Marie Farge’s research while affiliated with Ecole Normale Supérieure de Paris and other places

Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent and nondifferentiable, and we propose using an integral formulation in time. We analyze the existence and uniqueness of this weak form of adaptive Galerkin schemes and prove that nonsmooth projection operators can introduce energy dissipation, which is a crucial result for adaptive Galerkin schemes. To illustrate this, we study an adaptive Galerkin wavelet scheme which computes the time evolution of the inviscid Burgers equation in one dimension and of the incompressible Euler equations in two and three dimensions with a pseudospectral scheme, together with coherent vorticity simulation which uses wavelet denoising. With the help of the continuous wavelet representation we analyze the time evolution of the solution of the 1D inviscid Burgers equation: We first observe that numerical resonances appear when energy reaches the smallest resolved scale, then they spread in both space and scale until they reach energy equipartition between all basis functions, as thermal noise does. Finally we show how adaptive wavelet schemes denoise and regularize the solution of the Galerkin truncated inviscid equations, and for the inviscid Burgers case wavelet denoising even yields convergence towards the exact dissipative solution, also called entropy solution. These results motivate in particular adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws. This SIGEST article is a revised and extended version of the article [R. M. Pereira, N. Nguyen van yen, K. Schneider, and M. Farge, Multiscale Model. Simul., 20 (2022), pp. 1147--1166].

The state-of-the-art of insect flight research using advanced computational fluid dynamics techniques on supercomputers is reviewed, focusing mostly on the work of the present authors. We present a brief historical overview, discuss numerical challenges and introduce the governing model equations. Two open source codes, one based on Fourier, the other based on wavelet representation, are succinctly presented and a mass-spring flexible wing model is described. Various illustrations of numerical simulations of flapping insects at low, intermediate and high Reynolds numbers are presented. The role of flexible wings, data-driven modeling and fluid–structure interaction issues are likewise discussed.

Adaptive Galerkin methods for time-dependent partial differential equations are studied and shown to be dissipative. The adaptation implies that the subset of the selected basis function changes over time according to the evolution of the solution. The corresponding projection operator is thus time-dependent and non differentiable. We therefore propose to use an integral formulation in time. We analyze the existence and uniqueness of this weak form of the dynamical Galerkin scheme, and we then prove that the non smooth projection operator introduces energy dissipation, which is a crucial result for adaptive Galerkin methods, e.g., adaptive wavelet methods. Numerical examples for the inviscid Burgers equation in one dimension and the incompressible Euler equations in two and three spatial dimensions show that the selection of basis functions, for instance by filtering out weak wavelet coefficients from the solution, introduces energy dissipation. Moreover, for the Burgers case we can show that adaptive wavelet regularization yields convergence of the truncated Galerkin solution to the physically relevant entropy solution. These results motivate adaptive wavelet-based Galerkin schemes for nonlinear hyperbolic conservation laws.

Investigation of electrodynamic effects considering South American features is essential to extend understanding of middle- to low-latitude space weather phenomena. For retrieving magnetic contributions related to geomagnetically induced currents (GIC), a wavelet-based filtering method is verified and applied to magnetic records on the ground. The experimental data with one-minute resolution were acquired with magnetometers at two Brazilian sites, close to the South Atlantic Magnetic Anomaly, from Nov. 6 to 11, 2004. The signal intensities vary primarily under the influence of the geomagnetic disturbance periods. The performed wavelet analyses allow for a scale-dependent statistical characterization (including their cross-correlation) of the magnetospheric-ionospheric processes that affect the Earth’s surface. The non-stationary magnetic signals can thus be split into coherent events and background noise by the wavelet denoising technique. The statistics and physical features of both parts are analyzed, and it is shown that the proposed treatment yields a depurated GIC signal. As a complementary result, this procedure also establishes an objective-automatic computational method for the GIC calculation treatment.

In the first part of this article, I summarise two centuries of research on turbulence. I also critically discuss some of the interpretations that are still in use, as turbulence remains an inherently non-linear problem that is still unsolved to this day. In the second part, I tell the story of how Alex Grossmann introduced me to the continuous wavelet representation in 1983, and how he instantly convinced me that this is the tool I was looking for to study turbulence. In the third part, I present a selection of results I obtained in collaboration with several students and colleagues to represent, analyse and filter different turbulent flows using the continuous wavelet transform. I have chosen to present both these theories and results without the use of equations, in the hope that the reading of this article will be more enjoyable.

We study dynamical Galerkin schemes for evolutionary partial differential equations (PDEs), where the projection operator changes over time. When selecting a subset of basis functions, the projection operator is non-differentiable in time and an integral formulation has to be used. We analyze the projected equations with respect to existence and uniqueness of the solution and prove that non-smooth projection operators introduce dissipation, a result which is crucial for adaptive discretizations of PDEs, e.g., adaptive wavelet methods. For the Burgers equation we illustrate numerically that thresholding the wavelet coefficients, and thus changing the projection space, will indeed introduce dissipation of energy. We discuss consequences for the so-called `pseudo-adaptive' simulations, where time evolution and dealiasing are done in Fourier space, whilst thresholding is carried out in wavelet space. Numerical examples are given for the inviscid Burgers equation in 1D and the incompressible Euler equations in 2D and 3D.

Fluid-structure interactionKolomenskiy, Dmitry of the flapping wings of a hovering bumblebee is considered.Ravi, Sridhar Kinematic reconstruction of the wing motion using synchronized high-speed video recordings is described,Xu, Ru that provides the necessary input data for numerical modelling.Ueyama, Kohei Computational fluid dynamicsJakobi, Timothy (CFD) solver is combined withEngels, Thomas a dynamicalNakata, Toshiyuki model thatSesterhenn, Jörn describes the wing motion.Farge, Marie Results of a high resolutionSchneider, Kai numerical simulation are presented.Onishi, RyoLiu

We present a wavelet-based adaptive method for computing 3D flows in complex, time-dependent geometries, implemented on massively parallel computers. The incompressible fluid is modeled with an artificial compressibility approach in order to avoid the solution of elliptical problems. No-slip and in/outflow boundary conditions are imposed using volume penalization. The governing equations are discretized on a locally uniform Cartesian grid with centered finite differences, and integrated in time with a Runge--Kutta scheme, both of 4th order. The domain is partitioned into cubic blocks with equidistant grids and, for each block, biorthogonal interpolating wavelets are used as refinement indicators and prediction operators. Thresholding of wavelet coefficients allows to introduce dynamically evolving grids and the adaption strategy tracks the solution in both space and scale. Blocks are distributed among MPI processes and the global topology of the grid is encoded in a tree-like data structure. Analyzing the different physical and numerical parameters allows balancing their individual error contributions and thus ensures optimal convergence while minimizing computational effort. Different validation tests score accuracy and performance of our new open source code, \texttt{WABBIT} (Wavelet Adaptive Block-Based solver for Interactions with Turbulence), on massively parallel computers using fully adaptive grids. Flow simulations of flapping insects demonstrate its applicability to complex, bio-inspired problems.

It is shown that the wings of bumblebees during flapping undergo pitching (feathering angle) rotation that can be characterized as a fluid-structure interaction problem. Measurements of shape, size and inertial properties of the wings of bumblebees Bombus ignitus are described that provide the necessary input data for numerical modelling. A computational fluid dynamics (CFD) solver is combined with a dynamical model that describes the time evolution of the feathering angle. An example result of the numerical simulation is shown.