Academic literature on the topic 'A posteriori error estimate via stress reconstruction'

AbstractWe consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, {H(\mathrm{div)}}-conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold–Falk–Winther finite element spaces. We distinguish two stress reconstructions: one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with an analytical solution. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky–Mises and an isotropic damage model.

We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction–diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction–diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.

Purpose This study aims to propose a general and efficient adaptive strategy with local mesh refinement for two-dimensional (2D) finite element (FE) analysis based on the element energy projection (EEP) technique. Design/methodology/approach In view of the inflexibility of the existing global dimension-by-dimension (D-by-D) recovery method via EEP technique, in which displacements are recovered through element strips, an improved element D-by-D recovery strategy was proposed, which enables the EEP recovery of super-convergent displacements to be implemented mostly on a single element. Accordingly, a posteriori error estimate in maximum norm was established and an EEP-based adaptive FE strategy of h-version with local mesh refinement was developed. Findings Representative numerical examples, including stress concentration and singularity problems, were analyzed; the results of which show that the adaptively generated meshes reasonably reflect the local difficulties inherent in the physical problems and the proposed adaptive analysis can produce FE displacement solutions satisfying the user-specified tolerances in maximum norm with an almost optimal adaptive convergence rate. Originality/value The proposed element D-by-D recovery method is a more efficient and flexible displacement recovery method, which is implemented mostly on a single element. The EEP-based adaptive FE analysis can produce displacement solutions satisfying the specified tolerances in maximum norm with an almost optimal convergence rate and thus can be expected to apply to other 2D problems.

Summary 4D acoustic imaging via an array of 32 sources / 32 receivers is used to monitor hydraulic fracture propagating in a 250 mm cubic specimen under a true-triaxial state of stress. We present a method based on the arrivals of diffracted waves to reconstruct the fracture geometry (and fluid front when distinct from the fracture front). Using Bayesian model selection, we rank different possible fracture geometries (radial, elliptical, tilted or not) and estimate model error. The imaging is repeated every 4 seconds and provide a quantitative measurement of the growth of these low velocity fractures. We test the proposed method on two experiments performed in two different rocks (marble and gabbro) under experimental conditions characteristic respectively of the fluid lag-viscosity (marble) and toughness (gabbro) dominated hydraulic fracture propagation regimes. In both experiments, about 150 to 200 source-receiver combinations exhibit clear diffracted wave arrivals. The results of the inversion indicate a radial geometry evolving slightly into an ellipse towards the end of the experiment when the fractures feel the specimen boundaries. The estimated modelling error with all models is of the order of the wave arrival picking error. Posterior estimates indicate an uncertainty of the order of a millimeter on the fracture front location for a given acquisition sequence. The reconstructed fracture evolution from diffracted waves is shown to be consistent with the analysis of 90○ incidence transmitted waves across the growing fracture.

Les équipes d’ingénierie ont souvent recours aux simulations numériques par éléments finis pour étudier et analyser le comportement des ouvrages hydrauliques de grande dimension. Pour les ouvrages en béton, les modèles doivent être en mesure de prendre en compte le comportement non-linéaire des discontinuités aux diverses zones d’interfaces localisées en fondation, dans le corps du barrage ou à l’interface entre la structure et la fondation. Il faut non seulement être capable de représenter le comportement mécanique non-linéaire de ces interfaces (rupture, glissement, contact), mais également de prendre en compte l’écoulement hydraulique à travers ces ouvertures.Dans le cadre de cette thèse, nous nous focalisons d’abord sur la question du comportement des interfaces, que nous abordons à travers le modèle des zones cohésives (CZM). Ce dernier, introduit dans divers codes de calcul par éléments finis (avec éléments finis de joint), est une approche pertinente pour décrire la physique des problèmes de fissuration et de frottement au niveau de discontinuités géométriques. Bien que le CZM a été initialement introduit pour prendre en compte que le phénomène de rupture, nous montrons dans cette thèse que son utilisation peut être étendue aux problèmes de glissement en s'appuyant sur le formalisme élasto-plastique éventuellement couplé à l'endommagement. En outre, des lois de comportement hydromécaniques non-linéaires peuvent être introduites pour modéliser la notion d’ouverture de fissure et le couplage avec les lois d’écoulement fluide. Au niveau mécanique, nous travaillons dans le cadre des matériaux standard généralisés (SGM), qui fournit une classe de modèles qui satisfont d’une manière automatique des principes de la thermodynamique tout en possédant des bonnes propriétés mathématiques utiles pour la modélisation numérique robuste. Nous adaptons le formalisme SGM volumique à la description des zones d'interface. Dans cette première partie de la thèse, nous présentons nos développements faites dans l'hypothèse de SGM adaptée aux CZM, capable de reproduire les phénomènes physiques observés expérimentalement : rupture, frottement, adhésion.En pratique, les non-linéarités du comportement des zones d’interface sont dominées par la présence de contact, ce qui engendre des difficultés numériques importantes pour la convergence des calculs par élément fini. Le développement de méthodes numériques efficaces pour le problème de contact est donc une étape clé pour atteindre l’objectif de simulateurs numériques industriels robustes. Récemment, l’utilisation de techniques d’imposition faible des conditions de contact à la Nitsche a été proposée comme moyen pour réduire la complexité numérique. Cette technique présente plusieurs avantages, dont les plus importants pour nos travaux sont: 1) possibilité de gérer une vaste gamme de conditions (glissement avec ou sans frottement, non interpénétration, etc); 2) la technique se prête à une analyse d'erreur a posteriori rigoureuse. Ce schéma basé sur les conditions d’interface faibles représente le point de départ pour l’estimation d’erreur a posteriori par reconstruction équilibrée de la contrainte. Cette analyse est utilisée pour estimer les différentes composantes d’erreur (p.e., spatiale, non-linéaire), et pour mettre en place un algorithme de résolution adaptatif, ainsi que des critères d’arrêt pour les solveurs itératifs et le réglage automatique d’éventuels paramètres numériques.L'objectif principal de la thèse est donc de rendre robuste la simulation numérique par éléments finis des ouvrages présentant des discontinuités géométriques. On aborde cette question sous angle double : d’un côté on revisite les méthodes existantes de représentation de fissuration en travaillant sur la loi de comportement mécanique pour les joints ; de l’autre on introduit une nouvelle méthode a posteriori pour traiter le problème de contact et propose son adaptation pour les modèles d’interfaces génériques
Engineering teams often use finite element numerical simulations for the design, study and analysis of the behavior of large hydraulic structures. For concrete structures, models of increasing complexity must be able to take into account the nonlinear behavior of discontinuities at the various interfaces located in the foundation, in the body of the dam or at the interface between structure and foundation. Besides representing the nonlinear mechanical behavior of these interfaces (rupture, sliding, contact), one should also be able to take into account the hydraulic flow through these openings.In this thesis, we first focus on the topic of interface behavior modeling, which we address through the Cohesive Zone Model (CZM). This model was introduced in various finite element codes (with the joint elements), and it is a relevant approach to describe the physics of cracking and friction problems at the geometrical discontinuities level. Although initially the CZM was introduced to take into account the phenomenon of rupture, we show in this thesis that it can be extended to sliding problems by possibly relying on the elasto-plastic formalism coupled to the damage. In addition, nonlinear hydro-mechanical constitutive relations can be introduced to model the notion of crack opening and the coupling with the laws of fluid flow. At the mechanical level, we work in the Standard Generalized Materials (SGM) framework, which provides a class of models automatically satisfying some thermodynamical principles, while having good mathematical and numerical properties that are useful for robust numerical modeling. We adapt the formalism of volumetric SGM to the interface zones description. In this first part of the thesis, we present our developpements under the hypothesis of SGM adapted to CZM, capable of reproducing the physical phenomena observed experimentally: rupture, friction, adhesion.In practice, nonlinearities of behavior of interface zones are dominated by the presence of contact, which generates significant numerical difficulties for the convergence of finite element computations. The development of efficient numerical methods for the contact problem is thus a key stage for achieving the goal of robust industrial numerical simulators. Recently, the weak enforcement of contact conditions à la Nitsche has been proposed as a mean to reduce numerical complexity. This technique displays several advantages, among which the most important for our work are: 1) it can handle a wide range of conditions (slip with or without friction, no interpenetration, etc.); 2) it lends itself for a rigorous a posteriori error analysis. This scheme based on the weak contact conditions represents in this work the starting point for the a posteriori error estimation via equilibrated stress reconstruction. This analysis is then used to estimate the different error components (e.g., spatial, nonlinear), and to develop an adaptive resolution algorithm, as well as stopping criteria for iterative solvers and the automatic tuning of possible numerical parameters.The main goal of this thesis is thus to make the finite element numerical simulation of structures with geometrical discontinuities robust. We address this question from two angles: on one side, we revisit the existing methods for the crack representation working on the mechanical constitutive relation for joints; on the other, we introduce a new a posteriori method for the contact problem and we propose its adaptation for the generic interface models