Multiscale analysis of wave propagation in composite materials

Abstract

The multiscale solution of the Klein‐Gordon equations in the linear theory of (two‐phase) materials with microstructure is defined by using a family of wavelets based on the harmonic wavelets. The connection coefficients are explicitly computed and characterized by a set of differential equations. Thus the propagation is considered as a superposition of wavelets at different scale of approximation, depending both on the physical parameters and on the connection coefficients of each scale. The coarse level concerns with the basic harmonic trend while the small details, arising at more refined levels, describe small oscillations around the harmonic zero‐scale approximation.

Darbe nagrinejamas Kleino‐Gordono lygčiu tiesineje faziu mikrostruktūriniu medžiagu teorijoje daugiasluoksnio uždavinio sprendimas. Sprendiniui nustatyti naudojamasi bangeliu šeima, turinčia harmoniniu bangeliu prigimti. Jungties koeficientai tiksliai randami ir nusakomi diferencialiniu lygčiu rinkiniu. Bangos plitimas yra nagrinejamas kaip bangeliu skirtinguose sluoksniuose aproksimacijos superpozicija, priklausanti tiek nuo fizikiniu parametru, tiek nuo jungties koeficientu kiekviename sluoksnyje. Grubus priartejimo lygmuo nagrineja tik harmonines slinktis, kai, tuo tarpu, smulkios detales, atsirandančios subtilesniuose lygmenyse, aorašo smulkias osciliacijas aplink harmonine nulinio lygio aproksimacija.

Key words:

• Harmonic Wavelets
• Connection Coefficients
• Multiscale
• Klein‐Gordon Equations