Thomas K. DeLillo

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.

Schwarz-Christoffel mapping of multiply connected domains

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivitym≥2 thereby extending the results of Christoffel (1867) and Schwarz (1869) form=1 and Komatu (1945),m=2. A formula forf, the conformal map of the exterior ofm bounded disks to the exterior ofm bounded disjoint polygons, is derived. The derivation characterizes the global preSchwarzianf″ (z)/f′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on them boundary circles via the reflection principle and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a “method of images” infinite sequence of iterations of reflecting prevertex singularities from them boundary circles to the whole sphere.

Radial and circular slit maps of unbounded multiply connected circle domains

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

The accuracy of numerical conformal mapping methods: a survey of examples and results

This paper shows how the geometry of the region affects the conditioning and the accuracy of numerical conformal mapping methods for simply connected regions, especially Fourier series methods. Both explicit examples of popular test cases and more general estimates are discussed. The severe ill conditioning that is known as the crowding phenomenon is discussed and its effect on a conformally transplanted boundary value problem is illustrated. Remarks on various numerical methods are included.

Schwarz-Christoffel Mapping of Bounded, Multiply Connected Domains

A Schwarz-Christoffel formula for conformal maps from the exterior of a finite number of disks to the exterior of polygonal curves was derived by DeLillo, Elcrat, and Pfaltzgraff in [9] using the Reflection Principle. The derivative of the map is expressed as an infinite product. In this paper, the formula for the map from bounded circular domains to bounded polygonal domains is derived by the same method. Convergence of the resulting infinite product is proved for sufficiently well-separated domains. A formula for the bounded case was also derived by Crowdy in [5] using Schottky-Klein prime functions. We show that Crowdy’s formula can be reduced to ours. In addition, we discuss the relation of these formulae to the Poincaré theta series for functions automorphic under the Schottky group of Moebius transformations generated by reflections in circles. We also derive a formula for the map to circular slit domains.

Numerical conformal mapping of multiply connected regions by Fornberg-like methods

Summary. We develop a new algorithm for computing conformal maps from regions exterior to non-overlapping disks to unbounded multiply connected regions exterior to non-overlapping, smoothly bounded Jordan regions. The method is an extension of Fornbergs original Newton-like method for mapping of the disk to simply connected regions. A Fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case. Numerical examples are reported.

Computation of Multiply Connected Schwarz-Christoffel Maps for Exterior Domains

We have recently derived a Schwarz-Christoffel formula for the conformal mapping of the exterior of a finite number of disks to the exterior of a set of polygonal curves [5]. In this work we show how to formulate a set of equations for determining the parameters of such a map. A number of examples are computed, including exteriors of multiple slits. We also recall the derivation of the mapping formulae and give a new formula for the doubly connected case.

A Fornberg-like conformal mapping method for slender regions

Abstract A method is presented for approximating the conformal map from the interior of an ellipse to the interior of a simply-connected target region. The map is represented as a truncated Chebyshev series. Conditions that the mapping function be conformal are transplanted from the ellipse to an annulus with the Joukowski map. The resulting conditions on the Laurent coefficients then give a system of equations for the Newton update of the approximate boundary correspondence. This system is a generalization of Fornbergs system for maps from the disk and is solved similarly in O(N log N) operations by the conjugate gradient method. Our numerical experiments demonstrate that the maps from the ellipse to a slender target region of similar aspect ratio can be constructed with far fewer mesh points than are required for maps from the disk, thus circumventing the ill-conditioning due to crowding in these cases.

The Numerical Solution of the Biharmonic Equation by Conformal Mapping

The solution to the biharmonic equation in a simply connected region