Formulation and computation of dynamic, interfacecompatible Whitney complexes in three dimensions
Abstract
A discrete De Rham complex enables compatible, structurepreserving 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 interfaceconforming discrete De Rham complex. Both approaches start by dividing cut elements into interfaceconforming subelements but differ in how they build the finite element basis on these subelements. The first approach discards the existing nonconforming 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 interfaceconforming 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 (XFEMAC), respectively, across the De Rham complex.
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1469640
 Report Number(s):
 SAND20189818J
Journal ID: ISSN 00219991; 667673
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 359; Journal Issue: C; Journal ID: ISSN 00219991
 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, interfacecompatible Whitney complexes in three dimensions. United States: N. p., 2018.
Web. doi:10.1016/j.jcp.2017.12.037.
Kramer, Richard M. J., Siefert, Christopher M., Voth, Thomas E., & Bochev, Pavel B. Formulation and computation of dynamic, interfacecompatible Whitney complexes in three dimensions. United States. https://doi.org/10.1016/j.jcp.2017.12.037
Kramer, Richard M. J., Siefert, Christopher M., Voth, Thomas E., and Bochev, Pavel B. Tue .
"Formulation and computation of dynamic, interfacecompatible Whitney complexes in three dimensions". United States. https://doi.org/10.1016/j.jcp.2017.12.037. https://www.osti.gov/servlets/purl/1469640.
@article{osti_1469640,
title = {Formulation and computation of dynamic, interfacecompatible 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, structurepreserving 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 interfaceconforming discrete De Rham complex. Both approaches start by dividing cut elements into interfaceconforming subelements but differ in how they build the finite element basis on these subelements. The first approach discards the existing nonconforming 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 interfaceconforming 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 (XFEMAC), respectively, across the De Rham complex.},
doi = {10.1016/j.jcp.2017.12.037},
journal = {Journal of Computational Physics},
number = C,
volume = 359,
place = {United States},
year = {2018},
month = {1}
}
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