The eXtended Bridging Scale Method (XBSM)

Pattabhi Ramaiah

Institute of Structural MechanicsBauhaus University of Weimar

ECCOMAS 2012, September 1014, Vienna, Austria

Motivation The eXtended BSM for fracture Adaptivity Numerical Examples Conclusions

Outline

Motivation

Propagating defects (macroscale)

Governing equations of motionNonlinear behavior at core

Singularity at core

Atomistic modelsMonte Carlo

Computationally expensiveBulk amounts of data

Motivation

Propagating defects (macroscale)

Governing equations of motionNonlinear behavior at core

Singularity at core

Atomistic modelsMonte Carlo

Computationally expensiveBulk amounts of data

Multiscale

Overview of MM

Propagating defects Must track the defect path Adaptivity required

BDM Belytschko & Xiao (2003, 2004) Adaptive XBDM Gracie & Belytscko, 2011 BSM – Liu et al., (20032007) VAC+BSM – Qian (2003, 2004) Phantom node method Timon, Song (2006, 2008) Adaptive XBSM – Current work

XBDM and XBSM

XBDM XBSM

Overlapping domain decomposition scheme

No coupling domain Coupling realized by

displacement bc's.

Why XBSM?

XBDMXBSMCoarse scale

Everywhere Solve independently

Coupling Displacement bc's

on ghost atoms

Coarse scale Doesn't exist – fine region Adaptivity complicated

Coupling Linear energy weighting, Compatibility enforced by

Lagrange multipliers

Bridging Scale Method

u = ū+ũTotal displacement

Coarse scale solution

Fine scale displacements exists in the region of interest

XBSM Overview

Fine scale lattice (111) atoms of FCC crystal Coarse scale – VAC model with phantom node Coupling – Displacement bc's. Adaptivity implemented

XBSM Overview

Fine scale lattice (111) atoms of FCC crystal Coarse scale – VAC model with phantom node Coupling – Displacement bc's. Adaptivity implemented

Fine scale model : Lattice

LJ potential – to model atomatom interactions

FCC unit lattice

(111) plane atoms

Lattice parameters

Fine scale model: Molecular Statics

Minimization of PE

Residual forces

Total system potential energy

Energy minimization – CG method Linearization of residual – not required Stiffness matrix – not required

Virtual Atom Cluster model

Each atom has the same environment Possess 6 nearest neighbors Replace the original structure with VAC Energy density parameter

VAC : features

Not the locations in the original lattice Same lattice parameters & interatomic potential When VAC confirms underlying lattice – Full

scale atomistic models No continuum measure – no stress update Energy density parameter Improved computational efficiency

Coarse scale modelInternal energy

The total displacement

The phantom node method

Cracked element = two overlapping elements Theory Hansbo & Hansbo (2004) The phantom node method Song, Timon(2006,2008)

Coupled model : solution

Coarse scale model: Energy & internal forcesCoupled region – 2 atom setselse where – 1 atom set

Place the phantom nodes Apply bc's Solve for coarse scale solution Ghost atom positions as bc's to fine scale model Solve for fine scale solution Propagate the crack – based on the cracktip position

Adaptivity1. Adaptive refining – convert the coarse scale region into fine scale region

2. Adaptive coarsening – convert the fine scale region into coarse scale region

Adaptivity : steps

High energy elements

tolE – 1530% higher than the Eeqilibrium

Locate the cracktip Identify the elements to be refined Create and initialize the atoms Identify newly cracked elements – phantom nodes Propagate the crack If coarsening – delete the atoms Update the fine scale atoms

Example 1

LJ potential (111) plane atoms of fcc Nearest neighbors only Brittle fracture Neighbor list not updated Displacement load –

5.892 Ang Adaptively propagate the

Example 1 : steps

Coarse scale mesh

Example 1 : steps

Coarse scale mesh Create VAC

Example 1 : steps

Coarse scale mesh Create VAC

Example 1 : steps

Coarse scale mesh Create VAC Fine scale lattice

Example 1 : steps

Coarse scale mesh Create VAC Fine scale lattice Insert the crack

Example 1 : steps

Coarse scale mesh Create VAC Fine scale lattice Insert the crack Apply BC's

Solve coarse scaleSolve fine scale

Example 1 : steps

Solve coarse scaleSolve fine scale

Adaptivity required?

Example 1 : steps

Solve coarse scalesolve fine scale

Adaptivity required? Refine, coarsegrain

Example 1 : steps

Solve coarse scalesolve fine scale

Adaptivity required? Refine, coarsegrain Propagate the crack

Example 1 : summary

1 to n Step

Example 1 : comparison

Example 2 : kink crack propagation

LJ potential – 440 Ang x 180 Ang (111) plane atoms of fcc Nearest neighbors only Brittle fracture Displacement load – 24.798 Ang at 600 Crack not propagated in the coarse scale

Example 2 : BSM

Example 2 : initial BSM and XBSM

Example 2 : BSM and XBSM – 18th LS

Example 2 : Comparison – 18th & 24th LS

18th LS

Example 2 : Comparison – 18th & 24th LS

18th LS

24th LS

Conclusions and future work

MS model with crack propagation VAC based coarse scale model Coupled XBSM model Validated through two examples

Apply XBSM for coupled field problems Piezoelectric materials Three dimensional models

Collaboration

Timon Rabczuk Robert Gracie Stephane Bordas

Thank you for your attention

The eXtended Bridging Scale Method (XBSM) · 2014-03-06 · ECCOMAS 2012, September 1014, Vienna,...

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