Study of finite element discretizations for multi-physics problems

Multi-physics problems refer to complex systems or phenomena that involve the simultaneous interaction of multiple physical phenomena or disciplines such as fluid dynamics[1], heat transfer[2], structural mechanics[3], electromagnetic fields[4][5], and chemical reactions[6] and many more. These interacting, influences the overall behaviour and response of the system.[7] The Finite Element Method is well-suited for solving multi-physics problems due to its flexibility, adaptability, and coupling capabilities. FEM allows for the incorporation of diverse physical phenomena into a unified computational model, accommodating complex geometries and irregular domains. Its numerical accuracy, versatility in handling boundary conditions and material properties, and ability to facilitate optimization and design exploration make it an effective tool for analysing the interaction of multiple physical processes. This thesis consists of three differentiated parts with the aim of expanding Swan repository’s capabilities in solving multi-physics problems. In the first one, a study of the Ciarlet concept of finite elements has been carried out, reviewing the mathematical foundations of the field. The properties and characteristics of three finite elements types (Lagrange, Raviart-Thomas and Nédélec) have been deeply studied and the deduction of its shape functions has been implemented in a Matlab® code. In the second part of the thesis, the Lagrangian, Raviart-Thomas and Nédélec elements have been implemented in the Swan repository. Swan is a collaborative, open source program of finite elements developed by CIMNE used for solving a wide variety of problems (structural, fluids, material optimization. . . ). Bearing in mind the open-source nature of Swan, it also aspires to set the foundations for future expansions of the code. In order to do so, the object-oriented programming and clean-code philosophy of the repository have been followed. Finally, different physical problems have been solved using the implemented finite elements. This work, far from implementing the total solving of multi-physics problem in Swan; settles the basis for, in a hopefully close future, achieving this complicated and broad goal