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Title: Nonlocal and Mixed-Locality Multiscale Finite Element Methods

Abstract

In many applications the resolution of small-scale 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 lower-scale 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 mixed-locality multiscalemore » finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.« less

Authors:
 [1];  [2];  [2]
  1. Oregon State Univ., Corvallis, OR (United States). Department of Mathematics; Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1429643
Report Number(s):
SAND-2017-10571J
Journal ID: ISSN 1540-3459; 657420
Grant/Contract Number:  
AC04-94AL85000; NA0003525
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Multiscale Modeling & Simulation
Additional Journal Information:
Journal Volume: 16; Journal Issue: 1; Journal ID: ISSN 1540-3459
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 Mixed-Locality 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 Mixed-Locality Multiscale Finite Element Methods. United States. doi:10.1137/16M1090351.
Costa, Timothy B., Bond, Stephen D., and Littlewood, David J. Tue . "Nonlocal and Mixed-Locality Multiscale Finite Element Methods". United States. doi:10.1137/16M1090351.
@article{osti_1429643,
title = {Nonlocal and Mixed-Locality Multiscale Finite Element Methods},
author = {Costa, Timothy B. and Bond, Stephen D. and Littlewood, David J.},
abstractNote = {In many applications the resolution of small-scale 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 lower-scale 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 mixed-locality 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},
number = 1,
volume = 16,
place = {United States},
year = {Tue Mar 27 00:00:00 EDT 2018},
month = {Tue Mar 27 00:00:00 EDT 2018}
}

Journal Article:
Free Publicly Available Full Text
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