Naoshi Nishimura

A Fast Boundary Element Method for the Analysis of Fiber-Reinforced Composites Based on a Rigid-Inclusion Model

A new boundary element method (BEM) is developed for three-dimensional analysis of fiber-reinforced composites based on a rigid-inclusion model. Elasticity equations are solved in an elastic domain containing inclusions which can be assumed much stiffer than the host elastic medium. Therefore the inclusions can be treated as rigid ones with only six rigid-body displacements. It is shown that the boundary integral equation (BIE) in this case can be simplified and only the integral with the weakly-singular displacement kernel is present. The BEM accelerated with the fast multipole method is used to solve the established BIE. The developed BEM code is validated with the analytical solution for a rigid sphere in an infinite elastic domain and excellent agreement is achieved. Numerical examples of fiber-reinforced composites, with the number of fibers considered reaching above 5800 and total degrees of freedom above 10 millions, are solved successfully by the developed BEM. Effective Young’s moduli of fiber-reinforced composites are evaluated for uniformly and ‘‘randomly’’ distributed fibers with two different aspect ratios and volume fractions. The developed fast multipole BEM is demonstrated to be very promising for large-scale analysis of fiber-reinforced composites, when the fibers can be assumed rigid relative to the matrix materials. @DOI: 10.1115/1.1825436#

A fast multipole boundary integral equation method for crack problems in 3D

This paper discusses a three-dimensional fast multipole boundary integral equation method for crack problems for Laplaces equation. The proposed implementation uses collocation and piecewise constant shape functions to discretise the hypersingular boundary integral equation for crack problems. The resulting numerical equation is solved with GMRES (generalised minimum residual method) in connection with FMM (fast multipole method). It is found that the obtained code is faster than a conventional one when the number of unknowns is greater than about 1300.

Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D

Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large-scale problems. This paper discusses an application of FMM to three-dimensional boundary integral equation method for elastostatic crack problems. The boundary integral equation for many crack problems is discretized with FMM and Galerkins method. The resulting algebraic equation is solved with generalized minimum residual method (GMRES). The numerical results show that FMM is more efficient than conventional methods when the number of unknowns is more than about 1200 and, therefore, can be useful in large-scale analyses of fracture mechanics. Copyright

A regularized Boundary Integral Equation Method for Elastodynamic Crack Problems

This paper presents a double layer potential approach of elastodynamic BIE crack analysis. Our method regularizes the conventional strongly singular expressions for the traction of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. The manipulation is systematized by the use of the stress function representation of the differentiated double layer kernel functions. This regularization, together with the use of B-spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elastodynamics.

Recent Advances and Emerging Applications of the Boundary Element Method

Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Greens functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

Application of new fast multipole boundary integral equation method to crack problems in 3D

Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large scale problems. This paper discusses an application of the new version of FMM to three-dimensional boundary integral equation method (BIEM) for crack problems for the Laplace equation. The boundary integral equation is discretised with collocation method. The resulting algebraic equation is solved with generalised minimum residual method (GMRES). The numerical results show that the new version of FMM is more efficient than the original FMM.

A periodic FMM for Maxwell's equations in 3D and its applications to problems related to photonic crystals

This paper presents an FMM (Fast Multipole Method) for periodic boundary value problems for Maxwells equations in 3D. The effect of periodicity is taken into account with the help of the periodised moment to local expansion (M2L) transformation formula, which includes lattice sums. We verify the proposed method by comparing the obtained numerical results with analytic solutions for models of the multi-layered dielectric slab. We then apply the proposed method to scattering problems for periodic two-dimensional arrays of dielectric spheres and compare the obtained energy transmittances with those from the previous studies. We also consider scattering problems for woodpile crystals, where we find a passband related to a localised mode. Through these numerical tests we conclude that the proposed method is efficient and accurate.

An Improved Boundary Integral Equation Method for Crack Problems

This paper presents a double layer potential approach of BIE crack analysis. Our method regularises the conventional strongly singular expressions for the derivatives of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. This regularization, together with the use of B spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elatodynamics.

Regularization in 3D for Anisotropic Elastodynamic Crack and Obstacle Problems

The problems of wave scattering by obstacles or cracks appear very often in geophysics and in mechanics. In particular the linearized theory of elastodynamics for 3 dimensional elastic material is used frequently, because this theory keeps the analysis relatively simple. Even with this theory, however, a practical analysis is possible only with the use of some numerical methods. This has been the raison d’etre of many numerical experiments carried out in the engineering community. Among those numerical methods tested so far, the boundary integral equation (BIE) method has been accepted favourably by engineers, presumably because it can deal with scattered waves effectively in external problems. In particular the double layer potential representation is considered to be an efficient tool of numerical analysis for wave problems including cracks. The only inconvenience of the double layer potential approach, however, is the hypersingularity of the kernel, which does not permit the use of conventional numerical integration techniques. Hence we can take advantage of this approach only after weakening the hypersingularity of the kernel, or only after ‘regularizing’ it. As a matter of fact, some of such attemps can be found in the articles by Sladek & Sladek [11], Bui [5], Bonnet [4], Polch et.al [10], Nishimura & Kobayashi [8], [9] who used the collocation method and in Nedelec [7], Bamberger [1] where the variational method has been used.

Determination of cracks having arbitrary shapes with the boundary integral equation method

This paper discusses an application of the boundary integral equation method (BIEM) to an inverse problem of reconstructing a 2D curved crack from an experiment using a quantity governed by Laplaces equation. This problem is solved with the help of BIEM and a nonlinear programming technique. In this inverse problem we consider an infinite domain which contains one unknown crack in its interior. This cracked body is subjected to known far fields, and the resulting near fields are measured at several interior points. The most plausible crack is then determined as the minimiser of a fit-to-data cost function. This numerical process of minimisation, however, tends to become unstable as the degrees of freedom allowed for the unknown crack increase. However, a remedy of this ill-posedness based on Tikhonovs regularisation is shown to be effective. Some numerical examples are presented to confirm the applicability of the proposed method.