# Research: few lines and some pictures

## Chimera meshes - Overset grids

### Advection-Diffusion problems

Advection-diffusion problem with deforming foreground mesh. In particular, the deformation is defined by the solution itself.

Advection-diffusion problem with an overset multimesh setting. One foreground mesh follows the displacement of the rotating bar. The other forground mesh translates independently of both the solution and the other foreground mesh displacement.

### Fluid-structure interaction: the incompressible Navier-Stokes equations

Velocity magnitude of cylinder immersed in a fluid (Re = 550). The cylinder first moves with constant velocity. Successively, it is stopped.

Velocity magnitude of an incompressible fluid around a steady cylinder at Re = 200. Long-time simulation.

Sedimentation of a cylinder in a channel full of water and subject to vertical gravity. The displacement of the cylinder is not known a priori.

## Model order reduction

### Proper Generalized Decomposition in Electrocardiology

In this picture there is the diffusion of an electrical signal in a real canine ventricle. The proposed model, through a Proper Generalized Decomposition (PGD), is able to find a parametric solution wrt the cardiac conductivities.

### Hyper-Reduced ADER method for geometrically-parametric solutions of Advection-Diffusion problems on Chimera meshes

The above three videos represent the solution for an advection-diffusion problem dependent on the velocity of displacement of the internal cylinder. This solution is found via a Domain Decomposition (DD) approach. A Proper Orthogonal Decomposition method is performed over the foreground subdomain. Even though the original database is defined on vertical translation with constant velocity, it is possible to reconstruct solutions with cylinders whose movement is time-dependent and not trivial (video #3). The solution is chosen such that the proximity of the cylinder perturbes the solution.

## Other topics

### Well balanced schemes for conservation laws on manifolds: the shallow water equations

Sketch of the physical problem: the equipotential surface provided by the gravity field defines the manifold. The bottom topography can be also discontinuous.

Solutions over a plane (the manifold) with a discontinuous plains as bottom topography.

Solutions over a geoid (the manifold) with an immersed hill as bottom topography.

### Medical imaging

A mesh is built on an atlas image. Successively, through a map deformation, it is adapted on a similar image. This helps doctors not to build a new mesh over a new image for any new patient.