Formulation and computation of dynamic, interface-compatible Whitney complexes in three dimensions

Kramer, Richard M. J.; Siefert, Christopher M.; Voth, Thomas E.; Bochev, Pavel B.

doi:10.1016/j.jcp.2017.12.037

Title: Formulation and computation of dynamic, interface-compatible Whitney complexes in three dimensions

Abstract

A discrete De Rham complex enables compatible, structure-preserving discretizations for a broad range of partial differential equations problems. Such discretizations can correctly reproduce the physics of interface problems, provided the grid conforms to the interface. However, large deformations, complex geometries, and evolving interfaces makes generation of such grids difficult. We develop and demonstrate two formally equivalent approaches that, for a given background mesh, dynamically construct an interface-conforming discrete De Rham complex. Both approaches start by dividing cut elements into interface-conforming subelements but differ in how they build the finite element basis on these subelements. The first approach discards the existing non-conforming basis of the parent element and replaces it by a dynamic set of degrees of freedom of the same kind. The second approach defines the interface-conforming degrees of freedom on the subelements as superpositions of the basis functions of the parent element. These approaches generalize the Conformal Decomposition Finite Element Method (CDFEM) and the extended finite element method with algebraic constraints (XFEM-AC), respectively, across the De Rham complex.

Authors:

Kramer, Richard M. J. ^[1]; Siefert, Christopher M. ^[1]; Voth, Thomas E. ^[1]; Bochev, Pavel B. ^[1]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Publication Date:: 2018-01-02

Research Org.:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1469640

Report Number(s):: SAND2018-9818J
Journal ID: ISSN 0021-9991; 667673

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Journal Article: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 359; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats


                    Kramer, Richard M. J., Siefert, Christopher M., Voth, Thomas E., and Bochev, Pavel B. Formulation and computation of dynamic, interface-compatible Whitney complexes in three dimensions.  United States: N. p., 2018. 
        Web.  doi:10.1016/j.jcp.2017.12.037.

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                    Kramer, Richard M. J., Siefert, Christopher M., Voth, Thomas E., & Bochev, Pavel B. Formulation and computation of dynamic, interface-compatible Whitney complexes in three dimensions.  United States.  https://doi.org/10.1016/j.jcp.2017.12.037

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                    Kramer, Richard M. J., Siefert, Christopher M., Voth, Thomas E., and Bochev, Pavel B. 2018.  
        "Formulation and computation of dynamic, interface-compatible Whitney complexes in three dimensions".  United States.  https://doi.org/10.1016/j.jcp.2017.12.037.  https://www.osti.gov/servlets/purl/1469640.

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@article{osti_1469640,

  title        = {Formulation and computation of dynamic, interface-compatible Whitney complexes in three dimensions},

  author       = {Kramer, Richard M. J. and Siefert, Christopher M. and Voth, Thomas E. and Bochev, Pavel B.},

  abstractNote = {A discrete De Rham complex enables compatible, structure-preserving discretizations for a broad range of partial differential equations problems. Such discretizations can correctly reproduce the physics of interface problems, provided the grid conforms to the interface. However, large deformations, complex geometries, and evolving interfaces makes generation of such grids difficult. We develop and demonstrate two formally equivalent approaches that, for a given background mesh, dynamically construct an interface-conforming discrete De Rham complex. Both approaches start by dividing cut elements into interface-conforming subelements but differ in how they build the finite element basis on these subelements. The first approach discards the existing non-conforming basis of the parent element and replaces it by a dynamic set of degrees of freedom of the same kind. The second approach defines the interface-conforming degrees of freedom on the subelements as superpositions of the basis functions of the parent element. These approaches generalize the Conformal Decomposition Finite Element Method (CDFEM) and the extended finite element method with algebraic constraints (XFEM-AC), respectively, across the De Rham complex.},

  doi          = {10.1016/j.jcp.2017.12.037},

  url          = {https://www.osti.gov/biblio/1469640},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = C,

  volume       = 359,

  place        = {United States},

  year         = {2018},

  month        = {1}

}

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Figures / Tables:

Table 1.: Summary of the differential and boundary operators for the model PDEs, the spaces for their weak variational forms (22), the Whitney spaces necessary for the compatible spatial discretization (23) of the weak forms with boundary conditions (20) and the mesh entities containing the degrees of freedom of thesemore »

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