Robust and efficient solution of the drum problem via Nystrom approximation of the Fredholm determinant

The drum problem-finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition-has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary integral equations are appealing for large-scale problems, yet certain difficulties have limited their use. We introduce two ideas to remedy this: 1) We solve the resulting nonlinear eigenvalue problem using Boyd's method for analytic root-finding applied to the Fredholm determinant. We show that this is many times faster than the usual iterative minimization of a singular value. 2) We fix the problem of spurious exterior resonances via a combined field representation. This also provides the first robust boundary integral eigenvalue method for non-simply-connected domains. We implement the new method in two dimensions using spectrally accurate Nystrom product quadrature. We prove exponential convergence of the determinant at roots for domains with analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency, in a variety of domain shapes including ones with strong exterior resonances.Comment: 21 pages, 7 figures, submitted to SIAM Journal of Numerical Analysis. Updated a duplicated picture. All results unchange

Characterization of Alaskan Hot-Mix Asphalt containing Reclaimed Asphalt Pavement Material

In order to properly characterize Alaskan HMA materials containing RAP, this study evaluated properties of 3 asphalt binders typically used in Alaska, PG 52-28, PG 52-40, and PG 58-34, and 11 HMA mixtures containing up to 35% RAP that were either produced in the lab or collected from existing paving projects in Alaska. Various binder and mixture engineering properties were determined, including true high binder grades, complex modulus (|G*|), and phase angle (δ) at high performance temperatures, MSCR recovery rate and compliance, BBR stiffness and m-value, DTT failure stress and strain for binders, and dynamic modulus, flow number, IDT creep stiffness and strength for mixtures. Binder cracking temperatures were determined through Thermal Stress Analysis Routine (TSAR) software along with BBR and DTT data. Mixture cracking temperatures were determined with IDT creep stiffness and strength data. It was found that rutting may not be a concern with Alaskan RAP mix, while low-temperature cracking concerns may still exist in RAP mix in Alaska. A savings of $13.3/ton was estimated for a 25% RAP mix, with consideration of Alaskan situations. Many recommendations for future RAP practice and research are recommended based on testing results and cost analysis.Alaska Department of Transportation Statewide Research Offic