arXiv subjects

Nicolas Moës

Publications and source records attributed to Nicolas Moës.

10 recordsLinked to original sources

arXiv2026

The Tempered Finite Element Method

In this paper, we propose a new approach -- the Tempered Finite Element Method (TFEM) -- that extends the Finite Element Method (FEM) to classes of meshes that include zero-measure or nearly degenerate elements for which standard FEM approaches do not allow convergence. First, we review why the maximum angle condition [2] is not necessary for FEM convergence and what are the real limitations in terms of meshes. Next, we propose a simple modification of the classical FEM for elliptic problems that provably allows convergence for a wider class of meshes including bands of caps that cause locking of the solution in standard FEM formulations. The proposed method is trivial to implement in an existing FEM code and can be theoretically analyzed. We prove that in the case of exactly zero-measure elements it corresponds to mortaring. We show numerically and theoretically that what we propose is functional and sound. The remainder of the paper is devoted to extensions of the TFEM method to linear elasticity, mortaring of non-conforming meshes, high-order elements, and advection.

math.NA↗

arXiv2026

Taming Slivers: A Robust TFEM Framework for Reliable Computations on Degenerate Tetrahedral Meshes

Sliver elements are an intrinsic difficulty of three-dimensional tetrahedral mesh generation and remain costly, and sometimes impractical, to eliminate completely. Although isolated degenerate elements do not necessarily prevent finite element convergence, connected clusters or sheets of slivers may impose artificial constraints on the discrete solution, leading to locking and severe loss of accuracy. In this work, we revisit the effect of slivers from the viewpoint of the finite element solution and propose a robust solver-side treatment based on the Tempered Finite Element Method (TFEM). The method limits the singular contribution of degenerate elements by introducing a lower bound on the Jacobian determinant, which can be interpreted as a vanishing added-volume correction. The resulting formulation prevents the effective element volume from falling below a threshold while preserving the relevant physical modes of the solution. We analyze the stiffness matrices of degenerate tetrahedra, identify the mechanisms responsible for locking in sliver bands, and assess the method on a range of representative physical problems, including incompressible flow, Cahn--Hilliard phase-field dynamics, transient wave propagation, and vibro-acoustic fluid--structure interaction. The numerical results show that TFEM consistently recovers accurate and physically meaningful solutions on meshes for which standard FEM exhibits locking or loss of convergence, providing a simple and broadly applicable alternative to exhaustive geometric sliver removal.

math.NA↗

arXiv2026

DG = FEM + flat elements, Part I: Diffusion

We establish a simple, rigorous, and easy to implement connection between the classical continuous finite element method (FEM) and the discontinuous Galerkin (DG) method for Poisson's problem. The key idea is to insert a vanishing-thickness layer of "dummy" elements along cell interfaces. By modifying the diffusion coefficient on these elements to be proportional to their thickness, we prove the FEM formulation converges to Babuška-Zlámal DG with trapezoidal edge quadrature. The scheme is trivial to implement by (i) a mesh edit that introduces degenerate interface elements and (ii) a single Jacobian threshold in an otherwise unmodified FEM code to handle the degenerate elements via the tempered finite element (TFEM) framework. We provide a rigorous derivation of the resulting TFEM-DG scheme, prove optimal $H^1$ and $L^2$ error estimates, and present numerical experiments in 2D and 3D. The method allows for simple implementation of DG in a FEM code and even adaptive element-by-element switching between FEM and DG with minimal coding effort. The framework is readily extensible, as we will demonstrate in a companion paper dedicated to evolutionary nonlinear first-order hyperbolic systems.

math.NA↗

arXiv2025

Asymptotic meshes from $r$-variational adaptation methods for static problems in one dimension

We consider the minimization of integral functionals in one dimension and their approximation by $r$-adaptive finite elements. Including the grid of the FEM approximation as a variable in the minimization, we are able to show that the optimal grid configurations have a well-defined limit when the number of nodes in the grid is being sent to infinity. This is done by showing that the suitably renormalized energy functionals possess a limit in the sense of $Γ$-convergence. We provide numerical examples showing the closeness of the optimal asymptotic mesh obtained as a minimizer of the $Γ$-limit to the optimal finite meshes.

math.NA↗

arXiv2025

Phase-field and lip-field approaches for fracture with extreme mesh deformation (X-Mesh): a one-dimensional study

We consider a one-dimensional fracture problem modelled using either the phase-field or lip-field approach. In both cases, we optimise the incremental potential with respect to the displacement and damage fields and the nodal coordinates of the mesh. This is thus a variational mesh study. We observe that, as the damage reaches its maximum value, the optimisation drives the most damaged element to zero size as the damage reaches its maximum value. This peculiar element provides a precise displacement jump representation as the bar breaks. The overall solution is also shown to be much more accurate than the fixed mesh solution. This work forms part of an exploration into the capabilities of extreme meshes in computational mechanics (X-Mesh).

cs.CE↗

arXiv2025

Open Problems in Technical AI Governance

AI progress is creating a growing range of risks and opportunities, but it is often unclear how they should be navigated. In many cases, the barriers and uncertainties faced are at least partly technical. Technical AI governance, referring to technical analysis and tools for supporting the effective governance of AI, seeks to address such challenges. It can help to (a) identify areas where intervention is needed, (b) identify and assess the efficacy of potential governance actions, and (c) enhance governance options by designing mechanisms for enforcement, incentivization, or compliance. In this paper, we explain what technical AI governance is, why it is important, and present a taxonomy and incomplete catalog of its open problems. This paper is intended as a resource for technical researchers or research funders looking to contribute to AI governance.

cs.CY↗

arXiv2025

Solving the Porous Medium Equation with the eXtreme Mesh deformation approach (X-Mesh)

We introduce a new scheme for solving the non-regularized Porous Medium Equation. It is mass conserving and uses only positive unknown values. To address these typically conflicting features, we employ the eXtreme Mesh deformation approach (X-Mesh), specifically designed for problems involving sharp interfaces. The method ensures that the interface is always meshed, even in the face of complex topological changes, without the need for remeshing or altering the mesh topology. We illustrate the effectiveness of the approach through various numerical experiments.

math.NA↗

arXiv2024

X-Mesh: A new approach for the simulation of two-phase flow with sharp interface

Accurate modeling of moving boundaries and interfaces is a difficulty present in many situations of computational mechanics. We use the eXtreme Mesh deformation approach (X-Mesh) to simulate the interaction between two immiscible flows using the finite element method, while maintaining an accurate and sharp description of the interface without remeshing. In this new approach, the mesh is locally deformed to conform to the interface at all times, which can result in degenerated elements. The surface tension between the two fluids is added by imposing the pressure jump condition at the interface, which, when combined with the X-Mesh framework, allows us to have an exactly sharp interface. If a numerical scheme fails to properly balance surface tension and pressure gradients, it leads to numerical artefacts called spurious or parasitic currents. The method presented here is well balanced and reduces such currents down to the level of machine precision.

cs.CE↗

arXiv2023

A two-scale solver for linear elasticity problems in the context of parallel message passing

This paper pushes further the intrinsic capabilities of the GFEM$^{gl}$ global-local approach introduced initially in [1]. We develop a distributed computing approach using MPI (Message Passing Interface) both for the global and local problems. Regarding local problems, a specific scheduling strategy is introduced. Then, to measure correctly the convergence of the iterative process, we introduce a reference solution that revisits the product of classical and enriched functions. As a consequence, we are able to propose a purely matrix-based implementation of the global-local problem. The distributed approach is then compared to other parallel solvers either direct or iterative with domain decomposition. The comparison addresses the scalability as well as the elapsed time. Numerical examples deal with linear elastic problems: a polynomial exact solution problem, a complex micro-structure, and, finally, a pull-out test (with different crack extent). 1: C. A. Duarte, D.-J. Kim, and I. Babuška. A global-local approach for the construction of enrichment functions for the generalized fem and its application to three-dimensional cracks. In Advances in Meshfree Techniques, Dordrecht, 2007. Springer

math.NA↗

arXiv2022

Fragmentation analysis of a bar with the Lip-field approach

The Lip-field approach is a new way to regularize softening material models. It has already been tested in 1D quasistatic and 2D quasistatic: this paper extends it to 1D dynamics, on the challenging problem of dynamic fragmentation. The Lip-field approach formulates the mechanical problem to be solved as an optimization problem, where the incremental potential to be minimized is the non-regularized one. Spurious localization is prevented by imposing a Lipschitz constraint on the damage field. The displacement and damage field at each time step are obtained by a staggered algorithm, that is the displacement field is computed for a fixed damage field, then the damage field is computed for a fixed displacement field. Indeed, these two problems are convex, which is not the case of the global problem where the displacement and damage fields are sought at the same time. The incremental potential is obtained by equivalence with a cohesive zone model, which makes material parameters calibration simple. A non-regularized local damage equivalent to a cohesive zone model is also proposed. It is used as a reference for the Lip-field approach, without the need to implement displacement jumps. These approaches are applied to the brittle fragmentation of a 1D bar with randomly perturbed material properties to accelerate spatial convergence. Both explicit and implicit dynamic implementations are compared. Favorable comparison to several analytical, numerical and experimental references serves to validate the modeling approach.

cs.CE↗