## 1 Introduction  
The seismic response behavior of engineering soil field has been the subject of intense research in the area of soil dynamics and geotechnical earthquake engineering for a long time. The numerical results of the seismic response are vital for the seismic safety evaluation of soil fields. On the other hand, the seismic response computation of soils is a necessary step of soil-structure dynamic interaction analysis. Some underground structures are buried in saturated soils, such as immersed tubes in river beds or sea beds and subway tunnels and stations in estuarine deposits or semimetal basins, and it also has practical significance in the computation and analysis of the seismic response of saturated soils. Saturated soils in nature are fluid-saturated porous media (FSPM). They consist of a solid skeleton and pores. The solid skeleton is composed of soil grains with various shapes. The inter-grain pores are filled with fluid; e.g., water. When the wave motion or seismic response of the FSPM is investigated, the dynamic coupling between the solid skeleton and the pore fluid should be taken into account. The governing equations describing the dynamic behavior of the FSPM were originally proposed by [PERSON] [1] using the solid skeleton displacement \(\mathbf{u}\) and the pore fluid displacement \(\mathbf{U}\) as basic variables. [PERSON] [2] also proposed the governing equations for the description of the dynamic behavior of the FSPM using the solid skeleton displacement \(\mathbf{u}\) and the relative displacement of the pore fluid to the solid skeleton \(\mathbf{w}\) as basic variables. However, it is more convenient to use the \(\mathbf{u}\)-\(p\) dynamic formulation as the governing equations for problems in which high-frequency components are absent, such as those associated with earthquake formulation. The \(\mathbf{u}\)-\(p\) dynamic formulation was originally proposed by [PERSON] et al. [3] using the solid skeleton displacement \(\mathbf{u}\) and the pore fluid pressure \(p\) as essential variables based on some basic assumptions. As a mixed formulation including the vector and scalar essential variables, the \(\mathbf{u}\)-\(p\) dynamic formulation results in a smaller number of nodal unknowns for the finite element formulation in comparison to the \(\mathbf{u}\)-\(\mathbf{U}\) dynamic formulation including only the vector essential variables. On the other hand, the numerical results of the pore fluid pressure \(p\) can be obtained directly from the solution to the \(\mathbf{u}\)-\(p\) dynamic formulation, and this is important for some geotechnical earthquake engineering problems, such as soil liquefaction. The numerical solution procedure for the \(\mathbf{u}\)-\(p\) dynamic formulation can be implemented using the time-stepping implicit, semi-explicit or explicit algorithm. For the time-stepping implicit algorithm, the solid skeleton displacement \(\mathbf{u}\) and pore fluid pressure \(p\) are all solved in an implicit mode. This means that a set of coupling linear equations with regard to the solid skeleton displacement \(\mathbf{u}\) and pore fluid pressure \(p\) should be solved in each time step. The time-stepping implicit algorithm is unconditionally stable, but for dynamic problems with a large number of degrees of freedom, the computational effort and computer memory requirement are enormous when it is adopted. Staggered implicit-implicit algorithms for the solution to the \(\mathbf{u}\)-\(p\) dynamic formulation have been proposed by [PERSON] [4], [PERSON] et al. [5], [PERSON] et al. [6; 7] and [PERSON] et al. [8]. When the semi-explicit algorithm is adopted for the numerical solution to the \(\mathbf{u}\)-\(p\) dynamic formulation, one of the two essential variables, such as the solid skeleton displacement, is solved in an implicit mode, while the other variable is solved in an explicit mode. This means that the numerical results of this essential variable can be obtained in an iterative manner, and the computation efficiency is improved to some extent. [PERSON] et al.