Study of Finite Element Analysis Using the Shifted Boundary Method

Study of Finite Element Analysis Using the Shifted Boundary Method Since the previous century, the Finite Element Method (FEM) has continuously gained attention to solve multiple types of physical problems, however as its range of application has increased, many Engineers, Mathematicians, Physicists, and Programmers have realized the difficulties that it encompasses. The reasons why the FEM gained increasing attention as to become a standard to solve a wide range of situations in nowadays were two: Because it can practically solve any differential problem, and because of the implementation of computers which allowed to obtain and test solutions for such problems. The latter reason being the starting point for which the scientific community paid attention to such method, even though this also represents a drawback since computers only have a finite computational capacity. Generally speaking, a Finite Element Analysis (FEA) has 4 steps, which are the Discretization of the physical boundary, Choosing the shape and test functions, Building the system of equations, and Post-processing the results. Each step of the FEM has its own computational cost, however one of the most costly steps happens when discretizing the physical domain. The reason for this is because, when there are different shapes and sizes of finite elements along the physical domain, unique calculations have to be done for every unique finite element. This cost can be diminished by using a constant shape of finite element across the physical domain, however this is not always possible while reaching the physical boundary of such domain. This occurs only because there is information about boundary conditions at the physical boundary, but could be avoided if equivalent boundary conditions were translated into the boundaries of the finite elements. That is the fundamental idea of Shifted Boundary Method, which is to translate boundary conditions from the physical boundary to the boundary of the finite elements, thus diminishing the computational cost of the FEM while maintaining the same level of approximation.