Nicolas Valdivia

The detection of surface vibrations from interior acoustical pressure

We consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin. Mathematically this field satisfies the Helmholtz equation. In this paper we consider the three-dimensional case. We show that any regular solution of this equation admits a unique representation by a single-layer potential, so that the problem is equivalent to the solution of a linear integral equation of the first kind. We study uniqueness of reconstruction and obtain a sharp stability estimate and convergence rates for some regularization algorithms when the domain is a sphere. We have developed a boundary element code to solve the integral equation. We report numerical results with this code applied to three geometries: a sphere, a cylinder with spherical endcaps and a cylinder with a floor modelling the interior of an aircraft cabin. The exact test solution is given by a point source exterior to the surfaces with about 1% random noise added. Regularization methods using the truncated singular value decomposition with generalized cross validation and the conjugate gradient (cg) method with a stopping rule due to Hanke and Raus are compared. An interesting feature of the three-dimensional problem is the relative insensitivity of the optimal regularization parameter (number of iterations) for the cg method to the wavenumber and the multiplicity of the singular values of the integral operator.

The Detection of the Source of Acoustical Noise in Two Dimensions

We consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin. Mathematically this field satisfies the Helmholtz equation. In this paper we consider the model two-dimensional case. We show that any regular solution of this equation admits a unique representation by a single layer potential, so that the problem is reduced to the solution of a linear integral equation of the first kind. We prove uniqueness of reconstruction and obtain a sharp stability estimate. Finally, for two geometries and sources of noise simulating the cabin of the aircraft and two engines, we give results of the numerical solution of this integral equation, comparing regularization by the truncated singular value decomposition and the conjugate gradient method.

Study of the comparison of the methods of equivalent sources and boundary element methods for near-field acoustic holography

Boundary element methods (BEM) based near-field acoustic holography (NAH) has been used successfully in order to reconstruct the normal velocity on an arbitrarily shaped structure surface from measurements of the pressure field on a nearby conformal surface. An alternative approach for this reconstruction on a general structure utilizes the equivalent sources method (ESM). In ESM the acoustic field is represented by a set of point sources located over a surface that is close to the structure surface. This approach is attractive mainly for its simplicity of implementation and speed. In this work ESM as an approximation of BEM based NAH is studied and the necessary conditions for the successful application of this approach in NAH is discussed. A cylindrical fuselage surface excited by a point force as an example to validate the results is used.

Implicit methods of solution to integral formulations in boundary element method based nearfield acoustic holography

Two integral equation methods are considered which can be inverted to provide the surface velocity and/or pressure given a measurement of the pressure on an imaginary surface in the nearfield of a vibrating or scattering body. This problem is central to nearfield acoustical holography (NAH). The integral equations are discretized using boundary element methods (BEMs). The integral equation methods considered are (1) an indirect formulation method based on the single layer integral equation and (2) a direct formulation method based on a system of equations derived from the Helmholtz–Kirchhoff integral equation. The formation of integral equations from the mentioned methods will not involve the explicit inversion of matrices, but instead will require this inversion to be done implicitly. Since these methods back-track the sound field from the measurement surface to the surface of the source/vibrator they are ill-posed in nature and Tikhonov regularization are used to stabilize the reconstruction. Problems a...

Volumetric acoustic vector intensity imager

A new measurement system, consisting of a mobile array of 50 microphones that form a spherical surface of radius 0.2m, that images the acoustic intensity vector throughout a large volume is discussed. A simultaneous measurement of the pressure field across all the microphones provides time-domain holograms. Spherical harmonic expansions are used to convert the measured pressure into a volumetric vector intensity field on a grid of points ranging from the origin to a maximum radius of 0.4m. Displays of the volumetric intensity image are used to locate noise sources outside the volume. There is no restriction on the type of noise source that can be studied. An experiment inside a Boeing 757 aircraft in flight successfully tested the ability of the array to locate flow-noise-excited sources on the fuselage. Reference transducers located on suspected noise source locations can also be used to increase the ability of this device to separate and identify multiple noise sources at a given frequency by using the ...

Acoustic source identification using multiple frequency information

We consider the inverse problem of identifying the location and shape of a finitely supported acoustic source function, separable with respect to space and frequency, from measurements of the acoustic field on a closed surface for many frequencies. A simple uniqueness proof and an error estimate for the unknown source function are presented. From the uniqueness proof an efficient numerical algorithm for the solution is developed. The algorithm is tested using numerically generated data in dimensions 2 and 3.

Approximations of inverse boundary element methods with partial measurements of the pressure field

Boundary element methods (BEMs) based near-field acoustic holography (NAH) requires the measurement of the pressure field over a closed surface in order to recover the normal velocity on a nearby conformal surface. There are practical cases when measurements are available over a patch from the measurement surface in which conventional inverse BEM based NAH (IBEM) cannot be applied directly, but instead as an approximation. In this work two main approximations based on the indirect-implicit methods are considered: Patch IBEM and IBEM with Cauchy data. Patch IBEM can be applied with a continuation procedure, which as its predecessor patch NAH (a well known technique that can be used on separable geometries of the wave equation) continues the pressure field using an iterative procedure, or it can be applied by a direct procedure. On the other hand, IBEM with Cauchy data requires measurements over two conformal patches and it will be shown that this technique will be reliable regardless of the position of the source. The theory behind each method will be justified and validated using a cylindrical surface with numerical data generated by point sources, and using experimental data from a cylindrical fuselage excited by a point force.

Recovery of volatility coefficient by linearization

Abstract We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.

A frequency based constraint for a multi-frequency linear sampling method

The linear sampling method (LSM) has become a well established non-iterative technique for a variety of inverse scattering problems. The method offers a number of advantages over competing inverse scattering methods, mainly it is based on solving a linear problem while being able to account for multi-path effects. Unfortunately under the current framework the method is only effective when using a large number of multi-static data, and therefore may be impractical for many imaging applications. While primarily developed under a single frequency framework, recently the extension of the method to multi-banded data sets has been considered. It is known in general that the availability of multi-frequency data should compensate for reduced spatial diversity, but it is not clear how this can be accomplished for the LSM. In this work we take a step in this direction by considering a frequency based partial variation approach. We first establish that on bands absent of any corresponding Dirichlet eigenvalues the Herglotz density exhibits bounded variation. We then consider a regularization method incorporating this prior knowledge. The proposed approach exhibited a good estimate of the unknown Dirichlet eigenvalues of the obstacle in question when using reduced data. This observation also correlated with higher quality 3D reconstructions.

Surface decomposition method for near-field acoustic holography

Near-field acoustic holography reconstruction of the acoustic field at the surface of an arbitrarily shaped radiating structure from pressure measurements at a nearby conformal surface is obtained from the solution of a boundary integral equation. This integral equation is discretized using the equivalent source method and transformed into a matrix system that can be solved using iterative regularization methods that counteract the effect of noise on the measurements. This work considers the case when the resultant matrix system is so large that it cannot be explicitly formed and iterative methods of solution cannot be directly implemented. In this case the method of surface decomposition is proposed, where the measurement surface is divided into smaller nonoverlapping subsurfaces. Each subsurface is used to form a smaller matrix system that is solved and the result joined together to generate a global solution to the original matrix system. Numerically generated data are used to study the use of subsurface extensions to increase the continuity of the global solution, and investigate the size of the subsurfaces, as well as the distance between the measurement and the vibrating surface. Finally a vibrating ship hull structure is considered as a physical example to apply and validate the proposed methodology.