(English) Building (yet other) bridges between polytopal methods
Désolé, cet article est seulement disponible en Anglais Américain.
Catégorie : Nouvelles
Shape optimisation using Lipschitz functions
Philip Herbert: Thursday, 06th March 2025 at 10:30 Abstract: In this talk, we discuss a novel method in PDE constrained shape optimisation. While it is known that many shape optimisation problems have a solution, finding the solution, or an approximation of the solution, may prove non-trivial. A typical approach to minimisation is to use a first order method; this raises questions when handling shapes – what is a shape derivative, where does it live? It happens to be convenient to define the derivatives as linear functionals on $W^{1,\infty}$. We present an analysis of this in a discrete setting along with the existence of directions of steepest descent. Several numerical experiments will be considered and extensions discussed.
(English) Shape optimisation using Lipschitz functions
Désolé, cet article est seulement disponible en Anglais Américain.
Perspectives in structure-preserving numerical schemes
Martin Licht: Thursday, 06th February 2025 at 10:00 Abstract: Structure-preserving numerical methods have had a transformative impact on the numerical analysis of partial differential equations, reproducing the fundamental mathematical structures of numerous partial differential equations exactly at the numerical level. This talk gives an introduction and overview of structure-preserving finite element methods via “finite element exterior calculus” (FEEC) and explores some new directions in the field. FEEC is a comprehensive framework for mixed finite element methods, at the heart of which are finite element differential complexes. This comprehensive theory of mixed finite element methods connects geometry, topology, and classical numerical analysis. We highlight recent research developments, including mixed boundary conditions in FEEC and finite element methods over manifolds, and discuss some future directions in numerical electromagnetism, elasticity, and fluid dynamics.
(English) Perspectives in structure-preserving numerical schemes
Désolé, cet article est seulement disponible en Anglais Américain.
Discontinuous Galerkin finite element methods for the control constrained Dirichlet control problem governed by the diffusion equation.
Divay Garg: Thursday, 30th January 2025 at 10:30 Abstract: We utilize a unified discontinuous Galerkin approach to approximate the control constrained Dirichlet boundary optimal control problem using finite element method over simplicial triangulation. The continuous optimality system obtained from this method simplifies the control constraints into a simplified Signorini type problem, which is then coupled with boundary value problems for the state and co-state variables. The symmetric property of the discrete bilinear forms is required in order to derive the discrete optimality system. The main focus is to derive residual based a posteriori error estimates in the energy norm, where we address the reliability and efficiency of the proposed a posteriori error estimator. The suitable construction of auxiliary problems, continuous and discrete Lagrange multipliers, and intermediate operators are crucial in developing a posteriori error analysis. We have also established optimal a priori error estimates in the energy norm for all the optimal variables (state, co-state, and control) under the appropriate regularity assumptions. Theoretical findings are confirmed and illustrated through numerical results on both uniform and adaptive meshes.
(English) Discontinuous Galerkin finite element methods for the control constrained Dirichlet control problem governed by the diffusion equation.
Désolé, cet article est seulement disponible en Anglais Américain.
Adaptive homotopy continuation for relative permeability models in reservoir simulation
Peter Moritz von Schultzendorff: Monday, 13th January 2025 at 10:30 Accurate modeling of physical processes requires an appropriate selection of constitutive laws. In physics-based reservoir simulation, constitutive laws, e.g., relative permeabilities are often chosen to be, mathematically speaking, simple functions, not necessarily adhering to physics. The paradigm of hybrid modeling allows the integration of machine learned (ML) constitutive laws. Trained on lab, field, and fine-scale simulation data, ML models represent the underlying physics with high fidelity.Strong nonlinearities in classic (i.e., non-ML) relative permeability have been identified as one of the main sources for convergence issues of nonlinear solvers in reservoir simulation. This issue grows in severance for ML relative permeability models, as their high fidelity to real-world data compromises the mathematically desirable properties of simpler models.In this work, we employ the homotopy continuation (HC) method to recover nonlinear solver robustness for classic relative permeability models. The HC method improves nonlinear solver robustness by first solving a problem with simpler relative permeabilities and then iteratively traversing a solution curve towards the original, more complex problem. To efficiently trace the solution curve, we leverage a posteriori error estimates to design an adaptive HC algorithm that minimizes the total number of solver iterations.We show the current status of our work, both on the theoretical and implementation side, and give an outlook into the application to ML relative permeabilities.
(English) Adaptive homotopy continuation for relative permeability models in reservoir simulation
Désolé, cet article est seulement disponible en Anglais Américain.
Analyse du problème des oscillations parasites des méthodes de Volumes Finis pour des écoulements à faible nombre de Mach en mécanique des fluides.
Ibtissem Lannabi: Thursday, 23rd January 2025 at 11:00 Ce travail de recherche porte sur la simulation numérique d’écoulements de fluides à faibles nombres de Mach, modélisés par le système d’Euler compressible. Les solveurs fréquemment utilisés pour discrétiser ce modèle sont des solveurs de type Godunov. Cependant, ces solveurs se comportent très mal à bas nombre de Mach en termes d’efficacité et de précision.En effet, lorsque le nombre de Mach tend vers zéro, les ondes matérielles et acoustiques se propagent sur deux échelles de temps distinctes, rendant ainsi la discrétisation temporelle délicate.En particulier, un schéma explicite est stable sous critère CFL, qui dépend de la vitesse du son, rendant ainsi cette condition très contraignante. En ce qui concerne le problème de précision que l’on observe particulièrement dans le cas des grilles quadrangulaires, il s’agit du fait que la solution discrète ne converge pas vers la solution incompressible lorsque le nombre de Mach tend vers zéro.Pour s’affranchir de ce problème de précision, plusieurs correctifs ont été développés, consistant à modifier la diffusion numérique du schéma original. Ces correctifs permettent d’améliorer la précision des schémas compressibles lorsque le nombre de Mach tend vers zéro. Malheureusement, ils introduisent d’autres problèmes, tels que l’apparition de modes oscillants (mode en échiquier sur une grille cartésienne) dans la solution numérique ou l’extrême diffusion des ondes acoustiques à bas nombre de Mach. L’efficacité est également compromise car ces schémas sont stables sous une CFL encore plus restrictive que le schéma original. Dans cet exposé, nous proposons d’analyser le phénomène des oscillations qui affecte certains des correctifs proposés dans la littérature. Nous nous intéressons principalement aux correctifs basés sur le schéma de Roe, en particulier ceux qui réduisent la diffusion numérique sur la vitesse normale. L’analyse asymptotique de ces schémas conduit à une discrétisation du système des ondes pour…
