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The modified basis functions of element-free Galerkin (EFG) method generated by weight function control of moving least-squares interpolants for an axisymmetric problem are investigated. In the axisymmetric problem, a physical quantity of rotating angle direction on the symmetry axis should be zero. By using standard EFG, however, the values on symmetric axis are non zero because the axisymmetric is not considered in basis function. In order to clear up the above problem, the modified basis function of EFG is proposed. The numerical simulation of shielding current density in axisymmetric high-temperature superconductor is adopted for practical applications of modified axisymmetric EFG. The results of computations show that the shielding current density on symmetric axis vanish in both cases. In addition, the experimental critical current density measurement method is reproduced numerically by using the numerical code.