## 2 Problem Statement: Nonlinear Submerged Arches Design
### Geometrical Nonlinear Analysis and Optimal Design  
In practice, the arch bending moment cannot be completely suppressed. Therefore, the real buckling factors (\(\alpha_{crit}\)) may be lower than the ones obtained by the eigenvalue method. A more accurate stability analysis may be performed taking into account the geometrically nonlinear behaviour of the submerged arch. For this purpose, the updated Lagrangian formulation [42; 43; 44] may be applied, at which all variables are referred to the current (or updated) configuration of the arch. In this context, the equilibrium of the arch may be expressed through the principle of virtual displacements in the following terms:  
\[\int_{V(t)}\mathbf{S}(t+\Delta t):\delta e(t+\Delta t)dV(t)=R(t+\Delta t) \tag{7}\]  
where \(\mathbf{S}\) is the 2 nd Piola-Kirchhoff stress tensor and \(\epsilon\) is the Green-Lagrange strain tensor from the configuration at time \(t\) to the configuration at time \(t+\delta t\) and referred to the configuration at time \(t\), and \(R\) is the external virtual work. Equation (7) is a nonlinear one in the unknown incremental displacements. Nevertheless, higher-order terms may be neglected and the previous weak form may be linearized through the corresponding incremental decomposition of stresses and strains. Moreover, the resulting linearized expression of Equation (7) may be solved using the iso  
matrix), **F** is the vector of external nodal loads at time \(t+\Delta t\) and \(i\mathbf{F}_{r}\) is the vector of nodal point forces equivalent to the current element stresses (or the vector of restoring forces) at time \(t\). **U** is the vector of increments in nodal displacements, **T** is the Cauchy's stress tensor and **C** is the incremental material property matrix. The linearization of the equilibrium equations may introduce errors which ultimately result into solution instability. Due to this reason, an iterative strategy is necessary. To this aim, the design load may be divided into a set of steps. At each load step the out-of-balance load vector (i.e., the difference between the loads corresponding to the element stresses and the applied loads) is evaluated in order to check the convergence criteria. If these criteria are not satisfied, a new seed is evaluated and a new solution is obtained until the problem converges. Consequently, obtaining the second-order load-displacement history of the submerged arch it is a path-dependent problem at which loads must be applied through several steps in order to get a high accuracy. In the context of the Modified Newton's method (where the stiffness relation is only evaluated at the start of the load step: this method is implemented in the finite element software ANSYS [45]), the previous iterative process may be formulated as [46]  
\[ \begin{gathered} i\mathbf{K}\Delta\mathbf{U}^{k}=\mathbf{F}(t+ \Delta t)\ -\begin{matrix}t+\Delta t\\ t+\Delta t\end{matrix}\mathbf{F}_{r}^{k-1}\\ \text{with}\\ \mathbf{\Delta U}^{k}(t+\Delta t)=\Delta\mathbf{U}^{k-1}(t+\Delta t)+\sum_{ j=1}^{k}\Delta\mathbf{U}^{j},\ k=1,2,3,...\\ \mathbf{U}^{0}(t+\Delta t)=\mathbf{U}(t)\end{gathered}  \tag{9}\]  
where \(k\) is the number of iterations until verifying the convergence criteria. As was pointed out in the first epigraph of this section, two types of loads have been considered in this work: self-weight and hydrostatic pressure. The first has been introduced using the gravitational acceleration, whereas the hydrostatic action over the submerged arch has been introduced as a pressure load. In a large-deflection analysis, pressure loads present a singular behaviour (with regard to other load types as accelerations or nodal forces), as they change their original orientation during their application, acting always normal to the deflected element surface and \"following\" the finite element as it undergoes large rotations.