Search

This paper presents ISOPE’s 2020 comparative study on the interaction between focused waves and a fixed cylinder. The paper discusses the qualitative and quantitative comparisons between 20 different numerical solvers from various universities across the world for a fixed cylinder. The moving cylinder cases are reported in a companion paper as part B (Agarwal, Saincher, et al., 2021). The numerical solvers presented in this paper are the recent state of the art in the field, mostly developed in-house by various academic institutes. The majority of the participants used hybrid modeling (i.e., a combination of potential flow and Navier–Stokes solvers). The qualitative comparisons based on the wave probe and pressure probe time histories and spectral components between laminar, turbulent, and potential flow solvers are presented in this paper. Furthermore, the quantitative error analyses based on the overall relative error in peak and phase shifts in the wave probe and pressure probe of all the 20 different solvers are reported. The quantitative errors with respect to different spectral component energy levels (i.e., in primary, sub-, and superharmonic regions) capturing capability are reported. Thus, the paper discusses the maximum, minimum, and median relative errors present in recent solvers as regards application to industrial problems rather than attempting to find the best solver. Furthermore, recommendations are drawn based on the analysis.

International audience; In this paper, we develop a fully discrete entropy preserving ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we equip the space part of the method with entropy correction terms that balance the entropy production in space, inspired by the work of Abgrall. Whereas for the time-discretization we apply the relaxation approach introduced by Ketcheson that allows to modify the timestep to preserve the entropy to machineprecision. Up to our knowledge, it is the first time that a provable fully discrete entropy preserving ADER-DG scheme is constructed. We verify our theoretical results with various numerical simulations.

In this work we propose a simple but effective high order polynomial correction allowing to enhance the consistencyof all kind of boundary conditions for the Euler equations (Dirichlet, characteristic far-field and slip-wall), both in 2Dand 3D, preserving a high order of accuracy without the need of curved meshes. The method proposed is a simplifiedreformulation of the Shifted Boundary Method (SBM) and relies on acorrectionbased on theextrapolated valueof the in cell polynomialto the true geometry, thus not requiring the explicit evaluation of high order Taylor series.Moreover, this strategy could be easily implemented into any already existing finite element and finite volume code.Several validation tests are presented to prove the convergence properties up to order four for 2D and 3D simulationswith curved boundaries, as well as an effective extension to flows with shocks Marie Skłodowska-Curie Individual FellowshipSuPerMan, grant agreement No. 101025563

We present a spectral/hp element method for a depth-integrated Boussinsq model for the efficient simulation of nonlinear wave-body interaction. The model exploits a `unified' Boussinesq framework, i.e. the flow under the body is also treated with the depth-integrated approach, initially proposed by (Jiang, 2001) and more recently rigorously analysed by (Lannes, 2016). The choice of the Boussinesq equations allows the elimination of the vertical dimension, resulting in a wave-body model with an adequate precision for weakly nonlinear and dispersive waves expressed in horizontal dimensions only. The framework involves the coupling of two different domains with different flow characteristics. In this work we employ flux-based conditions for domain coupling, following the recipes provided by the discontinuous Galerkin spectral/hp element framework. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model. The spectral/hp element method allows to achieve high-order, possibly exponential, convergence for non-breaking waves and account for the nonlinear interaction with fixed and floating bodies. Our main contribution is to include floating surface-piercing bodies in the conventional depth-integrated Boussinesq framework and the use of a spectral/hp element method for high-order accurate numerical discretization in space. The model is validated against published results for wave-body interaction and confirmed to have excellent accuracy. The proposed nonlinear model is demonstrated to be relevant for the simulation of wave energy devices.

International audience; Despite the recognized impact of tidal bores on estuarine ecosystems, the large scale mechanism of bore formation in convergent alluvial estuaries is still under investigation. So far, field data exist only for a small number of estuaries, while numerical simulations employ the shallow water equations mainly focusing on the small-scale and local processes. In this work, firstly we apply the fully nonlinear weakly dispersive Serre-Green–Naghdi equations to simulate the tide propagation in a convergent estuary of idealized form, verifying that the local dispersion effects, responsible for the appearance of the secondary waves, do not influence the tidal bore onset, which only results from the large scale processes of amplification/damping and distortion of the incoming wave. In a second part, we numerically investigate (225 runs) the estuarine parameter space in order to identify the physical conditions that lead to tidal bore generation. In this parameter space, we determine a critical curve which divides estuaries according to tidal bore occurrence. As a result of this investigation we have shown that bore formation is controlled by the competition between two physical processes: (a) the knee-shaped distortion of the tidal wave, with flood dominance and eventually bore inception; (b) the dissipation of the tidal wave, which is unfavorable to bore formation. We also provide evidence that amplification due to topographic convergence is not a necessary condition for tidal bore generation and that there exist estuaries which display both wave damping and bore development. Finally, the validity of the results has been also assessed in the presence of freshwater river discharge, showing that for low river discharge, its effect on estuarine dynamics can be neglected.

International audience; We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations.

UnDiFi-2D, an open source (free software) Unstructured-grid, Discontinuity Fitting code, is presented. The aim of UnDiFi-2D is to model gas-dynamic discontinuities in two-dimensional (2D) flows as if they were true discontinuities of null thickness that bound regions of the flow-field where a smooth solution to the governing PDEs exists. UnDiFi-2D therefore needs to be coupled with an unstructured CFD solver that is used to discretize the governing PDEs within the smooth regions of the flow-field. Two different, in-house developed, CFD solvers are also included in the current distribution. The main features of the UnDiFi-2D software can be summarized as follows: Programming Language: UnDiFi-2D is written in standard Fortran 77/95; its design is highly modular in order to enhance simplicity of use, maintenance and allow coupling with virtually any existing CFD solver; Usability, Maintenance and Enhancement: In order to improve the usability, maintenance and enhancement of the code also the documentation has been carefully taken into account. The git distributed versioning system has been adopted to facilitate collaborative maintenance and code development; Copyrights: UnDiFi-2D is a free software that anyone can use, copy, distribute, change and improve under the GNU Public License version 3. The present paper is a manifesto of the first public release of the UnDiFi-2D code. It describes the currently implemented features, which are the result of more than a decade of still ongoing CFD developments. This work is focused on the computational techniques adopted and a detailed description of the main characteristics is reported. UnDiFi-2D capabilities are demonstrated by means of examples test cases. The design of the code allows to easily include existing CFD codes and is aimed at ease code reuse and readability., Comment: Keywords: CFD, Fortran, C, shock-fitting, shock-capturing,, unstructured grids

International audience; In this work we consider an e cient discretization of the Shallow Water Equations in spherical geometry for oceanographic applications. Instead of the classical 2d-covariant or 3d-Cartesian approaches, we focus on the mixed 3d/2d form of [Bernard et al., JCP 2009] which evolves the 2d momentum tangential to the sphere by projecting the 3d-Cartesian right-hand side on the sphere surface. Di↵erently from the last reference we consider the exact representation of the sphere instead of the finite element one, mixed with a covariant basis projection of the momentum. This leads to several simplifications of the Discontinuous Galerkin scheme: the local mass matrix goes back to the standard block-diagonal form; the Riemann Problem does not require any tensor or vector rotations to align the bases on the two sides of an edge. Second we consider well balancing corrections related to relevant equilibrium states for tsunami and storm surge simulations. These corrections allow to compensate for the inherent non-exactness of the quadrature induced by the non-polynomial nature of both the geometrical mapping and of the covariant basis. In other words, these corrections are the order of the cubature error. We show that their addition makes the scheme exactly well balanced, and is is equivalent to recasting the integral of the hydrostatic pressure term in strong form. The proposed method is validated on academic benchmarks involving both smooth and discontinuous solutions, and applied to realistic tsunami and an historical storm surge simulation.

We propose a new first-order hyperbolic system approach for dispersive partial differential equations.We apply a compact 4th-order RD scheme, and solve time dependent dispersive PDEs.We demonstrate the performance of the high-order RD schemes on the proposed hyperbolic system, including dispersive shock cases.We verify that the predicted solution, its gradient and Hessian have the same order of accuracy on randomly distributed nodes.

International audience; We propose an extension of the embedded boundary method known as "shifted boundary method" to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.

We provide a review of high order methods for CFD. Besides recalling some classical methods, we show a framework allowing on one hand to see and work with these methods under a different light, and on the other to provide a different path to construct numerical methods for flow equations. In particular, we focus on Residual Based techniques, and Residual Distribution methods, as a framework to construct schemes of arbitrary order. The somewhat classical second-order multidimensional upwind fluctuation-splitting/residual-distribution schemes are reviewed in the chapter by Deconinck and Ricchiuto.

International audience; We describe fully explicit residual based discretizations of the shallow water equations with friction on unstructured grids. The schemes are obtained by properly adapting the explicit construction proposed in Ricchiuto and Abgrall (2010) . In particular, previous work on well balanced integration (Ricchiuto, 2011 ) and preservation of the depth non-negativity (Ricchiuto and Bollermann, 2009 ) is reformulated in the context of a genuinely explicit time stepping still based on a weighted residual approximation. The paper discusses in depth how to achieve in this context an exact preservation of all the simple known steady equilibria, and how to obtain a super-consistent approximation for smooth non-trivial moving equilibria. The treatment of the wetting/drying interface is also discussed, giving formal conditions for the preservation of the non-negativity of the depth for a particular case, based on a nonlinear variant of a Lax–Friedrichs type scheme. The approach is analyzed and tested thoroughly. The quality of the numerical results shows the interest in the proposed approach over previously proposed schemes, in terms of accuracy and efficiency.

International audience; This article describes a discontinuous implementation of residual distribution for shallow-water flows. The emphasis is put on the space-time implementation of residual distribution for the time-dependent system of equations with discontinuity in time only. This lifts the time-step restriction that even implicit continuous residual distribution schemes invariably suffer from, and thus leads to an unconditionally stable discretisation. The distributions are the space-time variants of the upwind distributions for the steady-state system of equations and are designed to satisfy the most important properties of the original mathematical equations: positivity, linearity preservation, conservation and hydrostatic balance. The purpose of the several numerical examples presented in this article is twofold. First, to show that the discontinuous numerical discretisation does indeed exhibit all the desired properties when applied to the shallow-water equations. Second, to investigate how much the time step can be increased without adversely affecting the accuracy of the scheme and whether this translates into gains in computational efficiency. Comparison to other existing residual distribution schemes is also provided to demonstrate the improved performance of the scheme

In this paper, we investigate the nonlinear properties of Boussinesq models. In particular, we consider the wave shoaling obtained in physical regimes which go from linear to weakly nonlinear, to the wave breaking limit. For a given asymptotic accuracy in terms of dispersion and nonlinearity, we consider two families of models: the first depending on derivatives of the velocity, the second on derivatives of the volume flux. We show that, while linear dispersion and linear shoaling characteristics are strongly dependent on the type of dispersive terms introduced, when approaching the nonlinear regime the only influencing factor is whether the model is in amplitude-velocity of amplitude-flux form. We investigate these two alternative formulations of several known models, and propose a new model with a compact differential form, and the same linear characteristics of the model of Nwogu. The nonlinear shoaling properties of the models are investigated numerically showing that inside one given family, all the models have almost identical behaviour.

SUMMARY We are interested in the numerical approximation of steady scalar convection–diffusion problems by means of high order schemes called Residual Distribution schemes. In the inviscid case, one can develop nonlinear Residual Distribution schemes that are nonoscillatory, even in the case of very strong discontinuities, while having the most possible compact stencil, on hybrid unstructured meshes. This paper proposes and compare extensions of these schemes for the convection–diffusion problem. This methodology, in particular in terms of accuracy, is evaluated on problem with exact solutions. Its nonoscillatory behavior is tested against the Smith and Hutton problem. Copyright © 2012 John Wiley & Sons, Ltd.

We construct a third order multidimensional upwind residual distribution scheme for the system of the Navier–Stokes equations. The underlying approximation is obtained using standard P2 Lagrange finite elements. To discretise the inviscid component of the equations, each element is divided in sub-elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub-element. The residuals are distributed to the nodes of each sub-element in a multi-dimensional upwind way. To obtain a discretisation of the viscous terms consistent with this multi-dimensional upwind approach, we make use of a Petrov–Galerkin analogy. The analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms. The performance of this high-order method is tested on flows with high and low Reynolds number.

International audience; In this paper we consider the very high order approximation of solutions of the Euler equations. We present a systematic generalization of the residual distribution method of (Abgrall, J.ComputPhys 2006) to very high order of accuracy, by extending the preliminary work discussed in (Abgrall, Larat, Ricchiuto, Tave, Computers and Fluids 2009) to systems and hybrid meshes. We present extensive numerical validation for the third and fourth order cases with Lagrange finite elements. In particular, we demonstrate that we both have a non-oscillatory behavior, even for very strong shocks and complex flow patterns, and the expected accuracy on smooth problems.

In this paper, we construct spatially consistent explicit second order discretizations for time dependent hyperbolic problems, starting from a given residual distribution (RD) discrete approximation of the steady operator. We review the existing knowledge on consistent RD mass matrices and highlight the relations between different definitions. We then introduce our explicit approach which is based on three main ingredients: first recast the RD discretization as a stabilized Galerkin scheme, then use a shifted time discretization in the stabilization operator, and lastly apply high order mass lumping on the Galerkin component of the discretization. The discussion is particularly relevant for schemes of the residual distribution type [18,3] which we will use for all our numerical experiments. However, similar ideas can be used in the context of residual-based finite volume discretizations such as the ones proposed in [14,12]. The schemes are tested on a wide variety of classical problems confirming the theoretical expectations.

We propose a stabilized Residual Distribution (RD) scheme for the simulation of shallow water flows. The final discretization is obtained combining the stabilized RD approach proposed in (Abgrall, J. Comp. Phys. 214, 2006) and (Ricchiuto and Abgrall, ICCFD4, Springer-Verlag 2006), with the conservative formulation already used in (Ricchiuto et al., J. Comp. Phys. 222, 2007) to simulate shallow water flows. The scheme proposed is a nonlinear variant of a Lax-Friedrichs type discretization. It is well balanced, it actually yields second-order of accuracy in smooth areas, and it preserves the positivity of the height of the water in presence of dry areas. This is made possible by the residual character of the discretization, by properly adapting the stabilization operators proposed in (Abgrall, J. Comp. Phys. 214, 2006) and (Ricchiuto and Abgrall, ICCFD4, Springer-Verlag, 2006), and thanks to the positivity preserving character of the underlying Lax-Friedrichs scheme. We demonstrate the properties of the discretization proposed on a wide variety of tests.

International audience; We discuss preliminary results on the construction of uniformly high-order residual distribution $(\cal RD)$ type discretizations for steady advection-diffusion on unstructured grids. A properly designed scaling of the $(\cal RD)$ upwind stabilization with the physical viscosity allows to obtain schemes with uniform and arbitrary accuracy, on a very compact stencil. Second- and third-order examples are given to illustrate the potential of the approach.

We consider the second-order accurate numerical solution of general time-dependent hyperbolic conservation laws over unstructured grids in the framework of the Residual Distribution method. In order to achieve full conservation of the linear, monotone and first-order space-time schemes of (Csik et al., 2003) and (Abgrall et al., 2000), we extend the conservative residual distribution (CRD) formulation of (Csik et al., 2002) to prismatic space-time elements. We then study the design of second-order accurate and monotone schemes via the nonlinear mapping of the local residuals of linear monotone schemes. We derive sufficient and necessary conditions for the well-posedness of the mapping. We prove that the schemes obtained with the CRD formulation satisfy these conditions by construction. Thus the nonlinear schemes proposed in this paper are always well defined. The performance of the linear and nonlinear schemes are evaluated on a series of test problems involving the solution of the Euler equations and of a two-phase flow model. We consider the resolution of strong shocks and complex interacting flow structures. The results demonstrate the robustness, accuracy and non-oscillatory character of the proposed schemes. d schemes.

The class of multidimensional upwind residual distribution (RD) schemes has been developed in the past decades as an attractive alternative to the finite volume (FV) and finite element (FE) approaches. Although they have shown superior performances in the simulation of steady two-dimensional and three-dimensional inviscid and viscous flows, their extension to the simulation of unsteady flow fields is still a topic of intense research [ICCFD2, International Conference on Computational Fluid Dynamics 2, Sydney, Australia, 15–19 July 2002; M. Mezine, R. Abgrall, Upwind multidimensional residual schemes for steady and unsteady flows]. Recently the space–time RD approach has been developed by several researchers [Int. J. Numer. Methods Fluids 40 (2002) 573; J. Comput. Phys. 188 (2003) 16; A.G. Csik, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002; J. Comput. Phys. 188 (2003) 16; R. Abgrall; M. Mezine, Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems] which allows to perform second order accurate unsteady inviscid computations. In this paper we follow the work done in [Int. J. Numer. Methods Fluids 40 (2002) 573; A.G. Csik, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002]. In this approach the space–time domain is discretized and solved as a (d+1)-dimensional problem, where d is the number of space dimensions. In [Int. J. Numer. Methods Fluids 40 (2002) 573; A.G. Csik, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002] it is shown that thanks to the multidimensional upwinding of the RD method, the solution of the unsteady problem can be decoupled into sub-problems on space–time slabs composed of simplicial elements, allowing to obtain a true time marching procedure. Moreover, the method is implicit and unconditionally stable for arbitrary large time-steps if positive RD schemes are employed. We present further development of the space–time approach of [Int. J. Numer. Methods Fluids 40 (2002) 573; A.G. Csik, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002] by extending it to laminar viscous flow computations. A Petrov–Galerkin treatment of the viscous terms [Project Report 2002-06, von Karman Institute for Fluid Dynamics, Belgium, 2002; J. Dobes, Implicit space–time method for laminar viscous flow], consistent with the space–time formulation has been investigated, implemented and tested. Second order accuracy in both space and time was observed on unstructured triangulation of the spatial domain. The solution is obtained at each time-step by solving an implicit non-linear system of equations. Here, following [Int. J. Numer. Methods Fluids 40 (2002) 573; A.G. Csik, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002], we formulate the solution of this system as a steady state problem in a pseudo-time variable. We discuss the efficiency of an explicit Euler forward pseudo-time integrator compared to the implicit Euler. When applied to viscous computation, the implicit method has shown speed-ups of more than a factor 50 in terms of computational time.

This paper summarizes the cumulative work undertaken in the frame of the EU shared-cost action “ASTAR Project”—the current status and future perspectives in the field of advanced numerical simulation of three-dimensional two-phase flow processes. This 3-year running project, which started in September 2000, involves seven partner institutes from around Europe. Specific emphasis is given to the further development of characteristic-based upwind differencing (also called “hyperbolic”) numerical methods and their application to transient two-phase flow. The paper summarizes the common basis adopted for the physical and mathematical modelling of two-phase flow in the form of a single-pressure “two-fluid” model and the various numerical solution techniques developed by the partners. Several benchmark exercises are presented which have been used as verification and assessment procedures for comparing the different modelling and numerical approaches. Comments on the suitability, accuracy, numerical stability, algorithmic robustness and computational efficiency serve as indicators for the possible extension of these methods to future code development activities. Two further tasks of the ASTAR project dealt with the production of high quality experimental field data in the LINX facility of PSI, for the validation of CFD models for two-phase bubbly flow, and the coupling of a two-phase CFD module with a system code. Details of these tasks have been published separately, and will not be recalled in this paper.

In the present paper we consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method. We propose a new formulation of the first-order linear, optimal positive N scheme, relying on a contour integration of the convective fluxes over the boundaries of an element. Full conservation is achieved for arbitrary flux functions, while the robustness and the monotone shock capturing of the original N scheme is retained. The new variant of the N scheme is combined with the conservative second-order linear LDA scheme to obtain a nonlinear second-order monotone B scheme. The performance of the new residual distribution schemes is evaluated on problems governed by the Euler equations. As an application to a more complex system of conservation laws lacking an exact conservative linearization, we solve the ideal magnetohydrodynamics equations in two spatial dimensions.

In this paper we consider the very high order approximation of solutions of the Euler equations. We present a systematic generalization of the Residual Distribution method of [8] to very high order of accuracy, by extending the preliminary work discussed in [17]. We present extensive numerical validation for the third and fourth order cases with Lagrange finite elements. In particular, we demonstrate that we an both have a non oscillatory behavior, even for very strong shocks and complex flow patterns, and the expected accuracy on smooth problems. We also extend the scheme to laminar viscous problems.

Despites the progress made in CFD over the years, the methods still need improvement because the computed flow problems are becoming more and more complex. One of the way to improve methods is to increase accuracy. In this paper, we present a numerical scheme that can handle unstructured meshes, have a very compact stencil (hence easy to parallelize) and is non oscillatory. This paper presents the third order version of the method, but a priori any order of accuracy can be achieved. This goals can also be achieved in principle with the Discontinous Galerkin schemes; but our method is a non DG one. It uses conformal finite elements, hence enabling a considerable decrease of degrees of freedom, especially in 3D, with respects to the DG schemes. We first sketch design principles, then describe the scheme for the Euler and Navier Stokes equations. Two and three dimensional examples are given.

We consider the discretization of the time dependent hyperbolic problem $$\frac{\partial{u}}{\partial{t}} + \nabla \cdot \mathbf{\mathcal{F}}(\mathbf{u}) = {0} \mathrm{on} \mathit\Omega \times [0,t_f] \subset \mathbb{R}^2 \times \mathbb{R}^+$$ (1) on unstructured grids. We present residual distribution \((\mathcal{RD})\) schemes which (i) give non-oscillatory solutions, (ii) are second order accurate by construction, and (iii) lead to well-posed algebraic problems, that is, they ultimately lead to linear systems Ax = y, with A invertible. How to construct nonlinear \((\mathcal{RD})\) satisfying (i) and (ii) is known for some time [3]. However, it is the satisfaction of (iii) that ensures that a (unique) discrete solution exists, and that second order of accuracy is actually obtained in practice (convergence).

We review a class of compact methods to approximate steady solutions to $$\frac{\partial u}{\partial t}+\nabla\cdot{\cal F}(u)=\nabla\cdot(\nu\nabla u)\quad \forall(x,y)\in\Omega$$ (1)

We introduce monotone first order fluctuation splitting schemes for solving hyperbolic systems on arbitrary finite elements, thereby generalizing the N-scheme previously proposed for linear P1 triangles. Conservation is retained by relaxing on strict monotonicity, using a simple method based on contour integration over the element boundaries. Numerical examples are given for the Euler equations solved on Q1 elements for applications ranging from transonic to hypersonic regimes.

Excellent lifetimes make Self-healing Ceramic-Matrix Composites (CMCs) are promising candidates for jet engine hot parts. These composites are constituted of a 3D arrangement of SiC fiber tows infiltrated by a multilayer matrix. A pyrocarbon interphase acts as a crack deviator, SiC matrix layers bring stiffness, and boron-containing phases are able to produce above 450°C a liquid oxide preventing further oxidation by a diffusion barrier effect. This paper introduces an image-based numerical simulation of the self-healing mechanism under oxygen gas. Existing 0D or 1D models give the time evolution of the oxygen concentration in the weakest fiber and deduce from it a global lifetime through an oxygen-controlled slow crack growth rate law. We propose an approach in which the resolution domain is a 2D FE mesh directly obtained from transverse images of a tow containing the crack. Oxygen diffusion, carbon consumption around the fibers, and conversion of boron carbide into boron oxide are simulated. The model solves mass balances in terms of heights of oxygen (gaseous or dissolved), liquid oxide, pyrocarbon, and boron-containing phase. All the heights are considered perpendicular to the image plane (thin layer approximation), and represent the unit-volume (per square meter) occupied by each phase. Preliminary results on images containing several dozens of fibers and a multilayer matrix are discussed.; Une importante durée de vie fait des composites à matrice céramique auto cicatrisante (CMCs) les candidats parfaits pour utilisation dans les composantes à haute température des moteur aéronautiques du future. Ce type de composites sont constitués par des rangement 3D de fibres de SiC infiltrées par une matrice multicouche. Une interphase de pyrocarbone protège les fibres en déviant la fissure, des couches de SiC dans la matrice la rendent plus rigide, et des phases qui contiennent du bore produisent à 450°C un oxyde liquide qui protège les fibres par un effet de barrière diffusive. Ce papier présente une modélisation numérique de ce phénomène de auto cicatrisation en présence d'oxygène. Les modèles 0D et 1D qui existent en littérature donnent une approximation de l'évolution de la concentration d'oxygène en correspondance de la fibre la plus faible et permettent d'en déduire une durée de vie globale via un mécanisme de croissance de la fissure contrôlée par l'oxygène. On propose une approche dans laquelle le domaine de calcul est un maillage éléments finis 2D, obtenu directement par des images de fils fissurés. La diffusion de l'oxygène, l'oxydation du pyrocarbone et la production d'oxyde liquide sont simulés. Le modèle résout des bilans de masse en termes de hauteur d'oxygène, d'oxyde de bore et de pyrocarbone. Ces hauteurs sont considérées comme étant perpendiculaires à la fissure et représentent le volume unitaire (par unité de surface) occupé par chaque phase. Résultats préliminaires sur des images qui contiennent des dizaines de fibres et une matrice multicouche sont présentés.

International audience

International audience; This work has been performed by a French national consortium within the framework of the nationalproject Tandem, with aim to improve knowledge about tsunami risk on the French coasts. Workpackage#1 of this project was the opportunity to build a database of benchmark cases to assess the capabilitiesof 18 codes, solving various set of equations with different numerical methods. 14 test cases were definedfrom the existing literature with validation data from reference simulations, theoretical solutions or lab experiments.They cover the main stages of tsunami life: 1) generation, 2) propagation, 3) run-up and submersion,and 4) impact. For each case several of the numerical codes were compared in order to identify the forces andweaknesses of the models, to quantify the errors that these models may induce, to compare the various modellingmethods, and to provide users with recommendations for practical studies. In this paper, 3 representativecases are selected and presented with an analysis of the results.

In recent years, the Shifted Boundary Method (SBM) have gained interest due to the its ability to handlecomplex domains through embedded domain computations. The SBM address the problem of avoidingsmall cut cells and makes the meshing task close to trivial. The key feature - and the expense - ofthe SBM is how the boundary conditions (BCs) are applied on a surrogate/approximate boundary viathe use of Taylor expansions to ensure that the convergence rates of the overall discrete formulation ispreserved, see [1,2]. This original work by Main & Scovazzi was exploiting the classical - second-orderaccurate - Finite Element Method (FEM), however, higher-order contributions have recently been made,see [3,4]. One high-order numerical method is the Galerkin-formulated Spectral Element Method (SEM)[5] that can be viewed as a multi-domain version of the single-domain polynomial spectral method.We present a SEM-based model combined with the SBM for solving the Poisson equation in 2D ondifferent domains/geometries with various BCs. Convergence studies are performed to establish thelegitimacy of the work, including considerations of matrix conditioning when imposing Dirichlet BCsweakly via Nitsche’s variational form. Also, the presented work is free of higher-order derivatives fromthe aforementioned Taylor expansions, as the formulation utilizes the polynomial behaving nature of thebasis functions