A Sub-domain Element Free Galerkin Method For Solid Mechanics Problems

Computational solid mechanics have been widely applied to solve practical engineering problems.In the numerical methods of computational solid mechanics,Finite element method(FEM)and the meshfree methods based on Galerkin weak form occupy very important position.But the two methods both have some shortcomings.The calculation accuracy of FEM usually depends on the quality of grid division,while the computational efficiency of the meshfree methods is general low.Therefore,it is of great significance to develop a numerical method which could intergrate the advantages of FEM and the meshfree methods.To provide an effective alternative path to solve the problems of computational solid mechanics,a sub-domain meshfree Galerkin method is presented in this paper.The method not only could keep the adaptive ability of the meshfree methods,but also have the advantages of high efficiency of FEM.In this paper,the sub-domain meshfree Galerkin method is firstly proposed and analysed.Then,in order to further improve the adaptability,accuracy and efficiency of the method,various measures have been developed.The main work of this paper is as follows:(1)The sub-domain meshfree Galerkin method is proposed by constructing sub-domains,and the compatibility and completeness of the method are analysed.The influence of calculation parameters on accuracy and efficiency is studied and the ranges of these parameters are given.Under the premise of ensuring the accuracy,stability and convergence,the calculation efficiency of the method could be availably improved by constructing sub-domains and reducing the number of freedom,which is verified by numerical examples.(2)A sub-domian smoothed meshfree Galerkin method is proposed by introducing strain smoothing technique into the sub-domain meshfree Galerkin method.The method could satisfy linear exactness,and change the domain integration based on the background meshes to line integration based on the smoothing cells.The efficiency and adaptability of the method are further improved.Strain smoothing technology makes the method have the properties of upper-lower bounds,and does not influence the ranges of calculation parameters.The effect of the number of boundary nodes on computational accuracy is analysed,and the high convergence,stability,accuracy and efficiency of the method are verified by numerical examples.(3)The static and free vibration analysis of Reissner-Mindlin plates with different shapes and thicknes is performed under different boundary conditions.When the thickness of the plate approaches to zero,the shear locking phenomena could be avoided by modifying the material matrix.The calculation results verify the high convergence,stability and accuracy of the method.(4)In order to analyse the fracture mechanics problems better,a sub-domain smoothed extended meshfree Galerkin method is proposed by introducing the enriched shape functions to the sub-domain smoothed meshfree Galerkin method.When using the interaction integral to calculate the stress intensity factor in the method,the influence of the size of the integral domain on the calculation results is studied.By analysing the results of the mode I and mixed-mode I-II stationary crack problems obtained by the sub-domain smoothed extended meshfree Galerkin method,the accuracy and stability of the method for linear elastic crack problems are verified.(5)The sub-domain smoothed extended meshfree Galerkin method with the modified material matrix is used for free vibration analysis of cracked Reissner-Mindlin plate.Under different boundary conditions,the plates with various form crack and different geometrical sizes are calculated.The results are compared with the results obtained by other methods,which verifies that the method has high accuracy and stability for the analysis of cracked plate.The shapes of eigenmodes express exactly the real physical modes of the cracked plate,and it indicates the prospects for the application of the method to further solving the practical engineering problems.