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In this work, we have computed and implemented one-body integrals concerning Gaussian confinement potentials over Gaussian basis functions. Then, we have set an equivalence between Gaussian and Hooke atoms and we have observed that, according to singlet and triplet state energies, both systems are equivalent for large confinement depth for a series of even number of electrons n = 2, 4, 6, 8 and 10. Unlike with harmonic potentials, Gaussian confinement potentials are dissociative for small enough depth parameter; this feature is crucial in order to model phenomena such as ionization. In this case, in addition to corresponding Taylor-series expansions, the first diagonal and sub-diagonal Pade approximant were also obtained, useful to compute the upper and lower limits for the dissociation depth. Hence, this method introduces new advantages compared to others.

We outline an approach to calculating the quantum mechanical propagator in the presence of geometrically nontrivial Dirichlet boundary conditions. The method is based on a generalization of an integral transform of the propagator studied in previous work (the so-called “hit function”) and a convergent sequence of Padé approximants that exposes the limit of perfectly reflecting boundaries. In this paper the generalized hit function is defined as a many-point propagator, and we describe its relation to the sum over trajectories in the Feynman path integral. We then show how it can be used to calculate the Feynman propagator. We calculate analytically all such hit functions in D = 1 and D = 3 dimensions, giving recursion relations between them in the same or different dimensions and apply the results to the simple cases of propagation in the presence of perfectly conducting planar and spherical plates. We use these results to conjecture a general analytical formula for the propagator when Dirichlet boundary conditions are present in a given geometry, also explaining how it can be extended for application for more general, nonlocalized potentials. Our work has resonance with previous results obtained by Grosche in the study of path integrals in the presence of delta potentials. We indicate the eventual application in a relativistic context to determining Casimir energies using this technique.

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present work, a multivariate finite element approach is proposed, allowing for a fast evaluation of parametric responses. It is based on the construction of a reduced basis spanning a subspace able to capture rough variations of the response. The method consists in an extension of the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) to multivariate problems, by an appropriate choice of derivative sequences, and a selection of the most relevant basis components. It is validated and demonstrated for its potential on a semi-industrial sized 3D application involving coupled poroelasticand internal acoustic domains., QC 20200130, PadéSweep, VR Grant 2015- 04925, Centre for ECO2 Vehicle Design, VINNOVA Grant 2016- 05195

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present work, a multivariate finite element approach is proposed, allowing for a fast evaluation of parametric responses. It is based on the construction of a reduced basis spanning a subspace able to capture rough variations of the response. The method consists in an extension of the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) to multivariate problems, by an appropriate choice of derivative sequences, and a selection of the most relevant basis components. It is validated and demonstrated for its potential on a semi-industrial sized 3D application involving coupled poroelasticand internal acoustic domains., QC 20200130, PadéSweep, VR Grant 2015- 04925, Centre for ECO2 Vehicle Design, VINNOVA Grant 2016- 05195

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present work, a multivariate finite element approach is proposed, allowing for a fast evaluation of parametric responses. It is based on the construction of a reduced basis spanning a subspace able to capture rough variations of the response. The method consists in an extension of the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) to multivariate problems, by an appropriate choice of derivative sequences, and a selection of the most relevant basis components. It is validated and demonstrated for its potential on a semi-industrial sized 3D application involving coupled poroelasticand internal acoustic domains., QC 20200130, PadéSweep, VR Grant 2015- 04925, Centre for ECO2 Vehicle Design, VINNOVA Grant 2016- 05195

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present work, a multivariate finite element approach is proposed, allowing for a fast evaluation of parametric responses. It is based on the construction of a reduced basis spanning a subspace able to capture rough variations of the response. The method consists in an extension of the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) to multivariate problems, by an appropriate choice of derivative sequences, and a selection of the most relevant basis components. It is validated and demonstrated for its potential on a semi-industrial sized 3D application involving coupled poroelasticand internal acoustic domains., QC 20200130, PadéSweep, VR Grant 2015- 04925, Centre for ECO2 Vehicle Design, VINNOVA Grant 2016- 05195

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present work, a multivariate finite element approach is proposed, allowing for a fast evaluation of parametric responses. It is based on the construction of a reduced basis spanning a subspace able to capture rough variations of the response. The method consists in an extension of the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) to multivariate problems, by an appropriate choice of derivative sequences, and a selection of the most relevant basis components. It is validated and demonstrated for its potential on a semi-industrial sized 3D application involving coupled poroelasticand internal acoustic domains., QC 20200130, PadéSweep, VR Grant 2015- 04925, Centre for ECO2 Vehicle Design, VINNOVA Grant 2016- 05195

To increase the robustness of a Pade-based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Pade-based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Pade approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component-wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Pade approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural-acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed., QC 20180103

To increase the robustness of a Pade-based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Pade-based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Pade approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component-wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Pade approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural-acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed., QC 20180103

To increase the robustness of a Pade-based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Pade-based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Pade approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component-wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Pade approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural-acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed., QC 20180103

To increase the robustness of a Pade-based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Pade-based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Pade approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component-wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Pade approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural-acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed., QC 20180103

To increase the robustness of a Pade-based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Pade-based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Pade approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component-wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Pade approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural-acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed., QC 20180103

In this work, a solution strategy based on the use of Padé approximants is investigated for efficient solution of parametric finite element problems such as, for example, frequency sweep analyses. An improvement to the Padé-based expansion of the solution vector components is proposed, suggesting the advantageous a priori estimate of the poles of the solution. This allows for the intervals of approximation to be chosen a priori in connection with the Padé approximants to be used. The choice of these approximants is supported by the Montessus de Ballore theorem, proving the convergence of a series of approximants with fixed denominator degrees. An acoustic case study is presented in order to illustrate the potential of the approach proposed by the authors., QC 20151013

In this work, a solution strategy based on the use of Padé approximants is investigated for efficient solution of parametric finite element problems such as, for example, frequency sweep analyses. An improvement to the Padé-based expansion of the solution vector components is proposed, suggesting the advantageous a priori estimate of the poles of the solution. This allows for the intervals of approximation to be chosen a priori in connection with the Padé approximants to be used. The choice of these approximants is supported by the Montessus de Ballore theorem, proving the convergence of a series of approximants with fixed denominator degrees. An acoustic case study is presented in order to illustrate the potential of the approach proposed by the authors., QC 20151013

In the present paper the orthogonality relations, exhibited by both numerator and denominator polynomials of both [n/n] and [n−1/n] Pade approximants for power series expansion of e-x, given by regular C-fraction expansion, are described. The four orthogonal polynomials thus derived are shown to form classical orthogonal polynomials.

In this work, a solution strategy based on the use of Padé approximants is investigated for efficient solution of parametric finite element problems such as, for example, frequency sweep analyses. An improvement to the Padé-based expansion of the solution vector components is proposed, suggesting the advantageous a priori estimate of the poles of the solution. This allows for the intervals of approximation to be chosen a priori in connection with the Padé approximants to be used. The choice of these approximants is supported by the Montessus de Ballore theorem, proving the convergence of a series of approximants with fixed denominator degrees. An acoustic case study is presented in order to illustrate the potential of the approach proposed by the authors., QC 20151013

In this work, a solution strategy based on the use of Padé approximants is investigated for efficient solution of parametric finite element problems such as, for example, frequency sweep analyses. An improvement to the Padé-based expansion of the solution vector components is proposed, suggesting the advantageous a priori estimate of the poles of the solution. This allows for the intervals of approximation to be chosen a priori in connection with the Padé approximants to be used. The choice of these approximants is supported by the Montessus de Ballore theorem, proving the convergence of a series of approximants with fixed denominator degrees. An acoustic case study is presented in order to illustrate the potential of the approach proposed by the authors., QC 20151013

In this work, a solution strategy based on the use of Padé approximants is investigated for efficient solution of parametric finite element problems such as, for example, frequency sweep analyses. An improvement to the Padé-based expansion of the solution vector components is proposed, suggesting the advantageous a priori estimate of the poles of the solution. This allows for the intervals of approximation to be chosen a priori in connection with the Padé approximants to be used. The choice of these approximants is supported by the Montessus de Ballore theorem, proving the convergence of a series of approximants with fixed denominator degrees. An acoustic case study is presented in order to illustrate the potential of the approach proposed by the authors., QC 20151013

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat Friedmann-Lemaitre-Robertson-Walker cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the ’attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Pade approximants and compare with the slow-roll approximation.

In this work, a solution strategy is investigated for the resolution of multi-frequency structural-acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal-based reduction of the poroelastic domain and a Pade-based reconstruction approach. It thus takes advantage of the reduced-size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full-sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro-acoustic example., QC 20140228. QC 20160212

Differential matrix Riccati equations (DMREs) enable to model many physical systems appearing in different branches of science, in some cases, involving very large problem sizes. In this paper, we propose an adaptive algorithm for time-invariant DMREs that uses a piecewise-linearized approach based on the Padé approximation of the matrix exponential. The algorithm designed is based upon intensive use of matrix products and linear system solutions so we can seize the large computational capability that modern graphics processing units (GPUs) have on these types of operations using CUBLAS and CULATOOLS libraries (general purpose GPU), which are efficient implementations of BLAS and LAPACK libraries, respectively, for NVIDIA © GPUs. A thorough analysis showed that some parts of the algorithm proposed can be carried out in parallel, thus allowing to leverage the two GPUs available in many current compute nodes. Besides, our algorithm can be used by any interested researcher through a friendly MATLAB © interface.

Differential matrix Riccati equations (DMREs) enable to model many physical systems appearing in different branches of science, in some cases, involving very large problem sizes. In this paper, we propose an adaptive algorithm for time-invariant DMREs that uses a piecewise-linearized approach based on the Padé approximation of the matrix exponential. The algorithm designed is based upon intensive use of matrix products and linear system solutions so we can seize the large computational capability that modern graphics processing units (GPUs) have on these types of operations using CUBLAS and CULATOOLS libraries (general purpose GPU), which are efficient implementations of BLAS and LAPACK libraries, respectively, for NVIDIA © GPUs. A thorough analysis showed that some parts of the algorithm proposed can be carried out in parallel, thus allowing to leverage the two GPUs available in many current compute nodes. Besides, our algorithm can be used by any interested researcher through a friendly MATLAB © interface.

In the context of interior noise reduction, the present work aims at proposing Finite Element (FE) solution strategies for interior structural-acoustic applications including 3D modelling of homogeneous and isotropic poroelastic materials, under timeharmonic excitations, and in the low frequency range. A model based on the Biot-Allard theory is used for the poroelastic materials, which is known to be very costly in terms of computational resources. Reduced models offer the possibility to enhance the resolution of such complex problems. However, their applicability to porous materials remained to be demonstrated.First, this thesis presents FE resolutions of poro-elasto-acoustic coupled problems using modal-based approaches both for the acoustic and porous domains. The original modal approach proposed for porous media, together with a dedicated mode selection and truncation procedure, are validated on 1D to 3D applications.In a second part, modal-reduced models are combined with a Padé approximants reconstruction scheme in order to further improve the efficiency.A concluding chapter presents a comparison and a combination of the proposed methods on a 3D academic application, showing promising performances. Conclusions are then drawn to provide indications for future research and tests to be conducted in order to further enhance the methodologies proposed in this thesis., Dans le contexte de lutte contre les nuisances sonores, cette thèse porte sur le développement de méthodes de résolution efficaces par éléments finis, pour des problèmes de vibroacoustique interne avec interfaces dissipatives, dans le domaine des basses fréquences. L’étude se limite à l’utilisation de solutions passives telles que l’intégration de matériaux poreux homogènes et isotropes, modélisés par une approche fondée sur la théorie de Biot-Allard. Ces modèles étant coûteux en terme de résolution, un des objectifs de cette thèse est de proposer une approche modale pour la réduction du problème poroélastique, bien que l’adéquation d’une telle approche avec le comportement dynamique des matériaux poreux soit à démontrer.Dans un premier temps, la résolution de problèmes couplés élasto-poro-acoustiques par sous-structuration dynamique des domaines acoustiques et poreux est établie. L’approche modale originale proposée pour les milieux poroélastiques, ainsi qu’une procédure de sélection des modes significatifs, sont validées sur des exemples 1D à 3D.Une deuxième partie présente une méthode combinant l’utilisation des modèles réduits précédemment établis avec une procédure d’approximation de solution par approximants de Padé. Il est montré qu’une telle combinaison offre la possibilité d’accroître les performances de la résolution (allocation mémoire et ressources en temps de calcul).Un chapitre dédié aux applications permet d’évaluer et comparer les approches sur un problème académique 3D, mettant en valeur leurs performances encourageantes. Afin d’améliorer les méthodes établies dans cette thèse, des perspectives à ces travaux de recherche sont apportées en conclusion., QC 20120224, FP6 Marie-Curie Smart Structures, FP7 Marie-Curie Mid-Frequency

In the context of interior noise reduction, the present work aims at proposing Finite Element (FE) solution strategies for interior structural-acoustic applications including 3D modelling of homogeneous and isotropic poroelastic materials, under timeharmonic excitations, and in the low frequency range. A model based on the Biot-Allard theory is used for the poroelastic materials, which is known to be very costly in terms of computational resources. Reduced models offer the possibility to enhance the resolution of such complex problems. However, their applicability to porous materials remained to be demonstrated.First, this thesis presents FE resolutions of poro-elasto-acoustic coupled problems using modal-based approaches both for the acoustic and porous domains. The original modal approach proposed for porous media, together with a dedicated mode selection and truncation procedure, are validated on 1D to 3D applications.In a second part, modal-reduced models are combined with a Padé approximants reconstruction scheme in order to further improve the efficiency.A concluding chapter presents a comparison and a combination of the proposed methods on a 3D academic application, showing promising performances. Conclusions are then drawn to provide indications for future research and tests to be conducted in order to further enhance the methodologies proposed in this thesis., Dans le contexte de lutte contre les nuisances sonores, cette thèse porte sur le développement de méthodes de résolution efficaces par éléments finis, pour des problèmes de vibroacoustique interne avec interfaces dissipatives, dans le domaine des basses fréquences. L’étude se limite à l’utilisation de solutions passives telles que l’intégration de matériaux poreux homogènes et isotropes, modélisés par une approche fondée sur la théorie de Biot-Allard. Ces modèles étant coûteux en terme de résolution, un des objectifs de cette thèse est de proposer une approche modale pour la réduction du problème poroélastique, bien que l’adéquation d’une telle approche avec le comportement dynamique des matériaux poreux soit à démontrer.Dans un premier temps, la résolution de problèmes couplés élasto-poro-acoustiques par sous-structuration dynamique des domaines acoustiques et poreux est établie. L’approche modale originale proposée pour les milieux poroélastiques, ainsi qu’une procédure de sélection des modes significatifs, sont validées sur des exemples 1D à 3D.Une deuxième partie présente une méthode combinant l’utilisation des modèles réduits précédemment établis avec une procédure d’approximation de solution par approximants de Padé. Il est montré qu’une telle combinaison offre la possibilité d’accroître les performances de la résolution (allocation mémoire et ressources en temps de calcul).Un chapitre dédié aux applications permet d’évaluer et comparer les approches sur un problème académique 3D, mettant en valeur leurs performances encourageantes. Afin d’améliorer les méthodes établies dans cette thèse, des perspectives à ces travaux de recherche sont apportées en conclusion., QC 20120224, FP6 Marie-Curie Smart Structures, FP7 Marie-Curie Mid-Frequency

The exact expression for the maximum tension of a pendulum string is used to obtain a closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions. This approximate solution has a rational harmonic expression and depends on an unknown function, which must be determined. This unknown function is expanded using the Padé approximant and two new parameters are introduced which are determined by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. Using this approach, accurate approximate analytical expressions for the periodic solution are obtained. We also compared the Fourier series expansions of the approximate solutions and the exact ones. This allowed us to compare the coefficients for the different harmonics in these solutions. We also compared the approximate and exact solutions as a function of time for several oscillation amplitudes. Finally, in this procedure we used some of the approximate expressions for the simple pendulum frequency which can be found in the bibliography; however, the procedure can be applied using other approximate frequencies.

The exact expression for the maximum tension of a pendulum string is used to obtain a closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions. This approximate solution has a rational harmonic expression and depends on an unknown function, which must be determined. This unknown function is expanded using the Padé approximant and two new parameters are introduced which are determined by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. Using this approach, accurate approximate analytical expressions for the periodic solution are obtained. We also compared the Fourier series expansions of the approximate solutions and the exact ones. This allowed us to compare the coefficients for the different harmonics in these solutions. We also compared the approximate and exact solutions as a function of time for several oscillation amplitudes. Finally, in this procedure we used some of the approximate expressions for the simple pendulum frequency which can be found in the bibliography; however, the procedure can be applied using other approximate frequencies.

In the context of interior noise reduction, the present work aims at proposing Finite Element (FE) solution strategies for interior structural-acoustic applications including 3D modelling of homogeneous and isotropic poroelastic materials, under timeharmonic excitations, and in the low frequency range. A model based on the Biot-Allard theory is used for the poroelastic materials, which is known to be very costly in terms of computational resources. Reduced models offer the possibility to enhance the resolution of such complex problems. However, their applicability to porous materials remained to be demonstrated.First, this thesis presents FE resolutions of poro-elasto-acoustic coupled problems using modal-based approaches both for the acoustic and porous domains. The original modal approach proposed for porous media, together with a dedicated mode selection and truncation procedure, are validated on 1D to 3D applications.In a second part, modal-reduced models are combined with a Padé approximants reconstruction scheme in order to further improve the efficiency.A concluding chapter presents a comparison and a combination of the proposed methods on a 3D academic application, showing promising performances. Conclusions are then drawn to provide indications for future research and tests to be conducted in order to further enhance the methodologies proposed in this thesis., Dans le contexte de lutte contre les nuisances sonores, cette thèse porte sur le développement de méthodes de résolution efficaces par éléments finis, pour des problèmes de vibroacoustique interne avec interfaces dissipatives, dans le domaine des basses fréquences. L’étude se limite à l’utilisation de solutions passives telles que l’intégration de matériaux poreux homogènes et isotropes, modélisés par une approche fondée sur la théorie de Biot-Allard. Ces modèles étant coûteux en terme de résolution, un des objectifs de cette thèse est de proposer une approche modale pour la réduction du problème poroélastique, bien que l’adéquation d’une telle approche avec le comportement dynamique des matériaux poreux soit à démontrer.Dans un premier temps, la résolution de problèmes couplés élasto-poro-acoustiques par sous-structuration dynamique des domaines acoustiques et poreux est établie. L’approche modale originale proposée pour les milieux poroélastiques, ainsi qu’une procédure de sélection des modes significatifs, sont validées sur des exemples 1D à 3D.Une deuxième partie présente une méthode combinant l’utilisation des modèles réduits précédemment établis avec une procédure d’approximation de solution par approximants de Padé. Il est montré qu’une telle combinaison offre la possibilité d’accroître les performances de la résolution (allocation mémoire et ressources en temps de calcul).Un chapitre dédié aux applications permet d’évaluer et comparer les approches sur un problème académique 3D, mettant en valeur leurs performances encourageantes. Afin d’améliorer les méthodes établies dans cette thèse, des perspectives à ces travaux de recherche sont apportées en conclusion., QC 20120224, FP6 Marie-Curie Smart Structures, FP7 Marie-Curie Mid-Frequency

In the context of interior noise reduction, the present work aims at proposing Finite Element (FE) solution strategies for interior structural-acoustic applications including 3D modelling of homogeneous and isotropic poroelastic materials, under timeharmonic excitations, and in the low frequency range. A model based on the Biot-Allard theory is used for the poroelastic materials, which is known to be very costly in terms of computational resources. Reduced models offer the possibility to enhance the resolution of such complex problems. However, their applicability to porous materials remained to be demonstrated.First, this thesis presents FE resolutions of poro-elasto-acoustic coupled problems using modal-based approaches both for the acoustic and porous domains. The original modal approach proposed for porous media, together with a dedicated mode selection and truncation procedure, are validated on 1D to 3D applications.In a second part, modal-reduced models are combined with a Padé approximants reconstruction scheme in order to further improve the efficiency.A concluding chapter presents a comparison and a combination of the proposed methods on a 3D academic application, showing promising performances. Conclusions are then drawn to provide indications for future research and tests to be conducted in order to further enhance the methodologies proposed in this thesis., Dans le contexte de lutte contre les nuisances sonores, cette thèse porte sur le développement de méthodes de résolution efficaces par éléments finis, pour des problèmes de vibroacoustique interne avec interfaces dissipatives, dans le domaine des basses fréquences. L’étude se limite à l’utilisation de solutions passives telles que l’intégration de matériaux poreux homogènes et isotropes, modélisés par une approche fondée sur la théorie de Biot-Allard. Ces modèles étant coûteux en terme de résolution, un des objectifs de cette thèse est de proposer une approche modale pour la réduction du problème poroélastique, bien que l’adéquation d’une telle approche avec le comportement dynamique des matériaux poreux soit à démontrer.Dans un premier temps, la résolution de problèmes couplés élasto-poro-acoustiques par sous-structuration dynamique des domaines acoustiques et poreux est établie. L’approche modale originale proposée pour les milieux poroélastiques, ainsi qu’une procédure de sélection des modes significatifs, sont validées sur des exemples 1D à 3D.Une deuxième partie présente une méthode combinant l’utilisation des modèles réduits précédemment établis avec une procédure d’approximation de solution par approximants de Padé. Il est montré qu’une telle combinaison offre la possibilité d’accroître les performances de la résolution (allocation mémoire et ressources en temps de calcul).Un chapitre dédié aux applications permet d’évaluer et comparer les approches sur un problème académique 3D, mettant en valeur leurs performances encourageantes. Afin d’améliorer les méthodes établies dans cette thèse, des perspectives à ces travaux de recherche sont apportées en conclusion., QC 20120224, FP6 Marie-Curie Smart Structures, FP7 Marie-Curie Mid-Frequency

В данной работе рассматривается совершенная система функций, состоящая из двух экспонент, и для нее исследуется асимптотика поведения совместных аппроксимаций Паде. Полученные теоремы дополняют и обобщают результаты Эрмита, Е.М. Никишина, А.И. Аптекарева. =In the given article the perfect system of functions consisting of two exponents and is considered. The asymptotic of behavior of joint Pade approximants for such system of functions is researched. The received theorems supplement and generalize the known results of Ermite, E.M. Nikishin, A.I. Aptekarev.

Differential Matrix Riccati Equations (DMREs) appear in several branches of science such as applied physics and engineering. For example, these equations play a fundamental role in control theory, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a new method based on a theorem proved in this paper is described for solving DMREs by a piecewise-linearized approach. This method is applied for developing two block-oriented algorithms based on diagonal Padé approximants. MATLAB versions of the above algorithms are developed, comparing, under equal conditions, accuracy and computational costs with other piecewise-linearized algorithms implemented by the authors. Experimental results show the advantages of solving stiff or non-stiff DMREs by the implemented algorithms.

Differential Matrix Riccati Equations (DMREs) appear in several branches of science such as applied physics and engineering. For example, these equations play a fundamental role in control theory, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a new method based on a theorem proved in this paper is described for solving DMREs by a piecewise-linearized approach. This method is applied for developing two block-oriented algorithms based on diagonal Padé approximants. MATLAB versions of the above algorithms are developed, comparing, under equal conditions, accuracy and computational costs with other piecewise-linearized algorithms implemented by the authors. Experimental results show the advantages of solving stiff or non-stiff DMREs by the implemented algorithms.

Commonly used techniques to study non-perturbative aspects of the strong interactions have a deep connection with rational approximants, and in particular with Pack approximants to meromorphic functions. However, only recently this connection has been acknowledged and efforts at fully exploiting it are only starting. In this article I will briefly review the most prominent techniques used in non-perturbative strong interactions with special emphasis on its relation with Pade approximants. I will then concentrate on a set of open problems outside the scope of these conventional techniques where Pade approximants might be extremely useful.

A method is presented for estimating microstructural parameters from permittivity measurements of two-component composites. This structural information is described by a particular positive measure in the Stieltjes integral representation of the effective permittivity. The dependence on the geometrical structure can be reduced to the problem of calculating the moments of the measure. We present a method that uses measurement data at a set of distinct frequencies or temperatures to calculate bounds on several moments. These inverse bounds are improved when the volume fraction is known or the material is isotropic. Composites with known geometrical structure illustrate the method.

An efficient procedure for topology optimization of dynamics problems is proposed. The method is based on frequency responses represented by Padé approximants and analytical sensitivity analysis derived using the adjoint method. This gives an accurate approximation of the frequency response over wide frequency ranges and a formulation that allows for design sensitivities to be computed at low computational cost also for a large number of design variables. Two examples that deal with optimization of forced vibrations are included.

An efficient procedure for topology optimization of dynamics problems is proposed. The method is based on frequency responses represented by Padé approximants and analytical sensitivity analysis derived using the adjoint method. This gives an accurate approximation of the frequency response over wide frequency ranges and a formulation that allows for design sensitivities to be computed at low computational cost also for a large number of design variables. Two examples that deal with optimization of forced vibrations are included.

A method is presented for estimating microstructural parameters from permittivity measurements of two-component composites. This structural information is described by a particular positive measure in the Stieltjes integral representation of the effective permittivity. The dependence on the geometrical structure can be reduced to the problem of calculating the moments of the measure. We present a method that uses measurement data at a set of distinct frequencies or temperatures to calculate bounds on several moments. These inverse bounds are improved when the volume fraction is known or the material is isotropic. Composites with known geometrical structure illustrate the method.

This paper deals with the parabolic formulation of propagation equations of gravity waves in surface in terms of angular capability with respect to the privileged propagation direction. This parabolic formulation is obtained by splitting the Berkhoff equation operator into two parabolic operators representing progressive and regressive wave propagation. The use of Pade approximants permits to derive simultaneously parabolic equations for the transmitted waves and the reflected waves. Two well-known reference examples, which represent the propagation of the wave when a caustic occur, are studied numerically and the results are compared with those of the literature. (C) 2001 Ifremer/CNRS/IRD/Editions scientifiques et medicales Elsevier SAS., Cet article présente une formulation parabolique des équations de propagation, en profondeur finie, des ondes de gravité en surface en terme de capacité angulaire par rapport à la direction de propagation privilégiée. Cette formulation est obtenue par factorisation de l’opérateur de l’équation de Berkhoff en deux opérateurs représentant les ondes progressives et régressives. L’utilisation des approximants de Padé permet d’accéder simultanément à des équations paraboliques pour les ondes transmises et pour les ondes réfléchies. Deux exemples de propagation de la houle en présence d’une bosse circulaire et d’une bosse elliptique sont étudiés numériquement. La comparaison des résultats avec ceux de la littérature prouve la supériorité de cette équation parabolique vis-à-vis de celle de Radder

This paper deals with the parabolic formulation of propagation equations of gravity waves in surface in terms of angular capability with respect to the privileged propagation direction. This parabolic formulation is obtained by splitting the Berkhoff equation operator into two parabolic operators representing progressive and regressive wave propagation. The use of Pade approximants permits to derive simultaneously parabolic equations for the transmitted waves and the reflected waves. Two well-known reference examples, which represent the propagation of the wave when a caustic occur, are studied numerically and the results are compared with those of the literature. (C) 2001 Ifremer/CNRS/IRD/Editions scientifiques et medicales Elsevier SAS., Cet article présente une formulation parabolique des équations de propagation, en profondeur finie, des ondes de gravité en surface en terme de capacité angulaire par rapport à la direction de propagation privilégiée. Cette formulation est obtenue par factorisation de l’opérateur de l’équation de Berkhoff en deux opérateurs représentant les ondes progressives et régressives. L’utilisation des approximants de Padé permet d’accéder simultanément à des équations paraboliques pour les ondes transmises et pour les ondes réfléchies. Deux exemples de propagation de la houle en présence d’une bosse circulaire et d’une bosse elliptique sont étudiés numériquement. La comparaison des résultats avec ceux de la littérature prouve la supériorité de cette équation parabolique vis-à-vis de celle de Radder

This paper deals with the parabolic formulation of propagation equations of gravity waves in surface in terms of angular capability with respect to the privileged propagation direction. This parabolic formulation is obtained by splitting the Berkhoff equation operator into two parabolic operators representing progressive and regressive wave propagation. The use of Pade approximants permits to derive simultaneously parabolic equations for the transmitted waves and the reflected waves. Two well-known reference examples, which represent the propagation of the wave when a caustic occur, are studied numerically and the results are compared with those of the literature. (C) 2001 Ifremer/CNRS/IRD/Editions scientifiques et medicales Elsevier SAS., Cet article présente une formulation parabolique des équations de propagation, en profondeur finie, des ondes de gravité en surface en terme de capacité angulaire par rapport à la direction de propagation privilégiée. Cette formulation est obtenue par factorisation de l’opérateur de l’équation de Berkhoff en deux opérateurs représentant les ondes progressives et régressives. L’utilisation des approximants de Padé permet d’accéder simultanément à des équations paraboliques pour les ondes transmises et pour les ondes réfléchies. Deux exemples de propagation de la houle en présence d’une bosse circulaire et d’une bosse elliptique sont étudiés numériquement. La comparaison des résultats avec ceux de la littérature prouve la supériorité de cette équation parabolique vis-à-vis de celle de Radder

The paper discusses the formulation of high-order accurate time-stepping schemes for transient convection–diffusion problems to be combined with finite element methods of the least-squares type for a stable discretization of highly convective problems. Padé approximations of the exponential function are considered for deriving multi-stage time integration schemes involving first time derivatives only, thus easier to implement in conjunction with C0 finite elements than standard time-stepping schemes which incorporate higher-order time derivatives. After a brief discussion of the stability and accuracy properties of the multi-stage Padé schemes and having underlined the similarity between Padé and Runge–Kutta methods, the paper closes with the presentation of illustrative examples which indicate the effectiveness of the proposed methods., Peer Reviewed, Postprint (author’s final draft)

We present a simple series of high-order finite difference and finite element propagation procedures based on approximating the exponential of two noncommuting operators as a product of single-operator Pade approximants. We have previously generated a class of precise split-step fast Fourier transform propagation techniques from an expansion of eA+B, where A and B are noncommuting operators, into an alternating product of exponentials of the individual component operators. We here extend this formalism to split-operator finite difference and finite element alternating direction implicit procedures by defining the generalized Pade approximant of eA+B as a product of Pade approximants of each individual operator. We demonstrate the central fact that that high-order propagation methods often require only (1,1) component approximants. Our results, which are found numerically to apply in fourth and sixth order even to longitudinally varying refractive index distributions, are verified by analyzing light propagation through an integrated optic microlens., This Article is from 'Integrated Photonics Research'. Volume 10, AD-A255 423, p042-045

In recent years the properties of percolation models have been studied intensively. The purpose of our project was to develop a general theory of percolation and clustering between particles of arbitrary size and shape, with arbitrary correlations between them. The goal of such a theory includes the treatment of continuum percolation as well as a novel treatment of lattice percolation. We made substantial progress toward this goal. The quantities basic to a description of clustering, the mean cluster size, mean number of clusters, etc. were developed. Concise formulas were given for the terms in such series, and proved, at least for sufficiently low densities, that the series are absolutely convergent. These series can now be used to construct Pade approximants that will allow one to probe the percolation transition. A scaled- particle theory of percolation was developed which gives analytic approximants for the mean number of clusters in a large class of two and three dimensional percolation models. Although this quantity is essential in many applications, e. g., explaining colligative properties, and interpreting low-angle light- scattering data, no systematic studies of it have been done before this work. Recently carried out detailed computer simulations show that the mean number of clusters is given to high accuracy by several of there approximations. Extensions of this work will allow calculation of the complete cluster size distribution. This work should be of practical importance in studying systems ranging from colloidal dispersions to nano-scale metal clusters.

Aln analytical model was formulated for the calculation of unsymmetrical pressure distributions inside the cavities of a multi-stage labyrinth seal. The model is based on a set of linearized continuity and momentum equations for the throughflow, and can account for the effects of rotation, whirl, flow swirl, differential gap width and (limited) compressibility. Preliminary calculations show excellent agreement with non-whirling side-force data in the literature. Also, models are being constructed that will permit calculation of the effect of these seal forces on the running stability of flexible, highly loaded turbomachines, with the Space Shuttle Main Engine serving as the prototype of such machines.

This book contains the proceedings of a Symposium on Approximation Theory which was held in Austin on January 18-21, 1976. The program of the symposium included seven one-hour invited expository lectures, more than forty one-half-hour invited research talks, and a large number of contributed fifteen minute talks. The expository papers describe recent developments in some important special fields. For example, D. Braess discusses nonlinear approximation, C. de Boor surveys the central role of B-splines in the general theory of splines, R. DeVore discusses the degree of approximation by polynomials, trigonometric polynomials, and splines, obtaining the results from a relation between the moduli of smoothness and Peetre's K-functionals, A. G. Vituskin has a new point of view upon entropy-he indicates its applications to coding, G. P. Nevai describes recent results (by himself, G. Freud and others) in the theory of convergence of interpolation formulas and of orthogonal polynomials, C. K. Chui surveys Pade approximations of arbitrary analytic functions, and L. L. Schumaker discusses fitting surfaces., Proceedings of an International Symposium conducted by the Texas Univ. at Austin, 18-21 Jan 76. Library of Congress Catalog Card No. 73- 2083.

Active feedback control is applied to cantilever forward-swept wings (FSW) providing significant increases in the critical airspeeds of aeroelastic flutter and divergence. Feedback compensation improves the critical airspeed for a divergence critical FSW by 25% and improves the critical airspeed for a flutter critical FSW example by 30%. These improvements are limited by the emergence as most critical, instabilities that were only secondary without feedback. The method of analysis is linear control theory in the Laplace domain. A significant improvement is made in the Pade Approximant method for calculating damped unsteady aerodynamic forces in the Laplace domain. Keywords: Aeroelasticity, Swept forward wings, Aerodynamics, Feedback control. (Theses)

This report gives the iterative and Pade solutions of the propagating and evanescent wave modes of the dispersion relation for the linear water-wave theory. Their corresponding FORTRAN subroutines are also provided. The Pade solution is generally more efficient than the iterative one in most engineering applications. Keywords: Water waves--Dynamics; Dispersion relations; Pade approximant; Iterative methods (Mathematics).

Work completed under this grant on polar fluids includes an extension of perturbation theory to polar molecules with point octopoles and hexadecapoles as well as a preliminary investigation of molecules with non-spherical cores. In addition, the ordering of dipolar molecules near a flat electrified wall has been investigated in the linearized hypernetted chain (LHNC) and quadratic hypernetted chain (QHNC) approximations. This has led to a new study of electrostriction and non-linear effects in the polarization density of dipolar fluids. (Author)

The application of conformal mapping methods to the solution of free-surface flow problems is considered. Methods of numerical conformal mapping based on Fourier series are extended to handle efficiently problems with time-dependent boundaries. They are shown to be practicable only for moderately distorted geometries. Extensions of the Menikoff-Zemach method to breaking geometries are presented. This method works well at large distortions. (Author)

Many of the gaps in the scaling theory of percolation clusters for random systems have been filled during the past three years. These have led to a better understanding of the statistics of lattice animals, and of the nature of the percolation transition. Calculations have been undertaken to determine parameters which characterize the structure of the infinite percolating cluster. Investigations have been initiated of the properties of systems with correlated percolation (i.e., in which the probability of occupation of a given site influences its neighbours). Work has continued on high temperature expansions for the Ising and classical Heisenberg models in an attempt to remove the discrepancies with renormalization group calculations. Particular attention has been devoted to calculations in 4 dimensions because of the marginal significance of this dimension. Further investigations have been made of the statistical properties of a single polymer chain using the Domb-Joyce model, and of particular properties of self-avoiding walks. (Author)