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The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the centre of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm. • We construct a univariate interpolatory subdivision scheme that reproduces conic sections. • It is a geometric subdivision scheme based on point-normal pairs. • We demonstrate our new scheme on several examples. [ABSTRACT FROM AUTHOR]

We study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m , where m > 2 is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle at e 2 π ki / m = cos (2 k π / m) + i sin (2 k π / m) ↔ (cos (2 k π / m) , sin (2 k π / m)). Given a sequence of integers (s 0 , ... , s m) mod m , we connect consecutive values s j s j + 1 on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence (0 , 1 , ... , m) mod m , then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m -sided polygon and regular m -pointed stars that repeat with a period equal to the minimal integer k such that 2 k = ± 1 m o d m. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue 2 − 1 m o d m. We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m -sided polygons and the complete graphs of m -sided polygons, are fixed points of these subdivision procedures. [Display omitted] [ABSTRACT FROM AUTHOR]

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted average of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN C 2 subdivision schemes generalizing linear subdivision schemes that generate C 2 subdivision surfaces with flat extraordinary points can generate visually C 2 subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling. • We generalize linear subdivision schemes to nonlinear subdivision schemes by refining control points and control normals. • The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres. • PN subdivision schemes can have the same convergence and smoothness orders as linear subdivision schemes. • PN C2 subdivision schemes can generate visually curvature continuous surfaces with non-flat extraordinary points. [ABSTRACT FROM AUTHOR]

It is possible to define multiresolution by reversing the process of subdivision. One approach to reverse a subdivision scheme appropriates pure numerical algebraic relations for subdivision using the interaction of diagrams ( Bartels and Samavati, 2011; Samavati and Bartels, 2006 ). However, certain assumptions carried through the available work, two of which we wish to challenge: (1) the construction of multiresolutions for irregular meshes are reconsidered in the presence of any extraordinary vertex rather than being prepared beforehand as simple available relations and (2) the connectivity graph of the coarse mesh would have to be a subgraph of the connectivity graph of the fine mesh. 3 subdivision ( Kobbelt, 2000 ) lets us engage both of these concerns. With respect to (2), the 3 post-subdivision connectivity graph shares no interior edge with the pre-subdivision connectivity graph. With respect to (1), we observe that no subdivision produces an arbitrary connectivity graph. Rather, there are local regularities imposed by the subdivision on the fine mesh, which may be exploited to establish, in advance, the decomposition and reconstruction filters of a multiresolution for an initially given irregular coarse mesh. We provide indications that our proposed approach for 3 subdivision is potentially useful for other primal subdivision schemes by mentioning results for the Loop (1987) and Catmull–Clark subdivisions ( Catmull and Clark, 1978 ). Finally, after showing some results, we analyze the quality of the reversal matrix and present a technique using numerical optimization. [ABSTRACT FROM AUTHOR]

CAD models generally consist of multiple NURBS patches, both trimmed and untrimmed. There is a long-standing challenge that trimmed NURBS patches cause unavoidable gaps in the model. We address this by converting multiple NURBS patches to a single untrimmed NURBS-compatible subdivision surface in a three stage process. First, for each patch, we generate in domain space a quadrangulation that follows boundary edges of the patch and respects the knot spacings along edges. Second, the control points of the corresponding subdivision patch are computed in model space. Third, we merge the subdivision patches across their common boundaries to create a single subdivision surface. The converted model is gap-free and can maintain inter-patch continuity up to C 2 . [ABSTRACT FROM AUTHOR]

Subdivision is a well-known and established method for generating smooth curves and surfaces from discrete data by repeated refinements. The typical input for such a process is a mesh of vertices. In this work we propose to refine 2D data consisting of vertices of a polygon and a normal at each vertex. Our core refinement procedure is based on a circle average , which is a new non-linear weighted average of two points and their corresponding normals. The ability to locally approximate curves by the circle average is demonstrated. With this ability, the circle average is a candidate for modifying linear subdivision schemes refining points, to schemes refining point-normal pairs. This is done by replacing the weighted binary arithmetic means in a linear subdivision scheme, expressed in terms of repeated binary averages, by circle averages with the same weights. Here we investigate the modified Lane–Riesenfeld algorithm and the 4-point scheme. For the case that the initial data consists of a control polygon only, a naive method for choosing initial normals is proposed. An example demonstrates the superiority of the above two modified schemes, with the naive choice of initial normals over the corresponding linear schemes, when applied to a control polygon with edges of significantly different lengths. [ABSTRACT FROM AUTHOR]

In this paper a construction of a globally G 1 family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of Bézier patches, which are biquintic G 1 around extraordinary vertices and bicubic elsewhere; the smoothness around regular vertices neighboring EV with even valence will be C 1 , while it results to be C 2 in the other occurrences of regular vertices. Each Bézier point is computed using a locally defined mask around the neighboring mesh vertices. To define G 1 conditions, we assign quadratic gluing data around extraordinary vertices that depend solely on their valence and we use degree five patches to satisfy these G 1 constraints. We explore the space of possible solutions, considering several projections on the solution space leading to different explicit formulas for the masks. Certain control points are computed by means of degree elevation of the C 0 scheme of Loop and Schaefer (2008) , while for others, explicit masks are deduced by providing closed-form solutions of the G 1 conditions, expressed in terms of the masks. We come up with four different schemes and conduct curvature analysis on an extensive benchmark in order to assert the quality of the resulting surfaces and identify the ones that lead to the best result, both visually and numerically. We demonstrate numerically that the resulting surfaces converge quadratically to the CC limit when the mesh is subdivided. • We propose geometrically smooth approximation schemes for Catmull-Clark limit surfaces by means of biquintic Bézier patches • We assign quadratic gluing data to the mesh edges and provide general, closed form solutions to the G 1 constraint system • We develop explicit masks for the Bézier control points that involve the 2-ring of the mesh vertices. • The resulting surfaces are globally G 1 and converge quadratically to the Catmull-Clark limit surface. [ABSTRACT FROM AUTHOR]

In this paper, we study bivariate dual 2 -subdivision schemes using one-dimensional (1D) stencils and reproducing bivariate polynomials of high orders. We show that such dual 2 -subdivision schemes always possess the quasi-interpolating property (i.e., they interpolate bivariate polynomials of high orders) and are intrinsically linked to both 1D primal interpolating subdivision schemes and 1D masks having linear-phase moments. Using only 1D stencils, such subdivision schemes can be straightforwardly implemented on any quadrilateral meshes and there is no need to design special subdivision rules near extraordinary or boundary vertices. In this paper, we concentrate on a particular quasi-interpolating dual 2 -subdivision scheme using 4-point 1D stencils and interpolating all bivariate cubic polynomials. This dual 2 -subdivision scheme has C 2 smoothness near any ordinary vertex guaranteed by the smoothness of its underlying basis/refinable function, while it achieves C 1 smoothness near an extraordinary vertex by our numerical calculation using the known technique of analyzing characteristic maps. As an illustration, we apply the proposed dual 2 -subdivision scheme to several arbitrary polyhedra meshes to demonstrate the generated subdivision surfaces. • Quasi-interpolating bivariate dual 2 -subdivision schemes using 1D stencils are studied. • No special rules are required for extraordinary or boundary vertices. • The dual 2 -subdivision scheme has C 2 smoothness except for some extraordinary points. • The proposed scheme can reproduce bivariate polynomials of high degrees. [ABSTRACT FROM AUTHOR]

Abstract The aim of this short note is to provide a rigorous proof that the tensor product of two non-uniform linear convergent subdivision schemes converges and has the same regularity as the minimal regularity of the univariate schemes. It extends results that are known for the uniform linear case and are based on symbols, a notion which is no longer available in the non-uniform setting. [ABSTRACT FROM AUTHOR]

A new equivalence notion between non-stationary subdivision schemes, termed asymptotic similarity, which is weaker than asymptotic equivalence, is introduced and studied. It is known that asymptotic equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotic equivalence can be relaxed to asymptotic similarity. This result applies to a wide class of non-stationary schemes. [ABSTRACT FROM AUTHOR]

We introduce and analyze univariate, linear, and stationary subdivision schemes for refining noisy data by fitting local least squares polynomials. This is the first attempt to design subdivision schemes for noisy data. We present primal schemes, with refinement rules based on locally fitting linear polynomials to the data, and study their convergence, smoothness, and basic limit functions. Then, we provide several numerical experiments that demonstrate the limit functions generated by these schemes from initial noisy data. The application of an advanced local linear regression method to the same data shows that the methods are comparable. In addition, several extensions and variants are discussed and their performance is illustrated by examples. We conclude by applying the schemes to noisy geometric data. [ABSTRACT FROM AUTHOR]

This article deals with interpolatory subdivision schemes generalizing the tensor-product version of the Dubuc–Deslauriers 4-point scheme to quadrilateral meshes of arbitrary manifold topology. In particular, we focus our attention on an extension of the C 1 regular stencils that respectively exploits ( N + 2 ) -point and ( 2 N + 8 ) -point stencils for the computation of an edge-point and a face-point in the vicinity of an extraordinary vertex of valence N ≠ 4 not lying on a boundary. The aim of our work consists in identifying which constraints are required to be respected by the weights appearing in the above stencils in order to get closed limit surfaces that are C 1 -continuous at extraordinary points, have both principal curvatures bounded and at least one of them nonzero. The obtained constraints are used to easily check these features in the limit surfaces resulting from the application of special extraordinary rules proposed in the literature by different authors. Moreover, the conditions derived on the stencil weights are exploited to design new extraordinary rules that can produce closed limit surfaces of the same quality as the existing proposals, but at a reduced computational cost. [ABSTRACT FROM AUTHOR]

For an arbitrary degree Bézier curve B , we first establish sufficient conditions for its control polygon to become homeomorphic to B via subdivision. This is extended to show a subdivided control polygon that is ambient isotopic to B . We provide closed-form formulas to compute the corresponding number of iterations for equivalence under homeomorphism and ambient isotopy. The development of these a priori values was motivated by application to high performance computing (HPC), where providing estimates of total run time is important for scheduling. [ABSTRACT FROM AUTHOR]

This paper presents a biarc-based subdivision scheme for space curve interpolation. Given a sequence of space points, or a sequence of space points and tangent vectors, the scheme produces a smooth curve interpolating all input points by iteratively inserting new points and computing new tangent vectors. For each step of subdivision, the newly inserted point corresponding to an existing edge is a specified joint point of a biarc curve which interpolates the two end points and the tangents. A provisional tangent is also assigned to the newly inserted point. Each of the tangents for the following subdivision step is further updated as a linear blending of the provisional tangent and the tangent at the respective point of a circle passing through the local point and its two adjacent points. If adjacent four initial points and their initial reference tangent vectors are sampled from a spherical surface, the limit curve segment between the middle two initial points exactly lies on the same spherical surface. The biarc based subdivision scheme is proved to be G 1 continuous with a nice convexity preserving property. Numerical examples also show that the limit curves are G 2 continuous and fair. Several examples are given to demonstrate the excellent properties of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we introduce triangular subdivision operators which are composed of a refinement operator and several averaging operators, where the refinement operator splits each triangle uniformly into four congruent triangles and in each averaging operation, every vertex will be replaced by a convex combination of itself and its neighboring vertices. These operators form an infinite class of triangular subdivision schemes including Loop's algorithm with a restricted parameter range and the midpoint schemes for triangular meshes. We analyze the smoothness of the resulting subdivision surfaces at their regular and extraordinary points by generalizing an established technique for analyzing midpoint subdivision on quadrilateral meshes. General triangular midpoint subdivision surfaces are smooth at all regular points and they are also smooth at extraordinary points under certain conditions. We show some general triangular subdivision surfaces and compare them with Loop subdivision surfaces. [ABSTRACT FROM AUTHOR]

In order to construct a C 1 -quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell–Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme ( Dyn and Lyche, 1998 ). In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally C 3 and globally C 2 . For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage . Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements. [ABSTRACT FROM AUTHOR]

Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate C 1 and C 2 limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one. [ABSTRACT FROM AUTHOR]

• An algorithm to compute a conservative approximation of the kernel of freeform curves and surfaces is presented. • Inequality constraints for the interior of kernel domains are formulated and solved using a subdivision-based approach. • The proposed algorithm computes the kernel of various freeform curves and surfaces consisting of multiple, C 1 -discontinuous, and/or open curves and surfaces. We present an algorithm to compute a tight-as-needed conservative approximation of the kernel domain of freeform curves in R 2 and freeform surfaces in R 3. Inequality constraints to detect the interior of the kernel domain are formulated as multivariates, and solved with a subdivision-based approach to find the domains in R 2 or R 3 that satisfy the inequalities and are in the kernel. The convex hull of the computed domains is also included in the kernel, and adopted as the approximated kernel domain. We can apply the presented algorithm to detect the kernel domain of not only C 1 continuous closed regular curves and surfaces, but also the kernel domains of multiple piecewise C 1 continuous regular freeform curves and surfaces. Further, the presented algorithm can be applied to find the gamma-kernel as well as the kernel domain of open curves and surfaces, under some assumptions. We demonstrate our experimental result using various freeform curves and surfaces, and compare it with the kernel computation algorithm presented in Elber et al. (2006). [ABSTRACT FROM AUTHOR]

Convexity preserving condition of subdivision curve is the focus of computer aided geometric design. In this note, we point out that the four point interpolation subdivision curve is everywhere non-convex or non-concave unless it contains a polynomial curve of degree three or less. Convexity preserving condition of subdivision curve is the focus of computer aided geometric design and has been discussed extensively. In this note, we point out that the well-known four point interpolation subdivision curve is everywhere non-convex or non-concave unless it contains a cubic or less polynomial curve. [ABSTRACT FROM AUTHOR]

Abstract: A new binary four-point subdivision scheme is presented, which keeps the second-order divided difference at the old vertices unchanged when the new vertices are inserted. Using the symbol of the subdivision scheme, we show that the limit curve is at least continuous. Furthermore, the conditions imposed on the initial points are also discussed, thus the given limit functions are both monotonicity preserving and convexity preserving [Copyright &y& Elsevier]

Abstract: Net subdivision schemes recursively refine nets of univariate continuous functions defined on the lines of planar grids, and generate as limits bivariate continuous functions. In this paper a family of interpolatory net subdivision schemes related to the family of Dubuc–Deslauriers interpolatory subdivision schemes is constructed and analyzed. The construction is based on Gordon blending interpolants to nets of univariate functions, and on a particular class of blending functions with properties related to the Dubuc–Deslauriers schemes. The general analysis tools for net subdivision schemes, developed in a previous paper by the authors, together with the properties of the blending functions, lead to the proof of the convergence of these schemes to limit functions having the same integer smoothness as the limits of the corresponding Dubuc–Deslauriers schemes. These results are proved for net subdivision schemes corresponding to the first 84 members of the Dubuc–Deslauriers family, and conjectured for the rest. A concrete example of a family of piecewise polynomial blending functions is considered, together with the corresponding family of net subdivision schemes. The performance of the first two net subdivision schemes in this family is demonstrated by two examples. [Copyright &y& Elsevier]

Abstract: The prospect of applying spline spaces over T-subdivisions to adaptive isogeometric analysis is an exciting one. One major issue with spline spaces over T-subdivisions is in providing proper bases (shape functions) for finite element analysis. In this paper, we propose a method for the construction of hierarchical bases of a spline space with highest order smoothness over a consistent hierarchical T-subdivision. Our method is induced by the surjection condition, and this set of basis functions is hierarchically adaptive. We also present a concrete set of non-negative hierarchical bases over a T-subdivision and apply them in adaptive finite element analysis. [Copyright &y& Elsevier]

Abstract: This paper presents a new perspective for constructing interpolatory subdivision from primal approximating subdivision. The basic idea is constructing the subdivision rule for new inserted vertices of a new interpolatory subdivision scheme based on an approximating subdivision algorithm applied to a local configuration of the mesh with one vertex updated for interpolation of the vertex. This idea is demonstrated by presenting two new interpolatory subdivision schemes based on Catmull–Clark subdivision for an arbitrary polygonal mesh and Loop subdivision for a triangular mesh, respectively. These algorithms are simple and have a small stencil for computing new points. The new perspective also shows a link between those classic approximating and interpolatory subdivision algorithms such as cubic B-spline curve subdivision and the four-point interpolatory subdivision, Catmull–Clark subdivision and Kobbeltʼs interpolatory scheme, and Loop subdivision and the butterfly algorithm. [Copyright &y& Elsevier]

Abstract: We consider in this paper the convergence of Hermite subdivision schemes related to the irregular grids . It is reduced to the convergence of a scalar subdivision scheme, so the results concerning perturbation of convergent scalar subdivision schemes are applied. For interpolatory subdivision schemes , with the irregular grid which is equivalent to the regular grid in a suitable sense, we proved that the convergence of is implied by that of . Our arguments work in the settings of -convergence and uniform convergence. [Copyright &y& Elsevier]

Abstract: We introduce a non-uniform subdivision algorithm that partitions the neighborhood of an extraordinary point in the ratio , where . We call σ the speed of the non-uniform subdivision and verify continuity of the limit surface. For , the Catmull–Clark limit surface is recovered. Other speeds are useful to vary the relative width of the polynomial spline rings generated from extraordinary nodes. [Copyright &y& Elsevier]

Abstract: Recently the study and construction of quad/triangle subdivision schemes have attracted attention. The quad/triangle subdivision starts with a control net consisting of both quads and triangles and produces finer and finer meshes with quads and triangles. The use of the quad/triangle structure for surface design is motivated by the fact that in CAD modelling, the designers often want to model certain regions with quad meshes and others with triangle meshes to get better visual quality of subdivision surfaces. Though the smoothness analysis tool for regular quad/triangle vertices has been established and and quad/triangle schemes (for regular vertices) have been constructed, there is no interpolatory quad/triangle schemes available in the literature. The problem for this is probably that since the template sizes of the local averaging rules of interpolatory schemes for either quad subdivision or triangle subdivision are big, an interpolatory quad/triangle scheme will have large sizes of local averaging rule templates. In this paper we consider matrix-valued interpolatory quad/triangle schemes. In this paper, first we show that both scalar-valued and matrix-valued quad/triangle subdivision scheme can be derived from a nonhomogeneous refinement equation. This observation enables us to treat polynomial reproduction of scalar-valued and matrix-valued quad/triangle schemes in a uniform way. Then, with the result on the polynomial reproduction of matrix-valued quad/triangle schemes provided in our accompanying paper, we obtain in this paper a smoothness estimate for matrix-valued quad/triangle schemes, which extends the smoothness analysis of Levin–Levin from the scalar-valued setting to the matrix-valued setting. Finally, with this smoothness estimate established in this paper, we construct matrix-valued interpolatory quad/triangle scheme (for regular vertices) with the same sizes of local averaging rule templates as those of Stam–Loop''s quad/triangle scheme. We also obtain matrix-valued interpolatory quad/triangle scheme (for regular vertices) with reasonable sizes of local averaging rule templates. [Copyright &y& Elsevier]

Abstract: We present a 3-point approximating non-stationary subdivision scheme for designing curves. The weights of the masks of the scheme are defined in terms of some values of trigonometric B-spline basis functions. The scheme has some interesting properties and it is compared with the 2-point and 3-point schemes generating uniform trigonometric spline curves of order 3 and 5. The comparison and the performance of the scheme are demonstrated by examples. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to study and construct matrix-valued templates for interpolatory curve subdivision. Since our investigation of this problem was motivated by the need of such subdivision stencils as boundary templates for interpolatory surface subdivision, we provide both spline and non-spline templates that are necessarily symmetric, due to the lack of direction-orientation in carrying out surface subdivision in general. For example, the minimum-supported Hermite interpolatory cubic spline curve subdivision scheme, with the skew-symmetric basis function for interpolating first derivatives, does not meet the symmetry specification. Non-spline interpolatory templates constructed in this paper are particularly important, due to their smaller support needed to minimize undesirable surface oscillations, when adopted as boundary templates for interpolatory surface subdivision. The curve subdivision templates introduced in this paper are adopted as boundary stencils for interpolatory surface subdivision with matrix-valued templates. [Copyright &y& Elsevier]

• Discrete operators acting directly on meshes made of arbitrary polygons are proposed. • Vector fields are discretized as 1-forms and methods for their design are provided. • Lie advection of vector fields and functions on general polygonal meshes is presented. • Helmholtz-Hodge decomposition of vector fields on polygonal meshes is made possible. • Discrete exterior calculus is extended from triangle to general polygonal meshes. Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three 'basic operators' we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons. [ABSTRACT FROM AUTHOR]

Exponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to reproduce a certain space of exponential polynomials, then it must be level-dependent, with rules depending on the frequencies (and eventual multiplicities) defining the considered space. This work discusses a general strategy that exploits annihilating operators to locally detect those frequencies directly from the given data and therefore to choose the correct subdivision rule to be applied. This is intended as a first step towards the construction of self-adapting subdivision schemes able to locally reproduce exponential polynomials belonging to different spaces. An application of the proposed strategy is shown explicitly on an example involving the classical butterfly interpolatory scheme. This particular example is the generalization of what has been done for the univariate case in Donat and López-Ureña (2019) , which inspired this work. [ABSTRACT FROM AUTHOR]

• We define purely geometrical subdivision schemes generating curves on 2-dimensional unit sphere. • We derive sufficient conditions for the convergence and G 1 -Continuity of these spherical limit curves. • We give a spherical version of the planar incenter subdivision scheme. We define general geometric subdivision schemes generating curves on the 2-dimensional unit sphere by using geodesic polygons and the spherical distance. We show that a spherical interpolatory geometric subdivision scheme is convergent if the sequence of maximum edge lengths is summable and the limit curve is G 1 -continuous if in addition the sequence of maximum angular defects is summable. In particular, we study the case of bisector interpolatory schemes. Some experimental examples are given to demonstrate the excellent properties of these schemes. [ABSTRACT FROM AUTHOR]

• Polyhedral meshes are obtained through the use of tetrahedral subdivisions and dual cells. • Each polyhedral-dual cell is formulated as a finite element. • The resulting polyhedral discretizations are used for applications in nonlinear solid mechanics. • Verification examples are presented showing optimal convergence rates. • Results are comparable to those obtained using conventional hexahedral meshes. General polyhedral discretizations offer several advantages over classical approaches consisting of standard tetrahedra and hexahedra. These include increased flexibility and robustness in the meshing of geometrically complex domains and higher-quality solutions for both finite element and finite volume schemes. Currently, the use of general polyhedra is hampered by the lack of general-purpose polyhedral meshing algorithms and software. One approach for generating polyhedral meshes is the use of tetrahedral subdivisions and dual-cell aggregation. In this approach, each tetrahedron of an existing tetrahedral mesh is subdivided using one of several subdivision schemes. Polyhedral-dual cells may then be formed and formulated as finite elements with shape functions obtained through the use of generalized barycentric coordinates. We explore the use of dual-cell discretizations for applications in nonlinear solid mechanics using a displacement-based finite element formulation. Verification examples are presented that yield optimal rates of convergence. Accuracy of the methodology is demonstrated via several nonlinear examples that include large deformation and plasticity. [ABSTRACT FROM AUTHOR]