Boundary correspondence of planar domains for isogeometric analysis based on optimal mass transport

Zheng, Yongliu; Pan, Maodong; Chen, Falai

doi:10.1016/j.cad.2019.04.008

Cited by 19 publications

(2 citation statements)

References 32 publications

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“…The main difference in this case lies in the value for W; this is very similar for less complex shapes and much worse for the Inpaint technique when highly non convex domains are analyzed. Future work will be devoted to the study of more sophisticated loss functionals for the considered purpose by trying to achieve bijective mappings that have low distortions, see, e.g., [47,48] and to the extension to the non-planar case.…”

Section: Discussionmentioning

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: Discussionmentioning

confidence: 99%

Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

et al. 2023

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“…Alternatively, one may obtain a patch parameterization as the solution of a suitable PDE, such as in [20], in [21,22] using harmonic functions, or based on a quasi-conformal Teichmüller map as in [23]. The methods in [24,25] are developed using optimal mass transport. Other patch parameterization techniques include, e.g., lowrank parameterizations as in [26], swept volume constructions as in [27] or parameterizations based on offsetting as in [28].…”

Section: Introductionmentioning

confidence: 99%

IGA Using Offset-based Overlapping Domain Parameterizations

Kargaran

Jüttler

Takacs

2021

Computer-Aided Design

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Boundary parameter matching for isogeometric analysis using Schwarz–Christoffel mapping

Ji,

Möller,

et al. 2024

Engineering with Computers

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Isogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. Existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz–Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter matching method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.

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Boundary correspondence of planar domains for isogeometric analysis based on optimal mass transport

Cited by 19 publications

References 32 publications

Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

IGA Using Offset-based Overlapping Domain Parameterizations

Boundary parameter matching for isogeometric analysis using Schwarz–Christoffel mapping

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