This work introduces an unstructured anisotropic mesh adaptation strategy for high-order Discontinuous Galerkin (DG) schemes, with application to compressible fluid flow problems related to aeronautical engineering. A metric-based anisotropic remeshing technique is considered, where the metric field is computed from an accurate Hessian of the Mach number, using the optimal metric formulation from the multiscale error estimator theory. This Hessian, that is computed from a Discontinuous Galerkin weak formulation, is then rescaled using state-of-the-art DG error estimators for high-order computations. The method is validated and applied to configurations of increasing complexity, including transonic 2D RAE2822 airfoil and 3D M6 wing, as well as high-lift 2D three-element airfoil and 3D CRM. For all test cases considered, it is shown that this new strategy captures efficiently and automatically anisotropic flow phenomena such as shocks, boundary layers and wakes with a minimal number of degrees of freedom. In terms of integrated aerodynamic coefficients, the results show superior accuracy when increasing the polynomial order of the computations, and comparable or better accuracy compared to a variety of state-of-the art adaptive and non-adaptive flow solvers.

Abstract We develop a Galerkin discretization of the electric field integral equation based on a lowest-order virtual element approximation for surface meshes composed of polygons. This new boundary element discretization relies on the divergence-conforming virtual element counterparts of the classical Raviart–Thomas (RT) finite elements on simplices and is particularly suited to handle hanging nodes. We prove the well-posedness of the resulting numerical scheme by establishing the stable-uniform character of the discrete inf-sup condition in the natural norm for polyhedral surfaces. Moreover, we demonstrate through an a priori error analysis the quasi-optimal convergence of the scheme, leading to the same convergence rate as that of the classical RT boundary element scheme. Finally, numerical experiments involving scattering problems are presented in order to give more insight into the behavior of the virtual boundary element scheme in terms of $h$-convergence and accuracy as a function of the regularity of solutions and meshes.

Facing the need to solve time-harmonic wave problems in wide propagation domains, one needs to resort to iterative algorithms, to which classic numerical methods are usually poorly suited. The Trefftz method, based on the use of actual local solutions as basis functions, can then appear as a viable option for this kind of configurations.Thus, we first propose a general formalism allowing to gather different wave problems, so that a general Ultra Weak Variational Formulation can be defined for them at the same time. Yet, as its classic discretisation by plane waves is used to causing independence and accuracy limitations, we suggest to characterise local solutions by impedance boundary conditions, which is discretised thanks to piecewise polynomial fields. Unfortunately, analytic expressions of such functions are not practically derivable, and we resort to a local numerical method so as to provide approximations: these quasi-solutions are finally chosen as basis functions in the Trefftz formulation, which is then referred to as a quasi-Trefftz method.In the end, numerical experiments allow to highlight robust independence, convergence and iterative behaviour of this method, under calibration conditions of the local solver with respect to the polynomial BC discretisation.

In this paper, we present a method for constructing a quad mesh from an initial cross-field given by the user. The idea of this approach is to provide a framework to process any cross-field in order to generate a quad mesh and thus benefit from the properties of the field in the resulting mesh. With this point of view, we handle the case of non-simply connected domains and we also address the notion of singular point placement on the edge, which allows us to handle geometries of arbitrary index. Finally, we give an extension of the method to non-planar surface manifolds.

A numerical factorization method of the unidirectional propagation operators induced by the linearized Euler and Navier–Stokes equations is performed to construct a new One-Way approach to compute the sound radiation in a partially lined duct. The complex phenomena of reflection and transmission of incident waves originating from discontinuities in the duct wall are accurately taken into account by an iterative domain decomposition method. Furthermore, the proposed factorization approach allows both the derivation of the lined duct scattering matrix and an in-depth understanding of the impact of the acoustic liner on left- and right-going waves coming from a transmission or a reflection. Finally, the efficiency of the numerical method is shown based on a classical benchmark with two types of baseflows (laminar and turbulent Poiseuille flows) and by comparison with other numerical and experimental results. An unstable surface mode is observed at 1000 Hz, and we find good agreement with experimental data for the turbulent mean flow associated with the One-Way Navier–Stokes equations.

The Flux Reconstruction (FR) method is classically used in the Computational Fluid Dynamics field. However, its use for the simulation of electromagnetic wave propagation is not as developed yet. Following on from the development of a priori error estimates for the 1D wave equations, we introduce optimisation problems to allow an adaptation of the FR correction polynomial functions to the discretisation parameters. Showing notable accuracy gains in 1D, especially in the preasymptotic regime, we generalise this procedure to the 3D Maxwell's equations, leading to similar interesting possibilities to reduce the computational cost for a given accuracy.

Electromagnetic simulations on large domains require a huge memory consumption. Domain decomposition methods, based on Trefftz methods, could be an answer to this issue. In this paper, we associate to heterogeneous three-dimensional Maxwell equations one variational formulation which can be obtained either by upwind fluxes or Riemann traces. We associate to this variational formulation an iterative Trefftz Krylov solver. The poor conditioning due to the use of plane wave basis functions is bypassed thanks to a compression strategy. Moreover, the developed iterative solver is accelerated thanks to a left preconditioner. The considered numerical cases illustrate the performance of this basis reduction, which leads to the consideration of an industrial case of more than 750 millions of degrees of freedom.

We present an approach to discretize Maxwell's equations called Compatible Discrete Operators (CDO). It is a low-order discretization that belongs to the large family of mimetic schemes. This method is a generalization of the FDTD method. It discretizes the closure relations in a geometrically robust way that allows to deal with distorted and non-conformal polyhedral meshes. We briefly introduce the scheme, its link with the FDTD method and we give a first numerical result.

This paper presents a purely numerical factorization of the propagation operator for a generic hyperbolic equation, based on the work of Towne & Colonius in 2015, that does not require heavy analytical development. This method is applied to form one-way equations with the objective of computing the propagation of waves inside a medium. The main advantage of this formulation is that pseudo eigenvectors and eigenvalues matrices are built, leading to the possibility to use the one-way equations into a true amplitude formalism and/or inside a Bremmer series. These two methods allow an extension of the domain of application of the one-way equations when the medium of propagation presents variations along the privileged direction. In particular, these formulations allow to take into account the phenomena of reflection and refraction of the incident wave. Finally numerical results are presented on different 2D situations based on the linearized Euler equations and compared to the results obtained with a full wave resolution. The issues of both the accuracy and the requirements in computational resources of the one-way resolution are also addressed.

This study aimed to specialise a directional H2 (DH2) compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal DH2 approximation for low and high-frequency problems. We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a DH2 gives better computational efficiency than a classical BEM in the case of high-order finite elements and hp heterogeneous meshes. The results indicate that DG is suitable for an auto-adaptive context in integral equations.

In this report, we devise a discontinuous Galerkin method for the propagation of electromagnetic waves in a homogeneous medium. The numerical fluxes are deduced from the solution of a 1D electromagnetic problem. Numerical simulations illustrate the accuracy of the method.

We present an improved formulation of the numerical One-Way approximation that permits to diagonalize in a fully numerical way the propagation operator of a general hyperbolic system.In this paper, it is applied to the linearized Euler equations in the case of a duct with a partially lined section presenting a Poiseuille flow.Domain decomposition is developed in this paper since it is needed for the handling of the transverse boundary conditions discontinuity (hard-wall and impedance condition) which is at the origin of the refraction and reflection of the incident wave inside the computational domain.The results of the One-Way method are compared to experimental and numerical results and show a good accuracy for a low amount of computing resources.

This work is concerned with the construction and the hp non-conforming a priori error analysis of a Discontinuous Galerkin DG numerical scheme applied to the hypersingular integral equation related to the Helmholtz problem in 3D. The main results of this article are an error bound in a norm suited to the problem and in the L2-norm. Those bounds are quasi-optimal for the h-convergence and the p-convergence. Various formulation choices and penalty functions are theoretically discussed. In particular we show that a penalty function of the shape h2p leads to a quasi-optimal convergence of the scheme. Some numerical experiments confirm the expected rates of convergence and the effect of the penalty function.

Trefftz methods are known to be very efficient to reduce the numerical pollution when associated to plane wave basis. However, these local basis functions are not adapted to the computation of evanescent modes or corner singularities. In this article, we consider a two dimensional time-harmonic Maxwell system and we propose a formulation which allows to design an electromagnetic Trefftz formulation associated to local Galerkin basis computed thanks to an auxiliary Nédélec finite element method. The results are illustrated with numerous numerical examples. The considered test cases reveal that the short range and long range propagation phenomena are both well taken into account.

The purpose of this paper is the study of a stable method for coupling two types of schemes using or not numerical fluxes for the resolution of Maxwell's equations in time domain. The methods considered in the hybridization process are Finite Volume (FV), Finite Element (FEM), Finite Difference (FD) and Discontin-uous Galerkin (DG). We give the principle of the hybrid method and by the study of an energetic quantity, one specifies the conditions of stability of it. Finally, one example is given to validate the proposed hybrid method.

We propose a methodology to generate an accurate and efficient reconstruction of radiated fields based on high order interpolation. As the solution is obtained with the convolution by a smooth but potentially high frequency oscillatory kernel, our basis functions therefore incorporate plane waves. Directional interpolation is shown to be efficient for smart directions. An adaptive subdivision of the domain is established to limit the oscillations of the kernel in each element. The new basis functions, combining high order polynomials and plane waves, provide much better accuracy than low order ones. Finally, as standard visualization softwares are generally unable to represent such fields, a method to have a well-suited visualization of high order functions is used. Several numerical results confirm the potential of the method.