Arbitrary bi-dimensional finite strain crack propagation

In the past two decades numerous numerical procedures for crack propagation have been developed. Lately, enrichment methods (either local, such as SDA or global, such as XFEM) have been applied with success to simple problems, typically involving some intersections. For arbitrary finite strain propagation, numerous difficulties are encountered: modeling of intersection and coalescence, step size dependence and the presence of distorted finite elements. In order to overcome these difficulties, an approach fully capable of dealing with multiple advancing cracks and self-contact is presented (see Fig.1). This approach makes use of a coupled Arbitrary Lagrangian-Eulerian method (ALE) and local tip remeshing. This is substantially less costly than a full remeshing while retaining its full versatility. Compared to full remeshing, angle measures and crack paths are superior. A consistent continuationbased linear control is used to force the critical tip to be exactly critical, while moving around the candidate set. The critical crack front is identified and propagated when one of the following criteria reaches a material limiting value: (i) the stress intensity factor; or (ii) the element-ahead tip stress. These are the control equations. The ability to solve crack intersection and coalescence problems is shown. Additionally, the independence from crack tip and step size and the absence of blade and dagger-shaped finite elements is observed. Classic benchmarks are computed leading to excellent crack path and load-deflection results, where convergence rate is quadratic

Long-Time Behaviour and Self-Similarity in a Coagulation Equation with Input of Monomers

For a coagulation equation with Becker-Doring type interactions and time-independent monomer input we study the detailed long-time behaviour of nonnegative solutions and prove the convergence to a self-similar function.Comment: 30 pages, 5 Figures, now published in Markov Processes and Related Fields 12, 367-398, (2006

HST Observations of the Central-Cusp Globular Cluster NGC 6752. The Effect of Binary Stars on the Luminosity Function in the Core

We consider the effect of binary stars on the main-sequence luminosity functions observed in the core of globular clusters, with specific reference to NGC 6752. We find that mass segregation results in an increased binary fraction at fainter magnitudes along the main-sequence. If this effect is not taken into account when analyzing luminosity functions, erroneous conclusions can be drawn regarding the distribution of single stars, and the dynamical state of the cluster. In the core of NGC 6752, our HST data reveal a flat luminosity function, in agreement with previous results. However, when we correct for the increasing binary fraction at faint magnitudes, the LF begins to fall immediately below the turn-off. This effect appears to be confined to the inner core radius of the cluster.Comment: 10 pages, 3 figures Accepted to ApJ Lett Vol 513 Number

Element-wise fracture algorithm based on rotation of edges

We propose an alternative, simpler algorithm for FEM-based computational fracture in brittle, quasi-brittle and ductile materials based on edge rotations. Rotation axes are the crack front edges (respectively nodes in surface discretizations) and each rotated edge affects the position of only one or two nodes. Modified positions of the entities minimize the difference between the predicted crack path (which depends on the specific propagation theory in use) and the edge or face orientation. The construction of all many-to-many relations between geometrical entities in a finite element code motivates operations on existing entities retaining most of the relations, in contrast with remeshing (even tip remeshing) and enrichment which alter the structure of the relations and introduce additional entities to the relation graph (in the case of XFEM, enriched elements which can be significantly different than classical FEM elements and still pose challenges for ductile fracture or large amplitude sliding). In this sense, the proposed solution has algorithmic and generality advantages. The propagation algorithm is simpler than the aforementioned alternatives and the approach is independent of the underlying element used for discretization. For history-dependent materials, there are still some transfer of relevant quantities between meshes. However, diffusion of results is more limited than with tip or full remeshing. To illustrate the advantages of our approach, two prototype models are used: tip energy dissipation (LEFM) and cohesive-zone approaches. The Sutton crack path criterion is employed. Traditional fracture benchmarks and newly proposed verification tests are solved. These were found to be very good in terms of crack path and load/deflection accuracy

What are the Best Hierarchical Descriptors for Complex Networks?

This work reviews several hierarchical measurements of the topology of complex networks and then applies feature selection concepts and methods in order to quantify the relative importance of each measurement with respect to the discrimination between four representative theoretical network models, namely Erd\"{o}s-R\'enyi, Barab\'asi-Albert, Watts-Strogatz as well as a geographical type of network. The obtained results confirmed that the four models can be well-separated by using a combination of measurements. In addition, the relative contribution of each considered feature for the overall discrimination of the models was quantified in terms of the respective weights in the canonical projection into two dimensions, with the traditional clustering coefficient, hierarchical clustering coefficient and neighborhood clustering coefficient resulting particularly effective. Interestingly, the average shortest path length and hierarchical node degrees contributed little for the separation of the four network models.Comment: 9 pages, 4 figure

Curvature induced toroidal bound states

Curvature induced bound state (E < 0) eigenvalues and eigenfunctions for a particle constrained to move on the surface of a torus are calculated. A limit on the number of bound states a torus with minor radius a and major radius R can support is obtained. A condition for mapping constrained particle wave functions on the torus into free particle wave functions is established.Comment: 6 pages, no figures, Late

New Techniques for Relating Dynamically Close Galaxy Pairs to Merger and Accretion Rates : Application to the SSRS2 Redshift Survey

We introduce two new pair statistics, which relate close galaxy pairs to the merger and accretion rates. We demonstrate the importance of correcting these (and other) pair statistics for selection effects related to sample depth and completeness. In particular, we highlight the severe bias that can result from the use of a flux-limited survey. The first statistic, denoted N_c, gives the number of companions per galaxy, within a specified range in absolute magnitude. N_c is directly related to the galaxy merger rate. The second statistic, called L_c, gives the total luminosity in companions, per galaxy. This quantity can be used to investigate the mass accretion rate. Both N_c and L_c are related to the galaxy correlation function and luminosity function in a straightforward manner. We outline techniques which account for various selection effects, and demonstrate the success of this approach using Monte Carlo simulations. If one assumes that clustering is independent of luminosity (which is appropriate for reasonable ranges in luminosity), then these statistics may be applied to flux-limited surveys. These techniques are applied to a sample of 5426 galaxies in the SSRS2 redshift survey. Using close dynamical pairs, we find N_c(-21<M_B<-18) = 0.0226+/-0.0052 and L_c(-21<M_B<-18) = 0.0216+/-0.0055 10^{10} h^2 L_sun at z=0.015. These are the first secure estimates of low-z close pair statistics. If N_c remains fixed with redshift, simple assumptions imply that ~ 6.6% of present day galaxies with -21<M_B<-18 have undergone mergers since z=1. When applied to redshift surveys of more distant galaxies, these techniques will yield the first robust estimates of evolution in the galaxy merger and accretion rates. [Abridged]Comment: 26 pages (including 10 postscript figures) plus 3 gif figures. Accepted for publication in ApJ. Paper (including full resolution images) also available at http://www.astro.utoronto.ca/~patton/ssrs2, along with associated pair classification experiment (clickable version of Figure 5

Avoided crossings in mesoscopic systems: electron propagation on a non-uniform magnetic cylinder

We consider an electron constrained to move on a surface with revolution symmetry in the presence of a constant magnetic field $B$ parallel to the surface axis. Depending on $B$ and the surface geometry the transverse part of the spectrum typically exhibits many crossings which change to avoided crossings if a weak symmetry breaking interaction is introduced. We study the effect of such perturbations on the quantum propagation. This problem admits a natural reformulation to which tools from molecular dynamics can be applied. In turn, this leads to the study of a perturbation theory for the time dependent Born-Oppenheimer approximation