Nonlocal and MixedLocality Multiscale Finite Element Methods
Abstract
In many applications the resolution of smallscale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lowerscale problems that produce enriched multiscale basis functions. Here, in this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. Finally, we conclude with preliminary results on a mixedlocality multiscalemore »
 Authors:

 Oregon State Univ., Corvallis, OR (United States). Department of Mathematics; Sandia National Lab. (SNLNM), Albuquerque, NM (United States). Center for Computing Research
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States). Center for Computing Research
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1429643
 Report Number(s):
 SAND201710571J
Journal ID: ISSN 15403459; 657420
 Grant/Contract Number:
 AC0494AL85000; NA0003525
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Multiscale Modeling & Simulation
 Additional Journal Information:
 Journal Volume: 16; Journal Issue: 1; Journal ID: ISSN 15403459
 Publisher:
 SIAM
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; multiscale; fi nite elements method; nonlocal; mixed locality; peridynamics
Citation Formats
Costa, Timothy B., Bond, Stephen D., and Littlewood, David J. Nonlocal and MixedLocality Multiscale Finite Element Methods. United States: N. p., 2018.
Web. doi:10.1137/16M1090351.
Costa, Timothy B., Bond, Stephen D., & Littlewood, David J. Nonlocal and MixedLocality Multiscale Finite Element Methods. United States. doi:10.1137/16M1090351.
Costa, Timothy B., Bond, Stephen D., and Littlewood, David J. Tue .
"Nonlocal and MixedLocality Multiscale Finite Element Methods". United States. doi:10.1137/16M1090351. https://www.osti.gov/servlets/purl/1429643.
@article{osti_1429643,
title = {Nonlocal and MixedLocality Multiscale Finite Element Methods},
author = {Costa, Timothy B. and Bond, Stephen D. and Littlewood, David J.},
abstractNote = {In many applications the resolution of smallscale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lowerscale problems that produce enriched multiscale basis functions. Here, in this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. Finally, we conclude with preliminary results on a mixedlocality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.},
doi = {10.1137/16M1090351},
journal = {Multiscale Modeling & Simulation},
issn = {15403459},
number = 1,
volume = 16,
place = {United States},
year = {2018},
month = {3}
}
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