In this paper, the filter function in the ICM method and the penalty function in the variable density method are both referred as the mapping functions. Different forms of mapping functions have a significant impact on the convergence efficiency of structural topology optimization. Therefore, it is necessary to study how to construct a suitable mapping function for the optimization model. Aimed at this problem, how to construct and select a mapping function in the establishment of the structural topology optimization model is studied, and the influence of different mapping functions on the convergence efficiency of structural topology optimization is discussed. An approach is proposed to construct a mapping function to achieve high-efficiency convergence in structural topology optimization. Five common forms of mapping functions are also given. An optimization model and a solution algorithm matching the mapping function with highly efficient convergence (MFHEC) are proposed. Firstly, the convergence rates of the filter function and the quasi-filter function of the same form of mapping functions are compared. Then the convergence rates of the fast filter function of different forms of mapping functions are compared. Taking the structural topology optimization problem of minimizing structural volume under displacement constraints as an example, the ICM method is adopted to establish the models and solve the problems. The higher convergence efficiency of MFHEC is verified by the results of numerical comparison. The results show that the fast filter function has a faster convergence rate than other functions in the same form of mapping functions. Compared with five different forms of mapping functions, the filter function of power function form has the fastest convergence efficiency. Finally, it should be emphasized that the conclusions of the mapping function studied in this paper are equally applicable to the filter function of the ICM method and the penalty function of the variable density method. The proposed method for constructing MFHEC is very useful for improving the efficiency of the ICM method and the variable density method.