Parametric stitching for smooth coupling of subdomains with non-matching discretizations

Chen, Chun-Pei; Chen, Yaxiong; Subbarayan, Ganesh

doi:10.1016/j.cma.2020.113519

Cited by 4 publications

(2 citation statements)

References 59 publications

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“…This way, we can obtain a respective basis on the manifold M in the following. From Definition 3.1, we recall the symmetry number p, the shift vectors S i , the reflection maps M i , and the symmetry functions s i defined in (7). Note that the functions M i are well-defined on Z d by the matrix multiplication in (6).…”

Section: Fourier Series and The Dfs Methodsmentioning

confidence: 99%

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A double Fourier sphere method for $d$-dimensional manifolds

Mildenberger¹,

Quellmalz²

2023

Preprint

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The double Fourier sphere (DFS) method uses a clever trick to transform a function defined on the unit sphere to the torus and subsequently approximate it by a Fourier series, which can be evaluated efficiently via fast Fourier transforms. Similar approaches have emerged for approximation problems on the disk, the ball, and the cylinder. In this paper, we introduce a generalized DFS method applicable to various manifolds, including all the abovementioned cases and many more, such as the rotation group. This approach consists in transforming a function defined on a manifold to the torus of the same dimension. We show that the Fourier series of the transformed function can be transferred back to the manifold, where it converges uniformly to the original function. In particular, we obtain analytic convergence rates in case of Hölder-continuous functions on the manifold.

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Section: Fourier Series and The Dfs Methodsmentioning

confidence: 99%

“…With non-overlapping patches, e.g. in computer-aided design, non-linear smoothness conditions or complicated patchstitching methods [7] might be required.…”

Section: Introductionmentioning

confidence: 99%

A double Fourier sphere method for $d$-dimensional manifolds

Mildenberger¹,

Quellmalz²

2023

Preprint

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show abstract

A double Fourier sphere method for d-dimensional manifolds

Mildenberger¹,

Quellmalz²

2023

Sampl. Theory Signal Process. Data Anal.

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The double Fourier sphere (DFS) method uses a clever trick to transform a function defined on the unit sphere to the torus and subsequently approximate it by a Fourier series, which can be evaluated efficiently via fast Fourier transforms. Similar approaches have emerged for approximation problems on the disk, the ball, and the cylinder. In this paper, we introduce a generalized DFS method applicable to various manifolds, including all the above-mentioned cases and many more, such as the rotation group. This approach consists in transforming a function defined on a manifold to the torus of the same dimension. We show that the Fourier series of the transformed function can be transferred back to the manifold, where it converges uniformly to the original function. In particular, we obtain analytic convergence rates in case of Hölder-continuous functions on the manifold.

show abstract

Improved Dixon Resultant for Generating Signed Algebraic Level Sets and Algebraic Boolean Operations on Closed Parametric Surfaces

Vaitheeswaran

Subbarayan

2021

Computer-Aided Design

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Parametric stitching for smooth coupling of subdomains with non-matching discretizations

Cited by 4 publications

References 59 publications

A double Fourier sphere method for $d$-dimensional manifolds

A double Fourier sphere method for $d$-dimensional manifolds

A double Fourier sphere method for d-dimensional manifolds

Improved Dixon Resultant for Generating Signed Algebraic Level Sets and Algebraic Boolean Operations on Closed Parametric Surfaces

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