Many computational engineering problems deal with the mechanical response of a body with unfitted boundary. With the term "unfitted" we mean that its boundary does not coincide with the underlying mesh nodes. This might hap- pen if the body undergoes large displacements or large strain, just to make an example. Under these circumstances accurately imposing the boundary con- ditions can be a challenging task. This problem can be solved using methods derived from optimization theory such as the Penalty approach and Lagrange multipliers. The purpose of this thesis is to revise the mathematical and numerical for- mulations that solve the problem of nonconforming boundary conditions in the classical Finite Element Method (FEM) and to extend these approaches to a continuum-based particle method such as the Material Point Method (MPM). In fact, while the Finite Element Method is a widely accepted and well es- tablished techniques in many engineering fields, it shows some limitations when dealing with large strain regime. The Material Point Method, on the contrary, is an hybrid technique which uses two discretizations and it is espe- cially suited for this kind of problems. MPM can be seen as a special Finite Element Method where the quadrature points are moving material points. In this thesis we will focus in particular on appending boundary conditions by means of the Lagrange multipliers. Then, a comparison between the Penalty method and Lagrange multipliers in MPM is conducted. The convergence property is studied in details and compared with other existing techniques. For this purpose we simulate classical benchmark problems in computational mechanics. It was observed that the Lagrange multipliers method exhibits quadratic convergence even for coarse computational meshes. In contrast, this is not the case for Penalty method, which shows quadratic convergence only for fairly fine meshes.     «

Many computational engineering problems deal with the mechanical response of a body with unfitted boundary. With the term "unfitted" we mean that its boundary does not coincide with the underlying mesh nodes. This might hap- pen if the body undergoes large displacements or large strain, just to make an example. Under these circumstances accurately imposing the boundary con- ditions can be a challenging task. This problem can be solved using methods derived from optimization theory such as...     »