arXiv subjects

Vincent Le Chenadec

Publications and source records attributed to Vincent Le Chenadec.

9 recordsLinked to original sources

arXiv2026

A Cartesian Cut-Cell Two-Fluid Method for Two-Phase Diffusion Problems

We present a Cartesian cut-cell finite-volume method for sharp-interface two-phase diffusion problems in static geometries. The formulation follows a two-fluid approach: independent diffusion equations are discretized in each phase on a fixed Cartesian grid, while the phases are coupled through embedded interface conditions enforcing continuity of diffusive flux and a general jump law. Cut cells are treated by integrating the governing equations over phase-restricted control volumes and surfaces, yielding discrete divergence and gradient operators that are locally conservative within each phase. Interface coupling is achieved by introducing a small set of interfacial unknowns per cut cell on the embedded boundary; the resulting algebraic system involves only bulk and interfacial averages. A key feature of the method is the use of a reduced set of geometric information based solely on low-order moments (trimmed volumes, apertures and interface measures/centroids), allowing robust implementation without constructing explicitly cut-cell polytopes. The method supports steady (Poisson) and unsteady (diffusion) regimes and incorporates Dirichlet, Neumann, Robin boundary conditions and general jumps. We validate the scheme on one-, two- and three-dimensional single-phase and two-phase benchmarks, including curved embedded boundaries, Robin conditions and strong property/jump contrasts. The results demonstrate a superlinear convergence behavior, sharp enforcement of interfacial laws and excellent conservation properties. Extensions to moving interfaces and Stefan-type free-boundary problems are natural perspectives of this framework.

math.NA↗

arXiv2025

A space-time extension of a conservative two-fluid cut-cell method for moving diffusion problems

We present a space-time extension of a conservative Cartesian cut-cell finite-volume method for two-phase diffusion problems with prescribed interface motion. The formulation follows a two-fluid approach: one scalar field is solved in each phase with discontinuous material properties, coupled by sharp interface conditions enforcing flux continuity and jump laws. To handle moving boundaries on a fixed Cartesian grid, the discrete balance is written over phase-restricted space-time control volumes, whose geometric moments (swept volumes and apertures) are used as weights in the finite-volume operators. This construction naturally accounts for the creation and destruction of cut cells (fresh/dead-cell events) and yields strict discrete conservation. The resulting scheme retains the algebraic structure of the static cut-cell formulation while incorporating motion through local geometric weights and interface coupling operators. A series of verification and validation tests in two and three dimensions demonstrate super-linear accuracy in space, robust behavior under repeated topology changes and conservation across strong coefficient jumps and moving interfaces. The proposed space-time cut-cell framework provides a conservative building block for multiphase transport in evolving geometries and a foundation for future free-boundary extensions such as Stefan-type phase change.

physics.comp-ph↗

arXiv2025

Adjoint-based optimization of the Rayleigh-Bénard instability with melting boundary

In this work, we propose an adjoint-based optimization procedure to control the onset of the Rayleigh-Bénard instability with a melting front. A novel cut cell method is used to solve the Navier-Stokes equations in the Boussinesq approximation and the convection-diffusion equation in the fluid layer, as well as the heat equation in the solid phase. To track the interface we use the level set method where its evolution is simply governed by an advection equation. An incomplete continuous adjoint problem is then derived by treating the velocity field obtained from the forward problem as a known variable in the adjoint convection-diffusion equation, thereby avoiding the need to solve a Navier-Stokes adjoint in the fluid phase. To the best of our knowledge, this provides the first adjoint-based optimization framework for Rayleigh-Bénard instability with a melting boundary. Two optimization problems, together with a comparison against a derivative-free particle-swarm method, demonstrate that the proposed incomplete adjoint yields gradients accurate enough to control the front shape while reducing the number of expensive function evaluations by about an order of magnitude.

math-ph↗

arXiv2025

HFNO: an interpretable data-driven decomposition strategy for turbulent flows

Fourier Neural Operators (FNOs) have demonstrated exceptional accuracy in mapping functional spaces by leveraging Fourier transforms to establish a connection with underlying physical principles. However, their opaque inner workings often constitute an obstacle to physical interpretability. This work introduces Hierarchical Fourier Neural Operators (HFNOs), a novel FNO-based architecture tailored for reduced-order modeling of turbulent fluid flows, designed to enhance interpretability by explicitly separating fluid behavior across scales. The proposed architecture processes wavenumber bins in parallel, enabling the approximation of dispersion relations and non-linear interactions. Inputs are lifted to a higher-dimensional space, Fourier-transformed, and partitioned into wavenumber bins. Each bin is processed by a Fully Connected Neural Network (FCNN), with outputs subsequently padded, summed, and inverse-transformed back into physical space. A final transformation refines the output in physical space as a correction model, by means of one of the following architectures: Convolutional Neural Network (CNN) and Echo State Network (ESN). We evaluate the proposed model on a series of increasingly complex dynamical systems: first on the one-dimensional Kuramoto-Sivashinsky equation, then on the two-dimensional Kolmogorov flow, and finally on the prediction of wall shear stress in turbulent channel flow, given the near-wall velocity field. In all test cases, the model demonstrates its ability to decompose turbulent flows across various scales, opening up the possibility of increased interpretability and multiscale modeling of such flows.

physics.flu-dyn↗

arXiv2022

A Conservative Cartesian Cut Cell Method for the Solution of the Incompressible Navier-Stokes Equations on Staggered Meshes

The treatment of complex geometries in Computational Fluid Dynamics applications is a challenging endeavor, which immersed boundary and cut-cell techniques can significantly simplify by alleviating the meshing process required by body-fitted meshes. These methods however introduce new challenges, as the formulation of accurate and well-posed discrete operators becomes nontrivial. Here, a conservative cartesian cut cell method is proposed for the solution of the incompressible Navier--Stokes equation on staggered Cartesian grids. Emphasis is set on the structure of the discrete operators, designed to mimic the properties of the continuous ones while retaining a nearest-neighbor stencil. For convective transport, a divergence is proposed and shown to also be skew-symmetric as long as the divergence-free condition is satisfied, ensuring mass, momentum and kinetic energy conservation (the latter in the inviscid limit). For viscous transport, conservative and symmetric operators are proposed for Dirichlet boundary conditions. Symmetry ensures the existence of a sink term (viscous dissipation) in the discrete kinetic energy budget, which is beneficial for stability. The cut-cell discretization possesses the much desired summation-by-parts (SBP) properties. In addition, it is fully conservative, mathematically provably stable and supports arbitrary geometries. The accuracy and robustness of the method are then demonstrated with flows past a circular cylinder and an airfoil.

physics.comp-ph↗

arXiv2022

A Comprehensive Study of Adjoint-Based Optimization of Non-Linear Systems with Application to Burgers' Equation

In the context of adjoint-based optimization, nonlinear conservation laws pose significant problems regarding the existence and uniqueness of both direct and adjoint solutions, as well as the well-posedness of the problem for sensitivity analysis and gradient-based optimization algorithms. In this paper we will analyze the convergence of the adjoint equations to known exact solutions of the inviscid Burgers' equation for a variety of numerical schemes. The effect of the non-differentiability of the underlying approximate Riemann solver, complete vs. incomplete differentiation of the discrete schemes and inconsistencies in time advancement will be discussed.

physics.comp-ph↗

arXiv2022

Adjoint-based optimization of two-dimensional Stefan problems

A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimisation algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterised boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.

math-ph↗

arXiv2017

Experimenting with the p4est library for AMR simulations of two-phase flows

Many physical problems involve spatial and temporal inhomogeneities that require a very fine discretization in order to be accurately simulated. Using an adaptive mesh, a high level of resolution is used in the appropriate areas while keeping a coarse mesh elsewhere. This idea allows to save time and computations, but represents a challenge for distributed-memory environments. The MARS project (for Multiphase Adaptative Refinement Solver) intends to assess the parallel library p4est for adaptive mesh, in a case of a finite volume scheme applied to two-phase flows. Besides testing the library's performances, particularly for load balancing, its user-friendliness in use and implementation are also exhibited here. First promising 3D simulations are even presented.

math.NA↗

arXiv2014

Thermoacoustic instability - a dynamical system and time domain analysis

This study focuses on the Rijke tube problem, which includes features relevant to the modeling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source, and nonlinearities. This thermo-acoustic system features a complex dynamical behavior. In order to synthesize accurate time-series, we tackle this problem from a numerical point-of-view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness, in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by means of the projection method proposed by Jarlebring (2008), which alleviates the linearization with respect to the retarded time. The results are then compared to the analytical solution of the undamped system, and to Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We illustrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case, predicted by using Galerkin expansion, show spurious oscillations resulting from the Gibbs phenomenon. By comparing the modes of the linear to that of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching. Finally, time-series in the fully nonlinear regime, where a limit cycle is established, are analyzed and dominant modes are extracted. The analysis of the saturated limit cycles shows the presence of higher frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not captured when the coupling between different frequencies is not accounted for.

physics.comp-ph↗