L.J. Gray

Green's function for a two–dimensional exponentially graded elastic medium

The free–space Green function for a two–dimensional exponentially graded elastic medium is derived. The shear modulus Âμ is assumed to be an exponential function of the Cartesian coordinates (x,y), i.e. μ ≡ μ(x,y) = μ0e2(β1x+β2y), where μ0, β1, and β2 are material constants, and the Poisson ratio is assumed constant. The Green function is shown to consist of a singular part, involving modified Bessel functions, and a non–singular term. The non–singular component is expressed in terms of one–dimensional Fourier–type integrals that can be computed by the fast Fourier transform.

Boundary element method for regions with thin internal cavities. II

Abstract A new boundary element technique for treating thin cavities is presented. This method treats the cavity as an infinitely thin slit, but does not utilize fictitious internal boundaries to subdivide the domain. A non-singular system of equations is obtained by constructing two equations for each node on the boundary of the slit: the standard boundary element equation and the normal derivative of this equation. Although, derived for Laplaces equation (in three dimensions) and tested on electroplating problems, it is expected that the method can be applied in other areas, most notably crack problems in fracture mechanics.

Symmetric Galerkin fracture analysis

Abstract The implementation of a symmetric Galerkin boundary integral method for crack problems is described. The symmetric Galerkin procedure requires the derivative (hypersingular) equation on the crack surface, and thus a straightforward application of the dual equation fracture method leads to a non-symmetric matrix. By employing the jump across the fracture as the variable on the crack surface, the problem can be formulated with a symmetric, and smaller dimension, coefficient matrix. An important advantage of this Galerkin approximation is that it avoids the difficulties inherent with C 1 interpolations demanded by a collocation approximation for hypersingular equations. Moreover, by incorporating previously developed efficient analytical integration techniques, the computational cost of this algorithm is shown to be competitive with collocation methods.

Approximate Green's functions in boundary integral analysis

It is well known that employing a Greens function which satisfies the prescribed conditions on a part of the boundary is advantageous for boundary integral calculations. In this paper, it is shown that an approximate Greens function, one in which the known data is nearly reproduced, can also be highly beneficial in implementations of the boundary-element method. This approximate Greens function approach is developed herein for solving the Laplace equation, and applied to the modeling of void dynamics under electromigration conditions in metallic thin-film interconnects used in integrated circuits.

Regularized spectral multipole BEM for plane elasticity

A multipole algorithm for plane elasticity based on the direct boundary element method (BEM) is presented. The kernels in the BEM are approximated as truncated Taylor series with expansion points taken from a uniform grid. The algorithm replaces the usual BEM elemental summations with correlation sums on the regular grid in terms of the sampled kernel data and density moments. Far field influences are rapidly computed in the frequency domain using the fast Fourier transform (FFT). The resultant linear system of equations is solved with GMRES. The multipole method is extended to whole-body regularized forms of the standard displacement-BIE and the stress-BIE. Free-term coefficients which arise from regularization in the far field are also rapidly computed as correlation sums with the FFT. The algorithm is shown to be faster than the traditional BEM for models with over 400 quartic elements while maintaining an acceptably high level of accuracy.

Boundary element methods for functionally graded materials

Functionally graded materials (FGMs) possess a smooth variation of material properties due to continuous change in microstructural details. For example, the material gradation may change gradually from a pure ceramic to a pure metal. This work focuses on potential (both steady state and transient) and elasticity problems for inhomogeneous materials. The Green’s function(GF) for these materials (e.g. exponentially graded) are expressed as the GF for the homogeneous material plus additional terms due to material gradation. The numerical implementations are performed using a Galerkin (rather than collocation) approximation. A number of examples have been carried out. The results of some speciflc test problems agree within plotting accuracy with available analytical solutions.

Interior point evaluation in the boundary element method

Abstract In the boundary integral method, interior point values are calculated by means of an integration over the boundary. For interior points close to the boundary, some integrands will be nearly singular and standard numerical integration is highly inaccurate. In this paper, interior values are computed by means of analytical integration formulas obtained via symbolic computation. Numerical experiments for the two-dimensional Laplace equation ∇ 2 φ = 0 demonstrate that the exact formulas accurately evaluate the potential φ arbitrarily close to the boundary. The interior potential gradient ∇φ is more difficult, and accurate results very near the boundary are obtained by suitably modifying the potential function.

PVM implementation of the symmetric-Galerkin method

We report on initial progress towards a parallel virtual machine (PVM) implementation of the symmetric-Galerkin boundary integral method. We take advantage of software packages specifically designed to solve linear algebra problems on distributed memory parallel computers. In particular we use linear algebra routines from the ScaLAPACK, PBLAS and BLACS libraries. These routines assume a block cyclic decomposition of the matrix operands. The decomposition of the operands and its impact on the construction of the coefficient matrix are described. Computational results for solving the two-dimensional Laplace equation are presented. This program is being used to simulate the performance of a proximity sensor used in robotics and other applications.

On the expansion for surface displacement in the neighborhood of a crack tip

It is shown that in the expansion of the crack opening displacement vs distance from the tip, there is no linear term present. This should lead to improved accuracy of the near tip fields and improved stress intensity factor results. The two-dimensional discussion should be able to be carried over to three dimensions.