Interpolation Type Heat Conduction Problem Free Galerkin Method

In recent years, the meshless methods is gradually being widely appliedwith its unique advantages for science and engineering problems.The meshlessmethods only need the information related to nodes, and needn’t divide thesolving domain into a mesh. The simple pre-processing and highcalculation precision make it different from other numerical methods. TheElement-Free Galerkin (EFG) method based on the moving least-squares (MLS),one of the meshless methods, has acheived a wide range of research.The shape function in the moving least-squares (MLS) approximation doesnot satisfy the property of Kronecker delta function, so an interpolating movingleast-squares (IMLS) method is discussed, Then in this dissertation, on the basisof the EFG method and the interpolating moving least-square (IMLS)approximation, an interpolating Element-Free Galerkin (IEFG) method isdeveloped. The main advantage of this approach over the conventional meshlessmethods is that essential boundary conditions can be applied directly, and thenthe computational efficiency is improved greatly.The IEFG method is applied to steady-state heat conduction problems withheat generation and spatially varying conductivity. Combining the shapefunction constructed by the IMLS method and Galerkin weak form of thetwo-dimensional steady-state heat conduction problems, an interpolatingelement-free Galerkin (IEFG) method for steady-state heat conduction problemsis presented, and the corresponding formulae are obtained.The IEFG method is also applicable to transient heat conduction problem inthis paper. The traditional two-point difference method is used to discrete timedomain and IEFG method is used to discrete spatial domain. the essentialboundary conditions is enforced directly, and then the IEFG method for transientheat conduction problem is presented，and the corresponding discrete equation isdeduced from the weak form of governing equations. Compared with sometraditional meshless methods, the IEFG method has greater precision.In order to verify the efficiency of the IEFG method in the dissertation, the MATLAB codes of the IEFG methods related to heat conduction problemsabove have been compiled. Some numerical examples are provided, andnumerical results show that the IEFG method has high computational accuracycompared with EFG method and EFM.