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

Fractal scattering dynamics of the three-dimensional HOCl molecule

We compare the 2D and 3D classical fractal scattering dynamics of Cl and HO for energies just above dissociation of the HOCl molecule, using a realistic potential energy surface for the HOCl molecule and techniques developed to analyze 3D chaotic scattering processes. For parameter regimes where the HO dimer initially has small vibrational energy, only small intervals of initial conditions show fractal scattering behavior and the scattering process is well described by a 2D model. For parameter regimes where the HO dimer initially has large vibrational energy, the scattering process is fully 3D and is dominated by fractal behavior.Robert A. Welch Foundation F-1051CONACyT 79988DGAPA IN110110Physic

A portable device for studying the effects of fluid flow on degradation properties of biomaterials inside cell incubators.

A portable device was designed and constructed for studying the properties of biomaterials in physiologically relevant fluids under controllable flow conditions that closely simulate fluid flow inside the body. The device can fit entirely inside a cell incubator; and, thus, it can be used directly under standard cell culture conditions. An impedance-driven pump was built in the sterile flow loop to control the flow rates of fluids, which made the device small and portable for easy deployment in the incubator. To demonstrate the device functions, magnesium (Mg) as a representative biodegradable material was tested in the flow device for immersion degradation under flow versus static conditions, while the flow module was placed inside a standard cell incubator. The flow rate was controlled at 0.17 ± 0.06 ml/s for this study; and, the flow rate is adjustable through the controller module outside of incubators for simulating the flow rates in the ranges of blood flow in human artery (0.05 ∼0.43 ml/s) and vein (0.02 ∼0.08 ml/s). Degradation of Mg under flow versus static conditions was characterized by measuring the changes of sample mass and thickness, and Mg2+ ion concentrations in the immersion media. Surface chemistry and morphology of Mg after immersion under flow versus static conditions were compared. The portable impedance-driven flow device is easy to fit inside an incubator and much smaller than a peristaltic pump, providing a valuable solution for studying biomaterials and implants (e.g. vascular or ureteral stents) in body fluids under flow versus static conditions with or without cells