Florian De Vuyst

Adrien Le Coënt, Laurent Fribourg, Nicolas Markey et al.

We present a correct-by-design method of state-dependent control synthesis for linear discrete-time switching systems. Given an objective region R of the state space, the method builds a capture set S and a control which steers any element of S into R. The method works by iterated backward reachability from R. More precisely, S is given as a parametric extension of R, and the maximum value of the parameter is solved by linear programming. The method can also be used to synthesize a stability control which maintains indefinitely within R all the states starting at R. We explain how the synthesis method can be performed in a distributed manner. The method has been implemented and successfully applied to the synthesis of a distributed control of a concrete floor heating system with 11 rooms and 2^11 = 2048 switching modes.

Adrien Le Coënt, Florian De Vuyst, Ludovic Chamoin et al.

In this paper, we propose a symbolic control synthesis method for nonlinear sampled switched systems whose vector fields are one-sided Lipschitz. The main idea is to use an approximate model obtained from the forward Euler method to build a guaranteed control. The benefit of this method is that the error introduced by symbolic modeling is bounded by choosing suitable time and space discretizations. The method is implemented in the interpreted language Octave. Several examples of the literature are performed and the results are compared with results obtained with a previous method based on the Runge-Kutta integration method.

Florian De Vuyst, Marie Béchereau, Thibault Gasc et al.

In this paper, stable and "low-diffusive" multidimensional interface capturing (IC) schemes using slope limiters are discussed. It is known that direction-by-direction slope-limited MUSCL schemes create geometrical artifacts and thus return a poor accuracy. We here focus on this particular issue and show that the reconstruction of gradient directions are an important factor of accuracy. The use of a multidimensional limiting process (MLP) added with an adequate time integration scheme leads to an artifact-free and instability-free interface capturing (IC) approach. Numerical experiments like the reference Kothe-Rider forward-backward advection case show the accuracy of the approach. We also show that the approach can be extended to the more complex compressible multimaterial hydrodynamics case, with potentially an arbitrary number of fluids. We also believe that this approach is appropriate for multicore/manycore architecture because of its SIMD feature, which may be another asset compared to interface reconstruction approaches.

Florian De Vuyst

The Lagrange-Flux schemes are Eulerian finite volume schemes that make use of an approximate Riemann solver in Lagrangian description with particular upwind convective fluxes. They have been recently designed as variant formulations of Lagrange-remap schemes that provide better HPC performance on modern multicore processors, see~[De Vuyst et al., OGST 71(6), 2016]. Actually Lagrange-Flux schemes show several advantages compared to Lagrange-remap schemes, especially for multidimensional problems: they do not require the computation of deformed Lagrangian cells or mesh intersections as in the remapping process. The paper focuses on the entropy property of Lagrange-Flux schemes in their semi-discrete in space form, for one-dimensional problems and for the compressible Euler equations as example. We provide pseudo-viscosity pressure terms that ensure entropy production of order $O(|Δu|^3)$, where $|Δu|$ represents a velocity jump at a cell interface. Pseudo-viscosity terms are also designed to vanish into expansion regions as it is the case for rarefaction waves.

Florian De Vuyst, Asma Toumi

In this paper we present a mixed EIM-SVD tensor decomposition for bivariate functions. This method is composed, as its name suggests, of two main steps. The first one, provides an approximate representation of a function $f$ in separate form by the use of a Tensor Empirical Interpolation Method (TEIM). The second phase consists in applying the Singular Value Decomposition (SVD) with low-rank truncation to the separate form of $f$ resulting from the first phase. Error estimates of the developed TEIM as well as truncated SVD decomposition are derived. Numerical experiments confirm that the decomposition techniques are efficient in terms of stability and accuracy.

Florian De Vuyst, Thibault Gasc, Renaud Motte et al.

In a previous paper, we have achieved the performance analysis of staggered Lagrange-remap schemes, a class of solvers widely used for Hydrodynamics applications. This paper is devoted to the rethinking and redesign of the Lagrange-remap process for achieving better performance using today's computing architectures. As an unintended outcome, the analysis has lead us to the discovery of a new family of solvers -- the so-called Lagrange-flux schemes -- that appear to be promising for the CFD community.