Gradient Weighted Embedded Error Estimator for Mesh Adaptation

Error estimation and mesh adaptivity in incompressible viscous flows using a residual power approach

Advances in Adaptive Computational Methods in Mechanics

Unstructured anisotropic mesh adaptation is known to be an efficient way to control discretization errors in computational fluid dynamics simulations. Method verification is required to provide the confidence for routine use in production analysis. The current work aims at verification of anisotropic mesh adaptation for Reynolds-averaged Navier–Stokes simulations over the ONERA M6 wing. The present verification study is performed using four different flow solvers, three different implementations of the metric field, and three mesh mechanics packages. Two of the flow solvers use stabilized finite element discretizations (FUN3D-SFE and general geometry Navier–Stokes), one uses finite volume discretization (FUN3D-FV), and the last one uses mixed finite volume and finite element discretizations (Wolf). The mesh adaptation is based on an error estimator that aims to control the quadratic error term in the linear interpolation of the Mach number. Two sets of adaptations were performed; the first one controls the interpolation error in the norm, and the second one controls the interpolation error in the norm. Convergence studies were performed on the forces and the pitching moment using all four solvers, and the results are compared with previously verified convergence studies on fixed (nonadapted) meshes. Both the forces and pitching moment on adapted meshes are found to be converging to the fine-mesh values faster than those on fixed meshes. In addition to forces and moments, the convergence of surface pressure and skin-friction coefficients at various measurement locations on the wing are also presented. Adapted-mesh surface pressure distributions agree with the fine fixed mesh pressure distributions. Adapted-mesh skin-friction distributions contain high-frequency noise with mean values approaching the fixed mesh pressure skin-friction distributions.

A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials

Mesh quality strongly affects the solution accuracy in electromagnetic finite-element analysis. Hence, the realization of adequate mesh generation becomes a very important task. Several adaptive meshing methods for automatic adjustments of the mesh density in accordance with the shape and complexity of the analyzed problem have been proposed. However, the most of them are not enough robust, some are quite laborious and could not be universally used for adaptive meshing of complex analysis models. In this paper, a new adaptive mesh refinement method based on magnetic field conservation at the border between finite elements is proposed. The proposed error estimation method provides easy mesh refinements, and generates smaller element within regions with large curvature of the magnetic flux lines. The proposed adaptive mesh refinement method based on non-conforming edge finite elements, which could avoid generation of flat or ill-shaped elements, was applied to a simple magnetostatic permanent magnet model. To confirm the validity and accuracy, the obtained results were compared with those obtained by means of the Zienkiewicz-Zhu (ZZ) error estimator. The results show that the computational error using the proposed method was reduced down to 1% compared with that of the ZZ method, which yields error of 8.6%, for the same model.

This is the first of two articles concerning error estimation and adaptive refinement techniques applied to convective heat transfer problems. This study presents the detailed development of the proposed error estimator. The error estimator takes into account the coupling effects of the dependent variables (e.g., velocity and temperature) on the discretization error and, consequently, on the adaptive meshes. An averaging procedure is also proposed as a substitute for the smoothing/recovery procedure used to evaluate the gradients. The implementation of the averaging technique is simple and cost effective. Numerical experiments demonstrate that the proposed methodology results in a dramatic reduction in computational costs.

The intensity variation in a scanning electron microscope is a complex function of sample topography and composition. Measurement accuracy is improved when an explicit accounting for the relationship between signal and measurand is made. Because the determinants of the signal are many, the theoretical understanding usually takes the form of a simulator. For samples with nonconducting regions that charge, one phase of the simulation is finite element analysis to compute the electric field. The size of the finite element mesh, and consequently computation time, can be reduced through the use of adaptive mesh refinement. We present a new a posteriori local error estimator and adaptive mesh refinement algorithm for the scanning electron microscope simulation. This error estimate is designed to minimize the error in the electron trajectories, rather than the energy norm of the error that traditional error estimators minimize. Using a test problem with a known exact solution, we show that the adaptive mesh can achieve the same error in electron trajectories as a carefully designed hand-graded mesh while using 3.5 times fewer vertices and 2.25 times less computation time.

Numerical simulations and optimisation methods, such as mesh adaptation, rely on the accurate and inexpensive use of error estimation methods. Adjoint-based error estimation is the most accurate method, and generally the most costly. A strong contributor to this cost is the need to compute a higher resolution adjoint solution. Here, it is proposed to use super-resolution neural networks to super-resolve a fine adjoint solution from a lower-cost coarse adjoint solution: a superAdjoint. The method is compared to reference error estimators on an unsteady Burgers’ equation using the method of manufactured solutions. Two forms of the superAdjoint were implemented, a twice and a four times refining super-resolution neural network. These were used to demonstrate both the computational cost reduction and the potential for the reduction of the storage footprint of the primal problem. The first, referred to as 2×CNN, was able to reconstruct the spatially enriched adjoint solution, thus providing a robust and inexpensive local output error. The second, the 4×CNN, was able to demonstrate the reconstruction ability of super-resolution neural networks for higher upscaling factors. This was leveraged in order to subsample the primal solution in space, thus reducing substantially the storage footprint of the discrete primal solution. Both superAdjoints could achieve the desired level of accuracy when compared to the reconstruction of the refined adjoint-based error estimate. Moreover, superAdjoint was shown to be able to generalise to a new QoI that was untrained for. This gives great confidence in the use of super-resolution neural networks for the reduction of both computational cost and storage requirements of adjoint-based error estimation, and goal-oriented mesh adaptation.

A novel high-order hybrid staggered discontinuous Galerkin method for general meshes is proposed to solve general second order elliptic problems. Our new formulation is related to standard staggered discontinuous Galerkin method, but more flexible and cost effective: rough grids are allowed and the size of the final system is remarkably reduced thanks to the partial hybridization. Optimal convergence estimates for both the scalar and vector variables are developed. Moreover, superconvergent results with respect to discrete $$H^1$$ norm and $$L^2$$ norm for the scalar variable are proved and negative norm error estimates for both the scalar and vector variables are also developed. On the other hand, mesh adaptation is particularly simple since hanging nodes are allowed, which makes the proposed method well suited for adaptive mesh refinement. Therefore, we design a residual type a posteriori error estimator, and the reliability and local efficiency of the error estimator are proved. Numerical experiments confirm the theoretical findings.

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling. This is done in three steps. First, a new ap riorierror estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal mesh is then derive. 3D examples of steady flows around supersonic and transsonic aircrafts are presented to validate the current approach and to demonstrate its efficiency.

We present a finite volume adaptive mesh refinement method for solid-liquid phase change problems with convection. The refinement criterion consisted of three different error estimators for the solid-liquid interface, the flow field, and the temperature field respectively. For the solid-liquid interface, the cells undergoing phase change were refined based on the maximum difference in the liquid fraction over the cell faces. For the flow field and the temperature field, an error indicator was used based on the cell residual method. To maintain a high parallelization efficiency, a dynamic load balancing procedure was used. The adaptive mesh refinement strategy was verified through three different test cases, these are the gallium melting in both 2D and 3D cavities, and the molten salt reactor freeze valve. For all three cases, very good agreement was obtained between the adaptive mesh results and the reference solutions. In addition, more accurate results were obtained with the adaptive meshes compared to static meshes with a similar amount of mesh cells. This illustrated the potential of the current approach for developing computationally efficient numerical methods for solid-liquid phase change problems.

FDOT: A Fast, memory-efficient and automated approach for Discrete adjoint sensitivity analysis using the Operator overloading Technique

This thesis investigates the integration of machine learning techniques into computational fluid dynamics (CFD) to improve simulation accuracy and computational performance. It proposes the use of the Gaussian Mixture Model (GMM) and unsupervised clustering as novel tools for mesh adaptation and flow feature detection. The methodology applies GMM to segment the computational domain into regions with distinct physical behaviour, viscous/turbulent and inviscid/irrotational. This clustering-based p-adaptation approach removes the need for predefined thresholds, enabling a fully automated and localised adjustment of the solution's polynomial degree. As a result, mesh refinement becomes more efficient and precise. A series of numerical experiments, including simulations of flow past a cylinder at varying Reynolds numbers, demonstrate the ability of the method to balance computational cost and solution fidelity. The framework is also tested on real-world engineering cases, such as wind turbine wake simulations, confirming its scalability and robustness. The results show substantial gains in efficiency and adaptability, establishing this approach as a promising solution for a wide range of CFD applications. By embedding machine learning into high-order solvers, this work advances the state-of-the-art and opens new directions for adaptive simulation strategies. It also lays the groundwork for future research in error-driven refinement, 3D turbulence modeling, and the broader application of AI in fluid mechanics. Keywords: Computational fluid dynamics, machine learning, Gaussian Mixture Model, p-adaptation, flow feature detection, mesh refinement, high-order methods. RESUMEN Esta tesis examina la integración de técnicas de aprendizaje automático en la dinámica de fluidos computacional (CFD) para mejorar la precisión de las simulaciones y reducir el coste computacional. Se propone el uso del Modelo de Mezclas Gaussianas (GMM) y métodos de agrupamiento no supervisado para desarrollar nuevas estrategias de adaptación de malla y detección de regiones del flujo sin recurrir a umbrales predefinidos. La metodología aplica GMM para segmentar automáticamente el dominio en regiones con diferentes características físicas, como zonas viscosas/turbulentas y no viscosas/irrotacionales. Esta clasificación guía un proceso de p-adaptación, ajustando el orden polinomial de la solución numérica de forma localizada, lo que permite un refinamiento automático, preciso y eficiente. Diversos experimentos numéricos, incluyendo simulaciones del flujo alrededor de un cilindro para distintos números de Reynolds, demuestran que el enfoque propuesto equilibra de manera eficaz el coste computacional y la calidad de la solución. Además, su aplicabilidad se valida en casos reales como las estelas de aerogeneradores, evidenciando la escalabilidad del método. Los resultados confirman que estas técnicas mejoran la eficiencia y adaptabilidad de los solucionadores de alto orden, lo que las convierte en herramientas prometedoras para aplicaciones complejas en CFD. La tesis contribuye así al estado del arte al integrar aprendizaje automático con métodos numéricos avanzados, y sienta las bases para investigaciones futuras sobre refinamiento adaptativo basado en errores, simulaciones tridimensionales de turbulencia y uso extendido de IA en mecánica de fluidos. Palabras clave: dinámica de fluidos computacional, aprendizaje automático, Modelo de Mezclas Gaussianas, p-adaptación, detección de características, refinamiento de mallas, métodos de alto orden. RÉSUMÉ Cette thèse explore l’intégration des techniques d’apprentissage automatique dans la dynamique des fluides numérique (CFD) afin d’améliorer la précision des simulations et la surcharge de calcul. En utilisant le modèle de mélanges gaussiens (GMM) et des méthodes de regroupement non supervisé, cette recherche introduit des approches innovantes pour l’adaptation du maillage et la détection des structures d’écoulement. La méthodologie se concentre sur l’application du GMM pour partitionner les champs d’écoulement en régions distinctes, régions distinctes, différenciant les zones visqueuses/turbulentes des zones inviscides/irrotationnelles. Cette technique de p-adaptation basée sur le regroupement élimine le besoin de seuils prédéfinis, permettant ainsi un raffinement du maillage automatisé, efficace et précis. De nombreuses expériences numériques sur divers cas tests, notamment l’écoulement autour d’un cylindre à différents nombres de Reynolds, démontrent la capacité de la méthode proposée à équilibrer le coût computationnel et la fidélité de la solution. Des applications en ingénierie, telles que la simulation du sillage d’une éolienne, illustrent davantage l’évolutivité des méthodes proposées. Les résultats montrent des améliorations significatives en termes du coût computationnel et d’adaptabilité, rendant ces techniques particulièrement adaptées à un large éventail d’applications en CFD. Ce travail contribue à l’état de l’art en intégrant de manière fluide l’apprentissage automatique dans les solveurs CFD d’ordre élevé, ouvrant ainsi la voie à de futures avancées dans les méthodes de simulation adaptative. Les méthodologies proposées établissent une base pour l’exploration du raffinement basé sur l’erreur, la modélisation tridimensionnelle de la turbulence et l’adoption plus large de l’apprentissage automatique dans la recherche en dynamique des fluides.

In this work, we present an abstract error analysis framework for the approximation of linear partial differential equation (PDE) problems in weak formulation. We consider approximation methods in fully discrete formulation, where the discrete and continuous spaces are possibly not embedded in a common space. A proper notion of consistency is designed, and, under a classical inf-sup condition, it is shown to bound the approximation error. This error estimate result is in the spirit of Strang's first and second lemmas, but applicable in situations not covered by these lemmas (because of a fully discrete approximation space). An improved estimate is also established in a weaker norm, using the Aubin--Nitsche trick. We then apply these abstract estimates to an anisotropic heterogeneous diffusion model and two classical families of schemes for this model: Virtual Element and Finite Volume methods. For each of these methods, we show that the abstract results yield new error estimates with a precise and mild dependency on the local anisotropy ratio. A key intermediate step to derive such estimates for Virtual Element Methods is proving optimal approximation properties of the oblique elliptic projector in weighted Sobolev seminorms. This is a result whose interest goes beyond the specific model and methods considered here. We also obtain, to our knowledge, the first clear notion of consistency for Finite Volume methods, which leads to a generic error estimate involving the fluxes and valid for a wide range of Finite Volume schemes. An important application is the first error estimate for Multi-Point Flux Approximation L and G methods. In the appendix, not included in the published version of this work, we show that classical estimates for discontinuous Galerkin methods can be obtained with simplified arguments using the abstract framework.

An adaptive mesh refinement algorithm has been developed for non-linear computations in geomechanics, based on a smoothed stress–strain finite element formulation. This uses estimates of error in the incremental shear strain invariant to guide the regeneration of unstructured meshes at regular intervals during loading. Following each mesh-update, no re-analysis of previous increments with the new mesh is necessary. Algorithm performance has been investigated by analysing a passive earth pressure problem using a linear elastic-perfectly plastic Mohr–Coulomb soil model. Perfectly drained behaviour has been considered, as have partially drained situations using hydromechanical coupling, while undrained behaviour has been approximated using time steps close to zero. In all cases, mesh adaptivity has been successful in capturing regions of high strain gradient. The results have been compared with analytical solutions. Accurate computations of limit load and shear band orientation have been obtained for a wide range of material dilation angles. Copyright © 2000 John Wiley & Sons, Ltd.