Séminaire Calcul Scientifique et Modélisation

I will consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this problem can be modeled by mixed-dimensional coupled equations (3D for the fluid and 1D for the solid). Several challenges must be faced when dealing with this type of problems. From a mathematical point of view, these include defining well posed trace operators of codimension two. On the computational standpoint, the non-standard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixed-dimensional discrete formulation as compared to a fully resolved one. I define the continuous formulation using the Navier-Stokes equations for the fluid and a Timoshenko beam model for the structure. I complement these models with a mixed-dimensional version of the fluid-structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional Fourier space. One of the fundamental advantages of this approach is that it enables the approximation of the problem within the framework of the Finite Element Method (FEM). I establish the energy stability of the discrete formulation and provide extensive numerical evidence of the accuracy of the mixed-dimensional model, notably with respect to a fully resolved (ALE based) model.

Many physical or economical applications rely on Monte-Carlo (MC) codes to solve deterministic partial differential equations (PDEs). This is the case for example for (non-exhaustive list) neutronics and photonics. The Monte-Carlo resolution implies the sampling of the physical variables: x the position, v the velocity and t the time. The simulations are costly but the MC resolution is competitive with respect to other methods due to the high dimensional (7d) deterministic problem. The numerical parameter controling the accuracy is N_MC, the number of particles. The larger N_MC, the more accurate the results. The convergence rate obeys the central limit theorem: it is O(1/sqrt(N_MC)).

Obviously, propagating uncertainties (for sensitivity analysis etc.) with respect to different parameters X \in R^d is of great interest in every of the aforementioned applications (uncertain cross-sections etc.). In fact, in our physical applications, we would like to be able to perform systematic uncertainty propagations. As a consequence, we often face a 7+d dimensional problem. Non-intrusive methods are usually applied (use of black box codes). But it demands a high number N of evaluations. In our MC resolution context, each one of them is costly. One accurate run can take several hours on hundreds of processors.

When applying any non-intrusive method to propagate uncertainties through the linear Boltzmann equation solved with an MC code, basically, the physical space (x,t,v) and the uncertain space (X) are both explored thanks to two different MC experimental designs. The first one has particles to explore the space of physical variables (x, t, v), the second one has N runs for the space of the uncertain variable X. In this non-intrusive context, the two MC samplings are tensorised in the sense we process N_MC x N = 1e9 -- 1e15 particles for an overall O()1/sqrt(N_MC) error. An uncertainty propagation study is consequently costly. The main idea of the present work comes from the fact that MC experimental designs should allow avoiding the tensorisation of the N_MC particles and N runs [1,2,3,4]. For this, we sample the whole space relative to (x,t,v,X) within the same MC design. This implies sampling the uncertain parameters X within the code, hence the intrusiveness of the approach. In practice in [1], fast convergence rates have been observed with respect to the polynomial Chaos truncation order P: the method is efficient for the linear [2], nonlinear [4] Boltzman equation and keff computations [3]. The aim of the talk is to present the details of the uncertain MC solver.

[1] G. Poëtte. A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation. Journal of Computational Physics, 385:135 – 162, 2019

[2] G. Poëtte. Spectral convergence of the generalized Polynomial Chaos reduced model obtained from the uncertain linear Boltzmann equation. Preprint submitted to Mathematics and Computers in Simulation , 2019.

[3] G. Poëtte and E. Brun. Efficient uncertain k eff computations with the Monte Carlo resolution of generalised Polynomial Chaos Based reduced models. Preprint,November 2020.

[4] G. Poëtte. Efficient uncertainty propagation for photonics: combining Implicit Semi-analog Monte Carlo (ISMC) and Monte Carlo generalised Polynomial Chaos (MC-gPC). Preprint, 2020.