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Multi-Resolution Homogenization of Multi-Scale Laminates: Scale Dependent Parameterization or: Homogenization procedure that retains FINITE-scale-related physics. Ben Z. Steinberg School of Electrical Engineering Tel-Aviv University. Overview. Multi-Resolution Homogenization – MRH - PowerPoint PPT Presentation
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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
URSI EMT Symp., May 2004 4
An Experiment
Pulse bandwidth:
Micro-Scale:
Initially:
the filed is described on macro-scale onlyMicro-Scale Laminate
URSI EMT Symp., May 2004 5
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
URSI EMT Symp., May 2004 6
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:
URSI EMT Symp., May 2004 7
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)
URSI EMT Symp., May 2004 9
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
URSI EMT Symp., May 2004 10
Decomposition of the EMO
• We have
• Invoke Neumann series representation of the EMO
• The leading term
URSI EMT Symp., May 2004 11
For the general term:
Location
Scale
Feynman Diagram in Location-Scale space:
Scattering by h (multiplication)
Propagation
Interaction + Propagation
URSI EMT Symp., May 2004 12
Contribution of the leading term
• Assume micro-scale heterogeneity
• Then
• It is known that
• But we want
URSI EMT Symp., May 2004 13
Finally we get:
However, recall the original formulation:
dependencies of and combined (!):
URSI EMT Symp., May 2004 14
Scale dependent dispersion:
• The new expression for the effective LARGE SCALE heterogeneity:
• Scale-related dispersion via
URSI EMT Symp., May 2004 15
Numerical example
• Scale-dependent phase (Delay) as a signal traverses a laminate:
Bragg regime
Theory
URSI EMT Symp., May 2004 16
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.
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