations. The bi-level approach with two refinement levels of the same geometry is employed: a coarse level (design model) to update the parameterization and a fine level (analysis model) to perform the isogeometric simulation. Moreover, we develop several detailed algorithms for explaining the sensitivity propagation from the design model to the analysis model and analytically computing the sensitivity, which allows accurate calculation of sensitivity and enhances the robustness during a gradient-based optimization. Several examples and comparisons are presented to demonstrate the effectiveness and efficiency of the proposed method. As an application, we apply the proposed method to a two-dimensional linear heat transfer problem with a moving Gaussian heat source, which is a simplified model for the additive manufacturing application. The proposed r -adaptive technique effectively captures the thermal history of the problem. [Display omitted] • An r-adaptivity based on absolute principal curvature is proposed for isogeometric analysis. • Bi-level approach using coarse and fine refinements is employed to balance efficiency and accuracy. • Bijectivity is guaranteed by regularization technique and common line search criteria. • Two detailed algorithms are given to explain the sensitivity propagation and analytical sensitivity analysis. • A simple additive manufacturing application demonstrates the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Parameterization of computational domains is a crucial step in isogeometric analysis (IGA). Non-Uniform Rational B-Spline (NURBS) is a standard tool in the CAD/CAM industry due to its powerful capability for shape representation. In this paper, we propose several sufficient conditions and a necessary condition for injective NURBS parameterizations of computational domains, taking into account both the control points and weights. Based on these conditions, an algorithm for the injectivity checking of NURBS parameterization is proposed. By taking both the internal control points and weights as optimization variables, we present an effective and robust approach for parameterizing planar computational domains. With the internal control points and weights updating alternately, the resulting parameterization is constructed by solving an unconstrained optimization problem whose objective function is a weighted sum of corrected Winslow functional and uniformity functional. Finally, the proposed checking algorithm is applied to verify the injectivity of the resulting parameterizations. Numerical examples demonstrate the effectiveness and robustness of the proposed method and show superior parameterization quality performance over the state-of-the-art approaches. [ABSTRACT FROM AUTHOR]