arXiv subjects

Daniele Boffi

Publications and source records attributed to Daniele Boffi.

At least 19 recordsLinked to original sources

arXiv2026

A Comprehensive p-VEM Framework for Advanced Variable Stiffness Plates with Arbitrary Shapes

This paper presents a comprehensive, high-order (p-version) Virtual Element Method (VEM) framework for the structural analysis of innovative variable stiffness plates. VEM is particularly suited for complex configurations due to its ability to handle arbitrary polygonal meshes, including curved edges. However, its mathematical formulation may hinder its spread in the engineering community. This work illustrates a formulation with an accessible implementation using well-known FEM notation and integrating at the same time a set of new advanced capabilities. Specifically, both standard stabilized and advanced self-stabilized strategies are adopted. To further improve the robustness of VEM in the presence of variable coefficients, polynomial projections taking into account the coefficients are employed. This approach is referred to as Variable Coefficients-VEM approach (VC-VEM). This unified framework is applied to linear static, free-vibration, and buckling analyses, and validated against analytical solutions and numerical benchmarks. In particular, plates with cutouts and problems featuring high-gradient solutions are investigated, demonstrating that the proposed comprehensive approach provides a flexible and ready-to-implement tool for advanced structural design.

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arXiv2026

Quadrature error estimates on non-matching grids in a fictitious domain framework for fluid-structure interaction problems

We consider a fictitious domain formulation for fluid-structure interaction problems based on a distributed Lagrange multiplier to couple the fluid and solid behaviors. How to deal with the coupling term is crucial since the construction of the associated finite element matrix requires the integration of functions defined over non-matching grids: the exact computation can be performed by intersecting the involved meshes, whereas an approximate coupling matrix can be evaluated on the original meshes by introducing a quadrature error. The purpose of this paper is twofold: we prove that the discrete problem is well-posed also when the coupling term is constructed in approximate way and we discuss quadrature error estimates over non-matching grids.

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arXiv2026

Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity

In this work, we analyze the conforming and $C^0$-non-conforming Virtual Element Method for a fourth-order Steklov eigenvalue problem on a generally shaped, possibly nonconvex, polygonal domain. By employing an {\it enriching } operator, we derive the convergence analysis using the discrete $H^2$ seminorm, and the $H^1$ and $L^2$ norms. We use the Babuška--Osborn spectral theory \cite{BO} to prove that the numerical scheme approximates the spectrum without introducing any spurious eigenvalue. Moreover, we derive the optimal order of convergence for eigenfunctions and double order for eigenvalues. We assess the performance of the method on several numerical tests using different families of polygonal meshes.

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arXiv2026

Auxiliary Gradient-Flow Solvers for Generalized Newtonian Models

We introduce an auxiliary gradient-flow framework for variational problems with generalized Newtonian structure governed by an N-function. The key idea is to replace the nonlinear constitutive dependence on the gradient, or symmetric gradient, by an auxiliary scalar variable representing its squared magnitude. This shifts the nonlinearity from the state equation to the auxiliary variable, yielding a sequence of uniformly elliptic weighted linear problems. At the continuous level, we construct an auxiliary energy on a metric space adapted to the growth of the underlying N-function. In this topology, we prove lower semicontinuity, geodesic $λ$-convexity, and exponential convergence of the associated minimizing-movement scheme. At the finite element level, we derive a metric gradient flow through an explicit Riesz map, prove global well-posedness of the resulting semi-discrete ODE, and establish convergence to the finite element solution of the Euler--Lagrange equations of the generalized Newtonian energy. For the $p$-Laplacian and $p$-Stokes models, this gives a rigorous convergence result for $4/3\le p\le 4$, $p\ne2$, with asymptotic rate estimates beyond this range. We also propose practical time discretizations, including an operator-splitting scheme that gives the \kac iteration as a special case, and an adaptive pseudo-transient method that can be implemented using scalable linear solvers. Numerical experiments for power-law, Carreau--Yasuda, regularized Bingham, and optimal-design models demonstrate robustness, mesh-independent iteration counts in the tested regimes, and performance that matches or outperforms Newton's method.

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arXiv2026

Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?

The VEM approximation of eigenvalue problems usually involves the appropriate tuning of stabilization parameters, unless self-stabilizing or stabilization-free VEM are used. In this paper we prove that for elliptic self-adjoint eigenvalue problems the stabilization of the mass matrix is not necessary when lower order standard VEM spaces are adopted. Numerical evidence shows that also for higher order schemes the same result is true on various mesh sequences.

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arXiv2026

Automatic computation of the glycemic index: data driven analysis of the glucose standard

The Glycemic Index (GI) is a tool for classifying carbohydrates based on their impact on postprandial glycemia, useful for diabetes prevention and management. This study applies a mathematical model for a data driven simulation of the glycemic response following glucose ingestion. The analysis is performed on a dataset of 35 healthy subjects undergone a standard 50 g oral glucose test. The results reveal a direct correlation between glucose response profiles and parameters describing glucose absorption, enabling the classification of subjects into three groups based on the timing of their glycemic peak: <30 min, 30-50 min, >50 min. These findings highlight the ability of a physiology-based mathematical model to capture inter-individual variability in postprandial glucose dynamics and represent a step toward simulation-based approaches for GI estimation.

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arXiv2026

Benchmarking stabilized and self-stabilized p-virtual element methods with variable coefficients

Standard Virtual Element Methods (VEM) are based on polynomial projections and require a stabilization term to evaluate the contribution of the non-polynomial component of the discrete space. However, the stabilization term is not uniquely defined by the underlying variational formulation and is typically introduced in an ad hoc manner, potentially affecting the numerical response. Stabilization-free and self-stabilized formulations have been proposed to overcome this issue, although their theoretical analysis is still less mature. This paper provides an in-depth numerical investigation into different stabilized and self-stabilized formulations for the p-version of VEM. The results show that self-stabilized and stabilization-free formulations achieve optimal accuracy while suffering from worse conditioning. Moreover, a new projection operator, which explicitly accounts for variable coefficients, is introduced within the framework of standard virtual element spaces. Numerical results show that this new approach is more robust than the existing ones for large values of p.

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arXiv2026

On the stability and conditioning of a fictitious domain formulation for fluid-structure interaction problems

We consider a distributed Lagrange multiplier formulation for fluid-structure interaction problems in the spirit of the fictitious domain approach. This is an unfitted method, which does not require the construction of meshes conforming to the interface. We focus on the stationary problem arising from the time discretization and we analyze the behavior of the condition number with respect to mesh refinement. At the numerical level, the computation of the term coupling the fluid and the solid mesh requires the knowledge of the intersection between fluid and mapped solid elements and it might happen that a portion of the intersected elements is very small. We show that our formulation is stable independently of such intersections and that the conditioning is not affected by the interface position.

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arXiv2025

Advances on the finite element discretization of fluid-structure interaction problems

We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical findings concerning well-posedness, stability, and convergence of the numerical schemes, and discuss the related computational challenges. In the case of elliptic interface problems, we also present a posteriori error estimates.

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arXiv2025

Adaptive Mixed FEM for the Stokes eigenvalue problem

In this paper we discuss the optimal convergence of a standard adaptive scheme based on mixed finite element approximation to the solution of the eigenvalue problem associated with the Stokes equations. The proofs of the quasi-orthogonality and the discrete reliability are presented. Our numerical experiments confirm the efficacy of the proposed adaptive scheme.

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arXiv2025

Multigrid Preconditioning for FD-DLM Method in Elliptic Interface Problems

We investigate the performance of multigrid preconditioners for solving linear systems arising from finite element discretizations of elliptic interface problems using the Fictitious Domain with Distributed Lagrange Multipliers (FD-DLM) formulation. Numerical experiments are conducted using continuous and discontinuous finite element spaces for the Lagrange multiplier. Results indicate that multigrid is a promising preconditioner for problems in the FD-DLM formulation.

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arXiv2024

A nodal ghost method based on variational formulation and regular square grid for elliptic problems on arbitrary domains in two space dimensions

This paper focuses on the numerical solution of elliptic partial differential equations (PDEs) with Dirichlet and mixed boundary conditions, specifically addressing the challenges arising from irregular domains. Both finite element method (FEM) and finite difference method (FDM), face difficulties in dealing with arbitrary domains. The paper introduces a novel nodal symmetric ghost {method based on a variational formulation}, which combines the advantages of FEM and FDM. The method employs bilinear finite elements on a structured mesh and provides a detailed implementation description. A rigorous a priori convergence rate analysis is also presented. The convergence rates are validated with many numerical experiments, in both one and two space dimensions.

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arXiv2024

Reduced basis approximation of parametric eigenvalue problems in presence of clusters and intersections

In this paper we discuss reduced order models for the approximation of parametric eigenvalue problems. In particular, we are interested in the presence of intersections or clusters of eigenvalues. The singularities originating by these phenomena make it hard a straightforward generalization of well known strategies normally used for standards PDEs. We investigate how the known results extend (or not) to higher order frequencies.

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arXiv2024

A posteriori error estimator for elliptic interface problems in the fictitious formulation

A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error estimation plays a pivotal role in assessing the accuracy and reliability of computational solutions across various domains of science and engineering. This study delves into the theoretical underpinnings and computational considerations of a residual-based estimator. Theoretically, the estimator is studied for cases with constant coefficients which jump across an interface as well as generalized scenarios with smooth coefficients that jump across an interface. Theoretical findings demonstrate the reliability and efficiency of the proposed estimators under all considered cases. Numerical experiments are conducted to validate the theoretical results, incorporating various immersed geometries and instances of high coefficients jumps at the interface. Leveraging an adaptive algorithm, the estimator identifies regions with singularities and applies refinement accordingly. Results substantiate the theoretical findings, highlighting the reliability and efficiency of the estimators. Furthermore, numerical solutions exhibit optimal convergence properties, demonstrating resilience against geometric singularities or coefficients jumps.

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arXiv2024

A data-driven method for parametric PDE Eigenvalue Problems using Gaussian Process with different covariance functions

We use a Gaussian Process Regression (GPR) strategy that was recently developed [3,16,17] to analyze different types of curves that are commonly encountered in parametric eigenvalue problems. We employ an offline-online decomposition method. In the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then, we generate our GPR model using four different Matérn covariance functions. In the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters. We then illustrate how the choice of each covariance function influences the performance of GPR. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity.

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arXiv2024

A parallel solver for fluid structure interaction problems with Lagrange multiplier

The aim of this work is to present a parallel solver for a formulation of fluid-structure interaction (FSI) problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The fluid subproblem, consisting of the non-stationary Stokes equations, is discretized in space by $\mathcal{Q}_2$-$\mathcal{P}_1$ finite elements, whereas the structure subproblem, consisting of the linear or finite incompressible elasticity equations, is discretized in space by $\mathcal{Q}_1$ finite elements. A first order semi-implicit finite difference scheme is employed for time discretization. The resulting linear system at each time step is solved by a parallel GMRES solver, accelerated by block diagonal or triangular preconditioners. The parallel implementation is based on the PETSc library. Several numerical tests have been performed on Linux clusters to investigate the effectiveness of the proposed FSI solver.

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arXiv2024

On the stabilization of a virtual element method for an acoustic vibration problem

In this paper we introduce an abstract setting for the convergence analysis of the virtual element approximation of an acoustic vibration problem. We discuss the effect of the stabilization parameters and remark that in some cases it is possible to achieve optimal convergence without the need of any stabilization. This statement is rigorously proved for lowest order triangular element and supported by several numerical experiments.

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arXiv2023

Unfitted mixed finite element methods for elliptic interface problems

In this paper, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid structure interaction problems. Two finite elements schemes with piecewise constant Lagrange multiplier are proposed and their stability is proved theoretically. Numerical results compare the performance of those elements, confirming the theoretical proofs and verifying that the schemes converge with optimal rate.

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