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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

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URSI EMT Symp.,May 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

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URSI EMT Symp.,May 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 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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URSI EMT Symp.,May An Experiment Pulse bandwidth: Micro-Scale: Initially: the filed is described on macro-scale only Micro-Scale Laminate

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URSI EMT Symp.,May 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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URSI EMT Symp.,May 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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URSI EMT Symp.,May 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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URSI EMT Symp.,May Scaling functions, wavelets, and their fields…

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URSI EMT Symp.,May Major Results Previous MRD homogenization results are contained in [Steinberg et. al, SIAM J. Appl. Math., 60(3) 2000 pp ] 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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URSI EMT Symp.,May Decomposition of the EMO We have Invoke Neumann series representation of the EMO The leading term

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URSI EMT Symp.,May For the general term: Location Scale Feynman Diagram in Location-Scale space: Scattering by h (multiplication) Propagation Interaction + Propagation

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URSI EMT Symp.,May Contribution of the leading term Assume micro-scale heterogeneity Then It is known that But we want

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URSI EMT Symp.,May Finally we get: However, recall the original formulation: dependencies of and combined (!):

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URSI EMT Symp.,May Scale dependent dispersion: The new expression for the effective LARGE SCALE heterogeneity: Scale-related dispersion via

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URSI EMT Symp.,May Numerical example Scale-dependent phase (Delay) as a signal traverses a laminate: Bragg regime Theory

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URSI EMT Symp.,May 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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