استفاده از توابع پايه نمايي در حل معادلات ديفرانسيل محيط هاي متخلخل اشباع

چكيده انگليسي :

Exponential Basis Functions EBFs in Solution of Fluid Saturated Porous Media Differential Equations Mohammadreza Taghdirian m taghdirian@yahoo com 23 April 2011 Department of Civil Engineering Isfahan University of Technology Isfahan 84156 83111 IranDegree M Sc Language FarsiSupervisor Prof Bijan Boroomand boromand@cc iut ac ir AbstractThe theory of poroelasticity concerns with the analysis of a porous medium consisting of an elastic matrixcontaining interconnected fluid saturated pores Due to interaction between solid and fluid phases behavior of multiphase continuum is completely different from single phase continuum One of the firsttheories was proposed by Biot to deal with soil consolidation quasi static and wave propagation dynamic problems The governing differential equations in such problems are derived from constitutiveequations equations of motion continuity equation and Darcy s law Another approach to describe thebehavior of porous media known as the Theory of Porous Media TPM is based on the theory ofmixtures The theories of poroelasticity are currently applied to a large number of problems in geophysics petroleum industry soil mechanics hydrogeology and biomechanics Due to the complexity of thesolution in such problems efficient methods are needed to achieve sufficiently accurate solutions The useof boundary element method BEM and the method of fundamental solution MFS is recommended inthe literature however it is well understood that these methods need Green s functions or fundamentalsolutions which are not available for many problems In this dissertation exponential basis functions EBFs are evaluated as the bases for the solution ofproblems with porous materials based on Biot s theory BT The use of EBFs enables one to solve a widerange of poroelastic problems since the governing differential equations in these problems are of constantcoefficient type In the presented method the solution is split into two parts i e homogeneous andparticular parts The homogeneous part of equations is then approximated by linear combination of EBFs Introduction of the EBFs into the homogeneous governing differential equations leads to a characteristicequation through which the exponents of the EBFs are defined For many cases the characteristicequation possesses some multiple roots This makes the evaluation of the EBFs a rather tedious task Insuch situations polynomial functions are added to EBFs In this dissertation closed form expressions arefound for such bases while explaining the procedure to give insight to the problem After selection of theEBFs as the bases for the approximation the unknown coefficients of the series are determined by thesatisfaction of the boundary conditions in a collocation style through a discrete transformation technique While using the transformation the number of EBFs should not necessarily be equal to or less than thenumber of boundary information data The content of the EBF series plays an important role in reducingthe computational error and reaching reasonable accuracy in the results Therefore in this study a suitablestrategy is proposed for choosing the EBFs based on determining the contribution of the bases to theboundary conditions The particular part of the solution is constructed by a rather similar approach Withthe characteristics explained above the method falls in the category of Trefftz methods Advantages ofthe method versus other numerical methods such as FEM and BEM lie in the fact that it is free formmeshes numerical integrations Moreover unlike the MFS method there is no need to find singularfundamental solutions in this method In this thesis EBFs are evaluated for dynamic quasi static andstatic formulations in compressible and incompressible models The capabilities of the method have beenshown by comparing the results with analytical solutions The results show that method performsexcellently in solution of different poroelastic problems with various boundary conditions Key WordsExpon

تقديريان، محمدرضا

استفاده از توابع پايه نمايي در حل معادلات ديفرانسيل محيط هاي متخلخل اشباع

تئوري بيوت , تبديل ويژه , روش بدون شبكه