# Numerical modeling

• Rigid body model: Newton´s second law is used to obtain the dynamics of the structure as a rigid solid with six degrees of freedom: three translations and three rotations. Thus, it is necessary to know the inertia matrix of the platform and calculate all the forces and moments that act on it.
• Hydrostatic and hydrodynamic models: with the hydrostatic model, we obtain the combination of buoyancy and gravity forces acting on the device due to the stiffness matrix. For the calculation of hydrodynamic forces, linear potential theory is used, characterizing the response of the platform with hydrodynamic coefficients obtained with the boundary element method (BEM).
• Mooring system model: to model the lines used in the mooring system, two options are considered. On the one hand, a quasi-static model that ignores the dynamic effects on the lines, based on the catenary equation and significantly faster, on the other hand a more elaborate dynamic model, based on the finite element method (FEM).
• Oscillating water column model (OWC): this model is based on the hydrodynamic model mentioned above. In this case, the aperture on the top of the device is considered as a new degree of freedom: the free surface in the chamber. This also affects the rest of the hydrodynamic coefficients of the platform. Finally, the OWC’s energy extraction system (PTO) is modelled, so its energy production can be obtained.
• Aerodynamic model (Semisubmersible): this model is used to obtain the influence of the wind turbine acting on the platform. The turbine speed is studied, so the forces and the production are obtained too.

Figure 1. OWC. General coupled model

Figure 2. Semisubmersible. General coupled model.

Once all the submodels are defined, they are coupled together in a single model. To do this, at each time step, the position and the speed of the platform are used in the others submodels as boundary conditions to obtain its forces and momentum.

Calibration process

The hydrodynamic model, which is the base of the WEC motions solver, is based on potential flow theory. Because of that, there are some assumptions behind (e.g. inviscid flow, …) which are partially mitigated through empirical linear and non-linear damping coefficients acting as viscous forces that need to be calibrated through experimental results (Iturrioz et al 2017). This process has three steps for each configuration:

• STATIC OFFSET TESTS: used to verify that the fairleads loads and displacements obtained with the numerical model and in the laboratory match.

OWC Figure 3. OWC. Static offset: Positive Surge. Configuration 1. Force-Displacement Figure 4. OWC. Static offset: Positive Surge. Configuration 1. Mooring Force-Displacement

SEMISUBMERSIBLE Figure 5. Semisubmersible. Static offset: Negative Surge. Force-Displacement Figure 6. Semisubmersible. Static offset: Positive Surge. Mooring Force-Displacement

• DECAY FREE AND MOORED TESTS: used to verify that the fairleads loads and displacements obtained with the numerical model and in the laboratory match.

OWC

• Configuration C2: Closed chamber.

(click to enlarge) Figure 7. OWC. Calibration. Configuration C2. Decay free test. Heave

Figure 8. OWC. Calibration. Configuration C2. Decay free test. Roll

Figure 9. OWC. Calibration. Configuration C2. Decay free test. Pitch

Figure 10. OWC. Calibration. Configuration C2. Decay moored test. Surge

Figure 11. OWC. Calibration. Configuration C2. Decay moored test. Sway Figure 12. OWC. Calibration. Configuration C2. Decay moored test. Heave

Figure 13. OWC. Calibration. Configuration C2. Decay moored test. Roll

Figure 14. OWC. Calibration. Configuration C2. Decay moored test. Pitch

Figure 15. OWC. Calibration. Configuration C2. Decay moored test. Yaw

• Configuration C1: Open aperture. Figure 16. OWC. Calibration. Configuration C1. Decay free test. Heave

Figure 17. OWC. Calibration. Configuration C1. Decay free test. Roll

Figure 18. OWC. Calibration. Configuration C1. Decay free test. Pitch

Figure 19. OWC. Calibration. Configuration C1. Decay free test. Free surface

Figure 20. OWC. Calibration. Configuration C1. Decay moored test. Surge Figure 21. OWC. Calibration. Configuration C1. Decay moored test. Sway

Figure 22. OWC. Calibration. Configuration C1. Decay moored test. Heave

Figure 23. OWC. Calibration. Configuration C1. Decay moored test. Roll

Figure 24. OWC. Calibration. Configuration C1. Decay moored test. Pitch

Figure 25. OWC. Calibration. Configuration C1. Decay moored test. Yaw

SEMISUBMERSIBLE

(click to enlarge) Figure 26. Semisubmersible. Calibration. Decay free test. Heave

Figure 27. Semisubmersible. Calibration. Decay free test. Roll

Figure 28. Semisubmersible. Calibration. Decay free test. Pitch

Figure 29. Semisubmersible. Calibration. Decay moored test. Surge

Figure 30. Semisubmersible. Calibration. Decay moored test. Sway Figure 31. Semisubmersible. Calibration. Decay moored test. Heave

Figure 32. Semisubmersible. Calibration. Decay moored test. Roll

Figure 33. Semisubmersible. Calibration. Decay moored test. Pitch

Figure 34. Semisubmersible. Calibration. Decay moored test. Yaw

• TESTS WITH REGULAR WAVES: these tests are used to validate the calibration process, verifying that the calibration is correct.

OWC

(click to enlarge) Figure 35 OWC. Validation. Configuration C2. Regular waves: H=2.5m Tp=10s. Surge, Heave and Pitch Figure 36. OWC. Validation. Configuration C1. Regular waves: H=2.5m Tp=10s. Surge, Heave and Pitch Figure 37. OWC. Validation. Configuration C1. Regular waves: H=2.5m Tp=10s. Free surface and pressure inside the chamber

SEMISUBMERSIBLE

(click to enlarge) Figure 38. Semisubmersible. Validation. Regular waves: H=2.5m Tp=8s Figure 39. Semisubmersible. Validation. Regular waves: H=5m Tp=14s Figure 40. Semisubmersible. Validation. Wind. Fconstant =12m/s. Figure 41.Semisubmersible. Validation. Wind. Fconstant =12m/s. Mooring lines Figure 42. Semisubmersible. Validation. Waves + Wind. F =16m/s, H=2.5m, T=10s Figure 43. Semisubmersible. Validation. Waves + Wind. F =16m/s, H=2.5m, T=10s. Mooring lines