Quartic Box Spline Research Articles

The subdivision-based IGA-EIEQ numerical scheme for the binary surfactant Cahn–Hilliard phase-field model on complex curved surfaces

In this paper, we consider numerical approximations of the binary surfactant phase-field model on complex surfaces. Consisting of two nonlinearly coupled Cahn–Hilliard type equations, the system is solved by a fully discrete numerical scheme with the properties of linearity, decoupling, unconditional energy stability, and second-order time accuracy. The IGA approach based on Loop subdivision is used for the spatial discretizations, where the basis functions consist of the quartic box-splines corresponding to the hierarchic subdivided surface control meshes. The time discretization is based on the so-called explicit-IEQ method, which enables one to solve a few decoupled elliptic constant-coefficient equations at each time step. We then provide a detailed proof of unconditional energy stability along with implementation details, and successfully demonstrate the advantages of this hybrid strategy by implementing various numerical experiments on complex surfaces.

A novel hybrid IGA-EIEQ numerical method for the Allen–Cahn/Cahn–Hilliard equations on complex curved surfaces

We present an efficient fully discrete algorithm for solving the Allen–Cahn and Cahn–Hilliard equations on complex curved surfaces. The spatial discretization employs the recently developed IGA (isogeometric analysis) framework, where we adopt the strategy of Loop subdivision with the superior adaptability of any topological structure, and the basis functions are quartic box-splines used to define the subdivided surface. The time discretization is based on the so-called EIEQ (explicit-Invariant Energy Quadratization) approach, which applies multiple newly defined variables to linearize the nonlinear potential and realize the efficient decoupled type computation. The combination of these two methods can help us to gain a linear, second-order time accurate scheme with the property of unconditional energy stability, whose rigorous proof is given. We also develop a nonlocal splitting technique such that we only need to solve decoupled, constant-coefficient elliptic equations at each time step. Finally, the effectiveness of the developed numerical algorithm is verified by various numerical experiments on the complex benchmark curved surfaces such as bunny, splayed, and head surfaces.

Local Quartic C^2 Spline Quasi-interpolation on 3D Bounded Domains

Given a 3D bounded domain, in this paper we present new quasi-interpolating spline schemes, based on trivariate C^2 quartic box splines on type-6 tetrahedral partitions with approximation order four. They are of near-best type, i.e. with coefficient functionals obtained by minimizing an upper bound for their infinity norm. Such quasiinterpolants can be used for the reconstruction of gridded volume data and their higher smoothness is useful, for example, when functions have to be reconstructed with C^2 smoothness. Moreover, we give norm and error bounds. Finally, some numerical tests, illustrating the approximation properties of the proposed quasiinterpolants, and comparisons with other known spline methods are presented.

Fast and stable evaluation of splines and their derivatives generated by the seven-direction quartic box-spline

In this paper, we propose a fast and stable evaluation method for the analytic derivatives of splines generated by the 7-direction quartic box-spline. We can maintain the spline structure by determining the derivative functions that can be represented as finite differences of box-splines defined by the sub-directions. Thus, the evaluation overhead can be reduced. We demonstrate that the first and second derivative functions are composed of only three cubic and six quadratic polynomial formulas, respectively, owing to their symmetries. Moreover, for each derivative order, all of the required functions possess change-of-variables relation with each other. Therefore, additional formulas are not required. As a result, for a given point, we only need to evaluate one quartic, three cubic, and six quadratic polynomial formulas to evaluate its spline value, gradient, and Hessian, respectively. This reduction in cases is especially advantageous for graphics processing unit (GPU) kernels, where conditional statements significantly degrade performance. To verify our technique, we implemented a real-time curvature-based GPU isosurface raycaster. Compared with other implementations, our method (i) achieves superior accuracy, (ii) is more than four times faster, and (iii) requires less than 15% of memory.

Isogeometric analysis for surface PDEs with extended Loop subdivision

We investigate the isogeometric analysis for surface PDEs based on the extended Loop subdivision approach. The basis functions consisting of quartic box-splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly, and construct the solution space for dependent variables as well, which is consistent with the concept of isogeometric analysis. The subdivision process is equivalent to the h-refinement of NURBS-based isogeometric analysis. The performance of the proposed method is evaluated by solving various surface PDEs, such as surface Laplace-Beltrami harmonic/biharmonic/triharmonic equations, which are defined on the limit surfaces of extended Loop subdivision for different initial control meshes. Numerical experiments show that the proposed method has desirable performance in terms of the accuracy, convergence and computational cost for solving the above surface PDEs defined on both open and closed surfaces. The proposed approach is proved to be second-order accuracy in the sense of L2-norm with theoretical and/or numerical results, which is also outperformed over the standard linear finite element by several numerical comparisons.

Isogeometric finite element approximation of minimal surfaces based on extended loop subdivision

In this paper, we investigate the formulation of isogeometric analysis for minimal surface models on planar bounded domains by extended Loop surface subdivision approach. The exactness of the physical domain of interest is fixed on the coarsest level of the triangular discretization with any topological structure, which is thought of as the initial control mesh of Loop subdivision. By performing extended Loop subdivision, the control mesh can be repeatedly refined, and the geometry is described as an infinite set of quartic box-spline while maintaining its original exactness. The limit function representation of extended Loop subdivision forms our finite element space, which possesses C1 smoothness and the flexibility of mesh topology. We establish its inverse inequalities which resemble the ones of general finite element spaces. We develop the approximation estimate with the aid of H1 convergence property of the corresponding linear models. It enables us to overcome the difficulty of proving the boundedness of the gradient of finite element solutions appearing in the coefficient of minimal surface models. Numerical examples are given with the comparison to the classical linear finite element method which is consistent with our theoretical results.

On the construction of trivariate near-best quasi-interpolants based on [formula omitted] quartic splines on type-6 tetrahedral partitions

The construction of new quasi-interpolants (QIs) having optimal approximation order and small infinity norm and based on a trivariate C2 quartic box spline is addressed in this paper. These quasi-interpolants, called near-best QIs, are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the QIs. We show that this problem has always a unique solution, which is explicitly given. We also prove that the sequence of the resulting near-best quasi-interpolants converges in the infinity norm to the Schoenberg operator.

Characterization of bivariate hierarchical quartic box splines on a three-directional grid

We consider the adaptive refinement of bivariate quartic C2-smooth box spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. Given a bounded domain Ω⊂R2 and finite sequence (Gℓ)ℓ=0,…,N of dyadically refined grids, we obtain a hierarchical grid by selecting mutually disjoint cells from all levels such that their union covers the entire domain. Using a suitable selection procedure allows to define a basis spanning the hierarchical box spline space. The paper derives a characterization of this space. Under certain mild assumptions on the hierarchical grid, the hierarchical spline space is shown to contain all C2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. Thus, in this case we can give an affirmative answer to the completeness questions for the hierarchical box spline basis.

Near-best $$C^2$$ C 2 quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains

In this paper, we present new quasi-interpolating spline schemes defined on 3D bounded domains, based on trivariate $C^2$ quartic box splines on type-6 tetrahedral partitions and with approximation order four. Such methods can be used for the reconstruction of gridded volume data. More precisely, we propose near-best quasi-interpolants, i.e. with coefficient functionals obtained by imposing the exactness of the quasi-interpolants on the space of polynomials of total degree three and minimizing an upper bound for their infinity norm. In case of bounded domains the main problem consists in the construction of the coefficient functionals associated with boundary generators (i.e. generators with supports not completely inside the domain), so that the functionals involve data points inside or on the boundary of the domain. We give norm and error estimates and we present some numerical tests, illustrating the approximation properties of the proposed quasi-interpolants, and comparisons with other known spline methods. Some applications with real world volume data are also provided.

Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

In this paper we consider the space generated by the scaled translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals.We give weights and nodes of the above rules and we analyse their properties.Finally, some numerical tests and comparisons with other known integration formulas are presented.

Best bounds on the distance between 3-direction quartic box spline surface and its control net

We present the best bounds on the distance between 3-direction quartic box spline surface patch and its control net by means of analysis and computing for the basis functions of 3-direction quartic box spline surface. Both the local bounds and the global bounds are given by the maximum norm of the first differences or second differences or mixed differences of the control points of the surface patch.

Quartic Box-Spline Reconstruction on the BCC Lattice

This paper presents an alternative box-spline filter for the body-centered cubic (BCC) lattice, the seven-direction quartic box-spline M7 that has the same approximation order as the eight-direction quintic box-spline M8 but a lower polynomial degree, smaller support, and is computationally more efficient. When applied to reconstruction with quasi-interpolation prefilters, M7 shows less aliasing, which is verified quantitatively by integral filter metrics and frequency error kernels. To visualize and analyze distributional aliasing characteristics, each spectrum is evaluated on the planes and lines with various orientations.

Quasi-interpolation operators based on the trivariate seven-direction C 2 quartic box spline

This paper investigates the space \(\mathcal{S}(X)\) generated by the integer translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions that forms a regular partition of the space into tetrahedra.In \(\mathcal{S}(X)\) local spline quasi-interpolants are defined by linear combinations of integer translates of B {B α ,α∈ℤ3} and local linear functionals \(\{\mathbb{D} f(\alpha), \alpha\in\mathbb{Z}^{3} \}\). First a quasi-interpolant of differential type is proposed and its coefficient functionals are linear combinations of values of f with its partial derivatives at the center of the support of B α . Then, by convenient discretisations of the above differential coefficients, three quasi-interpolants of discrete type with different properties are defined. In this case the coefficient functionals are linear combinations of values of f at specific points in the support of B α .Upper bounds both for the norm of the discrete quasi-interpolation operators and for the approximation error are given.Finally, some numerical examples, illustrating the approximation properties of the proposed quasi-interpolants, are presented.

Approximation by GP box-splines on a four-direction mesh

On a four-directional uniform mesh of the plane, we define and study C 0 Gori–Pitolli (abbreviated as GP) quadratics and C 1 GP quartics generated by integer translates of GP box-splines generalizing respectively the classical C 1 quadratic box-spline and one of the C 2 quartic box-splines. More specifically, we study the approximation of a bivariate function f by some quasi-interpolants and interpolants with values in these generalized spline spaces.

Piecewise C1 Surfaces Based on Bivariate Quartic Box-Splines for Arbitrary Triangular Meshes

For arbitrary triangular control meshes, a surface algorithm based-on bivariate box-spline is developed. Bivariate 3-direction is a triangulation with the least directions. Box spline built on it is widely applied in CAGD. Its standard surface algorithm is only for normal control mesh in which every point has valence 6. Starting with bivariate 3-directional quartic box-splines, the paper proposes an algorithm for arbitrary triangular control meshes. The analysis of its properties especially continuity are presented in detail. The constructed surfaces by the algorithm are convex preserving, and they are piecewise C 1 . The algorithm can be easily applied for global or local