arXiv subjects

Simone Scacchi

Publications and source records attributed to Simone Scacchi.

26 records · Page 2Linked to original sources

arXiv2021

On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

The Virtual Element Method (VEM) is a very effective framework to design numerical approximations with high global regularity to the solutions of elliptic partial differential equations. In this paper, we review the construction of such approximations for an elliptic problem of order $p_1$ using conforming, finite dimensional subspaces of $ H^{p_2}(Ω)$, where $p_1$ and $p_2$ are two integer numbers such that $p_2 \geq p_1 \geq 1$ and $Ω\in R^2$ is the computational domain. An abstract convergence result is presented in a suitably defined energy norm. The space formulation and major aspects such as the choice and unisolvence of the degrees of freedom are discussed, also providing specific examples corresponding to various practical cases of high global regularity. Finally, the construction of the "enhanced" formulation of the virtual element spaces is also discussed in details with a proof that the dimension of the "regular" and "enhanced" spaces is the same and that the virtual element functions in both spaces can be described by the same choice of the degrees of freedom.

math.NA↗

arXiv2021

Qualitative analysis of a mathematical model for Xylella fastidiosa epidemics

In Southern Italy, since 2013, there has been an ongoing Olive Quick Decline Syndrome (OQDS) outbreak, due to the bacterium Xylella fastidiosa. In a couple of previous papers, the authors have proposed a mathematical approach for identifying possible control strategies for eliminating or at least reduce the economic impact of such event. The main players involved in OQDS are represented by the insect vector, Philaenus spumarius, its host plants (olive trees and weeds) and the bacterium, X. fastidiosa. A basic mathematical model has been expressed in terms of a system of ordinary differential equations; a preliminary analysis already provided interesting results about possible control strategies within an integrated pest management framework, not requiring the removal of the productive resource represented by the olive trees. The same conjectures have been later confirmed by analyzing the impact of possible spatial heterogeneities on controlling a X. fastidiosa epidemic. These encouraging facts have stimulated a more detailed and rigorous mathematical analysis of the same system, as presented in this paper. A clear picture of the possible steady states (equilibria) and their stability properties has been outlined, within a variety of different parameter scenarios, for the original spatially homogeneous ecosystem. The results obtained here confirm, in a mathematically rigorous way, what had been conjectured in the previous papers, i.e. that the removal of a suitable amount of weed biomass (reservoir of the juvenile stages of the insect vector of X. fastidiosa from olive orchards and surrounding areas is the most acceptable strategy to control the spread of the OQDS. In addition, as expected, the adoption of more resistant olive tree cultivars has been shown to be a good strategy, though less cost-effective, in controlling the pathogen.

q-bio.PE↗

arXiv2021

Myocardial ischemic effects on cardiac electro-mechanical activity

In this work, we investigated the effect of varying strength of Hyperkalemia and Hypoxia, in human cardiac tissue with a local ischemic subregion, on the electrical and mechanical activity of healthy and ischemic zones of the cardiac muscle. The Monodomain model in a deforming domain is taken with the addition of mechanical feedback and stretch-activated channel current coupled with the ten Tusscher human ventricular membrane model. The equations of finite elasticity are used to describe the deformation of the cardiac tissue. The resulting coupled electro-mechanical PDEs-ODEs non-linear system is solved numerically using finite elements in space and finite difference method in time. We examined the effect of local ischemia on cardiac electrical and mechanical activity in different cases. We concluded that the spread of Hyperkalemic or Hypoxic region alters the electro-mechanical coupling in terms of the action potential ($v$), intracellular calcium ion concentration $[Ca^{+2}]_i$, active tension, ($T_A$), stretch ($λ$), stretch rate ($ \frac{d λ}{dt}$). With the increase in the size of the ischemic region by a factor of five, approximately $45\%$ variation in the stretch rate $\frac{d λ}{dt}$ is noticed. It is also shown that ischemia affects the deformation (expansion and contraction) of the heart.

math.NA↗

arXiv2021

A review on arbitrarily regular conforming virtual element methods for elliptic partial differential equations

The Virtual Element Method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order $2p_1$, for any integer $p_1\geq 1$. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of $H^{p_2}(Ω)$, $Ω$ being the computational domain and $p_2\geq p_1$ another suitable integer number. In this study, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order continuity on $Ω$. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases $p_1=1$ and $2$, respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.

math.NA↗

arXiv2021

The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper, we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.

math.NA↗

arXiv2020

Cardiac kinematic parameters computed from video of $\textit{in situ}$ beating heart

Mechanical function of the heart during open-chest cardiac surgery is exclusively monitored by echocardiographic techniques. However, little is known about local kinematics, particularly for the reperfused regions after ischemic events. We report a novel imaging modality, which extracts local and global kinematic parameters from videos of $\textit{in situ}$ beating hearts, displaying live video cardiograms of the contraction events. A custom algorithm tracked the movement of a video marker positioned $\textit{ad hoc}$ onto a selected area and analyzed, during the entire recording, the contraction trajectory, displacement, velocity, acceleration, kinetic energy and force. Moreover, global epicardial velocity and vorticity were analyzed by means of Particle Image Velocimetry tool. We validated our new technique by i) computational modeling of cardiac ischemia, ii) video recordings of ischemic/reperfused rat hearts, iii) videos of beating human hearts before and after coronary artery bypass graft, and iv) local Frank-Starling effect. In rats, we observed a decrement of kinematic parameters during acute ischemia and a significant increment in the same region after reperfusion. We detected similar behavior in operated patients. This modality adds important functional values on cardiac outcomes and supports the intervention in a contact-free and non-invasive mode. Moreover, it does not require particular operator-dependent skills.

q-bio.TO↗

arXiv2016

On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study

It is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh, which can steer the minimization process towards local (and non-physical) minima, and by the accuracy of the method employed to discretize the underlying differential problem, which may not be able to correctly capture the physics of the problem. In light of the above remarks, in this paper we consider polygonal meshes and employ the virtual element method (VEM) to solve two classes of paradigmatic topology optimization problems, one governed by nearly-incompressible and compressible linear elasticity and the other by Stokes equations. Several numerical results show the virtues of our polygonal VEM based approach with respect to more standard methods.

math.NA↗

arXiv2015

A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes

In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn-Hilliard equation. The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semi-discrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests.

math.NA↗