The quantum-mechanical Coulomb propagator in an L2 function representation

The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L$$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

Symplectic P-stable additive Runge—Kutta methods

<p style='text-indent:20px;'>Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are <i>P-stable</i>, therefore suitable for application to highly oscillatory problems.</p>

Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack

In this study, singular integral solutions were studied to investigate scattering of Rayleigh waves by subsurface cracks. Defining a wave scattering model by objects, such as cracks, still can be quite a challenge. The model’s analytical solution uses five different numerical integration methods: (1) the Gauss–Legendre quadrature, (2) the Gauss–Chebyshev quadrature, (3) the Gauss–Jacobi quadrature, (4) the Gauss–Hermite quadrature and (5) the Gauss–Laguerre quadrature. The study also provides an efficient dynamic finite element analysis to demonstrate the viability of the wave scattering model with an optimized model configuration for wave separation. The obtained analytical solutions are verified with displacement variation curves from the computational simulation by defining the correlation of the results. A novel, verified model, is proposed to provide variations in the backward and forward scattered surface wave displacements calculated by different frequencies and geometrical crack parameters. The analytical model can be solved by the Gauss–Legendre quadrature method, which shows the significantly correlated displacement variation with the FE simulation result. Ultimately, the reliable analytic model can provide an efficient approach to solving the parametric relationship of wave scattering.

Modeling for Microslip Behavior of Lap Joints Based on Non-Gaussian Rough Surfaces

A general contact model for a lap joint interface based on non-Gaussian surfaces was proposed. The effect of surface topography parameters on microslip behavior in a lap joint interface was studied. Pearson system was applied to produce non-Gaussian surfaces. Combining the topographical-dependent Zhao–Maietta–Chang (ZMC) model with the physical-related Iwan model, the nonlinear constitutive relationship of a lap interface was constructed by using Masing hypothesis. Meanwhile, the probability density function of asperity heights of an infinitely smooth surface was mathematically proved to be a delta function, verifying that the calculated value of friction in the model conforms to the physical law. Gauss-Legendre quadrature was conducted to calculate contact relations of different Pearson distribution surfaces. Furthermore, numerical results of microslip loops under oscillating tangential forces were compared with the published experiments, indicating the present model considering non-Gaussian surfaces could agree well with the experiments.