Two-scale And Finite Element Algorithm Analysis Of One Class Of Piezoelectric Problem In Quasi-periodic Structure

Because of the inherent mechanic-electric coupling effect and strong piezoelec-tricity the piezoelectric materials been used widely in the sensor, aerospace and semi-conductor materials and other high-technology areas. It is important to describe theeffective physical and mechanic behavior of those materials. And it is one of the mostimportant research objects in new functional materials in periodic or quasi-periodicstructure of the piezoelectric composite materials. Because the periodicity of mate-rials, it is very difficult to get the analytic solutions of the differential equations forpiezoelectric problems, the key of research object is to discuss the effective and feasi-ble numerical simulation method.In this paper, the two-scale and finite element analysis for the numerical solutionof differential equations of one class of piezoelectric problem in quasi-periodic struc-ture is given. Using two-scale asymptotic method and constructing the appropriatecell functions, the higher order formal two-scale asymptotic expansions for the piezo-electric problem are presented. Based on the formal expansions, the homogenizationequations of describe equivalent piezoelectric behavior and the calculation methodof homogenization mechanics and piezoelectric constants of material are established.More the asymptotic error estimation for the two-scale asymptotic expansions is pre-sented. Then the two-scale expansions of approximate higher order are establishedtheoretically. Finally the corresponding two-scale finite element computational algo-rithm for the formal asymptotic expansions is given, and some simple two-scale and fi-nite element error estimations are established. The computing speed and efficiency ofcomputing the numerical solutions for piezoelectric problem in quasi-periodic struc-ture is improved by means of the algorithm.The results in this paper will have some theoretical guidance and applicationvalue in the process of development, design and applications of advanced compositematerials.