A Local Element-free Galerkin Method Based On Reduced Freedom Of Sub-domains

Being one of the most effective project numerical analysis tool established in the20th century,the finite element method(FEM) not only developed its own theorylalgely,but also solved a lot of important scientific and engineering problems.lt hasbeen widely promoted by the engineering circles. But with the development of thetimes, the finite element method has met a growing challenge.For some discontinuityproblems like dynamic crack propagation, damage and failure of material, phasetransition and large deformation,it will be very difficult to sove for the finite elementmethod. The uninterrupted creating mesh will increase the amount of calculation.Element-free Galerkin(EFG) method brings a new hope for the solution of theabove problems. EFG method is a new numerical method developed in the1990s.Different from FEM,the interpolation function of EFG method is based on discretepoints rather than meshes,as a result，this method has not need to dependence themeshes as FEM and it can solve the above problems well.Compared with FEM.Butthe EFG method’s the biggest shortcoming is the high computing cost.To get the samecalculation precision,the EFG method will cost much more time.In order to deal with the problem，this paper come up with the local element-freeGalerkin Method Based on reduced freedom of sub-domains.A local element-freeGalerkin method is proposed to simulate elasticity, in which Radial PointInterpolation Method(RPIM) with polynomial and Moving Kriging(MK) interpolationare used seperately and essential boundary conditions are carried out directly likefinite element method. In the scheme of present method, the calculation of each cell iscarried out by meshfree method with Galerkin weak form, and local search isimplemented in interpolation. Local discrete equations of weak form are simplified byreducing freedom, which transfers equations of inner nodes to equations of connectivenodes based on sub-domains. Sub-domains can be random shape, and it is differentfrom influence domain which is usually used in interpolation of mesh-free method.Compatibility of displacement in adjacent sub-domains and convergence of presentmethod are discussed. In contrast to meshfree method with Galerkin weak form,certain modifications are presented to increase its computational efficiency in thispaper. Numerical examples show that computational efficiency of local element-freeGalerkin method is higher than that of standard element-free Galerkin method, andgood accuracy, high convergence can also be obtained.