arXiv subjects

Stefano Scialò

Publications and source records attributed to Stefano Scialò.

15 recordsLinked to original sources

arXiv2026

Hybrid-Dimensional Biot Problem with an Optimization Based Domain Decomposition Approach

The present work proposes a numerical approach for solving coupled flow and mechanics problems in fractured porous media, represented as mixed-dimensional domains. In this formulation, the elements of the 3D mesh are allowed to arbitrarily intersect the fractures. Displacements are discontinuous across fractures through the use of the eXtended Finite Element Method (XFEM) on the 3D mesh. The mechanical problem is formulated as a saddle-point problem, in which Lagrange multipliers are used to enforce displacement continuity across the fractures. The resulting Lagrange multipliers represent the stress field acting on the fracture surfaces. Likewise, the pressure field is allowed to be discontinuous across fractures through the XFEM formulation on the non-conforming mesh and is computed using an optimization-based domain decomposition strategy specifically designed for mixed-dimensional problems. The fixed-stress splitting scheme is employed to decouple the flow and mechanics subproblems, while the mixed-dimensional pressure problem is solved at each fixed-stress iteration using the Conjugate Gradient (CG) method. The combination of the fixed-stress scheme and the CG solver proves to be highly effective for this class of problems.

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arXiv2026

A new operator splitting method for coupled PDE problems, with an application to poromechanics

The present work proposes a new operator splitting technique. The method leverages ideas used in PDE-constrained domain decomposition, adapting them to the objective of decoupling equations defined on the same geometric domain. The methodology is applied here to the resolution of coupled flow and deformation in the linear quasi-static Biot problem. An overview of its main characteristics is provided, also in comparison with the well established fixed-stress split for poroelasticity.

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arXiv2026

A Fictitious Domain Formulation with Bubble Enrichment

The fictitious domain method offers a powerful alternative to circumvent complex mesh generation process by embedding the complex physical domain into a simpler, non-matching background mesh. Among the various strategies to enforce the boundary condition on such a mesh, one possibility is to impose it weakly through a Lagrange multiplier. Existing approaches following this strategy typically require two independent uniform meshes for the domain and the boundary, whose mesh sizes are related by a prescribed ratio. In this work, we relax this requirement by constructing the boundary mesh directly from the trace of the background triangulation. To recover the uniform discrete inf-sup condition under this weaker mesh assumption, we enrich the discrete space of the solution with bubble functions. Our main result is the construction of a suitable operator that allows us to prove this uniform discrete inf-sup condition, thereby establishing the stability of the resulting scheme. As a consequence, we derive optimal a priori error estimates and provide a numerical experiment to validate the theoretical results.

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arXiv2026

Modeling tumor growth with variable mass and angiogenesis-driven perfusion through a 3D-1D coupled framework

Tumor growth beyond a critical size relies on the development of a functional vascular network, which ensures adequate oxygen and nutrient supply. In this work, we present a modeling framework based on an optimization-based 3D-1D coupling strategy to simulate perfusion in a tumoral tissue with growing mass, interacting with a dynamically evolving capillary network. The tumor is described as a multiphase system including tumor cells and interstitial fluid, governed by a non-linear PDE system for cell volume fraction, pressure, oxygen, and VEGF, and discretized via finite elements. Capillary growth is tackled using a continuous-discrete hybrid tip-tracking approach. The vascular geometry is updated over time according to angiogenic signals, and coupled to the tissue model through a constrained optimization formulation that enforces fluid and nutrient exchange via interface variables. A sensitivity analysis using the Morris elementary effect method identifies key parameters influencing system behavior. Results highlight the critical role of vascular development in regulating tissue perfusion and tumor progression. Overall, the proposed numerical approach provides a versatile tool for investigating tumor-vascular interactions and can support further quantitative analysis of angiogenesis and tumor perfusion dynamics.

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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.

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arXiv2024

A five field formulation for flow simulations in porous media with fractures and barriers via an optimization based domain decomposition method

The present work deals with the numerical resolution of coupled 3D-2D problems arising from the simulation of fluid flow in fractured porous media modeled via the Discrete Fracture and Matrix (DFM) model. According to the DFM model, fractures are represented as planar interfaces immersed in a 3D porous matrix and can behave as preferential flow paths, in the case of conductive fractures, or can actually be a barrier for the flow, when, instead, the permeability in the normal-to-fracture direction is small compared to the permeability of the matrix. Consequently, the pressure solution in a DFM can be discontinuous across a barrier, as a result of the geometrical dimensional reduction operated on the fracture. The present work is aimed at developing a numerical scheme suitable for the simulation of the flow in a DFM with fractures and barriers, using a mesh for the 3D matrix non conforming to the fractures and that is ready for domain decomposition. This is achieved starting from a PDE-constrained optimization method, currently available in literature only for conductive fractures in a DFM. First, a novel formulation of the optimization problem is defined to account for non permeable fractures. These are described by a filtration-like coupling at the interface with the surrounding porous matrix. Also the extended finite element method with discontinuous enrichment functions is used to reproduce the pressure solution in the matrix around a barrier. The method is presented here in its simplest form, for clarity of exposition, i.e. considering the case of a single fracture in a 3D domain, also providing a proof of the well posedness of the resulting discrete problem. Four validation examples are proposed to show the viability and the effectiveness of the method.

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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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arXiv2022

An optimization based 3D-1D coupling strategy for tissue perfusion and chemical transport during tumor-induced angiogenesis

A new mathematical model and numerical approach are proposed for the simulation of fluid and chemical exchanges between a growing capillary network and the surrounding tissue, in the context of tumor-induced angiogenesis. Thanks to proper modeling assumptions the capillaries are reduced to their centerline: a well posed mathematical model is hence worked out, based on the coupling between a three-dimensional and a one-dimensional equation (3D-1D coupled problem). Also the application of a PDE-constrained optimization formulation is here proposed for the first time for angiogenesis simulations. Under this approach no mesh conformity is required, thus making the method particularly suitable for this kind of application, since no remeshing is required as the capillary network grows. In order to handle both the evolution of the quantities of interest and the changes in the geometry, a discrete-hybrid strategy is adopted, combining a continuous modeling of the tissue and of the chemicals with a discrete tip-tracking model to account for the vascular network growth. The tip-tracking strategy, together with some proper rules for branching and anastomosis, is able to provide a realistic representation of the capillary network.

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arXiv2022

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient resolution strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.

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arXiv2021

A gradient based resolution strategy for a PDE-constrained optimization approach for 3D-1D coupled problems

Coupled 3D-1D problems arise in many practical applications, in an attempt to reduce the computational burden in simulations where cylindrical inclusions with a small section are embedded in a much larger domain. Nonetheless the resolution of such problems can be non trivial, both from a mathematical and a geometrical standpoint. Indeed 3D-1D coupling requires to operate in non standard function spaces, and, also, simulation geometries can be complex for the presence of multiple intersecting domains. Recently, a PDE-constrained optimization based formulation has been proposed for such problems, proving a well posed mathematical formulation and allowing for the use of non conforming meshes for the discrete problem. Here an unconstrained optimization formulation of the problem is derived and an efficient gradient based solver is proposed for such formulation. Some numerical tests on quite complex configurations are discussed to show the viability of the method.

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arXiv2020

The mixed virtual element method for grids with curved interfaces

In many applications the accurate representation of the computational domain is a key factor to obtain reliable and effective numerical solutions. Curved interfaces, which might be internal, related to physical data, or portions of the physical boundary, are often met in real applications. However, they are often approximated leading to a geometrical error that might become dominant and deteriorate the quality of the results. Underground problems often involve the motion of fluids where the fundamental governing equation is the Darcy law. High quality velocity fields are of paramount importance for the successful subsequent coupling with other physical phenomena such as transport. The virtual element method, as solution scheme, is known to be applicable in problems whose discretizations requires cells of general shape, and the mixed formulation is here preferred to obtain accurate velocity fields. To overcome the issues associated to the complex geometries and, at the same time, retaining the quality of the solutions, we present here the virtual element method to solve the Darcy problem, in mixed form, in presence of curved interfaces in two and three dimensions. The numerical scheme is presented in detail explaining the discrete setting with a focus on the treatment of curved interfaces. Examples, inspired from industrial applications, are presented showing the validity of the proposed approach.

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arXiv2020

The Mixed Virtual Element Method on curved edges in two dimensions

In this work, we propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional computational grids with curvilinear edge elements. The approximation by means of rectilinear edges of a domain with curvilinear geometrical feature, such as a portion of domain boundary or an internal interface, may introduce a geometrical error that degrades the expected order of convergence of the scheme. In the present work a suitable VEM approximation space is proposed to consistently handle curvilinear geometrical objects, thus recovering optimal convergence rates. The resulting numerical scheme is presented along with its theoretical analysis and several numerical test cases to validate the proposed approach.

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arXiv2019

Analysis of conforming, non-matching, and polygonal methods for Darcy and advection-diffusion-reaction simulations in discrete fracture networks

The aim of this study is to compare numerical methods for the simulation of single-phase flow and transport in fractured media, described here by means of the Discrete Fracture Network (DFN) model. A Darcy problem is solved to compute the advective field, then used in a subsequent time dependent transport-diffusion-reaction problem. The numerical schemes are benchmarked in terms of flexibility in handling geometrical complexity, mass conservation and stability issues for advection dominated flow regimes. To this end, three benchmark cases have been specifically designed and are here proposed, representing some of the most critical issues encountered in DFN simulations.

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arXiv2018

Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations

Simulations of fluid flow in naturally fractured rocks have implications for several subsurface applications, including energy storage and extraction, and waste storage. We are interested in flow in discrete fracture networks, which explicitly represent flow in fracture surfaces, but ignore the impact of the surrounding host rock. Fracture networks, generated from observations or stochastic simulations, will contain intersections of arbitrary length, and intersection lines can further cross, forming a highly complex geometry. As the flow exchange between fractures, thus in the network, takes place in these intersections, an adequate representation of the geometry is critical for simulation accuracy. In practice, the intersection dynamics must be handled by a combination of the simulation grid, which may or may not resolve the intersection lines, and the numerical methods applied on the grid. In this work, we review different classes of numerical approaches proposed in recent years, covering both methods that conform to the grid, and non-matching cases. Specific methods considered herein include finite element, mixed and virtual finite elements and control volume methods. We expose our methods to an extensive set of test cases, ranging from artificial geometries designed to test difficult configurations, to a network extruded from a real fracture outcrop. The main outcome is guidances for choice of simulation models and numerical discretization with a trade off on the computational cost and solution accuracy.

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