Near-optimal perfectly matched layers for indefinite Helmholtz problems

A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

International audienceThis paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal and polyhedral computational domains for the finite element solution of high-frequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two- and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle

Stability and Convergence Analysis of Time-domain Perfectly Matched Layers for The Wave Equation in Waveguides

International audienceThis work is dedicated to the proof of stability and convergence of the Bérenger's perfectly matched layers in the waveguides for an arbitrary L ∞ damping function. The proof relies on the Laplace domain techniques and an explicit representation of the solution to the PML problem in the waveguide. A bound for the PML error that depends on the absorption parameter and the length of the PML is presented. Numerical experiments confirm the theoretical findings

An Investigation into the Analysis of Truncated Standard Normal Distributions Using Heuristic Techniques

Standard normal distributions (SND) and truncated standard normal distributions (TSND) have been widely used and accepted methods to characterize the data sets in various engineering disciplines, financial industries, medical fields, management, and other mathematic and scientific applications. For engineering managers, risk managers and quality practitioners, the use of the standard normal distribution and truncated standard normal distribution have particular relevance when bounding data sets, evaluating manufacturing and assembly tolerances, and identifying measures of quality. In particular, truncated standard normal distributions are used in areas such as component assemblies to bound upper and lower process specification limits. This dissertation presents a heuristic approach for the analysis of assembly-level truncated standard normal distributions. This dissertation utilizes unique properties of a characteristic function to analyze truncated assemblies. Billingsley (1995) suggests that an inversion equation aids in converting the characteristic functions for a given truncated standard normal distribution to its corresponding probability density function. The heuristic for the inversion characteristics for a single doubly truncated standard normal distribution uses a known truncated standard normal distribution as a probability density function baseline. Additionally, a heuristic for the analysis of TSND assemblies building from the initial inversion heuristic was developed. Three examples are used to further demonstrate the heuristics developed by this dissertation. Mathematical formulation, along with correlation and regression analysis results, support the alternate hypotheses presented by this dissertation. The correlation and regression analysis provides additional insight into the relationship between the truncated standard normal distributions analyzed. Heuristic procedures and results from this dissertation will also serve as a benchmark for future research. This research contributes to the body of knowledge and provides opportunities for continued research in the area of truncated distribution analysis. The results and proposed heuristics can be applied by engineering managers, quality practitioners, and other decision makers to the area of assembly analysis