Mark Lyon

Microstructure-sensitive design of a compliant beam

We show that mechanical design can be conducted where consideration of polycrystalline microstructure as a continuous design variable is facilitated by use of a spectral representation space. Design of a compliant fixed-guided beam is used as a case study to illustrate the main tenets of the new approach, called microstructure-sensitive design (MSD). Selection of the mechanical framework for the design (e.g., mechanical constitutive model) dictates the dimensionality of the pertinent representation. Microstructure is considered to be comprised of basic elements that belong to the material set. For the compliant beam problem, these are uni-axial distribution functions. The universe of pertinent microstructures is found to be the convex hull of the material set, and is named the material hull. Design performance, in terms of specified design objectives and constraints, is represented by one or more surfaces (often hyperplanes) of finite dimension that intersect the material hull. Thus, the full range of microstructure, and concomitant design performance, can be exploited for any material class. Optimal placement of the salient iso-property surfaces within the material hull dictates the optimal set of microstructures for the problem. Extensions of MSD to highly constrained design problems of higher dimension is also described.

A Fast Algorithm for Fourier Continuation

A new algorithm is presented which provides a fast method for the computation of recently developed Fourier continuations (a particular type of Fourier extension method) that yield superalgebraically convergent Fourier series approximations of nonperiodic functions. Previously, the coefficients of an approximating Fourier series have been obtained by means of a regularized singular value decomposition (SVD)-based least-squares solution to an overdetermined linear system of equations. These SVD methods are effective when the size of the system does not become too large, but they quickly become unwieldy as the number of unknowns in the system grows. We demonstrate a novel decoupling of the least-squares problem which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system. Utilizing randomized algorithms, the low-rank system is reduced to a significantly smaller system of equations. This new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized SVD. The computational cost of the new algorithm in on the order of the cost of a single FFT multiplied by a slowly increasing factor that grows only logarithmically with the size of the system.

Windowed Green Function Method for Layered-Media Scattering

This paper introduces a new windowed Green function (WGF) method for the numerical integral-equation solution of problems of electromagnetic scattering by obstacles in the presence of dielectric or conducting half-planes. The WGF method, which is based on the use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free-space Green function, does not require evaluation of expensive Sommerfeld integrals. The proposed approach is fast, accurate, flexible, and easy to implement. In particular, straightforward modifications of existing (accelerated or unaccelerated) integral-equation solvers suffice to incorporate the WGF capability. The method relies on a certain integral equation posed on the union of the boundary of the obstacle and a small flat section of the interface between the penetrable media. Our analysis and numerical experiments demonstrate that both the near- and far-field errors resulting from the proposed approach decrease faster than any negative power of the window size. In the examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than a corresponding method based on use of Sommerfeld integrals.

Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

We present a comparison between the performance of solvers based on Nystrom discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems (1) the classical first kind integral equations for transmission problems, (2) the classical second kind integral equations for transmission problems, (3) the {\em single} integral equation formulations, and (4) certain direct counterparts of recently introduced Generalized Combined Source Integral Equations. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystrom solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.

Sobolev smoothing of SVD-based Fourier continuations

Abstract A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.

The Fourier approximation of smooth but non-periodic functions from unevenly spaced data

We develop an algorithm to extend, to the nonequispaced case, a recently-introduced fast algorithm for constructing spectrally-accurate Fourier approximations of smooth, but nonperiodic, data. Fast Fourier continuation algorithms, which allow for the Fourier approximation to be periodic in an extended domain, are combined with the underlying ideas behind nonequispaced fast Fourier transform (NFFT) algorithms. The result is a method which allows for the fast and accurate approximation of unevenly sampled nonperiodic multivariate data by Fourier series. A particular contribution of the proposed method is that its formulation avoids the difficulties related to the conditioning of the linear systems that must be solved in order to construct a Fourier continuation. The efficiency, essentially equivalent to that of an NFFT, and accuracy of the algorithm is shown through a number of numerical examples. Numerical results demonstrate the spectral rate of convergence of this method for sufficiently smooth functions. The accuracy, for sufficiently large data sets, is shown to be improved by several orders of magnitudes over previously published techniques for scattered data interpolation.