Stable Perfectly Matched Layers with Lorentz transformation for the convected Helmholtz equation

International audiencePerfectly Matched Layers (PMLs) appear as a popular alternative to non-reflecting boundary conditions for wave-type problems. The core idea is to extend the computational domain by a fictitious layer with specific absorption properties such that the wave amplitude decays significantly and does not produce back reflections. In the context of convected acoustics, it is well-known that PMLs are exposed to stability issues in the frequency and time domain. It is caused by a mismatch between the phase velocity on which the PML acts, and the group velocity which carries the energy of the wave. The objective of this study is to take advantage of the Lorentz transformation in order to design stable perfectly matched layers for generally shaped convex domains in a uniform mean flow of arbitrary orientation. We aim at presenting a pedagogical approach to tackle the stability issue. The robustness of the approach is also demonstrated through several two-dimensional high-order finite element simulations of increasing complexity

A Non-Overlapping Schwarz Domain Decomposition Method with High-Order Finite Elements for Flow Acoustics

International audienceA non-overlapping domain decomposition method is proposed to solve large-scale finite element models for the propagation of sound with a background mean flow. An additive Schwarz algorithm is used to split the computational domain into a collection of sub-domains, and an iterative solution procedure is formulated in terms of unknowns defined on the interfaces between sub-domains. This approach allows to solve large-scale problems in parallel with only a fraction of the memory requirements compared to the standard approach which is to use a direct solver for the complete problem. While domain decomposition techniques have been used extensively for Helmholtz problems, this is the first application to aero-acoustics. The optimized Schwarz formulation is extended to the linearized potential theory for sound waves propagating in a potential base flow. A high-order finite element method is used to solve the governing equations in each sub-domain, and well-designed interface conditions based on local approximations of the Dirichlet-to-Neumann map are used to accelerate the convergence of the iterative procedure. The method is assessed on an academic test case and its benefit demonstrated on a realistic turbofan engine intake configuration

Conditions aux limites non-réfléchissantes et méthodes de décomposition de domaine pour l'acoustique industrielle en présence d'écoulement

This PhD project is devoted to non-overlapping Schwarz domain decomposition methods for the resolution of high frequency flow acoustics problems of industrial relevance. Time-harmonic solvers are difficult to parallelize due to their high-oscillatory behaviour, and current solvers quickly reach an upper frequency limit dictated by the available computer memory. Non-overlapping Schwarz methods split the domain into subdomains at the continuous level and provide a suitable setting for distributed memory parallelization. The problem is solved iteratively on the interface unknowns, where the keystone for quick convergence relies on appropriate transmission conditions. The first part of this thesis is devoted to the design of transmission operators tailored to convected and heterogeneous time-harmonic wave propagation. To this end we study two non-reflecting boundary techniques that provide local approximations to the Dirichlet-to-Neumann operator. On the one hand, Absorbing Boundary Conditions are designed based on microlocal analysis and pseudodifferential calculus. On the other hand, the convected acoustic stability issue is addressed for Perfectly Matched Layers in convex domains with Lorentz transformation. The second part of this thesis describes how to adapt a generic domain decomposition framework to flow acoustics, and applies the newly designed transmission conditions to simple academic problems. We explain the relation between the non-overlapping Schwarz formulation and an algebraic block LU factorization of the problem. Finally we propose a parallel implementation of the method and show the benefit of the approach for the three-dimensional noise radiation of a high by-pass ratio turbofan engine intake.Ce travail de thèse est consacré aux méthodes de décomposition de domaine de Schwarz sans recouvrement pour la résolution de problèmes industriels hautes fréquences d'acoustique en écoulement. Les méthodes de résolution en régime harmonique sont difficiles à paralléliser en raison de leur caractère oscillatoire, si bien que les méthodes actuelles sont limitées par une fréquence maximale, imposée par la mémoire disponible de l'ordinateur. Les méthodes de Schwarz sans recouvrement divisent le domaine en sous-domaines d'un point de vue continu et fournissent un cadre approprié en vue d'une parallélisation à mémoire distribuée. Le problème est résolu de manière itérative sur les inconnues d'interface, où la convergence rapide repose sur des conditions de transmission appropriées. La première partie de cette thèse est consacrée à la conception d'opérateurs de transmission adaptés à la propagation d'ondes harmoniques en milieu convecté et hétérogène. Dans ce cadre nous étudions deux catégories de conditions aux limites non-réfléchissantes qui fournissent des approximations locales de l'opérateur Dirichlet-to-Neumann. Dans un premier temps, des conditions aux limites absorbantes sont conçues basées sur l'analyse microlocale et le calcul pseudodifférentiel. Dans un second temps, la problématique de la stabilité acoustique en écoulement des couches parfaitement adaptées est abordée pour des domaines convexes par la transformation de Lorentz. La deuxième partie de cette thèse étend une méthode générique de décomposition de domaine à des problèmes d'acoustique en écoulement, et applique les conditions de transmission préalablement étudiées à des problèmes académiques simples. Nous expliquons le lien entre la méthode de Schwarz sans recouvrement et une factorisation algébrique LU par blocs du problème. Enfin, nous proposons une mise en œuvre parallèle et montrons l'intérêt de l'approche au rayonnement acoustique tridimensionnel de l'avant d'un turboréacteur d'avion

Conditions aux limites non-réfléchissantes et méthodes de décomposition de domaine pour l'acoustique industrielle en présence d'écoulement

Ce travail de thèse est consacré aux méthodes de décomposition de domaine de Schwarz sans recouvrement pour la résolution de problèmes industriels hautes fréquences d'acoustique en écoulement. Les méthodes de résolution en régime harmonique sont difficiles à paralléliser en raison de leur caractère oscillatoire, si bien que les méthodes actuelles sont limitées par une fréquence maximale, imposée par la mémoire disponible de l'ordinateur. Les méthodes de Schwarz sans recouvrement divisent le domaine en sous-domaines d'un point de vue continu et fournissent un cadre approprié en vue d'une parallélisation à mémoire distribuée. Le problème est résolu de manière itérative sur les inconnues d'interface, où la convergence rapide repose sur des conditions de transmission appropriées. La première partie de cette thèse est consacrée à la conception d'opérateurs de transmission adaptés à la propagation d'ondes harmoniques en milieu convecté et hétérogène. Dans ce cadre nous étudions deux catégories de conditions aux limites non-réfléchissantes qui fournissent des approximations locales de l'opérateur Dirichlet-to-Neumann. Dans un premier temps, des conditions aux limites absorbantes sont conçues basées sur l'analyse microlocale et le calcul pseudodifférentiel. Dans un second temps, la problématique de la stabilité acoustique en écoulement des couches parfaitement adaptées est abordée pour des domaines convexes par la transformation de Lorentz. La deuxième partie de cette thèse étend une méthode générique de décomposition de domaine à des problèmes d'acoustique en écoulement, et applique les conditions de transmission préalablement étudiées à des problèmes académiques simples. Nous expliquons le lien entre la méthode de Schwarz sans recouvrement et une factorisation algébrique LU par blocs du problème. Enfin, nous proposons une mise en œuvre parallèle et montrons l'intérêt de l'approche au rayonnement acoustique tridimensionnel de l'avant d'un turboréacteur d'avion.This PhD project is devoted to non-overlapping Schwarz domain decomposition methods for the resolution of high frequency flow acoustics problems of industrial relevance. Time-harmonic solvers are difficult to parallelize due to their high-oscillatory behaviour, and current solvers quickly reach an upper frequency limit dictated by the available computer memory. Non-overlapping Schwarz methods split the domain into subdomains at the continuous level and provide a suitable setting for distributed memory parallelization. The problem is solved iteratively on the interface unknowns, where the keystone for quick convergence relies on appropriate transmission conditions. The first part of this thesis is devoted to the design of transmission operators tailored to convected and heterogeneous time-harmonic wave propagation. To this end we study two non-reflecting boundary techniques that provide local approximations to the Dirichlet-to-Neumann operator. On the one hand, Absorbing Boundary Conditions are designed based on microlocal analysis and pseudodifferential calculus. On the other hand, the convected acoustic stability issue is addressed for Perfectly Matched Layers in convex domains with Lorentz transformation. The second part of this thesis describes how to adapt a generic domain decomposition framework to flow acoustics, and applies the newly designed transmission conditions to simple academic problems. We explain the relation between the non-overlapping Schwarz formulation and an algebraic block LU factorization of the problem. Finally we propose a parallel implementation of the method and show the benefit of the approach for the three-dimensional noise radiation of a high by-pass ratio turbofan engine intake

Non-reflecting boundary conditions and domain decomposition methods for industrial flow acoustics

This PhD project is devoted to non-overlapping Schwarz domain decomposition methods for the resolution of high frequency flow acoustics problems of industrial relevance. Time-harmonic solvers are difficult to parallelize due to their high-oscillatory behaviour, and current solvers quickly reach an upper frequency limit dictated by the available computer memory. Non-overlapping Schwarz methods split the domain into subdomains at the continuous level and provide a suitable setting for distributed memory parallelization. The problem is solved iteratively on the interface unknowns, where the keystone for quick convergence relies on appropriate transmission conditions. The first part of this thesis is devoted to the design of transmission operators tailored to convected and heterogeneous time-harmonic wave propagation. To this end we study two non-reflecting boundary techniques that provide local approximations to the Dirichlet-to-Neumann operator. On the one hand, Absorbing Boundary Conditions are designed based on microlocal analysis and pseudodifferential calculus. On the other hand, the convected acoustic stability issue is addressed for Perfectly Matched Layers in convex domains with Lorentz transformation. The second part of this thesis describes how to adapt a generic domain decomposition framework to flow acoustics, and applies the newly designed transmission conditions to simple academic problems. We explain the relation between the non-overlapping Schwarz formulation and an algebraic block LU factorization of the problem. Finally we propose a parallel implementation of the method and show the benefit of the approach for the three-dimensional noise radiation of a high by-pass ratio turbofan engine intake

Construction and Numerical Assessment of Local Absorbing Boundary Conditions for Heterogeneous Time-Harmonic Acoustic Problems

International audienceThis article is devoted to the derivation and assessment of local Absorbing Boundary Conditions (ABCs) for numerically solving heterogeneous time-harmonic acoustic problems. To this end, we develop a strategy inspired by the work of Engquist and Majda to build local approximations of the Dirichlet-to-Neumann operator for heterogeneous media, which is still an open problem. We focus on three simplified but characteristic examples of increasing complexity to highlight the strengths and weaknesses of the proposed ABCs: the propagation in a duct with a longitudinal variation of the speed of sound, the propagation in a non-uniform mean flow using a convected wave operator, and the propagation in a duct with a transverse variation of the speed of sound and density. For each case, we follow the same systematic approach to construct a family of local ABCs and explain their implementation in a high-order finite element context. Numerical simulations allow to validate the accuracy of the ABCs, and to give recommendations for the tuning of their parameters

A Non-Overlapping Schwarz Domain Decomposition Method with High-Order Finite Elements for Flow Acoustics

A non-overlapping domain decomposition method is proposed to solve large-scale finite element models for the propagation of sound with a background mean flow. An additive Schwarz algorithm is used to split the computational domain into a collection of sub-domains, and an iterative solution procedure is formulated in terms of unknowns defined on the interfaces between sub-domains. This approach allows to solve large-scale problems in parallel with only a fraction of the memory requirements compared to the standard approach which is to use a direct solver for the complete problem. While domain decomposition techniques have been used extensively for Helmholtz problems, this is the first application to aero-acoustics. The optimized Schwarz formulation is extended to the linearized potential theory for sound waves propagating in a potential base flow. A high-order finite element method is used to solve the governing equations in each sub-domain, and well-designed interface conditions based on local approximations of the Dirichlet-to-Neumann map are used to accelerate the convergence of the iterative procedure. The method is assessed on an academic test case and its benefit demonstrated on a realistic turbofan engine intake configuration