Isogeometric Analysis for C1-continuous Mortar Method

Isogeometric Analysis (IgA) is a technique for the discretization of Partial Differential Equations that preserves the same exact description of the computational domain through the analysis process. The exact geometry representation is based on the isoparametric concept, for which the same basis functions used for geometrical CAD parametrization are then used to approximate the unknown PDE solution. In this context, B-splines and NURBS are the class of functions most widely used in engineering design, and hence they are commonly employed in Isogeometric Analysis. These spaces include as a special case the piecewise polynomial spaces used in Finite Element Method, thus IgA can be viewed as a generalization of classical FEM in which high-regular basis functions are involved. This implies improvements in accuracy of IgA methods compared to FEM and a more flexible ability to treat higher order differential operators. IgA is then the natural framework for the study of methods that guarantee regular solutions on complex geometries, which can be handled by domain decomposition techniques. This thesis is devoted to the study of Isogeometric mortar method for the approximation of the bilaplacian operator on multipatch geometries. Mortar method is a domain decomposition technique in which weak regularity conditions are imposed along the interfaces of the decomposition. In particular, we are interested to obtain solutions that are globally C1-continuous. After an introduction to the main features of spline spaces, in Chapter 1 we recall two different strategies to enforce regularity constraints over multipatch domains. Chapter 2 is then devoted to the definition of the model problem and to provide a priori error estimates for the Isogeometric mortar method with strong C2 constraints on the vertices of the decomposition and weak C1 constraints along interfaces. The same method is proved to be stable under the choice of an appropriate Lagrange multiplier space. In Chapter 3 are collected some numerical tests that enhance the optimal approximation properties of the method and its flexibility on the approximation of solutions with weakened conditions on the vertices and on more complex configurations.