L. J. Gray

On Green's function for a three-dimensional exponentially graded elastic solid

The problem of a point force acting in an unbounded, three–dimensional, isotropic elastic solid is considered. Kelvin solved this problem for homogeneous materials. Here, the material is inhomogeneous; it is ‘functionally graded’. Specifically, the solid is ‘exponentially graded’, which means that the Lamé moduli vary exponentially in a given fixed direction. The solution for the Greens function is obtained by Fourier transforms, and consists of a singular part, given by the Kelvin solution, plus a non–singular remainder. This grading term is not obtained in simple closed form, but as the sum of single integrals over finite intervals of modified Bessel functions, and double integrals over finite regions of elementary functions. Knowledge of this new fundamental solution for graded materials permits the development of boundary–integral methods for these technologically important inhomogeneous solids.

A coupled level set-boundary integral method for moving boundary simulations

A numerical method for moving boundary problems based upon level set and boundary integral formulations is presented. The interface velocity is obtained from the boundary integral solution using a Galerkin technique for post-processing function gradients on the interface. We introduce a new level set technique for propagating free boundary values in time, and couple this to a narrow band level set method. Together, they allow us to both update the function values and the location of the interface. The methods are discussed in the context of the well-studied two-dimensional nonlinear potential flow model of breaking waves over a sloping beach. The numerical results show wave breaking and rollup, and the algorithm is verified by means of convergence studies and comparisons with previous techniques. 1. Introduction and overview

Evaluation of the anisotropic Green's function in three dimensional elasticity

A perturbation expansion technique for approximating the three dimensional anisotropic elastic Greens function is presented. The method employs the usual series for the matrix (I−A)-1 to obtain an expansion in which the zeroth order term is an isotropic fundamental solution. The higher order contributions are expressed as contour integrals of matrix products, and can be directly evaluated with a symbolic manipulation program. A convergence condition is established for cubic crystals, and it is shown that convergence is enhanced by employing Voigt averaged isotropic constants to define the expansion point. Example calculations demonstrate that, for moderately anisotropic materials, employing the first few terms in the series provides an accurate solution and a fast computational algorithm. However, for strongly anisotropic solids, this approach will most likely not be competitive with the Wilson-Cruse interpolation algorithm.

Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method

We present a numerical method for tracking breaking waves over sloping beaches. We use a fully non-linear potential model for incompressible, irrotational and inviscid flow, and consider the effects of beach topography on breaking waves. The algorithm uses a Boundary Element Method (BEM) to compute the velocity at the interface, coupled to a Narrow Band Level Set Method to track the evolving air/water interface, and an associated extension equation to update the velocity potential both on and off the interface. The formulation of the algorithm is applicable to two- and three-dimensional breaking waves; in this paper, we concentrate on two-dimensional results showing wave breaking and rollup, and perform numerical convergence studies and comparison with previous techniques.

A Parallel Domain Decomposition BEM Algorithm for Three-Dimensional Exponentially Graded Elasticity

A parallel domain decomposition boundary integral algorithm for three-dimensional exponentially graded elasticity has been developed. As this subdomain algorithm allows the grading direction to vary in the structure, geometries arising from practical functionally graded material applications can be handled. Moreover, the boundary integral algorithm scales well with the number of processors, also helping to alleviate the high computational cost of evaluating the Greens functions. For axisymmetric plane strain states in a radially graded material, the numerical results for cylindrical geometries are in excellent agreement with the analytical solution deduced herein.

Computing Through Singularities in Potential Flow with Applications to Electrohydrodynamic Problems

Many interesting fluid interface problems, such as wave propagation and breaking, droplet and bubble break-up, electro-jetting, rain drops, etc. can be modeled using the assumption of potential flow. The main challenge, both theoretically and computationally, is due to the presence of singularities in the mathematical models. In all the above mentioned problems, an interface needs to be advanced by a velocity determined by the solution of a surface partial differential equation posed on this moving boundary. By using a level set framework, the two surface equations of the Lagrangian formulation can be implicitly embedded in PDEs posed on one higher dimension fixed domain. The advantage of this approach is that it seamlessly allows breakup or merging of the fluid domain and therefore provide a robust algorithm to compute through these singular events. Numerical results of a solitary wave breaking, the Rayleigh-Taylor instability of a fluid column, droplets and bubbles breaking-up and the electrical droplet distortion and subsequent jet emission can be obtained using this levelset/extended potential model.