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چكيده انگليسي :

Solution of Solid Mechanics Problems Using Generalized Basis Functions Nima Noormohammadi n noormohammadi@cv iut ac ir Date of Submission September 21 2011 Department of Civil Engineering Isfahan University of Technology Isfahan 84156 83111 Iran Degree M Sc Language FarsiSupervisor Prof Bijan Boroomand boromand@cc iut ac ir AbstractIn this thesis a new boundary point method is presented for the solution of problems in solid mechanics The methodis based on the use of basis functions satisfying the governing partial differential equations PDEs approximately The idea is in contrast with the approach usually employed in the conventional methods using fundamentalsolutions In this method the basis functions are found through satisfaction of the weighted residual of the PDEswritten in integral forms The primary bases and the weight functions used for obtaining the final basis functions are respectively considered as Chebyshev polynomials of the first kind and exponential functions The reason of thisselection is the consistency between the basis and the weight functions which gives good characteristics to themethod The integrals are taken on a fictitious rectangular domain circumscribing the actual domain This helps tocompute the multidimensional integrals by multiplication of simpler one dimensional ones This is possible becausethe basis and the weight functions can always be written as multiplication of a function of x into a function of y Such decomposition also speeds up the computational procedure significantly After forming all the required integrals they will be arranged in a matrix equation to compute the constantcoefficients of solution series By solving the matrix equation a set of secondary basis functions will be created thatsatisfy the homogeneous operator approximately The new bases form another solution series with new unknowncoefficients which should be obtained by applying the boundary conditions A special discrete transformation isused for this purpose which calculates each coefficient according to the projection of the corresponding basis on theboundary Different shapes of domain can be considered in the method The method is capable of solving PDEs with non constant coefficients A special technique is developed for this partso as to continue the use of separation method discussed for PDEs with constant coefficients In this technique the socalled Pascal triangle is used to rewrite the non constant coefficients of operators as the sum of simple polynomials Then the one dimensional integrals are calculated for each part of the series and combined through a special matrixoperation By using this technique the speed of calculation for equations with constant and non constant coefficientwill be of the same order Three partial differential operators are discussed in this research namely operators in engineering problems knownas Helmholtz elasto static and elastic wave and Kirchhoff plate Each operator is formulated in two approaches forthe integral equation known as the strong form and the weak form The latter has the advantage of removing alldifferential operators from the non constant coefficients of the equation which may have complicated behavior Theformulation of thesis is for two dimensional problems but it is possible to extend it to three dimensional cases whileholding all advantages addressed for two dimensional problems The examples solved throughout the text confirmthe capability and accuracy of the method in solution of PDEs with both constant and non constant coefficients Convergence study is given for all examples solved Key WordsPartial differential equation Weighted residual integral non constant coefficients Chebyshevpolynomials

نورمحمدي، نيما

حل مسائل مكانيك جامدات با استفاده از توابع پايه تعميم يافته

معادله هلمهولتز , معادلات الاستيسيته , موج الاستيك