MULTI-SCALE ANALYSIS OF WAVE PROPAGATION IN DAMAGED

S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006

Stefano Gonella, Massimo RuzzeneSchool of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, GA

MULTI-SCALE ANALYSIS OF WAVE PROPAGATION IN DAMAGED

HOMOGENIZED PERIODIC MEDIA

USNCTAM 06Boulder, CO - June 25-30 2006


Overview of Homogenization of Periodic Structures

Bridging Multi-scale Method: Theoretical Fundamentals

Longitudinal Wave Propagation in a Rod

Wave propagation in a Damaged Rectangular Plate

Integration of homogenization and multi-scale analysis for a Homogenized Bi-Material Rod with Localized Imperfections

Outline


HOMOGENIZATION : technique to derive equivalent continuum equations that approximate the behavior of a given periodic structure

The Homogenization Method

DIFFERENT LENGTH SCALES between the macroscopic structure and the repetitiveelement are described by parameter ε such that

= characteristic dimension of the repetitive element= characteristic dimension of the macroscopic system

Structure’s lengths, stiffness and mass terms, and forces are scaled through ε

ASSUMPTION : Wavelengths of considered deformation are much higher than the characteristic dimensions of the unit cell;

The solution of homogenized equations reduces the size of the problem and its computational cost


Elastodynamic equations (harmonic motion) for unit cell at location m:

Nodal degrees of freedom and forces

=dynamic stiffness operator= lattice dimensions

Given spatial periodicity, Discrete spatial Fourier Transform gives:

Transformed elastodynamic equation:

=“Symbol” of the system

where =location of neighboring cells

The symbol needs to be scaled through the small parameter

With the scaled symbol the solution in the Fourier domain becomes:

(*) Martinsson, P.G. “Fast Multiscale Methods for Lattice Equations” Ph.D Thesis University of Texas at Austin, 2002.

Homogenization of Lattice Equations (*)


A Taylor expansion of the symbol can be performed such that

Homogenization of Lattice Equations

It can be shown that for lattices the approximated inverse of the symbol assumes the form:

Taking the IDFT of this expression yields an equivalent differential equation:

Using in the solution for in the Fourier domain, yields:


Structure featuring complex periodic pattern

Coupling of the two meshed regions requires particular careIn order to avoid SPURIOUS NUMERICAL PHENOMENA

BRIDGING MULTI-SCALE METHOD (*)

(*) Kadowaki, H., Liu, W.K. “Bridging multi-scale method for localization problems”Computer Methods In Applied Mechanics And Engineering, 193 (2004), pp. 3267-3302

Bridging multi-scale method Introduction

Homogenization reduces computational cost

Homogenization assumes periodicity

Homogenization fails around discontinuity

Mesh refinement is needed around discontinuity


=

Computational domain Coarse scale (homogenization) Fine scale

Solution is found as:

Coarse solution Fine solution

Bridging multi-scale method Methodology Outline (I)

Fine scale eq.

Coarse scale eq.

Kinetic and strain energy:

Application of Lagrange’s equations:

+


= +

Computational domain Coarse scale Fine scale

Bridging multi-scale method Methodology Outline (II)

Fine scale is partitioned as:retainedDOFs

condensedDOFs

Assumptions:• Coarse scale is sufficient to capture behavior of entire domain;• Fine scale accounts for perturbation due to discontinuity;

Interface conditionfor condensed DOFs

Partitioning of fine scale eq. yields:


Ωc

Ωf(a)

Nf / Nc = 10x, u(x)

2

0)0,(⎟⎠⎞

⎜⎝⎛−

== σx

eutxu

x, u(x)

Coarse-scale initial conditionFine-scale initial condition

Application to aOne-Dimensional Rod

Schematic

Initial disturbance


0 20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

0.8

1Fine mesh

Bridging multi-scalemethod

Simple continuity ofdisplacement at the scale interface

One-Dimensional Rod:Energy Analysis

Normalized Energy of Fine-Scale Window

t [s]

ε


Application to a DamagedRectangular Plate

In-plane wave propagation of a free-free rectangular plate

Initial disturbance

with

y

x

y

x

STIFFINCLUSION


Application to a DamagedRectangular Plate

y

x

STIFFINCLUSIONFINE SCALE

WINDOWS

COARSE-SCALE MESHOF THE ENTIRE DOMAIN

FINE-SCALE WINDOW


Homogenized equation

Truncation at first order (p=1)

E2 , ρ2E1 , ρ1

whereSame resultsUsing the Ruleof Mixtures

Homogenization of a Bi-Material Rod


“First Order” Homogenized E.O.M. valid in long wavelength limit

The suggested homogenization technique is developed in the frequency domain.

Homogenization of a Bi-Material Rod

Homogenized Bi-material Rod

Coarse–Scale FE solution of the Homogenized E.O.M.

Wave equation with homogenized properties

Fine-Scale FE solution of the full Bi-Material Rod Dynamic Problem


Homogenization and Bridging Multi-scale Method

The e.o.m. captures a non-dispersive behavior whereas the bi-material rod is a dispersive medium

The homogenized e.o.m. is valid at low frequencies only

The initial disturbance is to be chosen such that higher frequencies are not excited

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

ω[r

ad/s

]

ξ [1/m]

x 10 4

Bi-material rodHomogenized Rod


Bi-material Bar with Localized Defect

REFERENCE SOLUTION

Bi-Material Rod withLocalized Defects

Crack-like discontinuity at x=0.22 m modeled as a region with lower local

Young’s modulus

Localized Defect


COARSE-SCALE FINE-SCALE

Bi-Material Rod withLocalized Defect

A thin portion of the rod has lower Young’s modulus (simulates crack)


COARSE-SCALE FINE-SCALE

Bi-Material Rod withWith Damaged Region

A considerable portion of the rod has lower Young’s modulus (simulates soft inclusion)

Multiple reflections at the boundaries of the soft inclusion are visible in both scales with different levels of accuracy

The potentials of the bridging method are stretched to fit a more complex dynamic scenario with several crossings of the scale interface in a limited time frame


Dependence upon theInitial Disturbance

0

0.2

0.4

0.6

0.8

1

E T(fi

ne s

cale

regi

on)

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4tf [s]

Fine-scaleReference solution

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

0

0.2

0.4

0.6

0.8

E T(fi

ne s

cale

regi

on)

1

tf [s]

Fine-scaleReference solution

Abrupt variations in the initial displacement excite a broader range of frequencies. The consequent dispersiveBehavior of the coarse scale affects the solution for the reflected wave

INITIAL EXCITATIONS


Dependence upon theInitial Disturbance


Conclusions

1. The bridging multi-scale method works as a tool to model localized defects without involving a detailed discretization of the whole structural domain

2. The bridging method can be coupled with the multi-scale homogenization for periodic media

3. The limitation of the coupled procedure is related to the compatibility between the original and the homogenized model.

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MULTI-SCALE ANALYSIS OF WAVE PROPAGATION IN DAMAGED