Multi-Resolution Homogenization of Multi-Scale Laminates: Scale Dependent Parameterization

or:Homogenization procedure that retains FINITE-scale-

related physics

Ben Z. SteinbergSchool of Electrical Engineering

Tel-Aviv University

URSI EMT Symp., May 2004 2

Overview

• Multi-Resolution Homogenization – MRH

– Basic Properties

– Formulation Outline

• Extending the Role of Homogenization (use a specific

example)

– Keeping more Micro-Scale Information:

• In a Macro-scale formulation

• Scale-related physics (vanishes in the limit of vanishing micro-scale?)

– Achieved by: Global Effective Operator Study/Correction

• Higher order collective effects (Back Scattering)

• Feynman diagrams interpretation

• Length-Scale Related Dispersion – Analytic results

• Numerical Simulation

URSI EMT Symp., May 2004 3

MRH Theory

• Use Multi Resolution analysis and wavelets to achieve an exact

decomposition of the governing formulation into a hierarchy of

scales.– Define your scales (Medium properties and field observables)

– Galerkin-type projection

• Derive exact self consistent formulation – governing only the Macro-Scale field.

– Effects of Micro-Scale heterogeneity on the Macro-Scale field are expressed via

the EMO.

• Study (neglect?) the EMO. Norm bounds and properties w/respect to: – Greens function properties (regularity @ origin, wavelength)

– Heterogeneity properties (regularity, scale-content, size)

• Turn back the crank; identify structure similarity w/respect to original

formulation

• Associate: identify new heterogeneity functions as the effective ones

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An Experiment

Pulse bandwidth:

Micro-Scale:

Initially:

the filed is described on macro-scale onlyMicro-Scale Laminate

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Later….

Micro-Scale Field

Macro-Scale Fields

While passing through the laminate:

• Undergoes distortion (slight?

Negligible?)

• Transfers energy to small scales

After it traverses the laminate:

• Transfers energy from small scales back to large scale

• Observed on Macro-Scale: Distortion + Delay (micro-

scale related)

• Hence: Effective Dispersion, observed on Macro-Scale

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Major Technical Steps

• The field is governed by

• Choose homogenization scale - the scale on which the solution is to be smoothed.• Usually • Cast the problem into an integral equation formulation

– Bounded operator

– BC are inherent in the formulation structure (kernel)

• Decompose into scales via MRD (i.e. Galerkin) of the integral operator:

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Major Technical Steps (Cont.)

• The result is:

• Where:

• Formulation governing macro-scale field component:

• Main analytical effort: study the EMO (e.g. structure & norm bounds

w/respect to physical parameters)

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Scaling functions, wavelets, and their fields…

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Major Results

• Previous MRD homogenization results are contained in [Steinberg et. al, SIAM J. Appl. Math., 60(3) 2000 pp 939-966]

• Valid for periodic and non-periodic structures

• Allows for a continuum of scales

• Classical homogenization results reconstructed as special cases

– EMO has been neglected (“approved” by norm bounds)

• New study:

– Decompose the EMO into a hierarchy of multiple interactions– Scale-related analysis of the leading term

New physics not contained in classical results: scale dependent dispersion

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Decomposition of the EMO

• We have

• Invoke Neumann series representation of the EMO

• The leading term

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For the general term:

Location

Scale

Feynman Diagram in Location-Scale space:

Scattering by h (multiplication)

Propagation

Interaction + Propagation

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Contribution of the leading term

• Assume micro-scale heterogeneity

• Then

• It is known that

• But we want

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Finally we get:

However, recall the original formulation:

dependencies of and combined (!):

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Scale dependent dispersion:

• The new expression for the effective LARGE SCALE heterogeneity:

• Scale-related dispersion via

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Numerical example

• Scale-dependent phase (Delay) as a signal traverses a laminate:

Bragg regime

Theory

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Conclusions

• Scales = Fun !

• MRH provides micro-scale-dependent

parameterization of effective macro-scale formulation

• Effective dispersion that depends on micro-scale has

been derived

• Micro-Scale dependent effective description of the

medium is materialized on LARGE SCALES only.