Film No Longer Telling a Story; Film Itself as the Story: Reflexive Constructions in Alfred Hitchcock and Jean-Luc Godard

Senior Project submitted to The Division of Arts of Bard Colleg

Well-posedness, energy and charge conservation for nonlinear wave equations in discrete space-time

We consider the problem of discretization for the U(1)-invariant nonlinear wave equations in any dimension. We show that the classical finite-difference scheme used by Strauss and Vazquez \cite{MR0503140} conserves the positive-definite discrete analog of the energy if the grid ratio is $dt/dx\le 1/\sqrt{n}$, where $dt$ and $dx$ are the mesh sizes of the time and space variables and $n$ is the spatial dimension. We also show that if the grid ratio is $dt/dx=1/\sqrt{n}$, then there is the discrete analog of the charge which is conserved. We prove the existence and uniqueness of solutions to the discrete Cauchy problem. We use the energy conservation to obtain the a priori bounds for finite energy solutions, thus showing that the Strauss -- Vazquez finite-difference scheme for the nonlinear Klein-Gordon equation with positive nonlinear term in the Hamiltonian is conditionally stable.Comment: 10 page

Toucher la corde sensible

National audienceContrairement aux apparences, le piano est un système acoustique et mécanique sophistiqué dont la facture demeure à ce jour très largement empirique. Les connaissances, très précises, des concepteurs et fabricants de pianos sont issues de siècles d'expérimentations, d'échecs, de succès... Et intriguent beaucoup les chercheurs en acoustique musicale. En utilisant des méthodes scientifiques pour modéliser le fonctionnement d'un piano et de ses différents éléments, il est possible de confirmer ou non ces connaissances et d'aller plus loin dans la compréhension des phénomènes mis en jeu

Simuler le son d'un piano

National audienceLa conception et la fabrication du piano sont très largement basées sur un savoir empirique issu de plusieurs siècles d'expérimentations, d'échecs, de succès... Les facteurs de piano ont acquis un ensemble de connaissances extrêmement précises mais cherchent désormais à rationaliser leur approche en utilisant des méthodes scientifiques, afin de donner raison ou tort à certaines affirmations, et à aller plus loin dans la compréhension des phénomènes mis en jeu

Derivation of high order absorbing boundary conditions for the Helmholtz equation in 2D

We present high order absorbing boundary conditions (ABC)for the Helmholtz equation in 2D, that can adapt to any regular shapedsurfaces. The new ABCs are derived by using the technique ofmicro-diagonalisation to approximate the Dirichlet-to-Neumann map.Numerical results on different shapes illustrate the behavior of thenew ABCs along with high-order finite elements

Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string

International audienceThis paper considers a general class of nonlinear systems, "nonlinear Hamiltonian systems of wave equations". The first part of our work focuses on the mathematical study of these systems, showing central properties (energy preservation, stability, hyperbolicity, finite propagation velocity, etc.). Space discretization is made in a classical way (variational formulation) and time discretization aims at numerical stability using an energy technique. A definition of "preserving schemes" is introduced, and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is trivial. A general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear Hamiltonian systems of wave equations class. The problem of the vibration of a piano string is taken as an example. Nonlinear coupling between longitudinal and transversal modes is modeled in the "geometrically exact model", or approximations of this model. Numerical results are presented

Energy preserving scheme for non linear systems of wave equations. Application to piano strings.

International audienceThe linear wave equation does not describe the com- plexity of the piano strings vibration enough for physics based sound synthesis. The nonlinear cou- pling between transversal and longitudinal modes has to be taken into account, as does the "geometrically exact" model. This system of equations can be clas- sified among a general energy preserving class of sys- tems. We present an implicit, centered, second order accurate, numerical scheme that preserves a discrete energy, leading to unconditional stability of the nu- merical scheme. The complete model takes into ac- count the bridge coupling the strings, and the ham- mer non linear attack on the strings

A comparison of a one-dimensional finite element method and the transfer matrix method for the computation of wind music instrument impedance

International audienceThis work presents a computation tool for the calculation of wind instrument input impedance in the context of linear planar wave propagation with visco-thermal losses. The originality of the approach lies in the usage of a specific and simple 1D finite element method (FEM). The popular Transfer Matrix Method (TMM) is also recalled and a seamless formulation is proposed which unifies the cases cylinders vs. cones. Visco-thermal losses, which are natural dissipation in the system, are not exactly taken into account by this method when arbitrary shapes are considered. The introduction of an equivalent radius leads to an approximation that we quantify using the FEM method. The equation actually solved by the TMM in this case is exhibited. The accuracy of the two methods (FEM and TMM) and the associated computation times are assessed and compared. Although the TMM is more efficient in lossless cases and for lossy cylinders, the FEM is shown to be more efficient when targeting a specific precision in the realistic case of a lossy trumpet. Some additional features also exhibit the robustness and flexibility of the FEM over the TMM. All the results of this article are computed using the open-source python toolbox OpenWind

Modeling and numerical simulation of a grand piano.

International audienceWe consider a complete model of a piano which accounts for the acoustical behavior of the instrument from excitation to soundand, and we propose a numerical discretisation. The model is described as well as the numerical methods used for its discretisation. Nonlinearities and couplings are treated in such a way that energy techniques ensure numerical stability. Numerical results are presented and compared to measurements

Résolution de l'équation de Galbrun avec un flot quelconque. Application à l'hélioséismologie

In this report, we are concerned with the solution of Galbrun’s equations in timeharmonicdomain for an arbitrary flow with high order finite element methods. Several equivalentformulations of Galbrun’s equations are proposed and discretized with Discontinuous Galerkinmethod. They are compared with a formulation adapted for continuous Galerkin discretization.Numerically, it has been observed that the tested discretization methods converge correctly foran uniform flow, but no longer for a non-uniform flow. Two kinds of stabilization are proposedin order to restore a nice convergence though original equations are modified. Finally, simplifiedGalbrun’s equations are proposed to coincide with original Galbrun’s equations when the flow isnull. Numerical illustrations are presented in the context of helioseismology.Dans ce document, nous nous intéressons à la résolution des équations de Galbrunen régime harmonique pour un écoulement quelconque par des méthodes d’éléments finis d’ordreélevé. Nous proposons plusieurs formulations équivalentes des équations de Galbrun discrétiséespar une méthode de Galerkin discontinue, que nous comparons à une formulation adaptée auxéléments finis continus. Nous observons numériquement que les différentes méthodes de discrétisationtestées convergent correctement pour un écoulement uniforme, mais ne convergent paspour un écoulement non-uniforme. Nous proposons deux types de stabilisation qui bien quemodifiant les équations initiales permettent de retrouver une convergence satisfaisante. Nousproposons aussi des équations de Galbrun simplifiées qui coïncident avec les équations de Galbrunoriginales lorsque l’écoulement est nul. Nous montrons des illustrations dans le contexte del’héliosismologie