## 4 An Improved Integral Response Deformation Method Considering Far-Field Effects  
In the IRDM, ideal boundary conditions should ensure that non-free-field displacements resulting from uncoordinated motion between the soil and structure undergo sufficient attenuation, thereby accurately capturing the confinement effects of the external infinite medium. On the basis of the preceding analysis, the lateral boundary distance (\(L_{S}\)) and vertical bottom boundary distance (\(H_{B}\)) must each reach three times the structural width (\(3\ L\)) and height (\(3H\)), respectively, to ensure the accuracy of the IRDM. However, for large-scale three-dimensional seismic analyses of underground structures, such boundary configurations lead to excessively large computational models, significantly increasing modeling complexity and computational costs. To address these challenges, boundary conditions that simulate the static resistance of semi-infinite foundations and far-field constraint effects, namely, static artificial boundaries, can be introduced into computational models, thereby reducing the required analysis domain. For dynamic problems, researchers have developed a variety of artificial boundary methods to consider the wave radiation in an infinite domain. Our research team has conducted extensive research and proposed several artificial boundary techniques, including viscoelastic artificial boundaries and viscoelastic artificial boundary elements. By degenerating the viscoelastic artificial boundary into static conditions, a static artificial boundary element suitable for the IRDM can be derived, as shown in Figure 9, with its shear modulus (\(\widetilde{G}\)), Young's modulus (\(\widetilde{E}\)), and Poisson's ratio (\(\widetilde{\mu}\)) determined as follows:  
\[\widetilde{G}=\alpha_{T}\ \mathrm{h}\frac{G}{R},\widetilde{E}=\alpha_{N}\ \mathrm{h}\frac{G}{R}\times\frac{(1+\widetilde{\mu})(1-2 \widetilde{\mu})}{1-\widetilde{\mu}},\widetilde{\mu}=\frac{\alpha-2}{2(\alpha -1)} \tag{2}\]  
Here, \(\mathrm{h}\) denotes the thickness of the equivalent element, \(\mathrm{R}\) represents the distance from the structure to the boundary node, \(\mathrm{G}\) represents the shear modulus of the geological medium, and \(\alpha=\alpha_{N}/\alpha_{T}\), \(\alpha_{T}\), and \(\alpha_{N}\) represent the tangential and normal artificial boundary parameters, respectively. By incorporating static artificial boundary elements into the IRDM, an improved IRDM considering far-field effects is proposed. The implementation processes are as follows:  
1. Free-Field Seismic Response Analysis  
Free-field seismic analysis is performed via standard finite element solvers (e.g., ANSYS 2022R1, ABAQUS 2022(64 bit), Plaxis 3D 2024, or FLAC 3D 3.0-261) to obtain free-field displacements and accelerations at the structural location under vertically propagating seismic waves. The critical time instants corresponding to peak seismic responses are extracted. 2. Equivalent seismic load calculation  
A free-field soil finite element model with an outer layer of static artificial boundary elements (material properties defined by Equation (2)) and fixed exterior nodes is constructed. The free-field displacements from Step 1 at the soil-structure interface nodes are applied, and free-field inertial forces are imposed on the internal nodes (Figure 10a). Then, the nodal reaction forces at the interface are computed as the equivalent seismic loads. 3. Structural inertial force calculation  
Determine the structural inertial forces at critical time instants via the free-field accelerations obtained in Step 1. 4. Integrated Static Analysis  
An SSI model with an outer layer of static artificial boundary elements (material properties defined by Equation (2)) and fixed exterior nodes is developed. The equivalent seismic loads and structural inertial forces at designated positions (Figure 10b) are applied to derive the seismic responses through static computation. Key adaptability notes for engineers:  
1.