# The Dynamic Response of an Arch Dam During a Record Low-Intensity Earthquake
## 3 Numerical Model
### Modeling of Hydrodynamic Interaction with the 3 DEC Program  
The explicit representation of the reservoir requires the inclusion of elements in the calculation mesh that adequately simulate the dynamic behavior of the fluid and its interaction with the dam and the foundation rock mass, as well as the application of appropriate dynamic boundary conditions. The standard deformable blocks in 3 DEC are discretized into first-degree tetrahedrons, used for representing the rock mass, and 20-node finite elements, used for discretizing the arch. The water in the reservoir is assumed to be a fluid material characterized by an isotropic bulk compressibility and zero shear modulus, thus it does not carry shear stresses. The reservoir mesh is composed of 8-node brick fluid elements following [PERSON]'s mixed discretization technique [25], which is implemented in 3 DEC. The pressure in each element is assumed to be uniform, with it being obtained from the average volumetric strain, calculated as the average of the strains in the tetrahedral mesh in which the brick element is divided. To keep the element symmetric, the two possible discretizations of the brick into five tetrahedra are considered, and their strains are averaged [25]. The mixed discretization formulation to represent the dynamic reservoir response was tested through comparison with analytical solutions for the case of a rigid dam with a vertical face subjected to uniform acceleration [25]. These tests demonstrated that the formulation sufficiently approximates the known parabolic pressure distribution diagram exerted by the fluid on the dam [26]. The dam-reservoir interface is represented by a joint with zero tangential stiffness so that only normal pressure is applied on the upstream face of the dam. This interface allows for the free vertical movement of the fluid surface under the dynamic action imposed by the dam. The vertical water displacementsmay be significant for large earthquakes, but in the present case of a low-intensity event, the relative dam-water shear displacement is very small. Similarly, an interface of the same type was introduced at the base of the reservoir to permit the fluid to slide over the rock mass without generating shear stresses. At the far end of the reservoir, the boundary conditions are required to allow for the radiation of energy to the part of the reservoir not included in the model, which is obtained by a standard viscous boundary formulation.  
Even for a low level action, the simulation of the dynamic response of the dam depends on a correct representation of the hydrodynamic interaction, taking into account the mass and stiffness effects of the reservoir on the structure. Therefore, numerical models have to be validated against experimental data. In this case, there is extensive data obtained from in situ tests and monitoring, as presented in the next section. The numerical model developed using this approach was validated during interpretation of the results of the forced vibration tests conducted on the Baixo Sabor dam for two water levels in the reservoir [9] (Figure 7a,b). Figure 7c shows details of the mesh of the dam crest and reservoir. The height of the foundation mesh is 200 m in the central zone of the valley. The global dimensions of the model are 660 \(\times\) 800 \(\times\) 320 m (perpendicular to the river, along the river, and vertical).  
The methodology implemented at the LNEC starts with determining the modal parameters of the structures for an initial reference situation. This reference situation typically corresponds to the end of construction, for which two tests are performed: one with the reservoir empty, or nearly empty, and the other when it reaches the full storage level during the first filling of the reservoir.