Design of certification tools for acoustics problems solved with the Variational Theory of Complex Rays

Despite railway transportation being considered as an environmentally friendly means of transport, it has the main drawback of being a considerable source of noise which can be problematic for its surroundings. In order to address this criticism, actors from the railway sector, both industrial and academic, are interested in the development of numerical tools capable of modelling accurately and rapidly the midfield acoustic radiation. These numerical tools are developed with the intent of establishing methods of virtual certification for railway equipment, which makes error control essential for the numerical calculations. The acoustic bandwidth for the railway sector being in the mid frequency domain (in the order of several kHz), the Variational Theory of Complex Rays (VTCR) is used in this work. This method, belonging to the family of Trefftz methods, relies on the choice of partitionning the domain into starred sub-domains and representing the pressure field in these sub-domains as a linear combination of plane waves oriented along multiple radial directions. These plane waves therefore serve as shape functions that satisfy, a priori, the Helmholtz equation inside the domain. The acoustic 2D VTCR code used in this work has been obtained by introducing a variational form defined from the boundary conditions on the boundaries of the domain and from the continuity conditions on the interfaces between sub-domains. The matrix system obtained is characterized by a matrix that is non symmetric, dense and generally ill-conditionned. In a first part of the PhD work, a Python code based on VTCR is developed and presented for the resolution of linear acoustic 2D problems. The convergence of VTCR is studied according to the number of plane wave directions selected and the type of solver used for the resolution of the matrix system obtained after discretization. Simple examples with analytical solutions enable the estimation of the error in potential energy over the whole domain. However, in most applications, having a precise estimation of the pressure at all points of the domain is not necessarily sought. As certification tests rely on measuring the pressure in certain specific areas defined by standardized tests, the convergence of a local quantity of interest (such as the mean pressure over a section of the boundary) is also studied. Therefore, in a second part of the work, we seek to define a goal-oriented error estimator based on the introduction of an adjoint problem in the VTCR context. This Dual Weighted Residual (DWR) estimator, allows us to obtain a reliable approximation of the error, even for a coarse discretization of the adjoint problem. By decomposing this estimator as the sum of its contributions along the adjoint problem's plane wave directions, it is possible to explore various refinement strategies and pave the way for future adaptive computation strategies. Finally, theoretical developments are put forward to able to take into account point quantities of interest inside the domain.