arXiv subjects

Gioana Teora

Publications and source records attributed to Gioana Teora.

12 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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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arXiv2023

Improving high-order VEM stability on badly-shaped elements

For the 2D and 3D Virtual Element Methods (VEM), a new approach to improve the conditioning of local and global matrices in the presence of badly-shaped polytopes is proposed. It defines the local projectors and the local degrees of freedom with respect to a set of scaled monomials recomputed on more well-shaped polytopes. This new approach is less computationally demanding than using the orthonormal polynomial basis. The effectiveness of our procedure is tested on different numerical examples characterized by challenging geometries of increasing complexity.

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arXiv2023

A first-order stabilization-free Virtual Element Method

In this paper, we introduce a new Virtual Element Method (VEM) not requiring any stabilization term based on the usual enhanced first-order VEM space. The new method relies on a modified formulation of the discrete diffusion operator that ensures stability preserving all the properties of the differential operator.

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