Adaptive Mesh Refinement Computation of Solidification Microstructures using Dynamic Data Structures

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase-field model at low undercooling. In particular we show that the computational complexity of solving any phase-boundary problem scales with the interface arclength when using an adapting mesh. Moreover, the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions, which would otherwise require lattices greater that $2^{17}\times 2^{17}$ elements. We examine the convergence properties of our algorithm. We also present two dimensional, time-dependent calculations of dendritic evolution, with and without surface tension anisotropy. We benchmark our results for dendritic growth with microscopic solvability theory, finding them to be in good agreement with theory for high undercoolings. At low undercooling, however, we obtain higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive

Zonotopal algebra

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This well-known line of study is particularly interesting in case n\eqbd\rank X \ll N. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXi

Is high-density amorphous ice simply a “derailed” state along the ice I to ice IV pathway?

The structural nature of high-density amorphous ice (HDA), which forms through low-temperature pressure-induced amorphization of the “ordinary” ice I, is heavily debated. Clarifying this question is important for understanding not only the complex condensed states of H2O but also in the wider context of pressure-induced amorphization processes, which are encountered across the entire materials spectrum. We first show that ammonium fluoride (NH4F), which has a similar hydrogen-bonded network to ice I, also undergoes a pressure collapse upon compression at 77 K. However, the product material is not amorphous but NH4F II, a high-pressure phase isostructural with ice IV. This collapse can be rationalized in terms of a highly effective mechanism. In the case of ice I, the orientational disorder of the water molecules leads to a deviation from this mechanism, and we therefore classify HDA as a “derailed” state along the ice I to ice IV pathway

Banks' total factor productivity growth in a developing economy: does globalisation matter?

The paper provides, for the first time, empirical evidence on the impact of economic globalisation on bank total factor productivity in a developing economy. By employing the Malmquist Productivity Index method, we compute the total factor productivity of the Malaysian banking sector during 1998–2007. Examining different dimensions of economic globalisation, we find evidence supporting for greater trade and capital account restrictions and cultural proximity. On the other hand, personal contacts, information flows, and political globalisation seem to exert significant (negative) influence on banks' total factor productivity levels

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems