Robustness, automation and reliability of integral formulations for wave propagation: a posteriori estimators and adaptivity. – RAFFINE

We first propose to explore several theoretical ways (residual-based or «goal oriented« estimators) that are now classical for domain finite element methods, assessing and adapting them to the specific difficulties raised by integral equations. These estimators will be used to define and implement (i) meshing or remeshing strategies, and (ii) stopping criteria for iterative linear solvers. We will then propose efficient algorithms that are implementable for large-scale boundary element models, especially by exploiting parallelization strategies and acceleration by the FMM. Finally, the tools developed will be assessed and validated on canonical cases, illustrating the typical difficulties encountered, and then on particularly demanding industrial cases.

At this early stage, the available results are of a preliminary nature, and consist in

(a) An implementation in a simplified framework (2D acoustic wave equation) of the numerical evaluation of residual-based estimators. Attention focused so far on the practical evaluation of such indicatiors (and in particular on the required numerical quadrature of singular integrals involved in the computation) and on preliminary tests.

(b) An adaptation to integral equations and boundary-based discretizations (for 3D elastic waves) of the metric-based error estimation methodology (under development at INRIA for domain-based discretizations). Here also, implementation and preliminary numerical tests have been conducted.

The main short-term perspective (first part of a PhD thesis starting fall 2013) concerns the choice, development and validation of an error estimation methodology that achieves a satisfactory compromise between mathematical justification, feasibility of implementation, and computational efficiency. This stage will then be followed by an industrialisation phase (implementation of the estimation method into the participating industrial codes, test against case studies of industrial complexity).

Communication « An adaptive fast multipole accelerated boundary element method fir 3D elastodynamics », S. Chaillat et A. Loseille, 2013 SIAM Conf. on Mathematical And Computational Issues in the Geosciences, Padoue (Italy), juin 2013.

The RAFFINE project involves the development of a posteriori estimators and adaptive methods for integral equations in the field of simulation of acoustic, electromagnetic and elastic waves. Integral methods are one of the major tools for the numerical simulation of wave propagation and scattering. Indeed, these methods have undergone major development, from both the theoretical and the numerical viewpoints, over the last two decades, particularly with the advent of the Fast Multipole Method (FMM). Paradoxically, very few practical tools exist that allow to control the quality of the discrete solution and the impact of the discretization error on quantities of interest (such as the Radar Cross Section or the scattering matrix). Our project aims to fill this gap. For domain finite element methods, error estimation and control is classically performed using a posteriori estimators of various kinds, about which an abundant literature is available, and tools for adaptivity having proven efficiency. For integral equations, the theoretical literature is much scarcer, and attempts to implement existing estimators in commercial or industrial codes are very rare. We first propose to explore several theoretical ways (residual-based or "goal oriented" estimators) that are now classical for domain finite element methods, assessing and adapting them to the specific difficulties raised by integral equations (nonlocal integral operators, singularity of the kernel). These estimators will be used to define and implement (i) meshing or remeshing strategies, and (ii) stopping criteria for iterative linear solvers. We will then propose efficient algorithms that are implementable for large-scale boundary element models, especially by exploiting parallelization strategies and acceleration by the FMM. Finally, the tools developed will be assessed and validated on canonical cases, illustrating the typical difficulties encountered, and then on particularly demanding industrial cases.