Luca Dedè

Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow

We present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective Cahn-Hilliard equation, a stiff, nonlinear, parabolic equation characterized by the presence of fourth-order spatial derivatives. Our numerical solution procedure is based on isogeometric analysis, an approximation technique for which basis functions of high-order continuity are employed. These basis functions allow us to directly discretize the advective Cahn-Hilliard equation without resorting to a mixed formulation. We present steady state solutions for rectangular domains in two-dimensions and, for the first time, in three-dimensions. We also present steady state solutions for the two-dimensional Taylor-Couette cell. To enforce periodic boundary conditions in this curved domain, we derive and utilize a new periodic Bezier extraction operator. We present an extensive numerical study showing the effects of shear rate, surface tension, and the geometry of the domain on the phase evolution of the binary fluid. Theoretical and experimental results are compared with our simulations.

Reduced Basis Method and A Posteriori Error Estimation for Parametrized Linear-Quadratic Optimal Control Problems

We propose the reduced basis method for the solution of parametrized optimal control problems described by parabolic partial differential equations in the unconstrained case. The method, which is based on an off-line-on-line decomposition procedure, allows at the on-line step large computational cost savings with respect to the “truth” approximation used for defining the reduced basis. An a posteriori error estimate is provided by means of the goal-oriented analysis, thus associating an error bound to each optimal solution of the parametrized optimal control problem and answering to the demand for a reliable method. An adaptive procedure, led by the a posteriori error estimate, is considered for the generation of the reduced basis space, which is set according to the optimal primal and dual solutions of the optimal control problem at hand.

Numerical approximation of a control problem for advection-diffusion processes

Two different approaches are proposed to enhance the efficiency of the numerical resolution of optimal control problems governed by a linear advection-diffusion equation. In the framework of the Galerkin-Finite Element (FE) method, we adopt a novel a posteriori error estimate of the discretization error on the cost functional; this estimate is used in the course of a numerical adaptive strategy for the generation of efficient grids for the resolution of the optimal control problem. Moreover, we propose to solve the control problem by adopting a reduced basis (RB) technique, hence ensuring rapid, reliable and repeated evaluations of input-output relationship. Our numerical tests show that by this technique a substantial saving of computational costs can be achieved.

Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem

In this paper, we propose the mathematical and finite element analysis of a second-order partial differential equation endowed with a generalized Robin boundary condition which involves the Laplace--Beltrami operator by introducing a function space

IGS: an IsoGeometric approach for Smoothing on surfaces

H^1(\Omega; \Gamma)

Isogeometric analysis and proper orthogonal decomposition for parabolic problems

of

Complex blood flow patterns in an idealized left ventricle: A numerical study

H^1(\Omega)

A Hele–Shaw–Cahn–Hilliard Model for Incompressible Two-Phase Flows with Different Densities

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A patient-specific aortic valve model based on moving resistive immersed implicit surfaces

H^1(\Gamma)

Numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network

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