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Challenges and Advancements in Virtual Elements

Periodic Reporting for period 1 - CAVE (Challenges and Advancements in Virtual Elements)

Reporting period: 2016-07-01 to 2017-12-31

"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."
"In this first ""18 month term"", together with my team, I developed the following aspects. 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 structural 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. We developed in practice and analyzed theoretically the method for magneto-static problems, that originate from the maxwell equations and can, for instance, describe the magnetic field generated by some electric current in different media."
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.