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# Challenges and Advancements in Virtual Elements

## Periodic Reporting for period 4 - CAVE (Challenges and Advancements in Virtual Elements)

Période du rapport: 2021-01-01 au 2021-06-30

The CAVE project is collocated in the area of computational/applied mathematics with a strong connection with engineering. A mathematical model is a description of some phenomena of interest (in turn associated to some application such as weather forecast, design of an aircraft, etc..) in terms of mathematical functions and "equations". The majority of models cannot be solved by pen and paper, and thus in order to be useful need to be discretized (approximated by a numerical method and translated into a form that is edible for a computer) and solved computationally. Mathematics has a key role in designing the numerical method and guaranteeing theoretically its effectiveness. Since the majority of problems are set in some space domain (such as a piece of a vessel if one is studying blood flow, or a building if one is investigating earthquake effects, etc..) a key step in essentially all methods is some kind of discretization (decomposition) of such domain (called mesh).

The project CAVE is focused on developing an innovative numerical method for the discretization of the above models, the Virtual Element Method, and thus carrying an impact on the related applications. The novel side with respect to the pre-existing technology is related to many aspects, the first one being a much larger freedom in building the mesh, as the scheme can use any general polyhedral decomposition of the domain. Other aspects are, for instance, the capability to realize exactly certain physical conditions of the solution (such as being incompressible, or needing a higher regularity). The project has a wide theoretical foundation to be explored, in addition to practical computations and applications (such as in structural mechanics and electro-cardiology). The final goal, in a wide sense, is to develop, analyze and deliver a new method that, in many problems/situations, can yield much better (more accurate, more efficient,..) results with respect to the existing ones.

The project CAVE was successful in its main objectives, by actually developing novel discretization approaches for a wide range of problems, going from solid and fluid mechanics to electro-magnetism, that offer novel advantages with respect to existing competitors. Such developments also include (and where boosted by) a full theoretical background, plus implementation of computer codes and numerical benchmark testing. Furthermore, the Virtual Element Method enjoyed a strong growth in terms of attention and community size, both in the mathematical and in the engineering areas. As expected at the start of the action, it is still too early for embedding into the industry.
I here describe briefly the major aspects developed during the project, together with my team. We built a three-dimensional code of the scheme, since (being the Virtual Element very recent) previous developed codes where all in two dimensions. We tackled for the first time complex problems in solid mechanics, that is problems in which some elastic body undergoes a very large deformation due to large forces (or other conditions) applied; the results show an improved robustness of the new approach with respect to the main existing technology for this kind of problems. We developed in practice and analyzed theoretically a version of the scheme that, by using high order polynomials, can yield very accurate results for problems related to diffusion of substances in some porous media. The scheme can also automatically adapt the mesh and the local polynomial degree, yielding an improved efficiency. We developed in practice and analyzed theoretically the VE method for problems in electro-magnetism, starting from the development of the discrete spaces needed for the numerical discretization, then tackling first magneto-statics and then the full time-dependent Maxwell equations. We developed the method for problems in fluid dynamics, in such a way that the incompressibility constraint (present in many fluids) is exactly represented at the numerical level, thus yielding a more faithful approximation of certain properties of the solution. We also explored in deep the discrete structure behind such important constraint and tackled model applicative problems such as that of a leaflet immersed in a fluid or the displacement of miscible fluids in porous media. Finally, we extended the methodology to the case of an exact curved geometry representation, so that the domain of interest does not need to be approximated by a facet-like description. Such exploration included also interesting applications in solid mechanics, in collaboration with engineers, such as contact problems and statistical homogenization of fiber-reinforced composites. The results above are published in open access on (often top level) international journals with peer review, and disseminated worldwide through talks at international congresses and seminars in external departments. Shareware codes (and introductory slides) are available at the group webpage and more are available by emailing directly the group members.
The expectations are to develop the Virtual Element Method as an innovative numerical construction for the discretization of models of interest, carrying an impact on the related applications. The scheme will allow to use any general polyhedral decomposition of the domain (a very important asset that yields a wide field of advantages), the capability to realize exactly certain physical conditions of the solution (such as being incompressible, or needing a higher regularity) and a potentially improved robustness to mesh distortions. The final goal, in a wide sense, is to develop, analyze and deliver a new method that, in many problems/situations, can yield much better (more accurate, more efficient,..) results with respect to the existing ones.