NANov 26, 2015
A numerical comparison of some Multiscale Finite Element approaches for convection-dominated problems in heterogeneous mediaClaude Le Bris, Frederic Legoll, François Madiot
The purpose of this work is to investigate the behavior of Multiscale Finite Element type methods for advection-diffusion problems in the advection-dominated regime. We present, study and compare various options to address the issue of the simultaneous presence of both heterogeneity of scales and strong advection. Classical MsFEM methods are compared with adjusted MsFEM methods, stabilized versions of the methods, and a splitting method that treats the multiscale diffusion and the strong advection separately.
47.5NAApr 20
Diffusion Synthetic Acceleration for polytopic discretisations of Boltzmann transportAnsar Calloo, Matthew Evans, François Madiot et al.
We present a computational study of diffusion synthetic acceleration (DSA) for the monoenergetic, isotropically scattering $S_N$ transport equations, discretised in space by a polytopic discontinuous Galerkin method. Using a discrete ordinates angular discretisation, we construct the DSA correction with an interior-penalty diffusion operator and compare a classical symmetric interior penalty (SIP) formulation with a modified interior penalty (MIP) variant, together with homogeneous Dirichlet and Marshak (Robin) diffusion boundary conditions imposed weakly in the DG framework. We quantify the observed convergence behaviour of the resulting source iteration across variations in optical thickness, scattering ratio, angular quadrature, mesh refinement, polynomial degree and mesh anisotropy on families of bounded Voronoi meshes. The results show that MIP-based DSA remains robust across the parameter ranges tested, whereas SIP-based DSA can lose robustness in the intermediate regime. In challenging optically thick, highly scattering settings, the observed convergence factors for the MIP-based schemes are typically below $0.6$.
3.1NAApr 27
On Physics-Based Loss Scaling for MF-PINNs applied to the neutron diffusion equationMinh-Hieu Do, François Madiot, Karim Ammar et al.
Physics-Based Loss Scaling (PBLS) is introduced for Mixed-Formulation PINNs (MF-PINNs) applied to the neutron diffusion equation. In particular, we propose a new \textit{scaled} loss function based on the material cross sections, which is equivalent to the classical MF-PINN loss, but accelerates the convergence and improves accuracy of MF-PINNs. Several numerical experiments on both the fixed source and the k-eigenvalue problem, from one-group to multigroup cases and from two-dimensional (2D) to three-dimensional (3D) configurations, illustrate the efficiency of the proposed scaling method.