Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
11
6
2013
Preface: MULTISCALE METHODS IN FRACTURE MECHANICS WITH EXTENDED/GENERALIZED FINITE ELEMENTS
vi-vii
10.1615/IntJMultCompEng.2013006839
Haim
Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA
Fracture Mechanics
Multiscale Methods
Extended/Generalized Finite Elements
XFEM/GFEM
ADAPTIVE ATOMISTIC-CONTINUUM MODELING OF DEFECT INTERACTION WITH THE DEBDM
505-525
10.1615/IntJMultCompEng.2013005705
Philip
Moseley
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois, USA
Jay
Oswald
School for Engineering of Matter, Transport and Energy, Arizona State University, Phoenix, Arizona, USA
Ted
Belytschko
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois, USA
adaptivity
concurrent multiscale
bridging domain method
extended finite element method
fracture
crack propagation
New procedures for modeling interactions among dislocations and nanosized cracks within the dynamically evolving bridging domain method (DEBDM) have been developed. The DEBDM is an efficient concurrent atomistic-to-continuum approach based on the bridging domain method, where the atomic domain dynamically adapts to encompass evolving defects. New algorithms for identifying and coarse graining dislocation-induced slip planes have been added to the method, which previously focused on fracture. Additional improvements include continuously varying BDM energy-weighting functions, which allow the fine-graining and coarse-graining transitions to occur smoothly over multiple timesteps, reducing the potential for nonphysical or unstable behavior. Several examples of interacting dislocations and nanocracks are presented to demonstrate the flexibility and efficiency of the method.
MOLECULAR DYNAMICS/XFEM COUPLING BY A THREE-DIMENSIONAL EXTENDED BRIDGING DOMAIN WITH APPLICATIONS TO DYNAMIC BRITTLE FRACTURE
527-541
10.1615/IntJMultCompEng.2013005838
Hossein
Talebi
Institute of Structural Mechanics, Bauhaus University-Weimar
Mohammad
Silani
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran
S. P. A.
Bordas
Institute of Mechanics and Advanced Materials, Theoretical and Computational Mechanics, Cardiff University, Cardiff, CF24 3AA, United Kingdom
Pierre
Kerfriden
Institute of Materials, Mechanics and Advanced Manufacturing, Cardiff University, 5 the
Parade, Cardiff, UK; Centre des Matériaux, MINES ParisTech/PSL University, 63-65 rue Henri Auguste
Desbruères, Corbeil-essonnes, France
Timon
Rabczuk
Institute of Structural Mechanics, Bauhaus-Universitat Weimar, Marienstr. 15, D-99423 Weimar, Germany
multiscale
atomistic simulation
extended finite elements
crack
We propose a method to couple a three-dimensional continuum domain to a molecular dynamics domain to simulate propagating cracks in dynamics. The continuum domain is treated by an extended finite element method to handle the discontinuities. The coupling is based on the bridging domain method, which blends the continuum and atomistic energies. The Lennard-Jones potential is used to model the interactions in the atomistic domain, and the Cauchy-Born rule is used to compute the material behavior in the continuum domain. To our knowledge, it is the first time that a three dimensional extended bridging domain method is reported. To show the suitability of the proposed method, a three-dimensional crack problem with an atomistic region around the crack front is solved. The results show that the method is capable of handling crack propagation and dislocation nucleation.
A TWO-SCALE STRONG DISCONTINUITY APPROACH FOR EVOLUTION OF SHEAR BANDS UNDER DYNAMIC IMPACT LOADS
543-563
10.1615/IntJMultCompEng.2013005506
Alireza
Tabarraei
Department of Mechanical Engineering, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, North Carolina 28223-0001, USA
Jeong-Hoon
Song
University of Colorado Boulder
Haim
Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA
strain localization
shear band
multiscale analysis
extended finite element method
phantom node method
strong discontinuity approach
A micro-macro two-scale method for modeling adiabatic shear bands in rate-dependent materials is presented. The phantom node method, which is a variant of the extended finite element method is used to model the shear band at the macroscale. The key contribution is the development of a micromodel which allows the extraction of tangential and normal traction-separation laws, i.e., cohesive laws. These extracted rate-dependent cohesive laws are then injected back into the macro scale to accurately model the postlocalization behavior. The results show good accuracy as compared to very fine finite element meshes but are orders of magnitude faster. Hence the scheme is attractive when tracking of shear bands is of greater importance than microscopic behavior.
AN XFEM BASED MULTISCALE APPROACH TO FRACTURE OF HETEROGENEOUS MEDIA
565-580
10.1615/IntJMultCompEng.2013005569
Mirmohammadreza
Kabiri
Department of Civil, Environmental and Architectural Engineering, Program of Material Science and Engineering, University of Colorado, Boulder, Colorado, USA
Franck J.
Vernerey
Department of Civil, Environmental and Architectural Engineering, Program of Material Science and Engineering, University of Colorado, Boulder, Colorado, USA
concurrent multiscale method
adaptive refinement
macro-micro coupling
This paper introduces a concurrent adaptive multiscale methodology in which both macroscopic and microscopic deformation fields strongly interact. The method is based on the balance between numerical and homogenization error; while the first type of error states that the element's should be refined in regions of high deformation gradients, the second implies that elements size may not be smaller than a threshold determined by the size of the representative volume element (RVE). In this context, we introduce a multiscale method in which RVEs can be embedded in the continuum region through appropriate macro-micro boundary coupling conditions. By combining the idea of adaptive refinement with the embedded RVE method, the methodology ensures that appropriate descriptions of the material are used adequately, regardless of the severity of deformations. We show that this method, in conjunction with the extended finite element method, is ideal to study the strong interactions between a crack and the microstructure of heterogeneous media. In particular, the method enables an explicit description of microstructural features near the crack tip, while a computationally inexpensive coarse scale continuum description is used in the rest of the domain. We illustrate the method with several examples showing its accuracy and relatively low computational cost and discuss its potential in relating microstructure to the fracture toughness of a diversity of heterogeneous media.
EXTENSIONS OF THE TWO-SCALE GENERALIZED FINITE ELEMENT METHOD TO NONLINEAR FRACTURE PROBLEMS
581-596
10.1615/IntJMultCompEng.2013005685
Varun
Gupta
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, Urbana, Illinois 61801, USA
Dae-Jin
Kim
Department of Architectural Engineering, Kyung Hee University, Engineering Building, 1 Sochon-Dong Kihung-Gu, Yongin, Kyunggi-Do, Korea 446-701
Armando
Duarte
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, Urbana, Illinois 61801, USA
generalized FEM
extended FEM
nonlinear fracture
plasticity
global-local analysis
This paper presents an extension of a two-scale generalized finite element method (GFEM) to three-dimensional fracture problems involving confined plasticity. This two-scale procedure, also known as the generalized finite element method with global-local enrichments (GFEMgl), involves the solution of a fine-scale boundary value problem defined around a region undergoing plastic deformations and the enrichment of the coarse-scale solution space with the resulting nonlinear fine-scale solution through the partition-of-unity framework. The approach provides accurate nonlinear solutions with reduced computational costs compared to standard finite element methods, since the nonlinear iterations are performed on much smaller problems. The efficacy of the method is demonstrated with the help of numerical examples, which are three-dimensional fracture problems with nonlinear material properties and considering small-strain, rate-independent J2 plasticity.
IMPROVED CRACK TIP ENRICHMENT FUNCTIONS AND INTEGRATION FOR CRACK MODELING USING THE EXTENDED FINITE ELEMENT METHOD
597-631
10.1615/IntJMultCompEng.2013006523
Nicolas
Chevaugeon
LUNAM Universite, GeM UMR6183, Ecole Centrale de Nantes, 1 Rue de la Noe, 44321 Nantes, France
Nicolas
Moes
LUNAM Universite, GeM UMR6183, Ecole Centrale de Nantes, 1 Rue de la Noe, 44321 Nantes, France
Hans
Minnebo
LUNAM Universite, GeM UMR6183, Ecole Centrale de Nantes, 1 Rue de la Noe, 44321 Nantes, France
X-FEM
cracks
singular function integration
LEFM
This paper focuses on two improvements of the extended finite element method (X-FEM) in the context of linear fracture mechanics. Both improve the accuracy and the robustness of the X-FEM. In a first contribution, a new enrichment strategy is proposed to take into account the singular stress field at the crack tip that is meant to replace the traditional four-crack-tip enrichment functions. The efficiency of the new approach is demonstrated on mesh convergence experiments for two-dimensional straight and curved crack problems, using first- and second-order shape functions, both in terms of convergence rates and in terms of condition number of the system to solve. The second contribution revisits the problem of the numerical integration of the stiffness operator when singular functions like the tip enrichment functions are used. An original algorithm to build accurate and fast integration rules for elements in the enrichment zone, touching the crack tip singularity, or not, is presented. The effects on convergence rate of the choice of the integration rule are illustrated on numerical examples.
AN ADAPTIVE DOMAIN DECOMPOSITION PRECONDITIONER FOR CRACK PROPAGATION PROBLEMS MODELED BY XFEM
633-654
10.1615/IntJMultCompEng.2013006012
Haim
Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA
Luc
Berger-Vergiat
Department of Civil Engineering & Engineering Mechanics, Columbia University, New York 10027, USA
XFEM
extended finite elements
domain decomposition
Schwarz preconditioner
fracture analysis
algebraic multigrid
smoothed aggregation multigrid
Application of an algebraic multigrid (AMG) solver to linear systems arising from fracture problems modeled by extended finite elements (XFEM) will often result in poor convergence. This is due to coarsening operators in AMG that disregard the discontinuous enrichment functions and automatically coarsen across cracks. To overcome the AMG coarsening limitation, we propose a multiplicative-Schwarz domain decomposition preconditioner to the generalized minimum residual method. In this approach the domain is decomposed into one uncracked subdomain and multiple cracked subdomains. A cracked subdomain is the domain containing the crack and its enrichment functions and the uncracked subdomain contains the rest of the domain with a one-element-layer overlap between the two. Within the preconditioning scheme, one AMG V-cycle is applied to the uncracked subdomain to obtain an approximate solution while the cracked subdomains (often much smaller compared to the uncracked part) are solved concurrently by a direct solver, thus resolving the error from the discontinuous fields exactly. Hence any black box AMG solver can be used for XFEM, and the need for development of special coarsening procedures that handle enriched degrees of freedom can be avoided. We consider multiple propagating cracks and develop an algorithm that adaptively updates the subdomains, following the cracks. This adaptive scheme can be obtained directly from level set values which are updated with crack growth or from close neighbor search algorithms. The level set update scheme is fast but does not guarantee tight subdomains, while a neighbor search is slower but gives optimal subdomains. The preconditioner is tested on structured and unstructured meshes with multiple propagating cracks and shows convergence rates that are significantly better than a brute force application of AMG to the entire domain.