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.
•Construction of a basis for bicubic and biquartic C1-smooth isogeometric functions on planar bilinear multi-patch domains.•The basis functions are described by simple explicit formulas for their spline coefficients.•Numerical experiments (e.g. solving the biharmonic equation) showed optimal rates of convergence.
We study the linear space of Cs-smooth isogeometric functions defined on a multi-patch domain Ω⊂R2. We show that the construction of these functions is closely related to the concept of geometric ...continuity of surfaces, which has originated in geometric design. More precisely, the Cs-smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order (Gs-smoothness) of their graph surfaces. This motivates us to call them Cs-smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of C1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing L2 approximation and for solving Poisson’s equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.
We study the space of C2-smooth isogeometric functions on bilinearly parameterized multi-patch domains Ω⊂R2, where the graph of each isogeometric function is a multi-patch spline surface of ...bidegree (d,d), d∈{5,6}. The space is fully characterized by the equivalence of the C2-smoothness of an isogeometric function and the G2-smoothness of its graph surface (cf. Groisser and Peters (2015), Kapl et al. (2015)). This is the reason to call its functions C2-smooth geometrically continuous isogeometric functions.
In particular, the dimension of this C2-smooth isogeometric space is investigated. The study is based on the decomposition of the space into three subspaces and is an extension of the work Kapl and Vitrih (2017) to the multi-patch case. In addition, we present an algorithm for the construction of a basis, and use the resulting globally C2-smooth functions for numerical experiments, such as performing L2 approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.
The construction of smooth surfaces of complex shapes is at the heart of computer-aided design (CAD). Many different approaches generating C1-smooth surfaces are available and well-studied. ...Isogeometric analysis (IGA) has sparked new interest in these methods, since it allows to incorporate CAD based parameterizations into numerical simulations. In IGA one can utilize shape functions of global C1 continuity (or of higher continuity) over multi-patch geometries. Such functions can then be used to discretize high order partial differential equations, such as the biharmonic equation. However, the requirements posed by the IGA simulation are often different from the requirements in CAD. The construction ofC1-smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations to ensure that the resulting C1 isogeometric spaces possess optimal approximation properties. For this purpose, we select so-called analysis-suitable G1 (AS-G1) parameterizations, proposed in Collin et al. (2016).
In this work, we show through examples that it is possible to construct AS-G1 multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-G1 multi-patch parameterization possessing the same boundary, the same vertices and the same first derivatives at the vertices, and which is as close as possible to this initial geometry. Our algorithm is based on a quadratic optimization problem with linear side constraints. Numerical tests also confirm that C1 isogeometric spaces over AS-G1 multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced.
•Algorithm to construct analysis-suitable (AS) G1 multi-patch parameterizations of planar domains.•AS-G1 parameterizations are needed to define C1 isogeometric spaces with optimal approximation properties.•Method is simple and requires only to solve a system of linear equations.•Several examples to demonstrate the potential of our algorithm and to show the flexibility of AS-G1 geometries.
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We analyze the spaces of trivariate C1–smooth isogeometric functions on two-patch domains. Our aim is to generalize the corresponding results from the bivariate (Kapl et al. (2015) ...25) to the trivariate case. In the first part of the paper, we introduce the notion of gluing data and use it to define glued spline functions on two-patch domains. Applying the fundamental observation that “matched Gk–constructions always yield Ck–continuous isogeometric elements”, see Groisser and Peters (2015) 14, to graph hypersurfaces in four-dimensional space, allows us to characterize C1–smooth geometrically continuous isogeometric functions as the push-forwards of these functions for suitable gluing data. The second part of the paper is devoted to various special classes of gluing data. We analyze how the generic dimensions depend on the number of knot spans (elements) and on the spline degree. Finally we show how to construct locally supported basis functions in specific situations.
We study the dimension and construct a basis for C1-smooth isogeometric function spaces over two-patch domains. In this context, an isogeometric function is a function defined on a B-spline domain, ...whose graph surface also has a B-spline representation. We consider constructions along one interface between two patches. We restrict ourselves to a special case of planar B-spline patches of bidegree (p,p) with p≥3, so-called analysis-suitable G1 geometries, which are derived from a specific geometric continuity condition. This class of two-patch geometries is exactly the one which allows, under certain additional assumptions, C1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016).
Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation, using the isogeometric method. In particular, we analyze the dimension of the C1-smooth isogeometric space and present an explicit representation for a basis of this space. Both the dimension of the space and the basis functions along the common interface depend on the considered two-patch parameterization. Such an explicit, geometry dependent basis construction is important for an efficient implementation of the isogeometric method. The stability of the constructed basis is numerically confirmed for an example configuration.
The space of C2-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 C2-smooth geometrically continuous isogeometric functions on such domains is presented. In addition, C2-smooth isogeometric functions are constructed to be used for performing L2-approximation and for solving triharmonic equation on different two-patch geometries. The numerical results indicate optimal approximation order.
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