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Ben Z. Steinberg School of Electrical Engineering Tel-Aviv University

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

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Ben Z. Steinberg School of Electrical Engineering Tel-Aviv University

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  1. Multi-Resolution Homogenization of Multi-Scale Laminates: Scale Dependent Parameterizationor:Homogenization procedure that retains FINITE-scale-related physics Ben Z. Steinberg School of Electrical Engineering Tel-Aviv University

  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 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 May 2004

  4. Micro-Scale Laminate An Experiment Pulse bandwidth: Micro-Scale: Initially: the filed is described on macro-scale only May 2004

  5. Micro-Scale Field Macro-Scale Fields Later…. • 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 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: 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) May 2004

  8. Scaling functions, wavelets, and their fields… 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 May 2004

  10. Decomposition of the EMO • We have • Invoke Neumann series representation of the EMO • The leading term May 2004

  11. Scale Location For the general term: Feynman Diagram in Location-Scale space: Scattering by h (multiplication) Propagation Interaction + Propagation May 2004

  12. Contribution of the leading term • Assume micro-scale heterogeneity • Then • It is known that • But we want May 2004

  13. Finally we get: However, recall the original formulation: dependencies of and combined (!): May 2004

  14. Scale dependent dispersion: • The new expression for the effective LARGE SCALE heterogeneity: • Scale-related dispersion via May 2004

  15. Numerical example • Scale-dependent phase (Delay) as a signal traverses a laminate: Bragg regime Theory 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. May 2004

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