There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

The stability of two-dimensional (2D) layers and membranes is subject of a long standing theoretical debate. According to the so called Mermin-Wagner theorem, long wavelength fluctuations destroy the long-range order for 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These dangerous fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes making that a two-dimensional membrane can exist but should present strong height fluctuations. The discovery of graphene, the first truly 2D crystal and the recent experimental observation of ripples in freely hanging graphene makes these issues especially important. Beside the academic interest, understanding the mechanisms of stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest for its unusual Dirac spectrum and electronic properties. Here we address the nature of these height fluctuations by means of straightforward atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon. We find that ripples spontaneously appear due to thermal fluctuations with a size distribution peaked around 70 \AA which is compatible with experimental findings (50-100 \AA) but not with the current understanding of stability of flexible membranes. This unexpected result seems to be due to the multiplicity of chemical bonding in carbon.

Intrinsic ripples in graphene

Stochastic differential equations and diffusion processes

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

https://www.mat.univie.ac.at/~stefanelli/papers/ars.pdf

A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity

This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals $${\mathcal{E}}$$
 and the dissipation distance $${\mathcal{D}}$$
 . For sequences $$({\mathcal{E}}_k)_{k\in {\mathbb{N}}}$$
 and $$({\mathcal{D}}_k)_{k\in {\mathbb{N}}}$$
 we address the question under which conditions the limits q
 ∞ of solutions $$q_k : [0, T]\to {\mathcal{Q}}$$
 satisfy a suitable limit problem with limit functionals $${\mathcal{E}}_\infty$$
 and $${\mathcal{D}}_\infty$$
 , which are the corresponding Γ-limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q
 ∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model.

https://www.mat.univie.ac.at/~stefanelli/papers/mrs.pdf

Γ-limits and relaxations for rate-independent evolutionary problems

The celebrated Brezis-Ekeland principle [C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), pp. Ai, A1197-A1198, Aii, and A971-A974] characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems. This note extends this characterization to doubly nonlinear evolution equations driven by convex potentials. The characterization is exploited in order to establish approximation results for gradient flows, doubly nonlinear equations, and rate-independent evolutions.

The Brezis-Ekeland Principle for Doubly Nonlinear Equations

http://www-2.unipv.it/compmech/publications/2009_2.pdf

A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties

We investigate ground state configurations for atomic potentials including both two- and three-body nearest-neighbor interaction terms. The aim is to prove that such potentials may describe crystallization in carbon nanostructures such as graphene, nanotubes, and fullerenes. We give conditions in order to prove that planar energy minimizers are necessarily honeycomb, namely graphene patches. Moreover, we provide an explicit formula for the ground state energy which exactly quantifies the lower-order surface energy contribution. This allows us to give some description of the geometry of ground states. By recasting the minimization problem in three-space dimensions, we prove that ground states are necessarily nonplanar and, in particular, rolled-up structures like nanotubes are energetically favorable. Eventually, we check that the C20 and C60 fullerenes are strict local minimizers, hence stable.

/pdf/crystallization-in-carbon-nanostructures-21snxk2914.pdf

Ulisse Stefanelli

Papers

A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity

Γ-limits and relaxations for rate-independent evolutionary problems

The Brezis-Ekeland Principle for Doubly Nonlinear Equations

A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties

Crystallization in Carbon Nanostructures