arXiv subjects

Stefano Berrone

Publications and source records attributed to Stefano Berrone.

At least 19 recordsLinked to original sources

arXiv2026

Hydro-mechanical Model for Slope Stability Assessment: A polygonal stabilization-free discretization

Rainfall-induced landslides are governed by the interaction between subsurface water flow and soil mechanics, requiring robust numerical methods for the simulation of variably saturated porous media. In this work, we consider a semi-coupled hydro-mechanical model based on Richards' equation and linear elasticity and propose a numerical framework based on a stabilization-free Virtual Element Method for its spatial discretization. The proposed approach naturally accommodates general polygonal meshes while avoiding problem-dependent stabilization terms, whose design may become challenging when heterogeneous and strongly non-linear coefficients are involved. The approach is combined with a mass-lumping technique to improve stability in the treatment of the storage term and with Nitsche's method to weakly impose seepage-face and infiltration boundary conditions, allowing for the automatic switching between Neumann and Dirichlet conditions. Time integration is performed using the backward Euler scheme, while non-linearities are handled through a Picard iteration. Numerical experiments demonstrate the stability and robustness of the proposed methodology and show its effectiveness in simulating rainfall infiltration and evaluating slope stability through the Local Factor of Safety.

math.NA↗

arXiv2026

Two continuous extensions of the Neural Approximated Virtual Element Method

We propose two globally continuous neural-based variants of the Neural Approximated Virtual Element Method (NAVEM), termed B-NAVEM and P-NAVEM. Both approaches construct local basis functions using pre-trained fully connected neural networks while ensuring exact continuity across adjacent mesh elements. B-NAVEM leverages a Physics-Informed Neural Network to approximately solve the local Laplace problem that defines the virtual element basis functions, whereas P-NAVEM directly enforces polynomial reproducibility via a tailored loss function, without requiring harmonicity within the element interior. Numerical experiments assess the methods in terms of computational cost, memory usage, and accuracy during both training and testing phases.

math.NA↗

arXiv2025

The Zipped Finite Element Method: High-order Shape Functions for Polygons

In this paper, we present a new polygonal finite element method, called the Zipped Finite Element Method, for star-shaped polygons. The proposed approach constructs high-order shape functions as linear combinations of standard finite element basis functions defined on a local trivial sub-triangulation of each element. This refinement is used solely for the construction of the shape functions and does not affect the final number of degrees of freedom. The resulting finite element space includes polynomials of the desired order and preserves conformity across elements. Consequently, the method inherits the convergence properties of the finite element framework under suitable mesh assumptions. Numerical experiments confirm the expected rates of convergence.

math.NA↗

arXiv2025

Machine Learning enhanced parametric Reynolds-averaged Navier-Stokes equations at the full- and reduced-order levels

In this contribution, we focus on the Reynolds-averaged Navier-Stokes (RANS) models and their exploitation to build reliable reduced-order models to further accelerate predictions for real-time applications and many-query scenarios. Specifically, we investigate how machine learning can be employed to enhance the predictive capabilities of the model, both at the full-order model (FOM) and reduced-order model (ROM) levels. We explore a novel integration of these two areas. We generate the FOM snapshots, essential for ROM construction, using a data-driven RANS model: the $\nu_t$-Vector Basis Neural Network. This is the first time that this machine learning procedure covers a large parametric variation, and we propose tailored training strategies to increase the accuracy of the FOM model. At the ROM level, we compare the results obtained by standard proper orthogonal decomposition (POD) in an intrusive Galerkin setting (PODG) and POD neural network approach (PODNN). The numerical validation is conducted on a classic turbulent square-duct flow benchmark, with the bulk Reynolds number as the sole varying parameter. Our investigation reveals that the PODG method proves to be unstable and inaccurate for turbulent flow prediction, while PODNN demonstrates superior performance in terms of accuracy and computational efficiency.

physics.flu-dyn↗

arXiv2025

General Order Virtual Element Approximation for the Smagorinsky turbulence model

In this paper, we investigate a Smagorinsky model in a virtual element framework to simulate convection-dominated Navier-Stokes equations. We conduct a two-dimensional numerical investigation to assess the performance of the general order virtual element approximation in this context. First, we examine numerically the convergence of the method with respect to the meshsize to certify the novel virtual element numerical discretization, which includes, for the first time, a discretization of the Smagorinsky term. Moreover, we present a numerical study of a lid-driven cavity for different Reynolds numbers (up to 10000) and meshes (uniform, anisotropic, and isotropic with hanging nodes). The results highlight the main advantage of using the virtual elements method in this context: the isotropic refinement with hanging nodes enhances the accuracy of the solution compared to the anisotropic mesh, uses fewer degrees of freedom with respect to the uniform mesh, and yields the most stable behavior in terms of convergence of the Newton solver.

math.NA↗

arXiv2025

The Neural Approximated Virtual Element Method for Elasticity Problems

We present the Neural Approximated Virtual Element Method to numerically solve elasticity problems. This hybrid technique combines classical concepts from the Finite Element Method and the Virtual Element Method with recent advances in deep neural networks. Specifically, it is a polygonal method in which the virtual basis functions are element-wise approximated by a neural network, eliminating the need for stabilization or projection operators typical of the standard virtual element method. We present the discrete formulation of the problem and provide numerical tests on both linear and non-linear elasticity problems, demonstrating the advantages of having a simple discretization, particularly in handling non-linearities.

math.NA↗

arXiv2025

A residual a posteriori error estimate for the Stabilization-free Virtual Element Method

In this work, we present the a posteriori error analysis of Stabilization-Free Virtual Element Methods for the 2D Poisson equation. The abscence of a stabilizing bilinear form in the scheme allows to prove the equivalence between a suitably defined error measure and standard residual error estimators, which is not obtained in general for stabilized virtual elements. Several numerical experiments are carried out, confirming the expected behaviour of the estimator in the presence of different mesh types, and robustness with respect to jumps of the diffusion term.

math.NA↗

arXiv2025

POLYDIM: A C++ library for POLYtopal DIscretization Methods

This paper introduces PolyDiM, an open-source C++ library tailored for the development and implementation of polytopal discretization methods for partial differential equations. The library provides robust and modular tools to support advanced numerical techniques, with a focus on the Virtual Element Method in both 2D and 3D settings. PolyDiM is designed to address a wide range of challenging problems, including those involving non-convex geometries, Discrete Fracture Networks, and mixed-dimensional coupling. It is integrated with the geometry library GeDiM, and offers interfaces for MATLAB and Python to enhance accessibility. Distinguishing features include support for multiple polynomial bases, advanced stabilization strategies, and efficient local-to-global assembly procedures. PolyDiM aims to serve both as a research tool and a foundation for scalable scientific computing in complex geometrical settings.

math.NA↗

arXiv2024

A 3D-1D Virtual Element Method for Modeling Root Water Uptake

An optimization-based strategy is proposed for coupling three-dimensional and one-dimensional problems (3D-1D coupling) in the context of soil-root interaction simulations. This strategy, originally designed to tackle generic 3D-1D coupled problems with discontinuous solutions, is here extended to the case of non-linear problems and applied, for the first time, along with a virtual element discretization of the 3D soil sample. This further enhances the capability of the method to handle geometrical complexities, allowing to easily mesh domains characterized, for instance, by the presence of stones and other impervious obstacles of arbitrary shape. A discrete-hybrid tip-tracking strategy is adopted to model both the root growth and the evolution in time of the water flux, the pressure head and the water content, both in the roots and in the surrounding soil sample. By choosing proper rules for the generation of branches, realistic root-network configurations are obtained. Several numerical examples are proposed, proving both the accuracy of the adopted method and its applicability in realistic and large scale simulations.

math.NA↗

arXiv2024

The lowest-order Neural Approximated Virtual Element Method on polygonal elements

The lowest-order Neural Approximated Virtual Element Method on polygonal elements is proposed here. This method employs a neural network to locally approximate the Virtual Element basis functions, thereby eliminating issues concerning stabilization and projection operators, which are the key components of the standard Virtual Element Method. We propose different training strategies for the neural network training, each correlated by the theoretical justification and with a different level of accuracy. Several numerical experiments are proposed to validate our procedure on general polygonal meshes and demonstrate the advantages of the proposed method across different problem formulations, particularly in cases where the heavy usage of projection and stabilization terms may represent challenges for the standard version of the method. Particular attention is reserved to triangular meshes with hanging nodes which assume a central role in many virtual element applications.

math.NA↗

arXiv2024

A stabilized three fields formulation for Discrete Fracture Networks

We propose a hybridized domain decomposition formulation of the discrete fracture network model, allowing for independent discretization of the individual fractures. A natural norm stabilization, obtained by penalizing the residual measured in the norm for the space where it naturally lives, is added to the local problem in the individual fracture so that no compatibility condition of inf-sup type is required between the Lagrange multiplier and the primal unknown, which can then be discretized independently of each other. Optimal stability and error estimates are proven, which are confirmed by numerical tests.

math.NA↗

arXiv2024

3D Adaptive VEM with stabilization-free a posteriori error bounds

The present paper extends the theory of Adaptive Virtual Element Methods (AVEMs) to the three-dimensional meshes showing the possibility to bound the stabilization term by the residual-type error estimator. This new bound enables a stabilization-free a posteriori control for the energy error. Following the recent studies for the bi-dimensional case, we investigate the case of tetrahedral elements with aligned edges and faces. We believe that the AVEMs can be an efficient strategy to address the mesh conforming requirements of standard three-dimensional Adaptive Finite Element Methods (AFEMs), which typically extend the refinement procedure to non-marked mesh cells. Indeed, numerical tests on the Fichera corner shape domain show that this method can reduce the number of three-dimensional cells generated in the refinement process by about 30% with compared to standard AFEMs, for a given error threshold.

math.NA↗

arXiv2024

Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy

In this paper, we introduce a Meshfree Variational-Physics-Informed Neural Network. It is a Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions. In order to generate the test space, we exploit an a posteriori error indicator and add test functions only where the error is higher. Four training strategies are proposed and compared. Numerical results show that the accuracy is higher than the one of a Variational-Physics-Informed Neural Network trained with the same number of test functions but defined on a quasi-uniform mesh.

math.NA↗

arXiv2024

Mesh Optimization for the Virtual Element Method: How Small Can an Agglomerated Mesh Become?

We present an optimization procedure for generic polygonal or polyhedral meshes, tailored for the Virtual Element Method (VEM). Once the local quality of the mesh elements is analyzed through a quality indicator specific to the VEM, groups of elements are agglomerated to optimize the global mesh quality. The resulting discretization is significantly lighter: we can remove up to 80$\%$ of the mesh elements, based on a user-set parameter, thus reducing the number of faces, edges, and vertices. This results in a drastic reduction of the total number of degrees of freedom associated with a discrete problem defined over the mesh with the VEM, in particular, for high-order formulations. We show how the VEM convergence rate is preserved in the optimized meshes, and the approximation errors are comparable with those obtained with the original ones. We observe that the optimization has a regularization effect over low-quality meshes, removing the most pathological elements. This regularization effect is evident in cases where the original meshes cause the VEM to diverge, while the optimized meshes lead to convergence. We conclude by showing how the optimization of a real CAD model can be used effectively in the simulation of a time-dependent problem.

math.NA↗

arXiv2024

Effective polygonal mesh generation and refinement for VEM

In the present work we introduce a novel refinement algorithm for two-dimensional elliptic partial differential equations discretized with Virtual Element Method (VEM). The algorithm improves the numerical solution accuracy and the mesh quality through a controlled refinement strategy applied to the generic polygonal elements of the domain tessellation. The numerical results show that the outlined strategy proves to be versatile and applicable to any two-dimensional problem where polygonal meshes offer advantages. In particular, we focus on the simulation of flow in fractured media, specifically using the Discrete Fracture Network (DFN) model. A residual a-posteriori error estimator tailored for the DFN case is employed. We chose this particular application to emphasize the effectiveness of the algorithm in handling complex geometries. All the numerical tests demonstrate optimal convergence rates for all the tested VEM orders.

math.NA↗

arXiv2023

The lowest-order Neural Approximated Virtual Element Method

We introduce the Neural Approximated Virtual Element Method, a novel polygonal method that relies on neural networks to eliminate the need for projection and stabilization operators in the Virtual Element Method. In this paper, we discuss its formulation and detail the strategy for training the underlying neural network. The efficacy of this new method is tested through numerical experiments on elliptic problems.

math.NA↗

arXiv2023

The Mixed Virtual Element Discretization for highly-anisotropic problems: the role of the boundary degrees of freedom

In this paper, we discuss the accuracy and the robustness of the mixed Virtual Element Methods when dealing with highly-anisotropic diffusion problems. In particular, we analyze the performances of different approaches which are characterized by different sets of both boundary and internal degrees of freedom in presence of a strong anisotropy of the diffusion tensor with constant or variable coefficients. A new definition of the boundary degrees of freedom is also proposed and tested.

math.NA↗

arXiv2023

Orthogonal polynomial bases in the Mixed Virtual Element Method

The use of orthonormal polynomial bases has been found to be efficient in preventing ill-conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly-shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill-conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector-polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly-shaped elements, is proposed to validate our procedures.

math.NA↗