Research
Activity
Qualitative
models and fuzzy logic in Medicine
Numerical
approximation of linear elasticity problems
Theoretical
and numerical study of electromagnetic problems
Macroscopic "bidomain" models of reaction-diffusion equations
Numerical approximation of fluid-mechanic problems
Basic
properties of discrete schemes (numerical linear algebra, eigenvalue problems, domain decomposition, stabilization techniques)
Qualitative models
and fuzzy logic in Medicine. The central issue of the research activity deals with the application of
qualitative modelling and simulation techniques, with the final goal to build a robust input-output model
of the nonlinear dynamics of complex systems which can not be described by means of ordinary or partial differential equations.
Qualitative models (QSIM) are integrated with fuzzy logic systems: in such a way we can exploit the available
structural knowledge, even if incomplete, and embed it in a fuzzy approximator. This approach allows us to automatically determine the optimal
structure of the fuzzy model.
From the methodological point of view, the main problems addressed are:
the mapping of the qualitative framework onto the fuzzy-based one;
the choice of the optimal shape of the membership functions;
the training algorithms applied to estimate the values of the parameters.
The main application field is Medicine because in this context the incompleteness of the
available structural knowledge prevents from formulating a classical quantitative model of complex systems. In particular glucose-insulin system and
thiamine kinetics have been studied.
Numerical approximation of linear elasticity problems.
Particular attention is devoted to plate bending problems, using the
Reissner-Mindlin's model, and thin shell problems,
according to the Koiter and Naghdi models.
The principal research lines are:
mixed finite element approximation of Naghdi's model for shell problems
domain decomposition methods for compressible
and incompressible elastic linear systems
mixed finite element techniques for laminated elastic thin structures.
Analysis of regularity and elliptic property for bending shells.
Theoretical and numerical study of Maxwell equations for electromagnetic
problems.
We have been working on the following topics:
approximation of eigenvalue problems arising in finite element approximation
of Maxwell's equations by finite elements
derivation of the "Eddy Currents" model as
quasi-static approximation of linear Maxwell'sequations, by using
asymptotic analysis
application of a domain decomposition method in the simulation
of electric machines and magneto-mechanical systems
theoretical study and face-edge finite element approximation
of mixed variational formulations of tridimensional magnetostatic problems in
non-trivial domains.
Macroscopic "bidomain" models of reaction-diffusion equations.
Theoretical and numerical study of macroscopic
"bidomain" models of reaction-diffusion type equations for the simulation
of the electric propagation in cardiac ventricular.
Developing and testing of numerical methods for eikonal equations
for describing the motion of electric excitation wavefronts in the
myocardial tissue.
Numerical approximation of fluid mechanics problems. Theoretical study and implementation of numerical methods for Stokes, Oseen and Navier-Stokes equations.
We are mainly interested in
domain decomposition methods for Stokes and Oseen problems
finite element methods in space and time for hyperbolic conservation laws.
Basic properties of discrete schemes. We have been working on
preconditioned iterative methods for linear elasticity and Stokes problems,
discretized by finite element or spectral element methods; the
considered preconditioners are based on block and domain decomposition
techniques
analysis of an automatic procedure based on the numerical control of the
inf-sup condition to evaluate the stability of numerical schemes for
transport-diffusion problems, both in one-dimension and multi-dimension cases
analysis of finite element approximation of eigenvalue problems
for partial differential equations, with particular attention to
(non-coercive) problems in mixed form;
unlike in the coercive case, here the stability of a method which
approximates a direct problem doesn't guarantee a good approximation of
eigenvalues
theoretical study of bubble function-based stabilization techniques
for problems with non-matching grids, carried out
in the context of three-field domain decomposition methods
domain decomposition methods for discretizing
Maxwell's equations by H(curl)-conforming finite elements
on non-matching grids
preconditioning of algebraic linear systems with
symmetric and indefinite matrices arising in the finite element
discretization of mixed formulations
discontinuous Galerkin finite element methods.