A variational multiscale immersed meshfree method for heterogeneous materials

Huang, Tsung-Hui; Chen, Jiun-Shyan; Tupek, Michael R.; Beckwith, Frank N.; Koester, Jacob J.; Fang, H. Eliot

doi:10.1007/s00466-020-01968-1

Title: A variational multiscale immersed meshfree method for heterogeneous materials

Journal Article · 07 March 2021 · Computational Mechanics

DOI: https://doi.org/10.1007/s00466-020-01968-1 · OSTI ID:1769525

Huang, Tsung-Hui;

; Tupek, Michael R.; Beckwith, Frank N.; Koester, Jacob J.; Fang, H. Eliot

Abstract We introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method.

View Journal Article

Cite

Export

Save

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1769525

Report Number(s):: SAND-2021-2891J; PII: 1968

Journal Information:: Computational Mechanics, Journal Name: Computational Mechanics Journal Issue: 4 Vol. 67; ISSN 0178-7675

Publisher:: Springer Science + Business MediaCopyright Statement

Country of Publication:: Germany

Language:: English

References (49)

EFG approximation with discontinuous derivatives Krongauz, Y.; Belytschko, T. International Journal for Numerical Methods in Engineering, Vol. 41, Issue 7 https://doi.org/10.1002/(SICI)1097-0207(19980415)41:7<1215::AID-NME330>3.0.CO;2-#	journal	April 1998
A stabilized conforming nodal integration for Galerkin mesh-free methods Chen, Jiun-Shyan; Wu, Cheng-Tang; Yoon, Sangpil International Journal for Numerical Methods in Engineering, Vol. 50, Issue 2 https://doi.org/10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A	journal	January 2000
Isogeometric Analysis Cottrell, J. Austin; Hughes, Thomas J. R.; Bazilevs, Yuri John Wiley & Sons, Ltd https://doi.org/10.1002/9780470749081	book	January 2009
Reproducing Kernel Particle Method for Solving Partial Differential Equations Chen, Jiun-Shyan; Liu, Wing Kam; Hillman, Michael C. Encyclopedia of Computational Mechanics Second Edition https://doi.org/10.1002/9781119176817.ecm2104	reference-book	December 2017
Reproducing kernel particle methods Liu, Wing Kam; Jun, Sukky; Zhang, Yi Fei International Journal for Numerical Methods in Fluids, Vol. 20, Issue 8-9 https://doi.org/10.1002/fld.1650200824	journal	April 1995
Nitsche's method for interface problems in computa-tional mechanics: Nitsche's method for interface problems in computa-tional mechanics Hansbo, Peter GAMM-Mitteilungen, Vol. 28, Issue 2 https://doi.org/10.1002/gamm.201490018	journal	November 2005
Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction Batra, R. C.; Porfiri, M.; Spinello, D. International Journal for Numerical Methods in Engineering, Vol. 61, Issue 14 https://doi.org/10.1002/nme.1156	journal	January 2004
Treatment of discontinuity in the reproducing kernel element method Lu, Hongsheng; Wan Kim, Do; Kam Liu, Wing International Journal for Numerical Methods in Engineering, Vol. 63, Issue 2 https://doi.org/10.1002/nme.1284	journal	January 2005
A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces Mourad, Hashem M.; Dolbow, John; Harari, Isaac International Journal for Numerical Methods in Engineering, Vol. 69, Issue 4 https://doi.org/10.1002/nme.1788	journal	January 2006
Meshfree and finite element nodal integration methods: NODAL INTEGRATION TECHNIQUES FOR MESHFREE AND FINITE ELEMENT METHODS Puso, M. A.; Chen, J. S.; Zywicz, E. International Journal for Numerical Methods in Engineering, Vol. 74, Issue 3 https://doi.org/10.1002/nme.2181	journal	September 2007
A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method Béchet, Éric; Moës, Nicolas; Wohlmuth, Barbara International Journal for Numerical Methods in Engineering, Vol. 78, Issue 8 https://doi.org/10.1002/nme.2515	journal	May 2009
Immersed particle method for fluid-structure interaction: PARTICLE METHOD FOR FLUID-STRUCTURE INTERACTION Rabczuk, Timon; Gracie, Robert; Song, Jeong-Hoon International Journal for Numerical Methods in Engineering, Vol. 81, Issue 1 https://doi.org/10.1002/nme.2670	journal	August 2009
An arbitrary order variationally consistent integration for Galerkin meshfree methods: A VARIATIONALLY CONSISTENT INTEGRATION FOR MESHFREE METHODS Chen, Jiun-Shyan; Hillman, Michael; Rüter, Marcus International Journal for Numerical Methods in Engineering, Vol. 95, Issue 5 https://doi.org/10.1002/nme.4512	journal	June 2013
An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics: ACCELERATED, CONVERGENT, STABLE NODAL INTEGRATION IN MESHFREE METHODS Hillman, Michael; Chen, Jiun-Shyan International Journal for Numerical Methods in Engineering, Vol. 107, Issue 7 https://doi.org/10.1002/nme.5183	journal	December 2015
The finite element method with Lagrangian multipliers Babuška, Ivo Numerische Mathematik, Vol. 20, Issue 3 https://doi.org/10.1007/BF01436561	journal	June 1973
Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind Nitsche, J. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Vol. 36, Issue 1 https://doi.org/10.1007/BF02995904	journal	July 1971
New Cartesian grid methods for interface problems using the finite element formulation Li, Zhilin; Lin, Tao; Wu, Xiaohui Numerische Mathematik, Vol. 96, Issue 1 https://doi.org/10.1007/s00211-003-0473-x	journal	November 2003
Mathematical foundations of the immersed finite element method Liu, Wing Kam; Kim, Do Wan; Tang, Shaoqiang Computational Mechanics, Vol. 39, Issue 3 https://doi.org/10.1007/s00466-005-0018-5	journal	December 2005
On finite element formulations for nearly incompressible linear elasticity Nakshatrala, K. B.; Masud, A.; Hjelmstad, K. D. Computational Mechanics, Vol. 41, Issue 4 https://doi.org/10.1007/s00466-007-0212-8	journal	August 2007
Extended particle difference method for weak and strong discontinuity problems: part I. Derivation of the extended particle derivative approximation for the representation of weak and strong discontinuities Yoon, Young-Cheol; Song, Jeong-Hoon Computational Mechanics, Vol. 53, Issue 6 https://doi.org/10.1007/s00466-013-0950-8	journal	November 2013
Extended particle difference method for weak and strong discontinuity problems: part II. Formulations and applications for various interfacial singularity problems Yoon, Young-Cheol; Song, Jeong-Hoon Computational Mechanics, Vol. 53, Issue 6 https://doi.org/10.1007/s00466-013-0951-7	journal	November 2013
RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations Huang, Tsung-Hui; Wei, Haoyan; Chen, Jiun-Shyan Computational Particle Mechanics, Vol. 7, Issue 2 https://doi.org/10.1007/s40571-019-00272-x	journal	August 2019
Effect of phase shape and size on the FEM analysis of indentation test of duplex stainless steel microstructures Dengiz, Cengiz Görkem; Yıldızlı, Kemal SN Applied Sciences, Vol. 2, Issue 4 https://doi.org/10.1007/s42452-020-2432-x	journal	March 2020
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations Brooks, Alexander N.; Hughes, Thomas J. R. Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3 https://doi.org/10.1016/0045-7825(82)90071-8	journal	September 1982
A new finite element formulation for computational fluid dynamics: V. Circumventing the babuška-brezzi condition: a stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations Hughes, Thomas J. R.; Franca, Leopoldo P.; Balestra, Marc Computer Methods in Applied Mechanics and Engineering, Vol. 59, Issue 1 https://doi.org/10.1016/0045-7825(86)90025-3	journal	November 1986
A new finite element formulation for computational fluid dynamics: VIII. The galerkin/least-squares method for advective-diffusive equations Hughes, Thomas J. R.; Franca, Leopoldo P.; Hulbert, Gregory M. Computer Methods in Applied Mechanics and Engineering, Vol. 73, Issue 2 https://doi.org/10.1016/0045-7825(89)90111-4	journal	May 1989
A relationship between stabilized finite element methods and the Galerkin method with bubble functions Brezzi, Franco; Bristeau, Marie-Odile; Franca, Leopoldo P. Computer Methods in Applied Mechanics and Engineering, Vol. 96, Issue 1 https://doi.org/10.1016/0045-7825(92)90102-P	journal	April 1992
Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods Hughes, Thomas J. R. Computer Methods in Applied Mechanics and Engineering, Vol. 127, Issue 1-4 https://doi.org/10.1016/0045-7825(95)00844-9	journal	November 1995
On some techniques for approximating boundary conditions in the finite element method Stenberg, Rolf Journal of Computational and Applied Mathematics, Vol. 63, Issue 1-3 https://doi.org/10.1016/0377-0427(95)00057-7	journal	November 1995
Mesh-free simulations of shear banding in large deformation Li, Shaofan; Hao, Wei; Liu, Wing Kam International Journal of Solids and Structures, Vol. 37, Issue 48-50 https://doi.org/10.1016/S0020-7683(00)00195-5	journal	November 2000
New boundary condition treatments in meshfree computation of contact problems Chen, Jiun-Shyan; Wang, Hui-Ping Computer Methods in Applied Mechanics and Engineering, Vol. 187, Issue 3-4 https://doi.org/10.1016/S0045-7825(00)80004-3	journal	July 2000
Treatment of material discontinuity in the Element-Free Galerkin method Cordes, L. W.; Moran, B. Computer Methods in Applied Mechanics and Engineering, Vol. 139, Issue 1-4 https://doi.org/10.1016/S0045-7825(96)01080-8	journal	December 1996
Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures Chen, Jiun-Shyan; Pan, Chunhui; Wu, Cheng-Tang Computer Methods in Applied Mechanics and Engineering, Vol. 139, Issue 1-4 https://doi.org/10.1016/S0045-7825(96)01083-3	journal	December 1996
The variational multiscale method—a paradigm for computational mechanics Hughes, Thomas J. R.; Feijóo, Gonzalo R.; Mazzei, Luca Computer Methods in Applied Mechanics and Engineering, Vol. 166, Issue 1-2 https://doi.org/10.1016/S0045-7825(98)00079-6	journal	November 1998
Homogenization of magnetostrictive particle-filled elastomers using an interface-enriched reproducing kernel particle method Wang, Dongdong; Chen, Jiun-Shyan; Sun, Lizhi Finite Elements in Analysis and Design, Vol. 39, Issue 8 https://doi.org/10.1016/S0168-874X(03)00058-1	journal	May 2003
A distributed Lagrange multiplier/fictitious domain method for particulate flows Glowinski, R.; Pan, T. -W.; Hesla, T. I. International Journal of Multiphase Flow, Vol. 25, Issue 5 https://doi.org/10.1016/S0301-9322(98)00048-2	journal	August 1999
Multiple scale meshfree methods for damage fracture and localization Liu, Wing Kam; Hao, Su; Belytschko, Ted Computational Materials Science, Vol. 16, Issue 1-4 https://doi.org/10.1016/S0927-0256(99)00062-2	journal	December 1999
An immersed particle modeling technique for the three-dimensional large strain simulation of particulate-reinforced metal-matrix composites Wu, C. T.; Wang, D.; Guo, Y. Applied Mathematical Modelling, Vol. 40, Issue 4 https://doi.org/10.1016/j.apm.2015.09.107	journal	February 2016
Imposing essential boundary conditions in mesh-free methods Fernández-Méndez, Sonia; Huerta, Antonio Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 12-14 https://doi.org/10.1016/j.cma.2003.12.019	journal	March 2004
Immersed finite element method Zhang, Lucy; Gerstenberger, Axel; Wang, Xiaodong Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 21-22 https://doi.org/10.1016/j.cma.2003.12.044	journal	May 2004
An implicit gradient model by a reproducing kernel strain regularization in strain localization problems Chen, Jiun-Shyan; Zhang, Xinwei; Belytschko, Ted Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 27-29 https://doi.org/10.1016/j.cma.2003.12.057	journal	July 2004
A variational framework for fluid–solid interaction problems based on immersed domains: Theoretical bases Blanco, P. J.; Feijóo, R. A.; Dari, E. A. Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 25-28 https://doi.org/10.1016/j.cma.2008.01.017	journal	April 2008
Residual-free bubbles for embedded Dirichlet problems Dolbow, J. E.; Franca, L. P. Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 45-48 https://doi.org/10.1016/j.cma.2008.02.033	journal	August 2008
The finite cell method for three-dimensional problems of solid mechanics Düster, A.; Parvizian, J.; Yang, Z. Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 45-48 https://doi.org/10.1016/j.cma.2008.02.036	journal	August 2008
Nodally integrated implicit gradient reproducing kernel particle method for convection dominated problems Hillman, Michael; Chen, Jiun-Shyan Computer Methods in Applied Mechanics and Engineering, Vol. 299 https://doi.org/10.1016/j.cma.2015.11.004	journal	February 2016
Conforming window functions for meshfree methods Koester, Jacob J.; Chen, Jiun-Shyan Computer Methods in Applied Mechanics and Engineering, Vol. 347 https://doi.org/10.1016/j.cma.2018.12.042	journal	April 2019
Numerical modeling of composite solids using an immersed meshfree Galerkin method Wu, C. T.; Guo, Y.; Askari, E. Composites Part B: Engineering, Vol. 45, Issue 1 https://doi.org/10.1016/j.compositesb.2012.09.061	journal	February 2013
A penalty method for the simulation of fluid - rigid body interaction Janela, João; Lefebvre, Aline; Maury, Bertrand ESAIM: Proceedings, Vol. 14 https://doi.org/10.1051/proc:2005010	journal	September 2005
A Meshfree procedure for the microscopic analysis of particle-reinforced rubber compounds Wu, C. T.; Koishi, M. Interaction and multiscale mechanics, Vol. 2, Issue 2 https://doi.org/10.12989/imm.2009.2.2.129	journal	June 2009

Similar Records

Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion

Journal Article · 2021 · SIAM Journal on Numerical Analysis · OSTI ID:1738919

Leng, Yu; Tian, Xiaochuan; Trask, Nathaniel; +1 more

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

Journal Article · 2018 · Journal of Computational Physics · OSTI ID:1425765

Gao, Kai; Fu, Shubin; Chung, Eric T.

A multiscale two-point flux-approximation method

Journal Article · 2014 · Journal of Computational Physics · OSTI ID:22382131

Møyner, Olav; Lie, Knut-Andreas

Related Subjects

36 MATERIALS SCIENCE
Heterogeneous material
Reproducing kernel particle method
Variational multiscale immersed method
Volumetric constraint

Title: A variational multiscale immersed meshfree method for heterogeneous materials

Citation Formats

References (49)

Similar Records

Related Subjects