72 research outputs found

    The Argyris isogeometric space on unstructured multi-patch planar domains

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    Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. We deal with a particular class of C0C^0 planar multi-patch spline parametrizations called analysis-suitable G1G^1 (AS-G1G^{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, C1C^1 isogeometric spaces with optimal approximation properties. It was demonstrated in (Kapl, Sangalli, Takacs; CAD, 2018) that AS-G1G^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 C1C^1 isogeometric spline space W\mathcal{W} over a given AS-G1G^1 multi-patch parametrization. We call the space W\mathcal{W} the Argyris isogeometric space, since it is C1C^1 across interfaces and C2C^2 at all vertices and generalizes the idea of Argyris finite elements to tensor-product splines. The considered space W\mathcal{W} is a subspace of the entire C1C^1 isogeometric space V1\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 V1\mathcal{V}^{1}, the dimension of W\mathcal{W} does not depend on the domain parametrization, and W\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 W\mathcal{W} and demonstrate the applicability of our approach for isogeometric analysis

    A comparison of smooth basis constructions for isogeometric analysis

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    In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modeling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-C1C^1, Analysis-Suitable G1G^1 and the Approximate C1C^1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C1C^1 and Analysis-Suitable G1G^1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-C1C^1 and D-Patch provide relatively easy construction on complex geometries. The Almost-C1C^1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C1C^1 and Analysis-Suitable G1G^1 applicable to more complex geometries

    Adaptive isogeometric methods with C1C^1 (truncated) hierarchical splines on planar multi-patch domains

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    Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of C1C^1 isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We introduce a new abstract framework for the definition of hierarchical splines, which replaces the hypothesis of local linear independence for the basis of each level by a weaker assumption. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by C1C^1 splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems

    Almost-C1C^1 splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems

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    Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-C1C^1 are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are C1C^1 smooth almost everywhere, that is, at all vertices and across most edges, and in addition almost (i.e. approximately) C1C^1 smooth across all other edges. Thus, the splines form H2H^2-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-C1C^1 splines are described in an explicit B\'ezier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behavior

    A family of C1C^1 quadrilateral finite elements

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    We present a novel family of C1C^1 quadrilateral finite elements, which define global C1C^1 spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by (Brenner and Sung, J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product degree p6p\geq 6, to all degrees p3p \geq 3. Thus, we call the family of C1C^1 finite elements Brenner-Sung quadrilaterals. The proposed C1C^1 quadrilateral can be seen as a special case of the Argyris isogeometric element of (Kapl, Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles. Just as for the Argyris triangle, we additionally impose C2C^2 continuity at the vertices. In this paper we focus on the lower degree cases, that may be desirable for their lower computational cost and better conditioning of the basis: We consider indeed the polynomial quadrilateral of (bi-)degree~55, and the polynomial degrees p=3p=3 and p=4p=4 by employing a splitting into 3×33\times3 or 2×22\times2 polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree pp. We show that the space provides optimal approximation order. Due to the interpolation properties, the error bounds are local on each element. In addition, we describe the construction of a simple, local basis and give for p{3,4,5}p\in\{3,4,5\} explicit formulas for the B\'{e}zier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed C1C^1 quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for p=5p=5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom

    A locally based construction of analysis-suitable G1G^1 multi-patch spline surfaces

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    Analysis-suitable G1G^1 (AS-G1G^1) multi-patch spline surfaces [4] are particular G1G^1-smooth multi-patch spline surfaces, which are needed to ensure the construction of C1C^1-smooth multi-patch spline spaces with optimal polynomial reproduction properties [16]. We present a novel local approach for the design of AS-G1G^1 multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS-G1G^1 multi-patch spline surface by approximating a given G1G^1-smooth but non-AS-G1G^1 multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS-G1G^1 multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, over them
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