MULTI-SCALE ANALYSIS OF WAVE PROPAGATION IN DAMAGED ...
Transcript of 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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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 (*)
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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:
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
=
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:
+
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
= +
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:
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
Application to a DamagedRectangular Plate
In-plane wave propagation of a free-free rectangular plate
Initial disturbance
with
y
x
y
x
STIFFINCLUSION
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
Application to a DamagedRectangular Plate
y
x
STIFFINCLUSIONFINE SCALE
WINDOWS
COARSE-SCALE MESHOF THE ENTIRE DOMAIN
FINE-SCALE WINDOW
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
Homogenized equation
Truncation at first order (p=1)
E2 , ρ2E1 , ρ1
whereSame resultsUsing the Ruleof Mixtures
Homogenization of a Bi-Material Rod
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
“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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
COARSE-SCALE FINE-SCALE
Bi-Material Rod withLocalized Defect
A thin portion of the rod has lower Young’s modulus (simulates crack)
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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
S.Gonella, M. Ruzzene USNCTAM 06 - Boulder, 06-26-2006
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.