Isogeometric analysis with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e966" altimg="si1275.svg"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>-smooth functions over multi-patch surfaces

Farahat, Andrea; Jüttler, Bert; Kapl, Mario; Takacs, Thomas

doi:10.1016/j.cma.2022.115706

Cited by 18 publications

(2 citation statements)

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“…When the shape of Ω is rather challenging or not convex, a segmentation in subdomains, called patches, with easier shape is usually advised and the description of Ω is thus obtained as an atlas whose charts are the individual bijective parameterizations for each patch. The main challenges of such approach consist in the identification of a suitable segmentation technique [16][17][18][19] and in the smooth transition between the patches [20,21].…”

Section: Introductionmentioning

confidence: 99%

Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

et al. 2023

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The generation of structured grids on bounded domains is a crucial issue in the development of numerical models for solving differential problems. In particular, the representation of the given computational domain through a regular parameterization allows us to define a univalent mapping, which can be computed as the solution of an elliptic problem, equipped with suitable Dirichlet boundary conditions. In recent years, Physics-Informed Neural Networks (PINNs) have been proved to be a powerful tool to compute the solution of Partial Differential Equations (PDEs) replacing standard numerical models, based on Finite Element Methods and Finite Differences, with deep neural networks; PINNs can be used for predicting the values on simulation grids of different resolutions without the need to be retrained. In this work, we exploit the PINN model in order to solve the PDE associated to the differential problem of the parameterization on both convex and non-convex planar domains, for which the describing PDE is known. The final continuous model is then provided by applying a Hermite type quasi-interpolation operator, which can guarantee the desired smoothness of the sought parameterization. Finally, some numerical examples are presented, which show that the PINNs-based approach is robust. Indeed, the produced mapping does not exhibit folding or self-intersection at the interior of the domain and, also, for highly non convex shapes, despite few faulty points near the boundaries, has better shape-measures, e.g., lower values of the Winslow functional.

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Section: Introductionmentioning

confidence: 99%

Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

et al. 2023

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“…Eg., in case of the Galerkin method, C 1 -smooth isogeometric functions are required for solving 4-th order PDEs, like the biharmonic equation, see e.g. [10,15,27,29,45,60], or the Kirchhoff-Love shell problem, see e.g. [16,19,36,39], and C 2 -smooth functions for solving 6-th order PDEs, like the triharmonic equation, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations

Kapl

Vitrih

2021

Adv Comput Math

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The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des. 52–53:75–89, 2017, Kapl et al. Comput. Aided Geom. Des. 69:55–75, 2019 and Kapl and Vitrih J. Comput. Appl. Math. 335:289–311, 2018, Kapl and Vitrih J. Comput. Appl. Math. 358:385–404, 2019 and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg. 360:112684, 2020 for the construction of C1-smooth and C2-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of Cs-smooth isogeometric multi-patch spline spaces of degree p, inner regularity r and of a smoothness s ≥ 1, with p ≥ 2s + 1 and s ≤ r ≤ p − s − 1. More precisely, we study for s ≥ 1 the space of Cs-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular Cs-smooth subspace of the entire Cs-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this Cs-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the Cs-smooth spline functions to perform L2 approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed Cs-smooth subspace.

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