arXiv subjects

Annalisa Buffa

Publications and source records attributed to Annalisa Buffa.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness and finite element approximation of the electrostatic shear Alfv\'en wave equations

The aim of this paper is to study the well-posedness and finite element approximation of the electrostatic shear Alfv\'en wave equations, a coupled system of two partial differential equations arising in plasma physics as a simplified sub-model of the drift-reduced Braginskii equations. To this end, anisotropic Sobolev spaces depending on the normalized magnetic field $\b$ are introduced, together with a Poincar\'e-type inequality along the integral curves of $\b$, which holds under a geometric directedness condition on the magnetic field. Using these tools, existence, uniqueness, and stability of a weak solution are established via the Faedo-Galerkin method. It is also shown that the geometric condition is satisfied in tokamak and stellarator configurations. A numerical scheme is then proposed, combining Lagrange finite elements in space with a Crank-Nicolson discretization in time. The scheme is shown to conserve a discrete energy exactly in the homogeneous case, and a priori error estimates are derived in the natural energy norm. Several numerical experiments are reported in two and three space dimensions, which confirm the theoretical results and indicate that the geometric condition on the magnetic field is necessary for the invertibility of the discrete system matrix.

math.NA↗

arXiv2026

A Recovery-Based Error Indicator for Finite Difference Methods

A novel recovery-based error indicator for high-order Finite Difference Methods, based on post-processing of the Finite Difference values is presented. The values obtained on the Finite Difference grid are interpolated into a suitable polynomial Finite Element space. A recovery-based error indicator, with the polynomial-preserving property, is then applied to estimate the gradient error. The performance and accuracy of the proposed error indicator are demonstrated through several numerical experiments, including the two-dimensional Poisson problem solved using second- and fourth-order finite difference schemes. Additional experiments are conducted on elliptic problems with discontinuous coefficients, as well as on the two and three-dimensional wave equation in homogeneous media with second- and fourth-order finite differences, and in heterogeneous media with second-order finite differences.

math.NA↗

arXiv2025

A Certified Goal-Oriented A Posteriori Defeaturing Error Estimator for Elliptic PDEs

Defeaturing, the process of simplifying computational geometries, is a critical step in industrial simulation pipelines for reducing computational cost. Rigorous a posteriori estimators exist for the global energy-norm error introduced by geometry simplifications. However, practitioners are usually more concerned with the accuracy of specific quantities of interest (QoIs) in the solution. This paper bridges that gap by developing mathematically certified, goal-oriented a posteriori defeaturing error estimators for Poisson's equation, linear elasticity, and Stokes flow. First, we derive new reliable energy-norm estimators for features subject to Dirichlet boundary conditions in linear elasticity and Stokes flow, based on existing results for Poisson's equation. Second, we formulate general energy-norm estimators for multiple negative features, subject to either Dirichlet or Neumann boundary conditions for the first time. Finally, we combine these estimators with the dual-weighted residual (DWR) method to obtain reliable estimates for linear QoIs and demonstrate their effectiveness across a range of numerical experiments.

math.NA↗

arXiv2025

On design, analysis, and hybrid manufacturing of microstructured blade-like geometries

With the evolution of new manufacturing technologies such as multi-material 3D printing, one can think of new type of objects that consist of considerably less, yet heterogeneous, material, consequently being porous, lighter and cheaper, while having the very same functionality as the original object when manufactured from one single solid material. We aim at questioning five decades of traditional paradigms in geometric CAD and focus at new generation of CAD objects that are not solid, but contain heterogeneous free-form internal microstructures. We propose a unified manufacturing pipeline that involves all stages, namely design, optimization, manufacturing, and inspection of microstructured free-form geometries. We demonstrate our pipeline on an industrial test case of a blisk blade that sustains the desired pressure limits, yet requires significantly less material when compared to the solid counterpart.

cs.GR↗

arXiv2025

Stabilization techniques for immersogeometric analysis of plate and shell problems in explicit dynamics

Finite element plate and shell formulations are ubiquitous in structural analysis for modeling all kinds of slender structures, both for static and dynamic analyses. The latter are particularly challenging as the high order nature of the underlying partial differential equations and the slenderness of the structures all impose a stringent constraint on the critical time step in explicit dynamics. Unfortunately, badly cut elements in immersed finite element discretizations further aggravate the issue. While lumping the mass matrix often increases the critical time step, it might also trigger spurious oscillations in the approximate solution thereby compromising the numerical solution. In this article, we extend our previous work in \cite{voet2025stabilization} to allow stable immersogeometric analysis of plate and shell problems with lumped mass matrices. This technique is based on polynomial extensions and restores a level of accuracy comparable to boundary-fitted discretizations.

math.NA↗

arXiv2025

Analysis-Aware Defeaturing of Dirichlet Features

Feature removal from computational geometries, or defeaturing, is an integral part of industrial simulation pipelines. Defeaturing simplifies the otherwise costly or even impossible meshing process, speeds up the simulation, and lowers its memory footprint. Current defeaturing operators are often based on heuristic criteria and ignore the impact of the simplifications on the PDE solution. This work extends the mathematically rigorous framework developed by Buffa, Chanon, and V\'azquez (2022) to features subject to Dirichlet boundary conditions in Poisson problems. We derive a posteriori error estimators for negative features in the interior or on the boundary of the computational domain. The estimators' dependence on the feature size is explicit, and their evaluation only involves boundary integrals over the feature boundary. Numerical experiments in two and three dimensions showcase the validity and efficiency of the estimators.

math.NA↗

arXiv2025

Adaptive refinement in defeaturing problems via an equilibrated flux a posteriori error estimator

An adaptive refinement strategy, based on an equilibrated flux a posteriori error estimator, is proposed in the context of defeaturing problems. Defeaturing consists of removing features from complex domains to simplify mesh generation and reduce the computational cost of simulations. It is a common procedure, for example, in computer aided design for simulation-based manufacturing. However, depending on the problem at hand, geometrical simplification may significantly deteriorate the accuracy of the solution. The proposed adaptive strategy is hence twofold: starting from a defeatured geometry, it performs both standard mesh refinement and geometrical refinement by selecting, at each step, which features must be reintroduced to significantly improve accuracy. Similar adaptive strategies have been previously developed using residual-based error estimators within an IGA framework. Here, instead, we extend a previously developed equilibrated flux a posteriori error analysis, designed for standard finite element discretizations, to make it fully applicable within the adaptive procedure. In particular, we address the assembly of the equilibrated flux estimator in presence of elements trimmed by the boundary of included features, adopting a CutFEM strategy to handle feature inclusion. The resulting estimator allows us to bound both the defeaturing and the numerical sources of error, with additional contributions accounting for the weak imposition of boundary conditions.

math.NA↗

arXiv2025

Mass lumping and stabilization for immersogeometric analysis

Trimmed (multi-patch) geometries are the state-of-the-art technology in computer-aided design for industrial applications such as automobile crashworthiness. In this context, fast solution techniques extensively rely on explicit time integration schemes in conjunction with mass lumping techniques that substitute the consistent mass with a (usually diagonal) approximation. For smooth isogeometric discretizations, Leidinger [1] first showed that mass lumping removed the dependency of the critical time-step on the size of trimmed elements. This finding has attracted considerable attention but has unfortunately overshadowed another more subtle effect: mass lumping may disastrously impact the accuracy of low frequencies and modes, potentially inducing spurious oscillations in the solution. In this article, we provide compelling evidence for this phenomenon and later propose a stabilization technique based on polynomial extensions that restores a level of accuracy comparable to boundary-fitted discretizations.

math.NA↗

arXiv2025

A spline-based hexahedral mesh generator for patient-specific coronary arteries

This paper presents a spline-based hexahedral mesh generator for tubular geometries commonly encountered in haemodynamics studies, in particular coronary arteries. We focus on techniques for accurately meshing vessels with stenoses and aneurysms, as well as non-planar bifurcations. Our approach incorporates several innovations, including a spline-based description of the vessel geometry in both the radial and the longitudinal directions, the use of Hermite curves for modeling non-planar bifurcations, and a generalization to non-planar n intersecting branches. This method eliminates the need for a concrete vessel surface, grid smoothing, and other post-processing. A technique to generate grids with boundary layers is also presented. We validate the generated meshes using commonly employed quality indices, compare them against state-of-the-art mesh generators and apply our method to complex coronary trees. Finally, we present finite element fluid flow simulations with physiological boundary conditions. To validate the proposed framework, a wall-shear-stress-based convergence test and computations of haemodynamic indices are also presented.

math.NA↗

arXiv2024

Mass lumping and outlier removal strategies for complex geometries in isogeometric analysis

Mass lumping techniques are commonly employed in explicit time integration schemes for problems in structural dynamics and both avoid solving costly linear systems with the consistent mass matrix and increase the critical time step. In isogeometric analysis, the critical time step is constrained by so-called "outlier" frequencies, representing the inaccurate high frequency part of the spectrum. Removing or dampening these high frequencies is paramount for fast explicit solution techniques. In this work, we propose mass lumping and outlier removal techniques for nontrivial geometries, including multipatch and trimmed geometries. Our lumping strategies provably do not deteriorate (and often improve) the CFL condition of the original problem and are combined with deflation techniques to remove persistent outlier frequencies. Numerical experiments reveal the advantages of the method, especially for simulations covering large time spans where they may halve the number of iterations with little or no effect on the numerical solution.

math.NA↗

arXiv2024

A theoretical analysis of mass scaling techniques

Mass scaling is widely used in finite element models of structural dynamics for increasing the critical time step of explicit time integration methods. While the field has been flourishing over the years, it still lacks a strong theoretical basis and mostly relies on numerical experiments as the only means of assessment. This contribution thoroughly reviews existing methods and connects them to established linear algebra results to derive rigorous eigenvalue bounds and condition number estimates. Our results cover some of the most successful mass scaling techniques, unraveling for the first time well-known numerical observations.

math.NA↗

arXiv2024

The Immersed Boundary Conformal Method for Kirchhoff-Love and Reissner-Mindlin shells

This work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze Kirchhoff-Love and Reissner-Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite the simplicity of the approach. The IBCM method introduces additional layers conformal to the boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high-degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with the inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff-Love and Reissner-Mindlin theories. These tests include the generation of conformal interfaces, the coupling of Kirchhoff-Love and Reissner-Mindlin theories, and the simulation of a damaged shell.

math.NA↗

arXiv2024

Adaptive analysis-aware defeaturing: the case of Neumann boundary conditions

Removing geometrical details from a complex domain is a classical operation in computer aided design. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. However, depending on the partial differential equation that one wants to solve, removing some important geometrical features may greatly impact the solution accuracy. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected or its evaluation is based on engineering expertise, only due to the lack of reliable tools. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phases. In this work, we consider as a model problem the Poisson equation on a geometry with Neumann features, we consider some finite element discretization of it, and we build an adaptive strategy that is twofold. Firstly, it is able to perform geometrical refinements, that is, to choose at each iteration step which geometrical feature is important to obtain an accurate solution. Secondly, it performs standard mesh refinements; since the geometry changes at each iteration, the algorithm is designed to be used with an immersed method. To drive this adaptive strategy, we introduce an a posteriori estimator of the energy error between the exact solution defined in the exact fully-featured geometry, and the numerical approximation of the solution defined in the defeatured geometry. The reliability of the estimator is proven for very general (potentially trimmed multipatch) geometric configurations, and in particular for IGA with hierarchical B-splines. Finally, numerical experiments are performed to validate the presented theory and to illustrate the capabilities of the proposed adaptive strategy.

math.NA↗

arXiv2024

An Interior Penalty coupling strategy for Isogeometric non-conformal Kirchhoff-Love shell patches

This work focuses on the coupling of trimmed shell patches using Isogeometric Analysis, based on higher continuity splines that seamlessly meet the $C^1$ requirement of Kirchhoff-Love-based discretizations. Weak enforcement of coupling conditions is achieved through the symmetric interior penalty method, where the fluxes are computed using their correct variationally consistent expression that was only recently proposed and is unprecedentedly adopted herein in the context of coupling conditions. The constitutive relationships account for generically laminated materials, although the proposed tests are conducted under the assumption of uniform thickness and lamination sequence. Numerical experiments assess the method for an isotropic and a laminated plate, as well as an isotropic hyperbolic paraboloid shell from the new shell obstacle course. The boundary conditions and domain force are chosen to reproduce manufactured analytical solutions, which are taken as reference to compute rigorous convergence curves in the $L^2$, $H^1$, and $H^2$ norms, that closely approach optimal ones predicted by theory. Additionally, we conduct a final test on a complex structure comprising five intersecting laminated cylindrical shells, whose geometry is directly imported from a STEP file. The results exhibit excellent agreement with those obtained through commercial software, showcasing the method's potential for real-world industrial applications.

math.NA↗

arXiv2024

A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.

math.NA↗

arXiv2023

An equilibrated flux a posteriori error estimator for defeaturing problems

An a posteriori error estimator based on an equilibrated flux reconstruction is proposed for defeaturing problems in the context of finite element discretizations. Defeaturing consists in the simplification of a geometry by removing features that are considered not relevant for the approximation of the solution of a given PDE. In this work, the focus is on Poisson equation with Neumann boundary conditions on the feature boundary. The estimator accounts both for the so-called defeaturing error and for the numerical error committed by approximating the solution on the defeatured domain. Unlike other estimators that were previously proposed for defeaturing problems, the use of the equilibrated flux reconstruction allows to obtain a sharp bound for the numerical component of the error. Furthermore, it does not require the evaluation of the normal trace of the numerical flux on the feature boundary: this makes the estimator well-suited for finite element discretizations, in which the normal trace of the numerical flux is typically discontinuous across elements. The reliability of the estimator is proven and verified on several numerical examples. Its capability to identify the most relevant features is also shown, in anticipation of a future application to an adaptive strategy.

math.NA↗

arXiv2023

Energy conservative isogeometric techniques for the wave equation

We analyze the wave equation in mixed form, with periodic and/or Dirichlet homogeneous boundary conditions, and nonconstant coefficients that depend on the spatial variable. For the discretization, the weak form of the second equation is replaced by a strong form, written in terms of a projection operator. The system of equations is discretized with B-splines forming a De Rham complex along with suitable commutative projectors for the approximation of the second equation. The discrete scheme is energy conservative when discretized in time with a conservative method such as Crank-Nicolson. We propose a convergence analysis of the method to study the dependence with respect to the mesh size $h$, with focus on the consistency error. Numerical results show optimal convergence of the error in energy norm, and a relative error in energy conservation for long-time simulations of the order of machine precision.

math.NA↗

arXiv2023

Stabilized isogeometric formulation of the Stokes problem on overlapping patches

We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.

math.NA↗