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Improved meshless local Petrov Galerkin methods Mohadese Ramezani mohadeseh ramezani@math iut ac ir 20 09 2014 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Reza Mokhtari mokhtari@cc iut ac ir Advisor Dr Davoud Mirzaei d mirzaei@sci ui ac ir 2014 MSC 65M99 Keywords Meshless local Petrov Galerkin method Generalized Moving least squares approxima tion Moving Kriging interpolation Transient heat conduction Fractional di erential equation Abstract Transient heat transfer problems have found many applications to plenty of engineering andscience problems Most of the transient heat conduction problems have been numerically solvedby meshless methods such as the element free Galerkin EFG method reproducing kernel parti cle method RKPM meshless local Petrov Galerkin MLPG and boundary element free method BFEM Among all the meshless methods the MLPG method has been widely used in solving heatconduction problems But there exists two shortcomings which are the shape functions based on themoving least square MLS approximation lack the Kronecker s delta property and numerical inte gration over the complicated MLS shape functions leads to an expensive numerical scheme In orderto eliminate these shortcomings of the MLS shape functions the moving Kriging MK interpola tion technique and the generalized MLS GMLS method can be employed instead of the traditionalMLS approximation So some improved meshless local Petrov Galerkin methods such as the directMLPG DMLPG method have been established Gu has rstly proposed the moving Kriging inter polation and has successfully developed a new moving Kriging interpolation based EFG for solvingone dimensional steady state heat conduction problems Later the meshless methods based on themoving Kriging interpolation have also been developed and studied for example the element freeGalerkin method using moving Kriging interpolation the meshless local Kriging loKriging method and the boundary node method based on the moving Kriging interpolation The DMLPG method originated by Mirzaei and Schaback has been successfully applied to heat conduction two and threedimensional potential problems and anisotropic di usion problems Besides e ciencies of the methodfor handling the elasticity problems must be mentioned In this thesis some preliminaries have been prepared and some materials about the moving Kriginginterpolation GMLS approximation and MLS approximation have been represented One of the im proved MLPG uses the moving Kriging interpolation techniques to construct meshless shape functions for a set of randomly distributed points which possess Kronecker s delta property and therefore es sential boundary conditions can be implemented easily The local weak form of partial di erentialequations PDEs is derived by the weighted residual formulation based on a simple shaped localdomain Additionally in implementation of the local weak form the Heaviside step function is usedas the test function In another improved MLPG method DMLPG method boundary conditions aredirectly imposed a GMLS method is used and numerical integrations involved in the local weak formsare shifted to on the polynomial basis functions instead of on the MLS shape functions Then theseimproved methods have been extended to the transient heat conduction problems and three numericalexamples solved by them In these examples both regular and Halton points are employed Thesemethods applied to some fractional di erential equations as well A numerical example has beensolved by the rst improved MLPG method and reported its e ciencies compared to the existingschemes Finally we present some numerical results with both regular and scattered quasi uniformmesh points to demonstrate the e ciencies of the DMLPG method in comparison with the classicalMLPG methods So the improved MLPG methods have been successfully applied to the transientheat conduction problems and fractional di erential equations

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