This paper introduces algorithms to select/design kernels in Gaussian process regression/kriging ... more This paper introduces algorithms to select/design kernels in Gaussian process regression/kriging surrogate modeling techniques. We adopt the setting of kernel method solutions in ad hoc functional spaces, namely Reproducing Kernel Hilbert Spaces (RKHS), to solve the problem of approximating a regular target function given observations of it, i.e. supervised learning. A first class of algorithms is kernel flow, which was introduced in the context of classification in machine learning. It can be seen as a cross-validation procedure whereby a “best” kernel is selected such that the loss of accuracy incurred by removing some part of the dataset (typically half of it) is minimized. A second class of algorithms is called spectral kernel ridge regression, and aims at selecting a “best” kernel such that the norm of the function to be approximated is minimal in the associated RKHS. Within Mercer’s theorem framework, we obtain an explicit construction of that “best” kernel in terms of the main features of the target function. Both approaches of learning kernels from data are illustrated by numerical examples on synthetic test functions, and on a classical test case in turbulence modeling validation for transonic flows about a two-dimensional airfoil.
INTER-NOISE and NOISE-CON Congress and Conference Proceedings, 2021
In the context of aircraft noise reduction in varied applications where a cold or hot shear grazi... more In the context of aircraft noise reduction in varied applications where a cold or hot shear grazing flow is present (i.e., engine nacelle, combustion chamber, jet pump, landing gear), improved acoustic liner solutions are being sought. This is particularly true in the low-frequency regime, where space constraints limit the efficiency of conventional liner technology. Therefore, liner design must take into account the dimensional and phenomenological characteristics of constituent materials, assembly specifications and industrial requirements involving multiphysical phenomena. To perform the single/multi-objective optimization of complex meta-surface liner candidates, a software platform coined OPAL (OPtimisation of Acoustic Liners) was developed. Its first goal is to allow the user to assemble a large panel of parallel/serial elementary acoustic layers along a given duct. Then, the physical properties of this liner can be optimized, relatively to weighted objectives, for a given flo...
We study the propagation of sound waves in a three-dimensional, infinite ambient flow with weak r... more We study the propagation of sound waves in a three-dimensional, infinite ambient flow with weak random fluctuations of the mean particle velocity and speed of sound. We more particularly address the regime where the acoustic wavelengths are comparable to the correlation lengths of the weak inhomogeneities--the so-called weak coupling limit. The analysis is carried on starting from the linearized Euler equations and the convected wave equation with variable density and speed of sound, which can be derived from the nonlinear Euler equations. We use a multi-scale expansion of the Wigner distribution of a velocity potential associated to the waves to derive a radiative transfer equation describing the evolution of the angularly resolved wave action in space/time phase space. The latter experiences convection, refraction and scattering when it propagates through the heterogeneous ambient flow, although the overall wave action is conserved. The convection and refraction phenomena are accounted for by the convective part of the transport equation and depend on the smooth variations of the ambient quantities. The scattering phenomenon is accounted for by the collisional part of the transport equation and depends on the cross-power spectral densities of the fluctuations of the ambient quantities at the wavelength scales. The refraction, phase shift, spectral broadening, and multiple scattering effects of the high-frequency regimes described in various previous publications are thus encompassed by the proposed model. The overall derivation is based on the interpretation of spatial-temporal Wigner transforms in terms of semiclassical operators in their standard quantization.
This paper is concerned with the development of imaging methods to localize sources or reflectors... more This paper is concerned with the development of imaging methods to localize sources or reflectors in inhomogeneous moving media with acoustic waves that have travelled through them. A typical example is the localization of broad-band acoustic sources in a turbulent jet flow for aeroacoustic applications. The proposed algorithms are extensions of Kirchhoff migration (KM) and coherent interferometry (CINT) which have been considered for smooth and randomly inhomogeneous quiescent media so far. They are constructed starting from the linearized Euler equations for the acoustic perturbations about a stationary ambient flow. A model problem for the propagation of acoustic waves generated by a fixed point source in an ambient flow with constant velocity is addressed. Based on this result imaging functions are proposed to modify the existing KM and CINT functions to account for the ambient flow velocity. They are subsequently tested and compared by numerical simulations in various configurations, including a synthetic turbulent jet representative of the main features encountered in actual jet flows.
Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation pro... more Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adap-tive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, we assume that the observational data arise from optimisation problems with partially controllable one-sided errors, and this setting alters several characteristics of the problem and opens natural algorithmic possibilities. Our algorithm uses imperfect evaluations of the target function to direct further evaluations of the target function either at new sites in the function's domain or to improve the quality of evaluations at already-evaluated sites. We establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method both to synthetic test cases and to a real-world example of uncertainty quantification for aerodynamic design.
We consider the scattering of acoustic waves emitted by an active source above a plane turbulent ... more We consider the scattering of acoustic waves emitted by an active source above a plane turbulent shear layer. The layer is modeled by a moving random medium with small spatial and temporal fluctuations of its mean velocity, and constant density and speed of sound. We develop a multi-scale perturbative analysis for the acoustic pressure field transmitted by the layer and derive its power spectral density when the correlation function of the velocity fluctuations is known. Our aim is to compare the proposed analytical model with some experimental results obtained for jet flows in open wind tunnels. We start with the Euler equations for an ideal fluid flow and linearize them about an ambient, unsteady inhomogeneous flow. We study the transmitted pressure field without fluctuations of the ambient flow velocity to obtain the Green's function of the unperturbed medium with constant characteristics. Then we use a Lippmann-Schwinger equation to derive an analytical expression of the transmitted pressure field, as a function of the velocity fluctuations within the layer. Its power spectral density is subsequently computed invoking a stationary-phase argument, assuming in addition that the source is time-harmonic and the layer is thin. We finally study the influence of the source tone frequency and ambient flow velocity on the power spectral density of the transmitted pressure field and compare our results with other analytical models and experimental data.
This paper is concerned with aircraft aeroelastic interactions and the propagation of parametric ... more This paper is concerned with aircraft aeroelastic interactions and the propagation of parametric uncertainties in numerical simulations using high-fidelity fluid flow solvers. More specifically, the influence of variable operational and structural parameters (random inputs) on the drag performance and deformation (outputs) of a flexible wing in transonic regime, is assessed. Because of the complexity of fluid flow solvers, non-intrusive uncertainty quantifica-tion techniques are favored. Polynomial surrogate models based on homogeneous chaos expansions in the random inputs are commonly considered in this respect. The polynomial expansion coefficients are constructed using either structured sampling sets of the input parameters, as Gauss quadrature nodes, or unstructured sampling sets, as in Monte-Carlo methods. In complex systems such as the advanced aeroelastic test case studied here, the output quantities of interest generally depend only weakly on the multiple cross-interactions between the random inputs. Consequently, only low-order polynomials significantly contribute to their surrogates, which thus have a sparse structure in the underlying polynomial bases. This feature prompts to use compressed sensing, or compressive sampling theory for the construction of the polynomial surrogates. The proposed methodology is non-adapted and considers unstructured sampling sets orders of magnitude smaller than the ones required by the usual techniques with structured sampling sets. It is illustrated in the present work for a moderately to high dimensional parametric space.
8th International Conference on Computational Stochastic Mechanics, 10-13 June 2018, Paros, Greece, 2019
In this paper we study aircraft aeroelastic interactions and the propagation of parametric uncert... more In this paper we study aircraft aeroelastic interactions and the propagation of parametric uncertainties in numerical simulations using high-fidelity fluid flow solvers. We more particularly address the influence of variable operational and structural parameters (random inputs) on the drag performance and shape (outputs) of a flexible wing in transonic regime. Polynomial surrogate models based on homogeneous chaos expansions in the random inputs are considered in this respect. The polynomial expansion coefficients are usually constructed by projection using either structured sampling sets of the input parameters, as Gauss quadrature nodes, or unstructured sampling sets, as in Monte-Carlo methods. However, in complex systems such as the advanced aeroelastic test case studied here, the output quantities of interest generally depend only weekly on the multiple cross-interactions between the random inputs. Consequently, only low-order polynomials significantly contribute to their surrogates, which thus have a sparse structure in the underlying polynomial bases. This feature prompts the use of compressed sensing for the construction of the polynomial surrogates by regression. This alternative methodology is non-adapted and considers unstructured sampling sets orders of magnitude smaller than the structured or unstructured sampling sets required in projection methods. It is illustrated in the present work for a moderately to high dimensional parametric space and an aeroelastic test case of industrial relevance.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
In this chapter the different polynomial chaos and stochastic collocation methodologies used with... more In this chapter the different polynomial chaos and stochastic collocation methodologies used within the UMRIDA project are compared. Guidelines for their use and applicability are formulated.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
This chapter describes the methodology used to construct Kriging-based surrogate models and their... more This chapter describes the methodology used to construct Kriging-based surrogate models and their possible application to uncertainty quantifica-tion and robust shape optimization. More particularly, a two-dimensional RAE 2822 airfoil at transonic speed is considered for which the shape of the baseline profile is altered by localized bumps of small amplitudes. The flow around the airfoil is then subjected to important changes compared to the baseline configuration. The aim of the surrogate is to assess their influence on the aerodynamic performance of the profile as quantified by its lift-to-drag ratio. An optimization analysis is subsequently carried out in order to extract the local extrema of this performance measure. Assigning some uncertainty to the bump amplitudes, it also reveals that the global maximum identified by a high-quality surrogate is not necessarily the most robust one. This example constitutes an interesting benchmark for testing uncertainty quantification and robust optimization strategies.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
In this chapter the basic principles of two methodologies for uncertainty quantification (UQ) are... more In this chapter the basic principles of two methodologies for uncertainty quantification (UQ) are discussed, namely the polynomial chaos method and the collocation method. UQ deals with the propagation of uncertainties through complex numerical models, and in the present context of the UMRIDA project, mostly computational fluid dynamics (CFD) codes. The focus is on non-intrusive methods implying that the model does not require any changes and can be used as a black box.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
This chapter is concerned with the construction of polynomial sur-rogates of complex configuratio... more This chapter is concerned with the construction of polynomial sur-rogates of complex configurations arising in computational fluid dynamics for the purpose of propagating uncertainties pertaining to geometrical and/or operational parameters. Generalized homogeneous chaos expansions are considered and different techniques for the non-intrusive reconstruction of the polynomial expansion coefficients are outlined. A sparsity-based reconstruction approach is more particularly emphasized since it benefits from the "sparsity-of-effects" trend commonly observed on global quantities of interest such as the aerodynamic coefficients of a profile. The overall framework is illustrated on a two-dimensional transonic turbulent flow around a RAE 2822 airfoil subjected to a variable free-stream Mach number, angle of attack, and relative thickness of the profile.
This paper is concerned with the construction of polynomial surrogates of complex models typicall... more This paper is concerned with the construction of polynomial surrogates of complex models typically arising in computational fluid dynamics for the purpose of propagating uncertainties pertaining to geometrical and/or operational parameters. Polynomial chaos expansions are considered and different techniques for the intrusive and non intrusive reconstruction of the polynomial expansion coefficients are outlined. A sparsity-based reconstruction approach is more particularly emphasized since it benefits from the "sparsity-of-effects" trend commonly observed on global quantities of interest such as the aerodynamic coefficients of a profile.
International Journal for Numerical Methods in Engineering, 2017
Because of the complexity of fluid flow solvers, non-intrusive uncertainty quantification techniq... more Because of the complexity of fluid flow solvers, non-intrusive uncertainty quantification techniques have been developed in aerodynamic simulations in order to compute the quantities of interest required in an optimization process, for example. The objective function is commonly expressed in terms of moments of these quantities, such as the mean, standard deviation, or even higher-order moments. Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect. The original approach of uncertainty quantification using polynomial chaos is however intrusive. It is based on a Galerkin-type formulation of the model equations to derive the governing equations for the polynomial expansion coefficients. Third-order, indeed fourth-order moments of the polynomials are needed in this analysis. Besides, both intrusive and non-intrusive approaches call for their computation provided that higher-order moments of the quantities of interest need be post-processed. In most applications they are evaluated by Gauss quadratures, and eventually stored for use throughout the computations. In this paper analytical formulas are rather considered for the moments of the continuous polynomials of the Askey scheme, so that they can be evaluated by quadrature-free procedures instead. Matlab codes have been developed for this purpose and tested by comparisons with Gauss quadratures.
An integral equation for the influence of spatial variability of the incident free field on seism... more An integral equation for the influence of spatial variability of the incident free field on seismic soil-structure interaction is proposed. The fundamental solution for the overall domain when the unit load is applied on the structure, is required. It is obtained by means of sub-structuring techniques and boundary integral equations using the Green tensors for homogeneous or horizon-tally stratified soil media. The effects of a non-stationary modulated random incident field are ad-dressed in terms of the instantaneous power spectral density of the structural response for a given coherency function of the free-field on the traction-free soil surface. For embedded foundations, this coherency function can be computed by a stochastic deconvolution method. 2 INTRODUCTION The analysis of the random linear vibrations of elastic structures under stationary stochastic loads is classically based on a discretization by the finite element method in order to compute a finite set of eigenmodes of...
This paper introduces algorithms to select/design kernels in Gaussian process regression/kriging ... more This paper introduces algorithms to select/design kernels in Gaussian process regression/kriging surrogate modeling techniques. We adopt the setting of kernel method solutions in ad hoc functional spaces, namely Reproducing Kernel Hilbert Spaces (RKHS), to solve the problem of approximating a regular target function given observations of it, i.e. supervised learning. A first class of algorithms is kernel flow, which was introduced in the context of classification in machine learning. It can be seen as a cross-validation procedure whereby a “best” kernel is selected such that the loss of accuracy incurred by removing some part of the dataset (typically half of it) is minimized. A second class of algorithms is called spectral kernel ridge regression, and aims at selecting a “best” kernel such that the norm of the function to be approximated is minimal in the associated RKHS. Within Mercer’s theorem framework, we obtain an explicit construction of that “best” kernel in terms of the main features of the target function. Both approaches of learning kernels from data are illustrated by numerical examples on synthetic test functions, and on a classical test case in turbulence modeling validation for transonic flows about a two-dimensional airfoil.
INTER-NOISE and NOISE-CON Congress and Conference Proceedings, 2021
In the context of aircraft noise reduction in varied applications where a cold or hot shear grazi... more In the context of aircraft noise reduction in varied applications where a cold or hot shear grazing flow is present (i.e., engine nacelle, combustion chamber, jet pump, landing gear), improved acoustic liner solutions are being sought. This is particularly true in the low-frequency regime, where space constraints limit the efficiency of conventional liner technology. Therefore, liner design must take into account the dimensional and phenomenological characteristics of constituent materials, assembly specifications and industrial requirements involving multiphysical phenomena. To perform the single/multi-objective optimization of complex meta-surface liner candidates, a software platform coined OPAL (OPtimisation of Acoustic Liners) was developed. Its first goal is to allow the user to assemble a large panel of parallel/serial elementary acoustic layers along a given duct. Then, the physical properties of this liner can be optimized, relatively to weighted objectives, for a given flo...
We study the propagation of sound waves in a three-dimensional, infinite ambient flow with weak r... more We study the propagation of sound waves in a three-dimensional, infinite ambient flow with weak random fluctuations of the mean particle velocity and speed of sound. We more particularly address the regime where the acoustic wavelengths are comparable to the correlation lengths of the weak inhomogeneities--the so-called weak coupling limit. The analysis is carried on starting from the linearized Euler equations and the convected wave equation with variable density and speed of sound, which can be derived from the nonlinear Euler equations. We use a multi-scale expansion of the Wigner distribution of a velocity potential associated to the waves to derive a radiative transfer equation describing the evolution of the angularly resolved wave action in space/time phase space. The latter experiences convection, refraction and scattering when it propagates through the heterogeneous ambient flow, although the overall wave action is conserved. The convection and refraction phenomena are accounted for by the convective part of the transport equation and depend on the smooth variations of the ambient quantities. The scattering phenomenon is accounted for by the collisional part of the transport equation and depends on the cross-power spectral densities of the fluctuations of the ambient quantities at the wavelength scales. The refraction, phase shift, spectral broadening, and multiple scattering effects of the high-frequency regimes described in various previous publications are thus encompassed by the proposed model. The overall derivation is based on the interpretation of spatial-temporal Wigner transforms in terms of semiclassical operators in their standard quantization.
This paper is concerned with the development of imaging methods to localize sources or reflectors... more This paper is concerned with the development of imaging methods to localize sources or reflectors in inhomogeneous moving media with acoustic waves that have travelled through them. A typical example is the localization of broad-band acoustic sources in a turbulent jet flow for aeroacoustic applications. The proposed algorithms are extensions of Kirchhoff migration (KM) and coherent interferometry (CINT) which have been considered for smooth and randomly inhomogeneous quiescent media so far. They are constructed starting from the linearized Euler equations for the acoustic perturbations about a stationary ambient flow. A model problem for the propagation of acoustic waves generated by a fixed point source in an ambient flow with constant velocity is addressed. Based on this result imaging functions are proposed to modify the existing KM and CINT functions to account for the ambient flow velocity. They are subsequently tested and compared by numerical simulations in various configurations, including a synthetic turbulent jet representative of the main features encountered in actual jet flows.
Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation pro... more Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adap-tive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, we assume that the observational data arise from optimisation problems with partially controllable one-sided errors, and this setting alters several characteristics of the problem and opens natural algorithmic possibilities. Our algorithm uses imperfect evaluations of the target function to direct further evaluations of the target function either at new sites in the function's domain or to improve the quality of evaluations at already-evaluated sites. We establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method both to synthetic test cases and to a real-world example of uncertainty quantification for aerodynamic design.
We consider the scattering of acoustic waves emitted by an active source above a plane turbulent ... more We consider the scattering of acoustic waves emitted by an active source above a plane turbulent shear layer. The layer is modeled by a moving random medium with small spatial and temporal fluctuations of its mean velocity, and constant density and speed of sound. We develop a multi-scale perturbative analysis for the acoustic pressure field transmitted by the layer and derive its power spectral density when the correlation function of the velocity fluctuations is known. Our aim is to compare the proposed analytical model with some experimental results obtained for jet flows in open wind tunnels. We start with the Euler equations for an ideal fluid flow and linearize them about an ambient, unsteady inhomogeneous flow. We study the transmitted pressure field without fluctuations of the ambient flow velocity to obtain the Green's function of the unperturbed medium with constant characteristics. Then we use a Lippmann-Schwinger equation to derive an analytical expression of the transmitted pressure field, as a function of the velocity fluctuations within the layer. Its power spectral density is subsequently computed invoking a stationary-phase argument, assuming in addition that the source is time-harmonic and the layer is thin. We finally study the influence of the source tone frequency and ambient flow velocity on the power spectral density of the transmitted pressure field and compare our results with other analytical models and experimental data.
This paper is concerned with aircraft aeroelastic interactions and the propagation of parametric ... more This paper is concerned with aircraft aeroelastic interactions and the propagation of parametric uncertainties in numerical simulations using high-fidelity fluid flow solvers. More specifically, the influence of variable operational and structural parameters (random inputs) on the drag performance and deformation (outputs) of a flexible wing in transonic regime, is assessed. Because of the complexity of fluid flow solvers, non-intrusive uncertainty quantifica-tion techniques are favored. Polynomial surrogate models based on homogeneous chaos expansions in the random inputs are commonly considered in this respect. The polynomial expansion coefficients are constructed using either structured sampling sets of the input parameters, as Gauss quadrature nodes, or unstructured sampling sets, as in Monte-Carlo methods. In complex systems such as the advanced aeroelastic test case studied here, the output quantities of interest generally depend only weakly on the multiple cross-interactions between the random inputs. Consequently, only low-order polynomials significantly contribute to their surrogates, which thus have a sparse structure in the underlying polynomial bases. This feature prompts to use compressed sensing, or compressive sampling theory for the construction of the polynomial surrogates. The proposed methodology is non-adapted and considers unstructured sampling sets orders of magnitude smaller than the ones required by the usual techniques with structured sampling sets. It is illustrated in the present work for a moderately to high dimensional parametric space.
8th International Conference on Computational Stochastic Mechanics, 10-13 June 2018, Paros, Greece, 2019
In this paper we study aircraft aeroelastic interactions and the propagation of parametric uncert... more In this paper we study aircraft aeroelastic interactions and the propagation of parametric uncertainties in numerical simulations using high-fidelity fluid flow solvers. We more particularly address the influence of variable operational and structural parameters (random inputs) on the drag performance and shape (outputs) of a flexible wing in transonic regime. Polynomial surrogate models based on homogeneous chaos expansions in the random inputs are considered in this respect. The polynomial expansion coefficients are usually constructed by projection using either structured sampling sets of the input parameters, as Gauss quadrature nodes, or unstructured sampling sets, as in Monte-Carlo methods. However, in complex systems such as the advanced aeroelastic test case studied here, the output quantities of interest generally depend only weekly on the multiple cross-interactions between the random inputs. Consequently, only low-order polynomials significantly contribute to their surrogates, which thus have a sparse structure in the underlying polynomial bases. This feature prompts the use of compressed sensing for the construction of the polynomial surrogates by regression. This alternative methodology is non-adapted and considers unstructured sampling sets orders of magnitude smaller than the structured or unstructured sampling sets required in projection methods. It is illustrated in the present work for a moderately to high dimensional parametric space and an aeroelastic test case of industrial relevance.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
In this chapter the different polynomial chaos and stochastic collocation methodologies used with... more In this chapter the different polynomial chaos and stochastic collocation methodologies used within the UMRIDA project are compared. Guidelines for their use and applicability are formulated.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
This chapter describes the methodology used to construct Kriging-based surrogate models and their... more This chapter describes the methodology used to construct Kriging-based surrogate models and their possible application to uncertainty quantifica-tion and robust shape optimization. More particularly, a two-dimensional RAE 2822 airfoil at transonic speed is considered for which the shape of the baseline profile is altered by localized bumps of small amplitudes. The flow around the airfoil is then subjected to important changes compared to the baseline configuration. The aim of the surrogate is to assess their influence on the aerodynamic performance of the profile as quantified by its lift-to-drag ratio. An optimization analysis is subsequently carried out in order to extract the local extrema of this performance measure. Assigning some uncertainty to the bump amplitudes, it also reveals that the global maximum identified by a high-quality surrogate is not necessarily the most robust one. This example constitutes an interesting benchmark for testing uncertainty quantification and robust optimization strategies.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
In this chapter the basic principles of two methodologies for uncertainty quantification (UQ) are... more In this chapter the basic principles of two methodologies for uncertainty quantification (UQ) are discussed, namely the polynomial chaos method and the collocation method. UQ deals with the propagation of uncertainties through complex numerical models, and in the present context of the UMRIDA project, mostly computational fluid dynamics (CFD) codes. The focus is on non-intrusive methods implying that the model does not require any changes and can be used as a black box.
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2019
This chapter is concerned with the construction of polynomial sur-rogates of complex configuratio... more This chapter is concerned with the construction of polynomial sur-rogates of complex configurations arising in computational fluid dynamics for the purpose of propagating uncertainties pertaining to geometrical and/or operational parameters. Generalized homogeneous chaos expansions are considered and different techniques for the non-intrusive reconstruction of the polynomial expansion coefficients are outlined. A sparsity-based reconstruction approach is more particularly emphasized since it benefits from the "sparsity-of-effects" trend commonly observed on global quantities of interest such as the aerodynamic coefficients of a profile. The overall framework is illustrated on a two-dimensional transonic turbulent flow around a RAE 2822 airfoil subjected to a variable free-stream Mach number, angle of attack, and relative thickness of the profile.
This paper is concerned with the construction of polynomial surrogates of complex models typicall... more This paper is concerned with the construction of polynomial surrogates of complex models typically arising in computational fluid dynamics for the purpose of propagating uncertainties pertaining to geometrical and/or operational parameters. Polynomial chaos expansions are considered and different techniques for the intrusive and non intrusive reconstruction of the polynomial expansion coefficients are outlined. A sparsity-based reconstruction approach is more particularly emphasized since it benefits from the "sparsity-of-effects" trend commonly observed on global quantities of interest such as the aerodynamic coefficients of a profile.
International Journal for Numerical Methods in Engineering, 2017
Because of the complexity of fluid flow solvers, non-intrusive uncertainty quantification techniq... more Because of the complexity of fluid flow solvers, non-intrusive uncertainty quantification techniques have been developed in aerodynamic simulations in order to compute the quantities of interest required in an optimization process, for example. The objective function is commonly expressed in terms of moments of these quantities, such as the mean, standard deviation, or even higher-order moments. Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect. The original approach of uncertainty quantification using polynomial chaos is however intrusive. It is based on a Galerkin-type formulation of the model equations to derive the governing equations for the polynomial expansion coefficients. Third-order, indeed fourth-order moments of the polynomials are needed in this analysis. Besides, both intrusive and non-intrusive approaches call for their computation provided that higher-order moments of the quantities of interest need be post-processed. In most applications they are evaluated by Gauss quadratures, and eventually stored for use throughout the computations. In this paper analytical formulas are rather considered for the moments of the continuous polynomials of the Askey scheme, so that they can be evaluated by quadrature-free procedures instead. Matlab codes have been developed for this purpose and tested by comparisons with Gauss quadratures.
An integral equation for the influence of spatial variability of the incident free field on seism... more An integral equation for the influence of spatial variability of the incident free field on seismic soil-structure interaction is proposed. The fundamental solution for the overall domain when the unit load is applied on the structure, is required. It is obtained by means of sub-structuring techniques and boundary integral equations using the Green tensors for homogeneous or horizon-tally stratified soil media. The effects of a non-stationary modulated random incident field are ad-dressed in terms of the instantaneous power spectral density of the structural response for a given coherency function of the free-field on the traction-free soil surface. For embedded foundations, this coherency function can be computed by a stochastic deconvolution method. 2 INTRODUCTION The analysis of the random linear vibrations of elastic structures under stationary stochastic loads is classically based on a discretization by the finite element method in order to compute a finite set of eigenmodes of...
Outline of the talk:
1. Introduction. Need for Uncertainty Quantification
2. Probability basics, ... more Outline of the talk: 1. Introduction. Need for Uncertainty Quantification 2. Probability basics, Monte-Carlo, surrogate-based Monte-Carlo 3. Non-intrusive polynomial methods for 1D / tensorial nD propagation 4. Introduction to Smolyak’s sparse quadratures 5. Examples of application
The maturity of Computational Fluid Dynamics (CFD) and the variability of operational and geometr... more The maturity of Computational Fluid Dynamics (CFD) and the variability of operational and geometrical parameters in fluid dynamics analysis and design lead to the development of Uncertainty Quantification (UQ). Among the numerous methods for UQ, these lecture notes describe the main features of Monte-Carlo and metamodel-based Monte-Carlo in Sect. 2, generalized Polynomial Chaos in Sect. 3, and Stochastic Collocation in Sect. 4. The broadly used sparse grid quadrature method of Smolyak is then presented Sect. 5. Lastly Sect. 6 introduces some concepts of variance analysis. Two applications of UQ to 2D and 3D RANS flows, that have been carried out at ONERA, are finally outlined in Sect. 7.
Lecture #7 of the elective course Advanced Structural Acoustics, taught in the Graduate Program "Modélisation et Simulation en Mécanique des Structures et Systèmes Couplés" (MS)2SC at University Paris-Saclay, 2018
In this lecture we outline the basics of structural acoustics. We derive the Rayleigh integral fo... more In this lecture we outline the basics of structural acoustics. We derive the Rayleigh integral for the radiation of acoustic waves by surfaces, and then specialize it to the case of infinite and finite plates with boundary and corner effects. Some fundamental notions are finally introduced: reflection and transmission of acoustic waves by slender structures, acoustic transparency, mass and stiffness laws.
Lecture #8 of the elective course Advanced Structural Acoustics, taught in the Graduate Program "Modélisation et Simulation en Mécanique des Structures et Systèmes Couplés" (MS)2SC at University Paris-Saclay, 2016
In this lecture we show how to compute the boundary and radiation impedance operators for arbitra... more In this lecture we show how to compute the boundary and radiation impedance operators for arbitrary structures coupled to an acoustic fluid at rest. More particularly, we address the issue of solving the external Neumann problem pertaining to the acoustic Helmholtz equation describing fluid motion in the frequency domain by boundary integral equations (BIEs). The difficulty raised by the irregular frequencies arising from the internal Neumann problem is fully accounted for by the proposed formulation. The symmetric BIE that is derived yields a constructive representation of the impedance operators added to the structure coupled to the outer fluid medium.
Lecture #5 of the elective course MG3409 - Statistical Energy Analysis, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2015
We establish the basic SEA (Statistical Energy Analysis) equations for coupled structural-acousti... more We establish the basic SEA (Statistical Energy Analysis) equations for coupled structural-acoustic systems from the results derived in the previous lectures. The underlying hypotheses are carefully reviewed, and we subsequently discuss some modeling issues. We focus on the definition of sub-systems in SEA and elaborate on the determination of the input parameters: modal densities, loss factors, coupling loss factors, and input powers.
Lecture #4 of the elective course MG3409 - Statistical Energy Analysis, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2015
The random vibrations of structural-acoustic systems are outlined with an emphasis on energetic i... more The random vibrations of structural-acoustic systems are outlined with an emphasis on energetic issues. The general case of two conservatively coupled sub-systems is first addressed. The interaction between an acoustic cavity and a vibrating structure is subsequently shown to be a particular case of the general situation examined beforehand. Secondly, the interaction of a structure with an external acoustic fluid at rest is reviewed. The three problems are analyzed with the objective to introduce the basic concepts of SEA for multiply coupled sub-systems, possibly including acoustic domains.
Lecture #3 of the elective course MG3409 - Statistical Energy Analysis, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2015
This lecture outlines the usual spectral analysis of the stationary response of continuous mechan... more This lecture outlines the usual spectral analysis of the stationary response of continuous mechanical systems excited by stationary random loads. Energetic issues are more particularly addressed, with an emphasis on the power balance.
Lecture #2 of the elective course MG3409 - Statistical Energy Analysis, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2015
Two elementary coupled problems studied in the early 60s are reviewed in order to introduce the b... more Two elementary coupled problems studied in the early 60s are reviewed in order to introduce the basic concepts of Statistical Energy Analysis (SEA). The first one considers conservatively coupled single oscillators. More particularly, the fundamental relation of SEA expressing the proportionality of the power flow between the oscillators with the difference of their mechanical energies, is derived. The second example considers a single oscillator in motion coupled to an acoustic fluid. The concept of radiation loss factor is introduced at this stage.
Lecture #1 part B of the elective course MG3409 - Statistical Energy Analysis, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2015
Power balance issues in the free, forced and evolutionary responses of a single DOF oscillator ex... more Power balance issues in the free, forced and evolutionary responses of a single DOF oscillator excited by deterministic or random loads.
Lecture #1 part A of the elective course MG3409 - Statistical Energy Analysis, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2015
We introduce the basic notions of probabilistic spaces, random variables, stochastic processes an... more We introduce the basic notions of probabilistic spaces, random variables, stochastic processes and the spectral analysis of mean-square stationary, second-order random processes (power spectral density, mean-square derivation, filtering).
Lecture #6 of the core course MG3401 - Stochastic Dynamics, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2011
We review Ito-Stratonovich stochastic calculus, diffusion processes and FKP equation, and some n... more We review Ito-Stratonovich stochastic calculus, diffusion processes and FKP equation, and some numerical schemes for stochastic differential equations with an emphasis on mechanical systems.
Lecture #5 of the core course MG3401 - Stochastic Dynamics, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2011
We review sampling and estimation techniques for Monte-Carlo simulation: inverse transform method... more We review sampling and estimation techniques for Monte-Carlo simulation: inverse transform method, acceptance-rejection, pseudo-random numbers, Markov chains, estimators, Cramér-Rao inequality, maximum likelihood.
Lecture notes of the elective course MG3409 - Statistical Energy Analysis, taught in the Graduate Program in Material and Structural Dynamics (DSMSC) at Ecole Centrale Paris, 2003
Le comportement vibratoire d’une structure industrielle complexe est très bien représenté aux bas... more Le comportement vibratoire d’une structure industrielle complexe est très bien représenté aux basses fréquences par ses modes propres de vibration. Cette représentation n’est en revanche plus du tout adaptée aux hautes fréquences, pour lesquelles la structure exhibe un comportement radicalement différent. Ces aspects ont une grande importance pratique notamment pour tout ce qui concerne l’acoustique interne et/ou externe des véhicules de transport : automobile, ferroviaire, domaine spatial... L’Analyse Statistique Énergétique, connue sous le signe SEA, est à ce jour la méthode la plus aboutie pour traiter de tels problèmes, bien qu’elle soit fondée sur une série d’hypothèses relativement restrictives. L’objectif de ce cours est double : d’une part présenter les principaux développements ayant conduit à la formulation SEA, d’autre part insister sur les hypothèses introduites au fur et à mesure et qui restreignent son champ d’application.
Plan
0. Introduction : vibration et vibro-acoustique hautes fréquences, exemples, besoins, stratégie de modélisation.
1. Aspects énergétiques de l’oscillateur simple, équivalences de moyenne.
2. Deux problèmes couplés élémentaires : système à 2 ddls, oscillateur immergé dans un fluide acoustique.
3. Vibrations aléatoires stationnaires des systèmes linéaires continus, aspects énergétiques.
4. Systèmes linéaires continus couplés.
5. Formulation de l’Analyse Statistique Énergétique.
Etienne Gay - PhD Thesis, Université de Paris, 2019
The present research is aimed at developing coherent interferometric (CINT) imaging algorithms to... more The present research is aimed at developing coherent interferometric (CINT) imaging algorithms to localize sources and reflectors in applications involving fluid flows. CINT imaging has been shown to be efficient and statistically stable in quiescent cluttered media where classical imaging techniques, such as Kirchhoff’s migration, may possibly fail due to their lack of statistical robustness. We aim at extending these methods to inhomo- geneous moving media, for it has relevance to aero-acoustics, atmospheric and underwater acoustics, infrasound propagation, or even astrophysics. In this thesis report we address both the direct problem of modeling the propagation of acoustic waves in a randomly het- erogeneous ambient flow, and the inverse problem of finding the position of sources or reflectors by the CINT algorithm implemented with the traces of the acoustic waves that have travelled through the flow.
Joan Staudacher - PhD Thesis, Ecole Centrale Paris, 2013
The present work focuses on the numerical resolution of the acoustic or elastic wave equation in ... more The present work focuses on the numerical resolution of the acoustic or elastic wave equation in a piece-wise homogeneous medium, splitted by interfaces. We are interested in a high-frequency setting, introduced by strongly oscillating initial conditions, for which one computes the distribution of the energy density by a so-called kinetic approach (based on the use of a Wigner transform). This problem then reduces to a Liouville-type transport equation in a piece-wise homogeneous medium, supplemented by reflection and transmission laws at the interfaces. Several numerical techniques and ranges of application are also reviewed. The transport equation which describes the evolution of the energy density in the phase space positions x wave vectors is numerically solved by finite differences. This technique raises several difficulties related to the conservation of the total energy in the medium and at the interfaces. They may be alleviated by dedicated numerical schemes allowing to reduce the numerical dissipation by either a global or a local approach. The improvements presented in this thesis concern the interpolation of the energy densities obtained by transmission on the grid of discrete wave vectors, and the correction of the difference of variation scales of the wave celerity on each side of the interfaces. The interest of the foregoing developments is to obtain conservative schemes that also satisfy the usual convergence properties of finite difference methods. The construction of such schemes and their effective implementation constitute the main achievement of the thesis. The relevance of the proposed methods is illustrated by several numerical simulations, that also emphasize their efficiency for rather coarse meshes.
Yves Le Guennec - PhD Thesis, Ecole Centrale Paris, 2013
This research is dedicated to the simulation of the transient response of beam trusses under impu... more This research is dedicated to the simulation of the transient response of beam trusses under impulse loads. The latter lead to the propagation of high-frequency waves in such built up structures. In the aerospace industry, that phenomenon may penalize the functioning of the structures or the equipments attached to them on account of the vibrational energy carried by the waves. It is also observed experimentally that high-frequency wave propagation evolves into a diffusive vibrational state at late times. The goal of this study is then to develop a robust model of high-frequency wave propagation within three-dimensional beam trusses in order to be able to recover, for example, this diffusion regime. On account of the small wavelengths and the high modal density, the modelling of high-frequency wave propagation is hardly feasible by classical finite elements or other methods describing the displacement fields directly. Thus, an approach dealing with the evolution of an estimator of the energy density of each propagating mode in a Timoshenko beam has been used. It provides information on the local behavior of the structures while avoiding some limitations related to the small wavelengths of high-frequency waves. After a comparison between some reduced-order beam kinematics and the Lamb model of wave propagation in a circular waveguide, the Timoshenko kinematics has been selected for the mechanical modelling of the beams. It may be shown that the energy densities of the propagating modes in a Timoshenko beam obey transport equations. Two groups of energy modes have been isolated: the longitudinal group that gathers the compressional and the bending energetic modes, and the transverse group that gathers the shear and torsional energetic modes. The reflection/transmission phenomena taking place at the junctions between beams have also been investigated. For this purpose, the power flow reflection/transmission operators have been derived from the continuity of the displacements and efforts at the junctions. Some characteristic features of a high-frequency behavior at beam junctions have been highlighted such as the decoupling between the rotational and translational motions. It is also observed that the energy densities are discontinuous at the junctions on account of the power flow reflection/transmission phenomena. Thus a discontinuous finite element method has been implemented, in order to solve the transport equations they satisfy. The numerical scheme has to be weakly dissipative and dispersive in order to exhibit the aforementioned diffusive regime arising at late times. That is the reason why spectral-like approximation functions for spatial discretization, and strong-stability preserving Runge-Kutta schemes for time integration have been used. Numerical simulations give satisfactory results because they indeed highlight the outbreak of such a diffusion state. The latter is characterized by the following: (i) the spatial spread of the energy over the truss, and (ii) the equipartition of the energy between the different modes. The last part of the thesis has been devoted to the development of a time reversal processing, that could be useful for future works on structural health monitoring of complex, multi-bay trusses.
This thesis is devoted to the study of the high frequency Dirichlet and Neumann problems for the ... more This thesis is devoted to the study of the high frequency Dirichlet and Neumann problems for the elasticity system. We study the reflection phenomenon at the boundary by means of two techniques: Gaussian beams summation and Wigner measures. In chapters 1 and 2, we start by studying the simpler problem of the scalar wave equation with one speed. Under some hypotheses on the initial conditions, we build an approximate solution by a Gaussian beams superposition. Justification of the asymptotics is based on norms estimate of some integral operators with complex phases. For more general initial conditions, we use Wigner measures to compute the microlocal energy density. We compute Wigner transforms of Gaussian beams integrals in an explicit way. The behaviour of the microlocal energy density for the exact solution is deduced from the one for the approximate solution. In chapter 3, we use the established results on infinite sums of Gaussian beams to build an approximate solution for the elasticity equations and to compute its microlocal energy density. We treat new difficulties arising from the existence of two different speeds in the elasticity system.
Eric Savin - PhD Thesis, Ecole Centrale Paris, 1999
Classical analysis of seismic soil-structure interaction phenomena for the design of civil engine... more Classical analysis of seismic soil-structure interaction phenomena for the design of civil engineering structures and lifelines are based on two simplifying assumptions~: lateral homogeneity of the soil underneath the foundation and representation of the ground motion (free field) as a pulse train of vertically or obliquely incident plane waves. For large structures founded on flexible spread footings, these hypotheses are not valid any longer. Furthermore, data recorded after the installation of dense seismograph arrays have shown that seismic ground motions can have a strong spatial variability, even over short distances and irrespective of deterministic wave passage effects. We develop in this thesis some theoretical tools for the modelization and numerical analysis of the influence of the free field and soil spatial varibilities on the structural response. A probabilistic approach is chosen, and the structural response of interest is given by an integral formulation which considers these two aspects simultaneously. Relevancy of numerical implementations are enhanced by introducing a spectral reduction of the random dimension of the fluctuating part - large or small - of the uncertain soil mechanical parameters through its Karhunen-Loeve expansion. This technique is applied to the free field as well. The numerical results obtained for various realistic cases display some specific features whose consideration seems essential in the process of industrial design. Notably, some usefull trends and sensitivity of the structural response with the free field and soil spatial variabilities are derived.
SIAM Conference on Computational Science and Engineering (CSE23), Amsterdam, February 26 - March 3, 2023, 2023
Optimal Uncertainty Quantification is a powerful mathematical tool which can be used to rigorousl... more Optimal Uncertainty Quantification is a powerful mathematical tool which can be used to rigorously bound the probability of exceeding a given performance threshold for uncertain operational conditions or system characteristics. This mathematical framework can be very computationally demanding. The use of a metamodel is highly desirable. Moreover, the robustness of the bounding obtained in this framework will strongly depend on the quality of this metamodel. Therefore, one needs to build a metamodel which is quick to evaluate and as accurate as possible. An algorithm, called Spectral Kernel Ridge Regression, is introduced to design kernels from available data in Gaussian process regression surrogate modeling techniques. This algorithm is illustrated on a aerodynamic numerical example.
On présente dans cet exposé un modèle de propagation d'ondes acoustiques dans un écoulement porte... more On présente dans cet exposé un modèle de propagation d'ondes acoustiques dans un écoulement porteur tri-dimensionnel présentant des petites fluctuations aléatoires de la vitesse particulaire et de la célérité des ondes. On s'intéresse plus particulièrement au régime pour lequel les longueurs d'onde acoustiques sont voisines des corrélations de ces structures - régime dit de couplage faible. L'analyse est conduite partant des équations d'Euler linéarisées et d'une équation d'ondes convectées en milieu hétérogène, obtenue à partir des équations d'Euler non linéarisées. Ainsi les phénomènes de convection, réfraction et diffusion des ondes dans l'écoulement porteur hétérogène sont décrits par une équation de transfert radiatif portant sur la densité d'action d'onde dans l'espace des phases position-temps. Le modèle exhibe également les effets de déphasage et d'étalement de spectre observés aux plus hautes fréquences, alors que l'action d'onde totale est conservée. Cette recherche est notamment motivée par des applications d'imagerie par corrélations croisées de champs diffus dans des milieux aléatoires, ici l'écoulement porteur.
We discuss the influence of material anisotropy on the possible depolarization and diffusion of e... more We discuss the influence of material anisotropy on the possible depolarization and diffusion of elastic waves in randomly heterogeneous media. Anisotropy is considered at two levels. The first one is related to the constitutive law of random materials, which may be handled by a random matrix theory for the elasticity tensor. The second level is related to the correlation structure of these random materials. Since the propagation of waves in such complex media cannot be described by deterministic models, a probabilistic framework shall be adopted based on a radiative transfer model in the so-called mesoscopic regime, when the wavelength and the correlation length are comparable.
The polynomial chaos (PC) expansion is a powerful tool for constructing a spectral-like represent... more The polynomial chaos (PC) expansion is a powerful tool for constructing a spectral-like representation of a stochastic process or a random field. A general methodology based on the Galerkin method has been proposed by Ghanem & Spanos (1991) for the solutions of partial differential equations (PDEs) with random inputs and/or parameters. This original approach is heavily intrusive and has prompted the development of non-intrusive schemes, especially for PDEs arising in fluid dynamic models (Le Maitre & Knio 2010). Two approaches for computing the coefficients of a PC expansion have usually been considered: (i) a projection approach, in which they are computed by structured (i.e. Gauss quadratures) or unstructured (i.e. Monte-Carlo or quasi Monte-Carlo sampling) quadratures; or (ii) a regression approach, minimizing some error measure. Both techniques suffer from some well identified shortcomings when the dimension of the random inputs, and the number of required model evaluations alike, increase. Collocation with sparse quadratures or adaptive regression strategies have been developed in order to circumvent this dimensionality concern. In this work we aim at benefiting from the sparsity of the output signal itself to reconstruct its PC representation in a non-adaptive way. For that purpose, we rely on the common observation that many cross-interactions between the input parameters are actually smoothened, or even negligible, once that have been propagated to some global quantities of interest processed from complex aerodynamic computations. We can therefore expect to achieve a successful signal recovery by the techniques known under the terminology of compressed sensing (Candès-Romberg-Tao 2006, Donoho 2006). The reconstruction of a sparse signal on a given, known basis requires only a limited number of evaluations at randomly selected points-at least significantly less than the a priori dimension of the basis. Here we illustrate the techniques of collocation on sparse grids and sparse reconstruction by compressed sensing on uncertainty quantification for an RAE2822 airfoil with random Mach number, angle of attack, and thickness-to-chord ratio.
The dynamic response of built-up structures to low-frequency excitations can be predicted efficie... more The dynamic response of built-up structures to low-frequency excitations can be predicted efficiently by reduced models derived from modal analyses. This representation is however much less relevant in their higher frequency (HF) ranges of vibration, when such systems exhibit a typical diffusive behavior. In this talk we will present some recent developments concerning the mathematical modeling and numerical simulations of the transient dynamics of engineering structures, based on the semiclassical analysis of HF solutions of wave systems. The transport models we consider are constructed adopting a kinetic point of view of wave propagation phenomena in heterogeneous and/or random media. The latter describes the asymptotic evolution properties of the underlying kinetic and strain energy densities. Nodal/spectral discontinuous « Galerkin » finite element methods are used for numerical simulations in order to account for their possible discontinuities at the boundaries and interfaces. The concurrence of the proposed framework with the engineering approach will be illustrated by several examples.
We use some recent mathematical results obtained for the high-frequency asymptotics of hyperbolic... more We use some recent mathematical results obtained for the high-frequency asymptotics of hyperbolic partial differential equations to derive exact transient power flow equations for bending vibrations of random, heterogeneous Mindlin plates. The theory shows that the phase-space angularly resolved energy densities of an heterogeneous elastic medium, properly projected on the eigenvectors of its dispersion operator, satisfy Liouville-type transport equations. The behavior of solutions of such equations is fundamentally different from that of the solution of a heat equation, although the latter one is often invoked in power flow analyses of vibrating elastic structures. The proposed approach considers random material properties, and can be extended to the fully coupled dynamic equations for compression, shear and bending of beams as well as plates and shells of arbitrary curvatures.
Dans cet article est présentée une formulation intégrale de l'interaction sismique sol-structure ... more Dans cet article est présentée une formulation intégrale de l'interaction sismique sol-structure qui permet de traiter les cas d'un champ incident de nature stochastique et d'un sol hétérogène également de nature aléatoire. Dans le premier cas les méthodes de l'analyse spectrale des vibrations aléatoires s'appliquent sans difficulté. Dans le deuxième cas, l'approche proposée étend à des volumes la notion d'interface pour les méthodes de sous-structuration.
Céléna Louis - MSc Thesis, Sorbonne Université, 2021
The objective of the proposed work is first to develop a numerical model of the wing aeroelastici... more The objective of the proposed work is first to develop a numerical model of the wing aeroelasticity effects based on a classical beam theory for flexible wings, and then to solve the inverse problem of finding the distributed thrust to choose to obtain a target shape of the wing. The direct model takes into account the propulsive forces and moments generated by the distributed electric fans, as well as the follower force effects. This thesis starts with a review of the existing aeroelastic models and concepts of distributed electric propulsion, then develops an adapted beam model and its finite element formulation, and finally implements the proposed numerical model in Matlab. It begins with a paragraph on what I have done during my internship followed by a description of ONERA. Afterward, I detail my work starting with some physic’s definition and some notations. Then, the 3D aeroelastic equations are explained followed by the 2D aeroelastic equations to continue on 2D resolution of the problem. This is followed by the optimization problem and finally the numerical results are given.
A profound shift in today’s aeronautics is ongoing where CFD and wind tunnel testing are going to... more A profound shift in today’s aeronautics is ongoing where CFD and wind tunnel testing are going to work much more closely to drive the certification process of new aircrafts at a reduced cost and time, keeping reliability at the same level as today’s ground testing approach or going beyond. One of the key features is to learn how to account for the various uncertainties that are intimately present in wind tunnel testing as well as in CFD analysis. In this internship one proposes a probabilistic approach based on an experimental dataset of inflow parameters to account for the uncertainties of the flow prediction. In this framework, from the CFD point of view, these varying data can be seen as uncertain inputs and one proposes to develop methods to simulate realizations of these unknown inflow data. Surrogate models based on orthogonal polynomial expansions, for example, will be used to propagate uncertainties in the CFD solver, here elsA, the main CFD code developed in the department of aerodynamics at ONERA. A preliminary section will be devoted to understand the problem, including the nature of the experimental data while a second one will be dedicated to the post-processing required for establishing the probabilistic model, the mathematical approach for the stochastic analysis, and the use of the CFD solver.
In this research the empirical cross-correlation of recorded signals is used to localize sources ... more In this research the empirical cross-correlation of recorded signals is used to localize sources or reflectors in a random medium. The case of a quiescent medium has been considered elsewhere (Borcea-Papanicolaou-Tsogka 2005, 2006), using a coherent interferometric (CINT) imaging functional. Here these results are extended to the case of a moving medium, for possible applications in aeroacoustics considering the linearized Euler equations.
Manuel Pena Rodríguez - MSc Thesis, Universidad Politecnica Madrid, 2015
In this thesis the Discontinuous Galerkin Finite Element Method is applied to solve the equations... more In this thesis the Discontinuous Galerkin Finite Element Method is applied to solve the equations governing the propagation of the high frequency vibrational energy in a thick plate. The plate is modelled using the Midlin theory and the resulting equation is transformed using the Wigner-Weyl transform. The high frequency limit is taken and, finally,the Wigner angular resolved energy of each vibration mode is governed by what are called Multigroup Radiative Transfer Equations, which are like the Liouville transport equation in a phase space containing the spatial coordinates and the wave numbers, with a forcing term taking into account the scattering produced by the random inhomogeneities of the medium. A simple case of different homogeneous plates and unit Poynting vector is considered. The DG-FEM is explained and its implementation for high polynomial degree on unstructured triangular meshes is described. Finally the numerical results, the problems encountered and some future improvements are considered.
This technical report deals with some of the methodologies used to construct polynomial surrogate... more This technical report deals with some of the methodologies used to construct polynomial surrogate models based on generalized polynomial chaos (gPC) expansions for applications to uncertainty quantification (UQ) and robust design (RD) in aerodynamic computations. A core ingredient in gPC expansions is the choice of a dedicated sampling strategy, so as to define the most significant scenarios to be considered for the construction of such metamodels. A desirable feature of the proposed rules shall be their ability to handle high dimensions (typically of the order of 10). Methods to identify the relative "importance" of variable or uncertain inputs shall be ideally investigated as well. The report is divided into two main parts: the first part is dedicated to the development of a sampling strategy based on fully-nested quadrature rules, and the second part is dedicated to the development of sampling strategies based on sparsity principles. The latter are more particularly assessed in the framework of the Basic Test Case BC-02 being part of UMRIDA Project's database for UQ and RD methods.
Regarding the nesting strategy developed in the first part, the goal is to alleviate the constraint of the classical 1D interpolatory nested quadratures that one should typically go from a set of N points to a set of 2N+1 (or 2N-1) points to benefit from the nesting property. Here a sequence of recursively included quadrature sets for all odd number of quadrature points is proposed to define new interpolatory rules. The corresponding weights are also derived. These rules are efficient for calculating integrals of very regular functions with a control of accuracy via the application of successive formulas of increasing order. Concerning the sparsity-based strategies, sparse multi-dimensional cubature rules based on general one-dimensional Gauss-Jacobi-type quadratures are first addressed. These sets are non nested, but they are well adapted to the probability density functions considered with BC-02. On the other hand, observing that the aerodynamic quantities of interest (outputs) depend only weakly on the cross-interactions between the variable inputs, it is argued that only low-order polynomials shall significantly contribute to their surrogates. This "sparsity-of-effects" trend prompts the use of reconstruction techniques benefiting from the sparsity of the outputs, such as compressed sensing (CS). CS relies on the observation that one only needs a number of samples proportional to the compressed size of the outputs, rather than their uncompressed size, to construct reliable surrogate models. The results obtained with BC-02 corroborate this expected feature.
We use some recent mathematical results obtained for the high-frequency asymptotics of hyperbolic... more We use some recent mathematical results obtained for the high-frequency asymptotics of hyperbolic partial differential equations to derive exact transient power flow equations for bending vibrations of random, heterogeneous Timoshenko beams and Mindlin plates. The theory shows that the phase-space angularly resolved energy densities of an heterogeneous elastic medium, properly projected on the eigenvectors of its dispersion operator, satisfy Liouville-type transport equations. The behavior of solutions of such equations is fundamentally different from that of the solution of a heat equation, although the latter one is often invoked in power flow analyses of vibrating elastic structures. The proposed approach considers random material properties, arbitrary boundary conditions or periodicity, and externally applied loads. It can be extended to the fully coupled dynamic equations for compression, shear and bending of beams as well as plates and shells of arbitrary curvatures.
The evolution properties of the high-frequency vibrational energy density in slender visco-elasti... more The evolution properties of the high-frequency vibrational energy density in slender visco-elastic structures such as Timoshenko beams or thick shells have been derived in previous works. The theory shows that this density satisfies a transport equation, or a so-called radiative transfer equation in the presence of random heterogeneities. The latter can be approached by a diffusion equation after long propagation times. The diffusive characteristics can also be obtained in the course of the derivation. This report presents a Galerkin discoutinuous finite element method to solve numerically the radiative transfer equation which characterizes the energy propagation in a curved Timoshenko beam. Comparisons are made with the analytical solution of a one-dimensional diffusion equation. In view of these results, the relevance of the vibrational conductivity analogy of high-frequency structural-acoustics invoked by some researchers is discussed.
In this study we extend the results given in a previous report regarding the propagation features... more In this study we extend the results given in a previous report regarding the propagation features of high-frequency energy in slender structures to the case of three-dimensional porous media. The mathematical theory of microlocal analysis of hyperbolic partial differential equations shows that the energy associated to their high-frequency solutions (in a sense to be precised) satisfies Liouville-type transport equations, or radiative transfer equations for randomly heterogeneous materials. For long propagation times these equations can be asymptotically approached by diffusion equations. Some classical results of the engineering mechanics literature about the heat conduction analogy and the statistical energy analysis of structural vibrations at high frequencies are recovered in this process. The purpose of this report is to focus on such a diffusive regime for isotropic porous media.
The mathematical theory of microlocal analysis of hyperbolic partial differential equations shows... more The mathematical theory of microlocal analysis of hyperbolic partial differential equations shows that the energy density associated to their high-frequency solutions satisfies Liouville-type transport equations, or radiative transfer equations for randomly heterogeneous materials with correlation lengths comparable to the (small) wavelength. The main limitation to date to the existing theory is the consideration of boundary or interface conditions for the energy and power flow densities. This report deals with the radiative transfer regime in a randomly heterogeneous two-plate system. First, we propose an analytical model for the derivation of high-frequency reflection/transmission coefficients for the power flows at the plates junction. These results are used in subsequent computations to solve numerically the radiative transfer equations for this system, including the interface conditions. A transport model is finally proposed for the high-frequency guided waves which could possibly propagate along the junction line.
Ce rapport présente les résultats des développements d’une méthode numérique de prise en compte d... more Ce rapport présente les résultats des développements d’une méthode numérique de prise en compte de la complexité structurale pour la dynamique des structures complexes en moyennes fréquences. Les notions de complexité structurale et moyennes fréquences sont ici définies sur la base d’essais de vibration réalisés sur une structure expérimentale complexe. Les simulations numériques sont confrontées aux mesures effectuées sur cette structure.
We study the propagation of high-frequency electromagnetic waves in randomly heterogeneous bianis... more We study the propagation of high-frequency electromagnetic waves in randomly heterogeneous bianisotropic media with dissipative properties. For that purpose we consider randomly fluctuating optical responses of such media with correlation lengths comparable to the typical wavelength of the waves. Although the fluctuations are weak, they induce multiple scattering over long propagation times and/or distances such that the waves end up travelling in many different directions with mixed polarizations. We derive the dispersion and evolution properties of the Wigner measure of the electromagnetic fields, which describes their angularly-resolved energy density in this propagation regime. The analysis starts from Maxwell's equations with general constitutive equations. We first ignore the random fluctuations of the optical response and obtain uncoupled transport equations for the components of the Wigner measure on the different propagation modes (polarizations). Then we use a multi-scale expansion of the Wigner mesure to obtain the radiative transfer equations satisfied by these components when the fluctuations are no longer ignored. The radiative transfer equations are coupled through their collisional parts, which account for the scattering of waves by the random fluctuations and their possible changes in polarization. The collisional kernels describing these processes depend on the power and cross-power spectral densities of the fluctuations at the wavelength scale. The overall derivation is based on the interpretation of Wigner transforms and Wigner measures in terms of semiclassical pseudo-differential operators in their standard quantization.
The uncertainty associated with the experimental inflow in a wind tunnel affects the prediction o... more The uncertainty associated with the experimental inflow in a wind tunnel affects the prediction of the flow of interest by numerical simulations. We evaluate this impact using uncertainty quantification. A method is developed and applied to the simulation of the drag generated by the flow past a cylinder installed in the transonic S3Ch ONERA mid-scale facility. The inflow uncertainty results from the imperfect knowledge and variability of the flow in the settling chamber. It is taken into account via the inlet boundary condition in the numerical companion setup and evaluated experimentally by measuring the inflow using a hot-wire rake. The propagation of the input uncertainties is carried out through a two-dimensional RANS model of the experiment. A polynomial surrogate model is developed to infer the uncertainty associated with the drag of the cylinder. Following observations of Gaussian inputs, the parameters of the stochastic model are constructed in two ways, first through a projection approach, based on the Gauss-Hermite quadrature rule, and then using a sparsity based regression approach, based on compressed sensing. The latter drastically reduces the number of deterministic numerical simulations. The drag is most influenced by the central part of the inflow but the overall uncertainty remains low.
Uploads
1. Introduction. Need for Uncertainty Quantification
2. Probability basics, Monte-Carlo, surrogate-based Monte-Carlo
3. Non-intrusive polynomial methods for 1D / tensorial nD propagation
4. Introduction to Smolyak’s sparse quadratures
5. Examples of application
Plan
0. Introduction : vibration et vibro-acoustique hautes fréquences, exemples, besoins, stratégie de modélisation.
1. Aspects énergétiques de l’oscillateur simple, équivalences de moyenne.
2. Deux problèmes couplés élémentaires : système à 2 ddls, oscillateur immergé dans un fluide acoustique.
3. Vibrations aléatoires stationnaires des systèmes linéaires continus, aspects énergétiques.
4. Systèmes linéaires continus couplés.
5. Formulation de l’Analyse Statistique Énergétique.
Regarding the nesting strategy developed in the first part, the goal is to alleviate the constraint of the classical 1D interpolatory nested quadratures that one should typically go from a set of N points to a set of 2N+1 (or 2N-1) points to benefit from the nesting property. Here a sequence of recursively included quadrature sets for all odd number of quadrature points is proposed to define new interpolatory rules. The corresponding weights are also derived. These rules are efficient for calculating integrals of very regular functions with a control of accuracy via the application of successive formulas of increasing order. Concerning the sparsity-based strategies, sparse multi-dimensional cubature rules based on general one-dimensional Gauss-Jacobi-type quadratures are first addressed. These sets are non nested, but they are well adapted to the probability density functions considered with BC-02. On the other hand, observing that the aerodynamic quantities of interest (outputs) depend only weakly on the cross-interactions between the variable inputs, it is argued that only low-order polynomials shall significantly contribute to their surrogates. This "sparsity-of-effects" trend prompts the use of reconstruction techniques benefiting from the sparsity of the outputs, such as compressed sensing (CS). CS relies on the observation that one only needs a number of samples proportional to the compressed size of the outputs, rather than their uncompressed size, to construct reliable surrogate models. The results obtained with BC-02 corroborate this expected feature.