## 2 Optimization Model for the Embedded Retaining Wall
### Objective Function and Geotechnical Constraints of the Optimization Model
Since in some cases the water flow is also taken into account, the change in water pressure \(\Delta u_{a,i}\) is determined by Equations (2s)-(2v). The individual water pressures at a certain depth \(u_{a,i}\) are further calculated using Equations (2z)-(2 ac). The resistant horizontal force \(H_{Rd}\), which represents the passive side of the embedded retaining structure, is described by Equation (2 ad) and is the sum of the resistances due to the soil \(H_{Rd,z,i}\) and the resistance due to the existing groundwater \(H_{Rd,u,i}\). The resisting horizontal force is therefore calculated using Equations (2 ae) and (2 af). It is important to note that these resistances are reduced by the partial safety factor \(SF_{Re}\). Equations (2 ag) and (2 ah) are used to calculate for the resistance provided by groundwater, if present. The design horizontal stresses on the passive side \(\sigma^{\prime}_{p,d,i}\) are determined using Equations (2 ai) and (2 aj) and the required vertical stresses \(\sigma^{\prime}_{v,k,i}\) using Equations (2 ak) and (2 al). In addition, seepage is considered on the passive side, which requires the calculation of the changes in water pressures \(\Delta u_{p,i}\) and initial water pressures \(u_{p,i}\) on the passive side described by Equations (2 am) and (2 an), further using Equations (2 ao) and (2 ap). As the model considers several possible scenarios (no groundwater, presence of groundwater (hydrostatic conditions), seepage), the equations in Table 1 are therefore  
If no groundwater is present, the unit of weight is defined as given in Equations (2 ar) and (2 as). If groundwater is present, Equations (2 at) and (2 au) are used to determine the unit weight of soil. If water flow is also considered, Equations (2 av) and (2 az) must be used. If seepage is considered, the hydraulic gradient on the active side, \(i_{a}\) (Equation (2 ba)), and on the passive side, \(i_{p}\) (Equation (2 bb)), must be calculated. The hydraulic gradients are calculated according to the standard DIN 4085:2011-05 [26]. In order to take into account the influence of seepage, the weight of the soil on the active side \(\gamma_{k,a}\) is increased with the calculation method, which reduces the water pressures. On the passive side, the unit weight \(\gamma_{k,p}\) is reduced in the calculation, which increases the water pressures. These calculation methods, which involve increasing or decreasing the unit weight of the soil, have a direct effect on the effective stresses in the soil. While this approach does not fully capture the true mechanical behavior, it provides a valid simplification for computational purposes, eliminating the need for determining flow nets and calculating actual pore water pressures. This concept is illustrated in Figure 2. Line 1 represents the original water pressures, while line 2 shows the water pressures after an increase on the passive side. On the active side, the situation is similar: line 4 shows the original water pressure distribution, and line 3 the water pressures after reduction by groundwater flow. The active and passive earth pressures are calculated in accordance with Eurocode 7-1 [4]. The coefficients for the active and passive earth pressure, which result from the self-weight of the soil \(K_{a\gamma h}\), \(K_{\rho\gamma h}\), the cohesion \(K_{ach}\), \(K_{pth}\) and the uniform surcharge load \(K_{aqh}\), are determined using Equations (2 bc)-(2 bh). The design values for the shear angle of the soil \(\varphi^{\prime}_{d}\), the friction angle between the structure and the soil \(\delta^{\prime}_{d}\) and the cohesion cd are obtained from Equations (2 bi)-(2 bk). Equation (3) outlines the condition for the verification of the moments due to active pressures \(M_{Ed}\) and passive pressures \(M_{Rd}\). The moment on the active side includes contributions from the weight of the soil and the applied load in the backfill \(M_{Ed,z,i}\) as well as the moments due to groundwater \(M_{Ed,w,i}\), which are calculated using Equations (3a)-(3i).