Sign Characteristics of Regular Hermitian Matrix Pencils under Generic Rank-1 and Rank-2 Perturbations

The spectral behavior of regular Hermitian matrix pencils is examined under certain structure-preserving rank-1 and rank-2 perturbations. Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form, it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. The observed effects are as follows: Under a rank-1 or rank-2 perturbation, generically the largest one or two, respectively, Jordan blocks at each eigenvalue lambda are destroyed, and if lambda is an eigenvalue of the perturbation, also one new block of size one is created at lambda. If lambda is real (or infinite), additionally all signs at lambda but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. Also, if the potential new block of size one is real, its sign is in most cases prescribed to be the sign that is attached to the eigenvalue lambda in the perturbation.

Magnetic field of rectangular current loop with sides parallel and perpendicular to the surface of high-permeability material

In this paper, method of current images is used to calculate magnetic field of a rectangular loop in presence of high-permeability material. We provide closed form description for the images of the current loop placed between two semi-infinite blocks of high-permeability material, in the cases when the plane of the loop is parallel or perpendicular to the surfaces of blocks. A case study of using the method of images to calculate magnetic field of rectangular loop inside cube made of high-permeability material is also provided.

Approximations to quasi-birth-and-death processes with infinite blocks

The numerical analysis of quasi-birth-and-death processes rests on the resolution of a matrix-quadratic equation for which efficient algorithms are known when the matrices have finite order, that is, when the number of phases is finite. In this paper we consider the case of infinitely many phases from the point of view of theoretical convergence of truncation and augmentation schemes, and we develop four different methods. Two methods rely on forced transitions to the boundary. In one of these methods, the transitions occur as a result of the truncation itself, while in the other method, they are artificially introduced so that the augmentation may be chosen to be as natural as possible. Two other methods rely on forced transitions within the same level. We conclude with a brief numerical illustration.

Approximations to quasi-birth-and-death processes with infinite blocks

The numerical analysis of quasi-birth-and-death processes rests on the resolution of a matrix-quadratic equation for which efficient algorithms are known when the matrices have finite order, that is, when the number of phases is finite. In this paper we consider the case of infinitely many phases from the point of view of theoretical convergence of truncation and augmentation schemes, and we develop four different methods. Two methods rely on forced transitions to the boundary. In one of these methods, the transitions occur as a result of the truncation itself, while in the other method, they are artificially introduced so that the augmentation may be chosen to be as natural as possible. Two other methods rely on forced transitions within the same level. We conclude with a brief numerical illustration.

A Method to Extract Interface Stress Intensity Factors Using Interlayers

A method is presented here to extract stress intensity factors for interface cracks in plane bimaterial fracture problems. The method relies on considering a companion problem wherein a very thin elastic interlayer is artificially inserted between the two material regions of the original bimaterial problem. The crack in the companion problem is located in the middle of the interlayer with its tip located within the homogeneous interlayer material. When the thickness of the interlayer is small compared with the other length scales of the problem, a universal relation can be established between the actual interface stress intensity factors at the crack tip for the original problem and the mode I and II stress intensity factors associated with the companion problem. The universal relation is determined by formulating and solving a boundary value problem. This universal relation now allows the determination of the stress intensity factors for a generic plane interface crack problem as follows. For a given interface crack problem, the companion problem is formulated and solved using the finite element method. Mode I and II stress intensity factors are obtained using the modified virtual crack closure method. The universal relation is next used to obtain the corresponding interface stress intensity factors for the original interface crack problem. An example problem involving a finite interface crack between two semi-infinite blocks is considered for which analytical solutions exist. It is shown that the method described above provides very acceptable results.