Métodos Variacionales Multiescala
Victor M. Calo,
(Universidad King Abdullah de Ciencia y Tecnología (KAUST), Arabia Saudí)
Miercoles 8 de junio 2011
A variational framework for modeling non-linear multi-physics problems is presented for standard and isogeometric finite element methods. The variational formulation is designed such that fundamental mathematical and physical features of the solution are built into the weak form, while the formulation is kept flexible and robust. The construction of the variational form is twofold. First, a multiscale decomposition of the solution into coarse and fine scales is introduced a priori. The coarse scales are identified with the finite element approximation, while the fine scales are identified with the subgrid scales and need to be modeled. Using the Navier-Stokes equations as a model problem, a residual-based approximation of the fine scales is made, and alternative approximations of the fine scales based on element subproblems are currently being explored. Second, weak imposition of boundary conditions and coupling is used to introduce physical modeling when necessary. To discuss in detail the different parts inherent to the methodology, variational multiscale decomposition is used to derive a turbulence model for Large Eddy Simulation (LES) and weak imposition of boundary conditions is used to build a consistent wall model for wall-bounded turbulent flows. Following, this variational framework is used to model drug delivery in idealized and patient-specific geometrical models of the cardiovascular system, where the variational multiscale method is used to stabilize the weak form and weak imposition of boundary conditions is used to model endothelial permeability. In the simulations presented NURBS-based isogeometric analysis is employed to describe the geometry and discretize the balance equations. To conclude a brief discussion of the opportunities available at King Abdullah University of Science and Technology will be presented.