View in EDS

Higher order Bernstein–Bézier and Nédélec finite elements for the relaxed micromorphic model.

Saved in:
Bibliographic Details
Title: Higher order Bernstein–Bézier and Nédélec finite elements for the relaxed micromorphic model.
Authors: Sky, Adam1 (AUTHOR) adam.sky@tu-dortmund.de, Muench, Ingo1 (AUTHOR) ingo.muench@tu-dortmund.de, Rizzi, Gianluca1 (AUTHOR) gianluca.rizzi@tu-dortmund.de, Neff, Patrizio2 (AUTHOR) patrizio.neff@uni-due.de
Source: Journal of Computational & Applied Mathematics. Mar2024, Vol. 438, pN.PAG-N.PAG. 1p.
Subjects: Automatic differentiation, Bernstein polynomials, Finite element method, Commercial space ventures
Abstract: The relaxed micromorphic model is a generalized continuum model that is well-posed in the space X = [ H 1 ] 3 × [ H (curl) ] 3 . Consequently, finite element formulations of the model rely on H 1 -conforming subspaces and Nédélec elements for discrete solutions of the corresponding variational problem. This work applies the recently introduced polytopal template methodology for the construction of Nédélec elements. This is done in conjunction with Bernstein–Bézier polynomials and dual numbers in order to compute hp-FEM solutions of the model. Bernstein–Bézier polynomials allow for optimal complexity in the assembly procedure due to their natural factorization into univariate Bernstein base functions. In this work, this characteristic is further augmented by the use of dual numbers in order to compute their values and their derivatives simultaneously. The application of the polytopal template methodology for the construction of the Nédélec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein–Bézier basis. We introduce the Bernstein–Bézier basis along with its factorization to univariate Bernstein base functions, the principle of automatic differentiation via dual numbers and a detailed construction of Nédélec elements based on Bernstein–Bézier polynomials with the polytopal template methodology. This is complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on the consistent coupling condition. The performance of the elements is shown in examples of the relaxed micromorphic model. • Kinematical reduction of the relaxed micromorphic model to antiplane shear. • Bernstein–Bézier polynomial basis and its factorization via the Duffy transformation. • Forward automatic differentiation via dual numbers for enhanced optimal complexity. • Higher order Nédélec elements based on Bernstein polynomials via polytopal templates. • Discrete consistent coupling condition for the relaxed micromorphic model. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Computational & Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
FullText Text:
  Availability: 0
Header DbId: egs
DbLabel: Engineering Source
An: 173050724
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Higher order Bernstein–Bézier and Nédélec finite elements for the relaxed micromorphic model.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Sky%2C+Adam%22">Sky, Adam</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> adam.sky@tu-dortmund.de</i><br /><searchLink fieldCode="AR" term="%22Muench%2C+Ingo%22">Muench, Ingo</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> ingo.muench@tu-dortmund.de</i><br /><searchLink fieldCode="AR" term="%22Rizzi%2C+Gianluca%22">Rizzi, Gianluca</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> gianluca.rizzi@tu-dortmund.de</i><br /><searchLink fieldCode="AR" term="%22Neff%2C+Patrizio%22">Neff, Patrizio</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> patrizio.neff@uni-due.de</i>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22Journal+of+Computational+%26+Applied+Mathematics%22">Journal of Computational & Applied Mathematics</searchLink>. Mar2024, Vol. 438, pN.PAG-N.PAG. 1p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Automatic+differentiation%22">Automatic differentiation</searchLink><br /><searchLink fieldCode="DE" term="%22Bernstein+polynomials%22">Bernstein polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Commercial+space+ventures%22">Commercial space ventures</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The relaxed micromorphic model is a generalized continuum model that is well-posed in the space X = [ H 1 ] 3 × [ H (curl) ] 3 . Consequently, finite element formulations of the model rely on H 1 -conforming subspaces and Nédélec elements for discrete solutions of the corresponding variational problem. This work applies the recently introduced polytopal template methodology for the construction of Nédélec elements. This is done in conjunction with Bernstein–Bézier polynomials and dual numbers in order to compute hp-FEM solutions of the model. Bernstein–Bézier polynomials allow for optimal complexity in the assembly procedure due to their natural factorization into univariate Bernstein base functions. In this work, this characteristic is further augmented by the use of dual numbers in order to compute their values and their derivatives simultaneously. The application of the polytopal template methodology for the construction of the Nédélec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein–Bézier basis. We introduce the Bernstein–Bézier basis along with its factorization to univariate Bernstein base functions, the principle of automatic differentiation via dual numbers and a detailed construction of Nédélec elements based on Bernstein–Bézier polynomials with the polytopal template methodology. This is complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on the consistent coupling condition. The performance of the elements is shown in examples of the relaxed micromorphic model. • Kinematical reduction of the relaxed micromorphic model to antiplane shear. • Bernstein–Bézier polynomial basis and its factorization via the Duffy transformation. • Forward automatic differentiation via dual numbers for enhanced optimal complexity. • Higher order Nédélec elements based on Bernstein polynomials via polytopal templates. • Discrete consistent coupling condition for the relaxed micromorphic model. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Journal of Computational & Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=173050724
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1016/j.cam.2023.115568
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 1
        StartPage: N.PAG
    Subjects:
      – SubjectFull: Automatic differentiation
        Type: general
      – SubjectFull: Bernstein polynomials
        Type: general
      – SubjectFull: Finite element method
        Type: general
      – SubjectFull: Commercial space ventures
        Type: general
    Titles:
      – TitleFull: Higher order Bernstein–Bézier and Nédélec finite elements for the relaxed micromorphic model.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Sky, Adam
      – PersonEntity:
          Name:
            NameFull: Muench, Ingo
      – PersonEntity:
          Name:
            NameFull: Rizzi, Gianluca
      – PersonEntity:
          Name:
            NameFull: Neff, Patrizio
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 03
              Text: Mar2024
              Type: published
              Y: 2024
          Identifiers:
            – Type: issn-print
              Value: 03770427
          Numbering:
            – Type: volume
              Value: 438
          Titles:
            – TitleFull: Journal of Computational & Applied Mathematics
              Type: main
ResultId 1