We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally ...C4-smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the L2-norm as well as to equivalents of the Hs-seminorms for 1≤s≤4.
In the studied case of spline degree p=9, the numerical results indicate in case of the Greville points a convergence of order O(hp−3) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order O(hp−2) for all (semi)norms except for the equivalent of the H4-seminorm, where the order O(hp−3) is anyway optimal.
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 Dual–Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for non-conforming multi-patch discretizations of a generalized Poisson problem. We realize the coupling between the ...patches using a symmetric interior penalty discontinuous Galerkin (SIPG) approach. Previously, we have assumed that the interfaces between patches always consist of whole edges. In this paper, we drop this requirement and allow T-junctions. This extension is vital for the consideration of sliding interfaces, for example between the rotor and the stator of an electrical motor. One critical part for the handling of T-junctions in IETI-DP solvers is the choice of the primal degrees of freedom. We propose to add all basis functions that are non-zero at any of the vertices to the primal space. Since there are several such basis functions at any T-junction, we call this concept “fat vertices”. For this choice, we show a condition number bound that coincides with the bound for the conforming case.
We present an isogeometric framework based on collocation to construct a C2-smooth approximation of the solution of the Poisson’s equation over planar bilinearly parameterized multi-patch domains. ...The construction of the used globally C2-smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi-patch configurations. The basis of the C2-smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case 1, which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. 2–4) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.
•Isogeometric collocation method for solving Poisson’s equation over planar multi-patch domains.•As discretization space a globally C2-smooth isogeometric spline space is developed and used.•Two different choices of collocation points are numerically investigated.•Numerical experiments demonstrate the potential of the collocation method.
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 present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a ...globally C2-smooth isogeometric spline space which is used as discretization space. The generated C2-smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and numerical examples indicate that they are well-conditioned. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are obtained by computing the null space of a small system of linear equations. Several examples demonstrate the potential of our approach for solving the triharmonic equation.
Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. Typically, quadrilateral patches are adopted in both frameworks. We consider ...the particular class of multi-patch parametrizations that are analysis-suitable G1 (AS-G1), which is a specific geometric continuity definition which allows to construct, on the multi-patch domain, C1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016). It was demonstrated in Kapl et al. (2018) that AS-G1 multi-patch parametrizations are suitable for modeling complex planar multi-patch domains.
We construct a local basis, and an associated dual basis, for a specific C1 isogeometric spline space A over a given AS-G1 multi-patch parametrization. The space A is C1 across interfaces and C2 at all vertices, and is therefore a subspace of the entire C1 isogeometric space V1. At the same time, A allows optimal approximation of traces and normal derivatives along the interfaces and reproduces all derivatives up to second order at the vertices. In contrast to V1, the dimension of A does not depend on the domain parametrization.
This paper also contains numerical experiments which exhibit the optimal approximation order in L2 and L∞ of the isogeometric space A and demonstrate the applicability of our approach for isogeometric analysis.
•We define a C1-smooth isogeometric subspace over multi-patch domains.•The isogeometric subspace is C1 across edges and C2 at vertices.•We present a basis construction for the isogeometric subspace.•The isogeometric subspace displays optimal approximation properties.
A spline space suitable for Isogeometric Analysis (IgA) on multi-patch domains is presented. Our construction is motivated by emerging requirements in isogeometric simulations. In particular, IgA ...spaces should allow for adaptive mesh refinement and they should guarantee the optimal smoothness of the discretized solution, even across interfaces of adjacent patches.
Given a domain manifold M consisting of individual patches (isomorphic to the unit square or cube) that are glued together along interfaces, we present a construction of multi-patch B-splines defined on them. Their smoothness is enhanced by locally modifying or merging basis functions around the boundary of each patch. The resulting multi-patch B-splines with enhanced smoothness (MPBES) possess the property of local linear independence and form a non-negative partition of unity. Moreover, their span can be characterized as the linear space of all piecewise polynomial functions on the domain manifold that possess certain smoothness properties.
Subsequently, adaptively refined MPBES are obtained by generalizing the construction of truncated hierarchical (TH) B-splines. More precisely, a nested sequence of spaces spanned by MPBES is considered, corresponding to steps of local enrichment. In addition, an inversely nested sequence of subdomains (which are submanifolds of M) is used to specify the local refinement level of functions in these spaces. Finally, truncated hierarchical MPBES are obtained by means of the selection and truncation mechanism of THB-splines. The desired properties of linear independence and convex partition of unity are maintained.
The paper presents several numerical examples which demonstrate potential applications of the new basis in isogeometric analysis.
As a remarkable difference to the existing CAD technology, where shapes are represented by their boundaries, FEM-based isogeometric analysis typically needs a parameterization of the interior of the ...domain. Due to the strong influence on the accuracy of the analysis, methods for constructing a good parameterization are fundamentally important. The flexibility of single patch representations is often insufficient, especially when more complex geometric shapes have to be represented. Using a multi-patch structure may help to overcome this challenge.
In this paper we present a systematic method for exploring the different possible parameterizations of a planar domain by collections of quadrilateral patches. Given a domain, which is represented by a certain number of boundary curves, our aim is to find the optimal multi-patch parameterization with respect to an objective function that captures the parameterization quality. The optimization considers both the location of the control points and the layout of the multi-patch structure. The latter information is captured by pre-computed catalogs of all available multi-patch topologies. Several numerical examples demonstrate the performance of the method.
•A method for finding a multi-patch parameterization for a domain is presented.•The control points as well as the layout of the multi-patch structure are considered.•Pre-computed catalogs of all available multi-patch topologies are used.
Isogeometric Analysis is a high-order discretization method for boundary value problems that uses a number of degrees of freedom which is as small as for a low-order method. Standard isogeometric ...discretizations require a global parameterization of the computational domain. In non-trivial cases, the domain is decomposed into patches having separate parameterizations and separate discretization spaces. If the discretization spaces agree on the interfaces between the patches, the coupling can be done in a conforming way. Otherwise, non-conforming discretizations (utilizing discontinuous Galerkin approaches) are required. The author and his coworkers have previously introduced multigrid solvers for Isogeometric Analysis for the conforming case. In the present paper, these results are extended to the non-conforming case. Moreover, it is shown that the multigrid solves get even more powerful if the proposed smoother is combined with a (standard) Gauss–Seidel smoother.
•We consider conforming and non-conforming multi-patch Isogeometric Analysis.•We propose multigrid solvers that are robust in the spline degree and the grid size.•Hybrid smoothers appear to reduce the effect of the geometry on the convergence.•We give convergence theory that substantiate our findings.
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