arXiv subjects

Pierre Kerfriden

Publications and source records attributed to Pierre Kerfriden.

At least 19 recordsLinked to original sources

arXiv2025

A predictor-corrector scheme for approximating signed distances using finite element methods

In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction approach, involving first the solution of a linear diffusion-based prediction problem, followed by a nonlinear minimization-based correction problem associated with the Eikonal equation. The prediction step efficiently generates a suitable initial guess, significantly facilitating convergence of the nonlinear correction step. A key strength of our approach is its ability to handle complex interfaces and initial level set functions with arbitrary steep or flat regions, a notable challenge for existing techniques. Through several representative examples, including classical geometries and more complex shapes such as star domains and three-dimensional tori, we demonstrate the accuracy, efficiency, and robustness of the method, validating its broad applicability for reinitializing diverse level set functions.

cs.CE↗

arXiv2025

The Bayesian Finite Element Method in Inverse Problems: a Critical Comparison between Probabilistic Models for Discretization Error

When using the finite element method (FEM) in inverse problems, its discretization error can produce parameter estimates that are inaccurate and overconfident. The Bayesian finite element method (BFEM) provides a probabilistic model for the epistemic uncertainty due to discretization error. In this work, we apply BFEM to various inverse problems, and compare its performance to the random mesh finite element method (RM-FEM) and the statistical finite element method (statFEM), which serve as a frequentist and inference-based counterpart to BFEM. We find that by propagating this uncertainty to the posterior, BFEM can produce more accurate parameter estimates and prevent overconfidence, compared to FEM. Because the BFEM covariance operator is designed to leave uncertainty only in the appropriate space, orthogonal to the FEM basis, BFEM is able to outperform RM-FEM, which does not have such a structure to its covariance. Although inferring the discretization error via a model misspecification component is possible as well, as is done in statFEM, the feasibility of such an approach is contingent on the availability of sufficient data. We find that the BFEM is the most robust way to consistently propagate uncertainty due to discretization error to the posterior of a Bayesian inverse problem.

math.NA↗

arXiv2025

Effects of Interpolation Error and Bias on the Random Mesh Finite Element Method for Inverse Problems

Bayesian inverse problems are an important application for probabilistic solvers of partial differential equations: when fully resolving numerical error is computationally infeasible, probabilistic solvers can be used to consistently model the error and propagate it to the posterior. In this work, the performance of the random mesh finite element method (RM-FEM) is investigated in a Bayesian inverse setting. We show how interpolation error negatively affects the RM-FEM posterior, and how these negative effects can be diminished. In scenarios where FEM is biased for a quantity of interest, we find that RM-FEM struggles to accurately model this bias.

math.NA↗

arXiv2024

A time-discontinuous elasto-plasticity formalism to simulate instantaneous plastic flow bursts

Plastic flow is conventionally treated as continuous in finite element (FE) codes, whether in isotropic, anisotropic plasticity, or crystal plasticity. This approach, derived from continuum mechanics, contradicts the intermittent nature of plasticity at the elementary scale. Understanding crystal plasticity at micro-scale opens the door to new engineering applications, such as microscale machining. In this work, a new approach is proposed to account for the intermittence of plastic deformation while remaining within the framework of continuum mechanics. We introduce a material parameter, the plastic deformation threshold, denoted as $\Delta p_{min}$, corresponding to the plastic deformation carried by the minimal plastic deformation burst within the material. The incremental model is based on the traditional predictor-corrector algorithm to calculate the elastoplastic behavior of a material subjected to any external loading. The model is presented within the framework of small deformations for von Mises plasticity. To highlight the main features of the approach, the plastic strain increment is calculated using normality rule and consistency conditions, and is accepted only if it exceeds $\Delta p_{min}$. To achieve this, a time-discontinuous generalization of the Karush-Kuhn-Tucker (KKT) conditions is proposed. The simulations show that the introduction of the plastic threshold allows for the reproduction of the spatiotemporal intermittence of plastic flow, capturing the self-organization of plastic flow in complex loading scenarios within an FE model.

physics.comp-ph↗

arXiv2024

A multi-scale probabilistic methodology to predict high-cycle fatigue lifetime for alloys with process-induced pores

A multi-scale methodology is developed in conjunction with a probabilistic fatigue lifetime model for structures with pores whose exact distribution, i.e. geometries and locations, is unknown. The method takes into account uncertainty in fatigue lifetimes in structures due to defects at two scales: micro-scale heterogeneity & meso-scale pores. An element-wise probabilistic strain-life model with its criterion modified for taking into account multiaxial loading is developed for taking into account the effect of micro-scale defects on the lifetime. Meso-scale pores in the structure are taken into account via statistical modelling of the expected pore populations via a finite element method, based on tomographic scans of a small region of porous material used to make the structure. A previously implemented Neuber-type plastic correction algorithm is used for fast full-field approximation of the strain-life criterion around the statistically generated pore fields. The probability of failure of a porous structure is obtained via a weakest link assumption at the level of its constituent finite elements. The fatigue model can be identified via a maximum likelihood estimate on experimental fatigue data of structures containing different types of pore populations. The proposed method is tested on an existing high-cycle fatigue data-set of an aluminium alloy with two levels of porosity. The model requires lesser data for identification than traditional models that consider porous media as a homogeneous material, as the same base material is considered for the two grades of porous material. Numerical studies on synthetically generated data show that the method is capable of taking into account the statistical size effect in fatigue, and demonstrate that fatigue properties of subsurface porous material are lower than that of core porous material, which makes homogenisation of the method non-trivial.

cs.CE↗

arXiv2024

A Bayesian Approach to Modeling Finite Element Discretization Error

In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated based on observations from a finite element discretization. To avoid the computation of intractable integrals, a second, finer, discretization is introduced that is assumed sufficiently dense to represent the true solution field. A prior distribution is assumed over the fine discretization, which is then updated based on observations from the coarse discretization. This yields a posterior distribution with a mean that serves as an estimate of the solution, and a covariance that models the uncertainty associated with this estimate. Two particular choices of prior are investigated: a prior defined implicitly by assigning a white noise distribution to the right-hand side term, and a prior whose covariance function is equal to the Green's function of the partial differential equation. The former yields a posterior distribution with a mean close to the reference solution, but a covariance that contains little information regarding the finite element discretization error. The latter, on the other hand, yields posterior distribution with a mean equal to the coarse finite element solution, and a covariance with a close connection to the discretization error. For both choices of prior a contradiction arises, since the discretization error depends on the right-hand side term, but the posterior covariance does not. We demonstrate how, by rescaling the eigenvalues of the posterior covariance, this independence can be avoided.

math.NA↗

arXiv2024

A plastic correction algorithm for full-field elasto-plastic finite element simulations : critical assessment of predictive capabilities and improvement by machine learning

This paper introduces a new local plastic correction algorithm that is aimed at accelerating elasto-plastic finite element (FE) simulations for structural problems exhibiting localised plasticity (around e.g. notches, geometrical defects). The proposed method belongs to the category of generalised multi-axial Neuber-type methods, which process the results of an elastic prediction point-wise in order to calculate an approximation of the full elasto-plastic solution. The proposed algorithm relies on a rule of local proportionality, which, in the context of J2 plasticity, allows us to express the plastic plastic correction problem in terms of the amplitude of the full mechanical tensors only. This lightweight correction problem can be solved for numerically using a fully implicit time integrator that shares similarities with the radial return algorithm. The numerical capabilities of the proposed algorithm are demonstrated for a notched structure and a specimen containing a distribution of spherical pores, subjected to monotonic and cyclic loading. As a second point of innovation, we show that the proposed local plastic correction algorithm can be further accelerated by employing a simple meta-modelling strategy, with virtually no added errors. At last, we develop and investigate the merits of a deep-learning-based corrective layer designed to the approximation error of the plastic corrector. A convolutional architecture is used to analyse the neighbourhoods of material points and outputs a scalar correction to the point-wise Neuber-type predictions. This optional brick of the proposed plastic correction methodology relies on the availability of a set of full elasto-plastic finite element solutions to be used as training data-set.

cs.CE↗

arXiv2023

A graph-based probabilistic geometric deep learning framework with online enforcement of physical constraints to predict the criticality of defects in porous materials

Stress prediction in porous materials and structures is challenging due to the high computational cost associated with direct numerical simulations. Convolutional Neural Network (CNN) based architectures have recently been proposed as surrogates to approximate and extrapolate the solution of such multiscale simulations. These methodologies are usually limited to 2D problems due to the high computational cost of 3D voxel based CNNs. We propose a novel geometric learning approach based on a Graph Neural Network (GNN) that efficiently deals with three-dimensional problems by performing convolutions over 2D surfaces only. Following our previous developments using pixel-based CNN, we train the GNN to automatically add local fine-scale stress corrections to an inexpensively computed coarse stress prediction in the porous structure of interest. Our method is Bayesian and generates densities of stress fields, from which credible intervals may be extracted. As a second scientific contribution, we propose to improve the extrapolation ability of our network by deploying a strategy of online physics-based corrections. Specifically, we condition the posterior predictions of our probabilistic predictions to satisfy partial equilibrium at the microscale, at the inference stage. This is done using an Ensemble Kalman algorithm, to ensure tractability of the Bayesian conditioning operation. We show that this innovative methodology allows us to alleviate the effect of undesirable biases observed in the outputs of the uncorrected GNN, and improves the accuracy of the predictions in general.

cs.CE↗

arXiv2023

Local uncovering of unresolved physics in structural mechanics: seamless choice of modelling resolution using a CutFEM level-set approach

In this paper, we present a robust and efficient unfitted concurrent multiscale method for continuum-continuum coupling, based on the Cut Finite Element Method (CutFEM). The computational domain is defined using approximate signed distance functions over a fixed background mesh and is decomposed into microscale and macroscale regions using a novel zooming technique. The zoom interface is represented by a signed distance function intersecting the computational mesh arbitrarily. The mesh inside the zoomed region is hierarchically refined to resolve the microstructure. In the examples considered, the microstructure may contain voids and hard inclusions, and its geometry is defined implicitly by a signed distance function interpolated over the refined mesh. Our zooming technique allows the zoom interface to intersect the microstructure interface in an arbitrary fashion, enabling greater flexibility and accuracy in the modelling of complex geometries. The micro and macro regions are coupled using Nitsche's method, ensuring stability and accuracy in the solution. Ghost penalty terms are utilised to ensure the stability of cut elements along the zoom interface and the microstructure interface. To demonstrate the effectiveness of our framework, we apply it for modelling several heterogeneous structures with both linear elasticity and plasticity constitutive behaviours. The results show that our framework is robust and efficient, producing accurate and reliable solutions for such problems. Our proposed method provides a highly versatile and effective approach to multiscale modeling of structures with complex microstructures, and has the potential to be extended to problems requiring seamless moving of zooming region(s) during the simulation, such as damage growth and fracture propagation.

cs.CE↗

arXiv2022

Concurrent multiscale analysis without meshing: Microscale representation with CutFEM and micro/macro model blending

In this paper, we develop a novel unfitted multiscale framework that combines two separate scales represented by only one single computational mesh. Our framework relies on a mixed zooming technique where we zoom at regions of interest to capture microscale properties and then mix the micro and macroscale properties in a transition region. Furthermore, we use homogenization techniques to derive macro model material properties. The microscale features are discretized using CutFEM. The transition region between the micro and macroscale is represented by a smooth blending function. To address the issues with ill-conditioning of the multiscale system matrix due to the arbitrary intersections in cut elements and the transition region, we add stabilization terms acting on the jumps of the normal gradient (ghost-penalty stabilization). We show that our multiscale framework is stable and is capable to reproduce mechanical responses for heterogeneous structures in a mesh-independent manner. The efficiency of our methodology is exemplified by 2D and 3D numerical simulations of linear elasticity problems.

cs.CE↗

arXiv2022

A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features

Multiscale computational modelling is challenging due to the high computational cost of direct numerical simulation by finite elements. To address this issue, concurrent multiscale methods use the solution of cheaper macroscale surrogates as boundary conditions to microscale sliding windows. The microscale problems remain a numerically challenging operation both in terms of implementation and cost. In this work we propose to replace the local microscale solution by an Encoder-Decoder Convolutional Neural Network that will generate fine-scale stress corrections to coarse predictions around unresolved microscale features, without prior parametrisation of local microscale problems. We deploy a Bayesian approach providing credible intervals to evaluate the uncertainty of the predictions, which is then used to investigate the merits of a selective learning framework. We will demonstrate the capability of the approach to predict equivalent stress fields in porous structures using linearised and finite strain elasticity theories.

cs.CE↗

arXiv2021

Addressing materials' microstructure diversity using transfer learning

Materials' microstructures are signatures of their alloying composition and processing history. Therefore, microstructures exist in a wide variety. As materials become increasingly complex to comply with engineering demands, advanced computer vision (CV) approaches such as deep learning (DL) inevitably gain relevance for quantifying microstrucutures' constituents from micrographs. While DL can outperform classical CV techniques for many tasks, shortcomings are poor data efficiency and generalizability across datasets. This is inherently in conflict with the expense associated with annotating materials data through experts and extensive materials diversity. To tackle poor domain generalizability and the lack of labeled data simultaneously, we propose to apply a sub-class of transfer learning methods called unsupervised domain adaptation (UDA). These algorithms address the task of finding domain-invariant features when supplied with annotated source data and unannotated target data, such that performance on the latter distribution is optimized despite the absence of annotations. Exemplarily, this study is conducted on a lath-shaped bainite segmentation task in complex phase steel micrographs. Here, the domains to bridge are selected to be different metallographic specimen preparations (surface etchings) and distinct imaging modalities. We show that a state-of-the-art UDA approach surpasses the naïve application of source domain trained models on the target domain (generalization baseline) to a large extent. This holds true independent of the domain shift, despite using little data, and even when the baseline models were pre-trained or employed data augmentation. Through UDA, mIoU was improved over generalization baselines from 82.2%, 61.0%, 49.7% to 84.7%, 67.3%, 73.3% on three target datasets, respectively. This underlines this techniques' potential to cope with materials variance.

cond-mat.mtrl-sci↗

arXiv2018

A CutFEM method for two-phase flow problems

In this article, we present a cut finite element method for two-phase Navier-Stokes flows. The main feature of the method is the formulation of a unified continuous interior penalty stabilisation approach for, on the one hand, stabilising advection and the pressure-velocity coupling and, on the other hand, stabilising the cut region. The accuracy of the algorithm is enhanced by the development of extended fictitious domains to guarantee a well defined velocity from previous time steps in the current geometry. Finally, the robustness of the moving-interface algorithm is further improved by the introduction of a curvature smoothing technique that reduces spurious velocities. The algorithm is shown to perform remarkably well for low capillary number flows, and is a first step towards flexible and robust CutFEM algorithms for the simulation of microfluidic devices.

math.NA↗

arXiv2018

A CutFEM method for Stefan-Signorini problems with application in pulsed laser ablation

In this article, we develop a cut finite element method for one-phase Stefan problems, with applications in laser manufacturing. The geometry of the workpiece is represented implicitly via a level set function. Material above the melting/vaporisation temperature is represented by a fictitious gas phase. The moving interface between the workpiece and the fictitious gas phase may cut arbitrarily through the elements of the finite element mesh, which remains fixed throughout the simulation, thereby circumventing the need for cumbersome re-meshing operations. The primal/dual formulation of the linear one-phase Stefan problem is recast into a primal non-linear formulation using a Nitsche-type approach, which avoids the difficulty of constructing inf-sup stable primal/dual pairs. Through the careful derivation of stabilisation terms, we show that the proposed Stefan-Signorini-Nitsche CutFEM method remains stable independently of the cut location. In addition, we obtain optimal convergence with respect to space and time refinement. Several 2D and 3D examples are proposed, highlighting the robustness and flexibility of the algorithm, together with its relevance to the field of micro-manufacturing.

math.NA↗

arXiv2018

Isogeometric analysis with local adaptivity based on a posterior error estimation for elastodynamics

This paper presents a novel methodology of local adaptivity for the frequency-domain analysis of the vibrations of Reissner-Mindlin plates. The adaptive discretization is based on the recently developed Geometry Independent Field approximaTion (GIFT) framework, which may be seen as a generalisation of the Iso-Geometric Analysis (IGA). Within the GIFT framework, we describe the geometry of the structure exactly with NURBS (Non-Uniform Rational B-Splines), whilst independently employing Polynomial splines over Hierarchical T-meshes (PHT)-splines to represent the solution field. The proposed strategy of local adaptivity, wherein a posteriori error estimators are computed based on inexpensive hierarchical $h-$refinement, aims to control the discretisation error within a frequency band. The approach sweeps from lower to higher frequencies, refining the mesh appropriately so that each of the free vibration mode within the targeted frequency band is sufficiently resolved. Through several numerical examples, we show that the GIFT framework is a powerful and versatile tool to perform local adaptivity in structural dynamics. We also show that the proposed adaptive local $h-$refinement scheme allows us to achieve significantly faster convergence rates than when using a uniform $h-$refinement.

math.NA↗

arXiv2017

A stable and optimally convergent LaTIn-Cut Finite Element Method for multiple unilateral contact problems

In this paper, we propose a novel unfitted finite element method for the simulation of multiple body contact. The computational mesh is generated independently of the geometry of the interacting solids, which can be arbitrarily complex. The key novelty of the approach is the combination of elements of the CutFEM technology, namely the enrichment of the solution field via the definition of overlapping fictitious domains with a dedicated penalty-type regularisation of discrete operators, and the LaTIn hybrid-mixed formulation of complex interface conditions. Furthermore, the novel P1-P1 discretisation scheme that we propose for the unfitted LaTIn solver is shown to be stable, robust and optimally convergent with mesh refinement. Finally, the paper introduces a high-performance 3D level-set/CutFEM framework for the versatile and robust solution of contact problems involving multiple bodies of complex geometries, with more than two bodies interacting at a single point.

math.NA↗

arXiv2016

CAD model simplification error estimation for electrostatics problems

Simplifying the geometry of a CAD model using defeaturing techniques enables more efficient discretisation and subsequent simulation for engineering analysis problems. Understanding the effect this simplification has on the solution helps to decide whether the simplification is suitable for a specific simulation problem. It can also help to understand the functional effect of a geometry feature. The effect of the simplification is quantified by a user-defined quantity of interest which is assumed to be (approximately) linear in the solution. A bound on the difference between the quantity of interest of the original and simplified solutions based on the energy norm is derived. The approach is presented in the context of electrostatics problems, but can be applied in general to a range of elliptic partial differential equations. Numerical results on the efficiency of the bound are provided for electrostatics problems with simplifications involving changes inside the problem domain as well as changes to the boundaries.

math.NA↗

arXiv2014

Certification of projection-based reduced order modelling in computational homogenisation by the Constitutive Relation Error

In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model.

math.NA↗