The two-dimensional free-space Green’s function and its derivatives in a tangent-normal system

The two-dimensional free-space Green’s function, G⁽²⁾$G^{\left(2\right)}$, and its derivatives, are used extensively in the formulation of scattering and diffraction problems through its presence in single- and double-layer potentials, and their use in integral equations. The vast majority of the results from elementary classical mathematical physics for G⁽²⁾$G^{\left(2\right)}$ is based on Cartesian coordinate-space, either directly as a Hankel function in coordinate-space or through a transform, such as the Weyl transform, also based on Cartesian coordinate-space. However, if the geometry of the problem is not Cartesian, for example in scattering from a rough surface, there are difficulties in using a transform representation for G⁽²⁾$G^{\left(2\right)}$ which depends on Cartesian geometry, as the standard Weyl transform does. Here we formulate transform-space representations using a tangent-normal coordinate system. The result for G⁽²⁾$G^{\left(2\right)}$ is a new Weyl-type tangent-normal transform representation from which the results for the vector derivatives of the single-layer potential, the double-layer potential, and the vector derivatives of the double-layer potential follow quite simply. The latter three results can be expressed in terms of two new spectral functions in tangent-normal space, _S1$S_1$ and _S2$S_2$. The overall results are new representations for G⁽²⁾$G^{\left(2\right)}$ and its derivatives which may be useful in integral equation formulations of scattering problems for non-Cartesian geometries.     

On the efficient representation of the half-space impedance Green’s function for the Helmholtz equation

A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green’s function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for representing this Green’s function are the Sommerfeld integral and the (closely related) method of complex images. The former is extremely efficient when the source is at some distance from the half-space boundary, but involves an unwieldy range of integration as the source gets closer and closer. Complex image-based methods, on the other hand, can be quite efficient when the source is close to the boundary, but they do not easily permit the use of the superposition principle since the selection of complex image locations depends on both the source and the target. We have developed a new, hybrid representation which uses a finite number of real images (dependent only on the source location) coupled with a rapidly converging Sommerfeld-like integral. While our method applies in both two and three dimensions, we restrict the detailed analysis and numerical experiments here to the two-dimensional case.     

Analysis of an interfacial crack in a piezoelectric bi-material via the extended Green’s functions and displacement discontinuity method

Based on the extended Stroh formalism, we first derive the extended Green’s functions for an extended dislocation and displacement discontinuity located at the interface of a piezoelectric bi-material. These include Green’s functions of the extended dislocation, displacement discontinuities within a finite interval and the concentrated displacement discontinuities, all on the interface. The Green’s functions are then applied to obtain the integro-differential equation governing the interfacial crack. To eliminate the oscillating singularities associated with the delta function in the Green’s functions, we represent the delta function in terms of the Gaussian distribution function. In so doing, the integro-differential equation is reduced to a standard integral equation for the interfacial crack problem in piezoelectric bi-material with the extended displacement discontinuities being the unknowns. A simple numerical approach is also proposed to solve the integral equation for the displacement discontinuities, along with the asymptotic expressions of the extended intensity factors and J-integral in terms of the discontinuities near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, and the influence of different extended loadings on the interfacial crack behaviors is further investigated.     

An efficient and energy stable scheme for a phase‐field model for the moving contact line problem

In this paper, we propose for the first time a linearly coupled, energy stable scheme for the Navier–Stokes–Cahn–Hilliard system with generalized Navier boundary condition. We rigorously prove the unconditional energy stability for the proposed time discretization as well as for a fully discrete finite element scheme. Using numerical tests, we verify the accuracy, confirm the decreasing property of the discrete energy, and demonstrate the effectiveness of our method through numerical simulations in both 2‐D and 3‐D. Copyright © 2015 John Wiley & Sons, Ltd.     

Three‐dimensional computation for flow‐induced vibrations in in‐line and cross‐flow directions of a circular cylinder

We present numerical results for in‐line and cross‐flow vibrations of a circular cylinder, which is immersed in a uniform flow and is elastically supported by damper‐spring systems to compute vibrations of a rigid cylinder. In the case of a circular cylinder with a low Scruton number, it is well‐known that two types of self‐excited vibrations appear in the in‐line direction in the range of low reduced velocities. On the other hand, a cross‐flow vibration of the circular cylinder can be excited in the range of high reduced velocities. Therefore, we compute the flow‐induced vibrations of the circular cylinder in the wide range of the reduced velocities at low and high Scruton numbers and discuss about excitation mechanisms in the in‐line and cross‐flow directions. Copyright © 2011 John Wiley & Sons, Ltd.