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A sharp degree bound on G 2 -refinable multi-sided surfaces
Abstract
Refinement of a space of splines should yield additional degrees of freedom for modeling and engineering analysis, both along boundaries and in the interior. Yet such additional flexibility fails to materialize for multi-sided G2 surface constructions when the polynomial degree is too low.
This paper establishes a tight lower bound on the polynomial degree of flexibility-increasing refinable multi-sided G2 surface constructions within a C2 spline complex – by ruling out bi-5 constructions and by exhibiting a multi-sided bi-6 construction that yields good highlight line and curvature distributions. The bi-6 construction consists of one 2 × 2 macro-patch for each of the n sectors that join to form the multi-sided surface.
... This paper may be viewed as a prequel and complementary to the recent publication [6] that established that flexibly G 2 -refinable multi-sided surfaces for inclusion into C 2 bi-cubic splines require degree no less than bi-6; and presented a bi-6 construction. Compared to the present G 1 scenario, the G 2 flexibility bounds, proofs and the corresponding construction of multi-sided surfaces are technically considerably more complicated and so, potentially, obscure some of the underlying principles. ...
... The tiny cap uses the layout of the once-refined main construction, see Fig. 14a, and the linear b 0 (u) := b(u) and quadratic b 1 (u) := b(u) are split into b k (u) as in Section 5, i.e. so that by (6) ...
Geometrically smooth spline surfaces, generalized to include n-sided facets or configurations of n≠4 quads, can exhibit a curious lack of additional degrees of freedom for modelling or engineering analysis when refined.
This paper establishes a minimal polynomial degree for smooth constructions of multi-sided surfaces that guarantees more flexibility in all directions under refinement. Degree bi-4 is both necessary and sufficient for flexibility-increasing G1-refinability within a bi-quadratic C1 spline complex. Sufficiency is proven by two alternative flexibly G1- refinable constructions exhibiting good highlight line distributions.
This survey of piecewise polynomial surface constructions for filling multi-sided holes in a smooth spline complex focusses on a class of hybrid constructions that, while heterogeneous, combines all the practical advantages of state-of-the-art for modelling and analysis: good shape, easy implementation and simple refinability up a pre-defined level. After reviewing the three ingredients—subdivision, G-spline and guided surfaces—the hybrid is defined to consist essentially of one macro-patch for each of n sectors, leaving just a tiny n-sided central hole to be filled by a G-spline construction. Here tiny means both geometrically small, e.g., two orders of magnitude smaller than pieces of the spline complex, and small in its contribution to engineering analysis, i.e., it is unlikely to require further refinement to express additional geometric detail or resolve a function on the surface, such as the solution of a partial differential equation. Each macro-patch has the local structure of a subdivision surface near, but excluding the central, extraordinary point: all internal transitions are the identity or scale according to contraction speed toward the extraordinary point. Both the number of pieces of the macro-patches and the speed can be chosen application-dependent and adaptively.
Analysis-suitable T-splines (AST-splines) are a promising candidate to achieve a seamless integration between the design and the analysis of thin-walled structures in industrial settings. In this work, we allow multiple extraordinary points per face, i.e., we remove the restriction of preceding works that required extraordinary points to be at least four rings apart from each other. We do so by mathematically showing that AST-splines with multiple extraordinary points per face are linearly independent and their polynomial basis functions form a non-negative partition of unity. This extension of the subset of AST-splines drastically increases the flexibility to build geometries using AST-splines; e.g., much coarser meshes can be constructed around small holes. The AST-spline spaces detailed in this work have C1 inter-element continuity near extraordinary points and C2 inter-element continuity elsewhere. For the convergence studies performed in this paper involving second- and fourth-order linear elliptic problems with manufactured solutions, we have not found any drawback caused by allowing multiple EPs per face in either the first refinement levels or the asymptotic behavior. To illustrate a possible isogeometric framework that is already available, we design the B-pillar and the side outer panel of a car using T-splines with the commercial software Autodesk Fusion360, import the control nets into our in-house code to build AST-splines, and import the Bézier extraction information into the commercial software LS-DYNA to solve eigenvalue problems. The results are compared with conventional finite elements and good agreement is found between AST-splines and conventional finite elements.
This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson’s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, particularly for isogeometric constructions based on Catmull–Clark elements, the need to carefully choose quadrature formulas. The seven methods include two that are new to isogeometric analysis. Both new methods yield O(h3) convergence in the L2 norm, also when points are included where n 6≠ 4 pieces meet. One construction is based on a polar, singular parameterization; the other is a G1 tensor-product construction.
We present an algorithm for completing a C
2 surface of up to degree bi-6 by capping an n-sided hole with polar layout. The cap consists of n tensor-product patches, each of degree 6 in the periodic and degree 5 in the radial direction. To match the polar layout,
one edge of these patches is collapsed.
We explore and compare with alternative constructions, based on more pieces or using total-degree, triangular patches.
Compared to G
k
continuity, C
k
continuity simplifies the construction of functions on surfaces and their refinement, e.g. to solve differential equations on the surface. The new class of almost everywhere parametrically C2 free-form surfaces provide such a parameterization. For example, a new bi-6 construction combines a fast-contracting C2 guided subdivision surface with a tiny multi-sided G1 cap. The cap is chosen to be smaller than any refinement anticipated for geometric modeling or computing on surfaces. Fast contraction means that one subdivision step shrinks the remaining hole more than three steps of Catmull-Clark subdivision. This yields smooth surfaces consisting of a finite number of pieces that are suitable for engineering practice. Both the subdivision construction and the cap are guided by a reference surface. This guide conveys the basic shape, but has a different structure and lower smoothness.
Tools of subdivision theory allow constructing surfaces that are effectively C² and have a good highlight line distribution also near irregularities where more or fewer than four quadrilateral patches meet. Here, effectively C² means that transitions that are only first-order smooth are confined to a tiny multi-sided cap. The cap can be chosen smaller than any refinement required for geometric modeling or computing on surfaces. The remainder of the surface parameterization is C². The resulting surface is of degree bi-5 and consists of three or fewer surface rings that are rapidly converging towards the tiny central cap.
To be directly useful both for shape design and a thin shell analysis, a surface representation has to satisfy three properties: (1) be compatible with CAD surface representations, (2) yield generically a good highlight distribution, and (3) offer a refinable space of functions on the surface. Here we propose a new construction, based on a number of recently-developed techniques, that satisfies all three criteria. The construction converts quad meshes with irregularities, where more or fewer than four quads meet, to C¹ (or, at the cost of more pieces, C²) bi-4 surface consisting of subdivision rings for the main body completed by a tiny G¹ cap.
For two high-quality piecewise polynomial geometrically smooth ( ) surface constructions, explicit functions are derived that form the basis of a functions space on the surfaces. The spaces are refinable and nested, i.e. the functions can be re-represented at a finer level. By choosing all basis functions to be first order smooth a maximal set of degrees of freedom is obtained that have small support and near-uniform layout.
The space of -smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is considered. The investigation of the dimension of the spaces of biquintic and bisixtic -smooth geometrically continuous isogeometric functions on such domains is presented. In addition, -smooth isogeometric functions are constructed to be used for performing -approximation and for solving triharmonic equation on different two-patch geometries. The numerical results indicate optimal approximation order.
We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions, encoded in a so-called isogeometric analysis suitable extraction operator, yield smooth polar splines for any . We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for are presented. Optimal approximation behavior is observed numerically, and examples of applications to free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.
This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
We generate a basis of the space of bicubic and biquartic C1-smooth geometrically continuous isogeometric functions on bilinear multi-patch domains Ω⊂R2. The basis functions are obtained by suitably combining C1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains (cf. [18]). They are described by simple explicit formulas for their spline coefficients.
These C1-smooth isogeometric functions possess potential for applications in isogeometric analysis, which is demonstrated by several examples (such as the biharmonic equation). In particular, the numerical results indicate optimal approximation power.
One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p-degree splines (and extensions, such as NURBS), they enjoy up to Cp−1 continuity within each patch. However, global continuity beyond C° on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only C° at the patch interface. On such domains we study the h-refinement of C¹-continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C¹-continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently (Kapl et al., 2015b) has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C¹ splines. This is the starting point of our study. We introduce the class of analysis-suitable G¹ geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C¹ isogeometric spaces over analysis-suitable G¹ parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C¹ isogeometric spaces is prevented.
This paper addresses a gap in the arsenal of curvature continuous tensor-product spline constructions: an algorithm is provided to fill n-sided holes in C2 bi-3 spline surfaces using n patches of degree bi-6. Numerous experiments illustrate good highlight line and curvature distribution on the resulting surfaces.
Quad meshes can be interpreted as tensor-product spline control meshes as long as they form a regular grid, locally. We present a new option for complementing bi-3 splines by bi-4 splines near irregularities in the mesh layout, where less or more than four quadrilaterals join. These new generalized surface and IGA (isogeometric analysis) elements have as their degrees of freedom the vertices of the irregular quad mesh. From a geometric design point of view, the new construction distinguishes itself from earlier work by a notably better distribution of highlight lines. From the IGA point of view, increased smoothness and reproduction at the irregular point yield fast convergence.
(geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched constructions for geometry and analysis automatically yield isogeometric elements. This provides a formal framework for the existing and any future isogeometric elements based on geometric continuity.
Shape optimal design of an elastic structure is formulated using a design element technique. It is shown that Bezier and B-spline curves, typical of the CAD philosophy, are well suited to the definition of design elements. Complex geometries can be described in a very compact way by a small set of design variables and a few design elements. Because of the B-splines flexibility, it is no longer necessary to piece design elements together in order to agree with the shape complexity, nor to restrict the shape variations. Moreover, the additional optimization constraints that are most often needed to avoid unrealistic designs when the shape variables are the nodal coordinates of a finite element mesh, are automatically taken into account in the new formulation. An analytical derivation of the sensitivity analysis will be established, giving rise to numerical efficiency. It will be seen that the resulting optimization problem does not involve highly nonlinear functions with respect to the shape variables, so that simple mathematical programming algorithms can be applied to solve it. Some numerical examples are offered to demonstrate the power and generality of the new approach presented in this paper.
We present a second order smooth filling of an n-valent Catmull-Clark spline ring with n biseptic patches. While an underdetermined biseptic solution to this problem has appeared previously, we make several advances in this paper. Most notably, we cast the problem as a constrained minimization and introduce a novel quadratic energy functional whose absolute minimum of zero is achieved for bicubic polynomials. This means that for the regular 4-valent case, we reproduce the bicubic B-splines. In other cases, the resulting surfaces are aesthetically well behaved. We extend our constrained minimization framework to handle the case of input mesh with boundary. Categories and Subject Descriptors (according to ACM CCS): I.3.5 (Computer Graphics): Curve, surface, solid, and object representations
This paper presents some research results obtained recently using the of the finite element method (FEM) for shape optimal design. The use of Bezier and B-spline curves to define design elements has proven to be an excellent way to model the geometry of the design problem. The 2D elastic element was extended to employ part of a Bezier or B-spline curve as its element side for this purpose. This new element has been tested successfully with the patch test. Moreover, it is compatible, has no preferred direction, and contains all the required rigid-body modes (three zero eigenvalues are found in the element stiffness matrix).There are several advantages in using the FEM for shape optimal design. The analysis and design models are often identical. In the FEM, the stresses along an element side are as accurate as those inside the element. This feature is important because usually the critical stress constraints are found along the design element boundary. The final design can very easily be reanalyzed with a different element order to check the accuracy of the result. Some classical shape optimal design problems have been solved using the Conlin optimizer. The results indicate that similar optimal shapes can be obtained with fewer degrees of freedom than when compared with the FEM. As with the , less than ten structural analyses are sufficient for convergence in most of the problems.
The isoparametric spline finite strip methods for Mindlin plate bending, plane stress and plane strain problems are presented. The essence of the method lies in the use of uniform cubic curves in the modelling of the geometry and representation of the displacement field. In this paper, the general theory of isoparametric spline finite strip for analysis of plane structures is outlined. The method, when applied to most problems, yields a relatively narrow band matrix and requires little computational effort. Solutions of this method are compared with other available experimental, analytical and numerical solutions, and in all cases very good agreement is observed.
This paper presents a geometry based approach for coupling CAD with finite element methods. Non uniform rational B-splines (NURBs), known from computer aided geometric design, are used to describe the shape of a structure. NURB curves and surfaces and Gordon surfaces defined by NURB curves are introduced for design description. These geometries are used directly for the mapping of finite elements and for parameterization for the optimization of the structural shape. The cross-sectional torsion problem is employed to demonstrate the proposed mapping techniques for finite elements. Solid and thin walled cross-sectional shapes are treated differently. Hierarchical shape functions will be used for the approximation of the warping function to solve the torsion problem. The proposed mapping techniques are applied to shape optimization. A sensitivity analysis using NURB defined geometries is developed. Two optimization solutions are given to complete the presentation.
This paper derives strong relations that boundary curves of a smooth complex of patches have to obey when the patches are computed by local averaging. These relations restrict the choice of reparameterizations for geometric continuity.In particular, when one bicubic tensor–product B-spline patch is associated with each facet of a quadrilateral mesh with n-valent vertices and we do not want segments of the boundary curves forced to be linear, then the relations dictate the minimal number and multiplicity of knots: For general data, the tensor–product spline patches must have at least two internal double knots per edge to be able to model a G1-connected complex of C1 splines. This lower bound on the complexity of any construction is proven to be sharp by suitably interpreting an existing surface construction. That is, we have a tight bound on the complexity of smoothing quad meshes with bicubic tensor–product B-spline patches.
Many engineering design applications require geometric modeling and mechanical simulation of thin flexible structures, such as those found in the automotive and aerospace industries. Traditionally, geometric modeling, mechanical simulation, and engineering design are treated as separate modules requiring different methods and representations. Due to the incompatibility of the involved representations the transition from geometric modeling to mechanical simulation, as well as in the opposite direction, requires substantial effort. However, for engineering design purposes efficient transition between geometric modeling and mechanical simulation is essential. We propose the use of subdivision surfaces as a common foundation for modeling, simulation, and design in a unified framework. Subdivision surfaces provide a flexible and efficient tool for arbitrary topology free-form surface modeling, avoiding many of the problems inherent in traditional spline patch based approaches. The underlying basis functions are also ideally suited for a finite-element treatment of the so-called thin-shell equations, which describe the mechanical behavior of the modeled structures. The resulting solvers are highly scalable, providing an efficient computational foundation for design exploration and optimization. We demonstrate our claims with several design examples, showing the versatility and high accuracy of the proposed method.
It is shown that parametrical smoothness conditions are sufficient for modeling smooth spline surfaces of arbitrary topology if degenerate surface segments are accepted. In general, degeneracy, i.e., vanishing partial derivatives at extraordinary points, is leading to surfaces with geometrical singularities. However, if the partial derivatives of higher order satisfy certain conditions, the existence of a regular smooth reparametrization can be guaranteed. So, degeneracy is no fundamental obstacle to generating surfaces which are smooth in the sense of differential geometry. Besides its striking simplicity the approach presented here admits the construction of smooth spline spaces which have a natural refinement property. Thus, various algorithms based on subdivision of tensor product B-spline surfaces become available for surfaces of general type.
The concept of isogeometric analysis is proposed. Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h- and p-refinement schemes are presented and a new, more efficient, higher-order concept, k-refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. A k-refinement strategy is shown to converge toward monotone solutions for advection–diffusion processes with sharp internal and boundary layers, a very surprising result. It is argued that isogeometric analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.
In this paper we show how one can construct a regularly parametrized G2-spline surface of arbitrary topology from one control net. The surface is piecewise bisextic around extraordinary points and bicubic elsewhere. Furthermore, the bisextic representation of the surface allows for subdivision algorithms. The underlying ideas can also be used to construct subdividable Gk-splines of bidegree 2k + 2.
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Isogeometric analysis for subdivision surfaces
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