Coupled thermomechanical dynamics of phase transitions in shape memory alloys and related hysteresis phenomena

In this paper the nonlinear dynamics of shape memory alloy phase transformations is studied with thermomechanical models based on coupled systems of partial differential equations by using computer algebra tools. The reduction procedures of the original model to a system of differential-algebraic equations and its solution are based on the general methodology developed by the authors for the analysis of phase transformations in shape memory materials with low dimensional approximations derived from center manifold theory. Results of computational experiments revealing the martensitic-austenitic phase transition mechanism in a shape-memory-alloy rod are presented. Several groups of computational experiments are reported. They include results on stress- and temperature-induced phase transformations as well as the analysis of the hysteresis phenomenon. All computational experiments are presented for Cu-based structures

On power series solutions for the Euler equation, and the Behr-Necas-Wu initial datum

We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the initial datum. We then turn the attention to a paper by Behr, Necas and Wu in ESAIM: M2AN 35 (2001) 229-238; here, the authors chose a very simple Fourier polynomial as an initial datum for the Euler equation and analyzed the power series in time for the solution, determining the first 35 terms by computer algebra. Their calculations suggested for the series a finite convergence radius \tau_3 in the H^3 Sobolev space, with 0.32 < \tau_3 < 0.35; they regarded this as an indication that the solution of the Euler equation blows up. We have repeated the calculations of Behr, Necas and Wu, using again computer algebra; the order has been increased from 35 to 52, using the symmetries of the initial datum to speed up computations. As for \tau_3, our results agree with the original computations of Behr, Necas and Wu (yielding in fact to conjecture that 0.32 < \tau_3 < 0.33). Moreover, our analysis supports the following conclusions: (a) The finiteness of \tau_3 is not at all an indication of a possible blow-up. (b) There is a strong indication that the solution of the Euler equation does not blow up at a time close to \tau_3. In fact, the solution is likely to exist, at least, up to a time \theta_3 > 0.47. (c) Pade' analysis gives a rather weak indication that the solution might blow up at a later time.Comment: 34 pages, 8 figure

Dimensional analysis using toric ideals: Primitive invariants

© 2014 Atherton et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.Classical dimensional analysis in its original form starts by expressing the units for derived quantities, such as force, in terms of power products of basic units M, L, T etc. This suggests the use of toric ideal theory from algebraic geometry. Within this the Graver basis provides a unique primitive basis in a well-defined sense, which typically has more terms than the standard Buckingham approach. Some textbook examples are revisited and the full set of primitive invariants found. First, a worked example based on convection is introduced to recall the Buckingham method, but using computer algebra to obtain an integer K matrix from the initial integer A matrix holding the exponents for the derived quantities. The K matrix defines the dimensionless variables. But, rather than this integer linear algebra approach it is shown how, by staying with the power product representation, the full set of invariants (dimensionless groups) is obtained directly from the toric ideal defined by A. One candidate for the set of invariants is a simple basis of the toric ideal. This, although larger than the rank of K, is typically not unique. However, the alternative Graver basis is unique and defines a maximal set of invariants, which are primitive in a simple sense. In addition to the running example four examples are taken from: a windmill, convection, electrodynamics and the hydrogen atom. The method reveals some named invariants. A selection of computer algebra packages is used to show the considerable ease with which both a simple basis and a Graver basis can be found.The third author received funding from Leverhulme Trust Emeritus Fellowship (1-SST-U445) and United Kingdom EPSRC grant: MUCM EP/D049993/1

Rotation and scale space random fields and the Gaussian kinematic formula

We provide a new approach, along with extensions, to results in two important papers of Worsley, Siegmund and coworkers closely tied to the statistical analysis of fMRI (functional magnetic resonance imaging) brain data. These papers studied approximations for the exceedence probabilities of scale and rotation space random fields, the latter playing an important role in the statistical analysis of fMRI data. The techniques used there came either from the Euler characteristic heuristic or via tube formulae, and to a large extent were carefully attuned to the specific examples of the paper. This paper treats the same problem, but via calculations based on the so-called Gaussian kinematic formula. This allows for extensions of the Worsley-Siegmund results to a wide class of non-Gaussian cases. In addition, it allows one to obtain results for rotation space random fields in any dimension via reasonably straightforward Riemannian geometric calculations. Previously only the two-dimensional case could be covered, and then only via computer algebra. By adopting this more structured approach to this particular problem, a solution path for other, related problems becomes clearer.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1055 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org