The research activities are organized in three tasks that are presented below. For the first two years of execution of the project, we outline next the work performed in each task and the corresponding key findings.
Task 1 - Theoretical foundations: Elaboration of theoretical foundations that give robust support to DDCM through: i) the choice of a setting that is as simple as possible, but general enough to combine continuous and/or discrete mechanical models and data sets; ii) the formulation and mathematical analysis of optimization problems that are tractable and solvable; and, iii) the determination of admissibility conditions on data sets.
During the reporting period, we were able to formulate DDCM problems considering both the finite- and infinite-dimensional settings, e.g. elasticity problems defined on closed segments of the real line, reaction-diffusion problems defined on closed subsets of two- and three-dimensional ambient space, and discrete networks such as electrical circuits. For the DDCM problems in the finite-dimensional setting, we established the existence of solutions. For few specific DDCM problems in the infinite-dimensional setting, we also established the existent of solutions. In addition, we were able to combine the DDCM with FEM to formulate stable approximations and to enhance DDCM for contact problems. The work performed and the key findings achieved in this task are focusing the points i) and ii). The point iii) will be developed in the last two years of the project.
Task 2 - Computational procedures: Development of computational procedures to solve DDCM problems through: i) the design of a highly- efficient numerical solution strategy; ii) taking full advantage of the abstract structures available in the data set; and, iii) inversely employing the DDCM approach to identify data sets that are complete and consistent.
During the reporting period, we were able to investigate how DDCM problems can be solved. We considered three strategies. The first one is to strictly solve the discretized or discrete DDCM problem with state-of-the-art solvers. However, this approach can be used for very small problems involving few degrees of freedom and few data points. The second one is to approximately solve the DDCM problem by considering a smooth manifold replacing the constitutive data. This works very well and efficiently, but it depends on an offline step to determine the constitutive manifold, does not use directly the data and does not provide a strict solution to the original DDCM. The third one is based on the alternating-direction method. Even when this approach presents some shortcomings (locality or even partial locality), it resulted to be an effective approach balancing computational expenditures and quality of the solution achieved. A key aspect here is to have a good initialization. Therefore, we considered two type of initializations, one based on the mathematical structure of the constraints (relying on the pseudo inverse of the constraints’ Jacobian matrix) and another relying on the constitutive manifold. Once any of the two initialization is conducted, the ADM subsequent leaded to global optimality or almost-global optimality in most of the cases. Again, the work performed and the key findings achieved in this task are focusing the points i) and ii). The point iii) will be developed in the last one and a half years of the project.
Task 3 - DDCM-based aerohydroelastic analysis for OWTs: Empowering the aerohydroelastic analysis of OWTs through: i) the application of new foundations and procedures to the aerohydroelastic analysis of OWTs; ii) a reformulation of the DDCM setting for some methods of solid and fluid mechanics previously developed by the PI, which have already been successfully employed in the past; and, iii) taking advantage of the constructive features of the components of an OWT to further simplify the associated DDCM problems. These activities will be materialized in the form of a digital twin.
During the reporting period, we were able to start with the reformulation in the DDCM setting of the geometrically exact beam model with which the structure of the OWTs will be modeled. We performed some preparatory work for the mooting lines including contact with the seabed. We started with preparatory work for the fluid models such as mesh generation of floating OWTs and improvements on the aerodynamic model. We developed strategies for the time step adaptation of aerohydroelastic simulations such that the simulation runtime can be reduced and the robustness of the time integration scheme can be enhanced. Also, for a simple discrete system (an electric circuit) we created a concept for a digital twin model that is fed with data sets whose size is varying in time and also, we investigated how the system reacts to nonlinearity implicitly contained by the data. In the future, this digital twin concept will be extended to an OWTs. The work performed and the key findings achieved in this task are focusing the points i), ii) and iii).
