arXiv subjects

Mario Kapl

Publications and source records attributed to Mario Kapl.

23 records · Page 2Linked to original sources

arXiv2018

The Argyris isogeometric space on unstructured multi-patch planar domains

Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. We deal with a particular class of $C^0$ planar multi-patch spline parametrizations called analysis-suitable $G^1$ (AS-$G^{1}$) multi-patch parametrizations (Collin, Sangalli, Takacs; CAGD, 2016). This class of parametrizations has to satisfy specific geometric continuity constraints, and is of importance since it allows to construct, on the multi-patch domain, $C^1$ isogeometric spaces with optimal approximation properties. It was demonstrated in (Kapl, Sangalli, Takacs; CAD, 2018) that AS-$G^1$ multi-patch parametrizations are suitable for modeling complex planar multi-patch domains. In this work, we construct a basis, and an associated dual basis, for a specific $C^1$ isogeometric spline space $\mathcal{W}$ over a given AS-$G^1$ multi-patch parametrization. We call the space $\mathcal{W}$ the Argyris isogeometric space, since it is $C^1$ across interfaces and $C^2$ at all vertices and generalizes the idea of Argyris finite elements to tensor-product splines. The considered space $\mathcal{W}$ is a subspace of the entire $C^1$ isogeometric space $\mathcal{V}^{1}$, which maintains the reproduction properties of traces and normal derivatives along the interfaces. Moreover, it reproduces all derivatives up to second order at the vertices. In contrast to $\mathcal{V}^{1}$, the dimension of $\mathcal{W}$ does not depend on the domain parametrization, and $\mathcal{W}$ admits a basis and dual basis which possess a simple explicit representation and local support. We conclude the paper with some numerical experiments, which exhibit the optimal approximation order of the Argyris isogeometric space $\mathcal{W}$ and demonstrate the applicability of our approach for isogeometric analysis.

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arXiv2017

Dimension and basis construction for $C^{2}$-smooth isogeometric spline spaces over bilinear-like $G^{2}$ two-patch parameterizations

A particular class of planar two-patch geometries, called bilinear-like $G^{2}$ two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like $G^2$ two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of $C^{2}$-smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the $C^2$-smoothness of an isogeometric function and the $G^2$-smoothness of its graph surface (cf. [12, 20]). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like $G^{2}$ two-patch parameterizations. Numerical results obtained by performing $L^{2}$-approximation indicate that already the subspace possesses optimal approximation properties.

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arXiv2017

Construction of analysis-suitable $G^1$ planar multi-patch parameterizations

Isogeometric analysis allows to define shape functions of global $C^{1}$ continuity (or of higher continuity) over multi-patch geometries. The construction of such $C^{1}$-smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations, so-called analysis-suitable $G^{1}$ (in short, AS-$G^{1}$) parameterizations, to ensure that the resulting $C^{1}$ isogeometric spaces possess optimal approximation properties, cf. [7]. In this work, we show through examples that it is possible to construct AS-$G^{1}$ multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-$G^{1}$ 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 $C^{1}$ isogeometric spaces over AS-$G^{1}$ multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced.

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arXiv2017

Space of $C^2$-smooth geometrically continuous isogeometric functions on planar multi-patch geometries: Dimension and numerical experiments

We study the space of $C^{2}$-smooth isogeometric functions on bilinearly parameterized multi-patch domains $Ω\subset \mathbb{R}^{2}$, where the graph of each isogeometric function is a multi-patch spline surface of bidegree $(d,d)$, $d \in \{5,6 \}$. The space is fully characterized by the equivalence of the $C^2$-smoothness of an isogeometric function and the $G^2$-smoothness of its graph surface, cf. (Groisser and Peters,2015; Kapl et al.,2015). This is the reason to call its functions $C^{2}$-smooth geometrically continuous isogeometric functions. In particular, the dimension of this $C^{2}$-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 $C^{2}$-smooth functions for numerical experiments, such as performing $L^{2}$ approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.

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arXiv2017

Dimension and basis construction for analysis-suitable $G^1$ two-patch parameterizations

We study the dimension and construct a basis for $C^1$-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 $p \geq 3$, so-called analysis-suitable $G^1$ 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, $C^1$ isogeometric spaces with optimal approximation properties (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 $C^1$-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.

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