Modeling Metamaterials with a Time-Domain Perfectly Matched Layer Formulation
INTRODUCTION: Perfectly matched layers (PML) have been widely used for simulating wave propagation in unbounded media to effectively avoid spurious wave reflections from the computational domain boundaries. Time-domain PML formulations, especially for elastic waves, usually use a complex system of first-order equations. Compact second-order time-domain formulations are particularly desired for finite element method (FEM) simulations in COMSOL Multiphysics® software. Such PML formulations were introduced by the authors [1, 2] for modeling wave propagation in unbounded fluid, solid, as well as for coupled fluid-solid media. In this work, we demonstrate the use of a time-domain PML formulation for the complex case of wave propagation in acoustic/elastic metamaterials. In particular, the time-domain superfocusing of acoustic waves in water by a slab of a solid metamaterials lens (see Zhou et al. [3]) is presented, enabling many interesting characteristics of metamaterials to be observed. The limitations of the PML in modeling metamaterials will be addressed, particularly the use of periodic structures within the PML, as considered in [4]. USE OF COMSOL MULTIPHYSICS®: Our PML formulations were reformulated in a weak form to facilitate the use of the mathematical models using the PDE interfaces in the COMSOL software. The fluid-solid coupling boundary conditions were presented in a form that enables these conditions to be easily integrated into the weak formulation. Second order Lagrange finite elements were used for space discretization, while the generalized alpha method was used for time discretization. RESULTS: The solid metamaterial lens proposed by Zhou et al. [3], was used in our simulations. It consists of a periodic structure of brass cylinders and cavities embedded in Al-SiC foam, with a unit cell as shown in the top right panel of Figure 1. The left column of Figure 1 shows snapshots of the simulations for the case were the slab of solid consists only of the foam matrix with water on both sides, while the right column shows the corresponding snapshots when the metamaterial periodic structure is included, but well within the physical domain. The very slow group velocity, reversed phase velocity, and other theoretically expected properties of metamaterials are observed in the simulations. Figure 2 shows results corresponding to those of Figure 1, but with the periodic structure extending to the inner edge of the PML. The way the total energy evolves in time within the solid slab is presented in Figure 3 for the three simulations in the first two figures. Figure 4 shows the simulations for the case where the periodic structure extends inside the PML. It is clear that the solution blow up marking a limitation of the PMLs in this case. CONCLUSION: The use of compact second-order time-domain formulations in modeling the complex and interesting case metamaterials, which are usually modeled in frequency domain, is presented. The limitation of the PML in the case of periodic metamaterials is addressed. Some proposed future work to overcome these limitations for periodic and homogenized metamaterials will be discussed.
Acoustic superfocusing by solid phononic crystals
We propose a solid phononic crystal lens capable of acoustic superfocusing beyond the diffraction limit. The unit cell of the crystal is formed by four rigid cylinders in a hosting material with a cavity arranged in the center. Theoretical studies reveal that the solid lens produces both negative refraction to focus propagating waves and surface states to amplify evanescent waves. Numerical analyses of the superfocusing effect of the considered solid phononic lens are presented with a separated source excitation to the lens. In this case, acoustic superfocusing beyond the diffraction limit is evidenced. Compared to the fluid phononic lenses, the solid lens is more suitable for ultrasonic imaging applications.
Igneous Rock Associations 23. The Bushveld Complex, South Africa: New Insights and Paradigms
SUMMARYThe Bushveld Complex has continued to serve as the basis for study into the fundamental nature of petrological processes for layered intrusion formation and for oxide and sulphide hosted Platinum Group Element (PGE)–Cu–Ni ore deposits. These studies have included discoveries in terms of the physical extent of Bushveld magmatism, both laterally and internally. Lateral variations in the mafic to ultramafic Rustenburg Layered Suite of the Northern Lobe of the complex have also revealed petrologically distinctive Upper Critical Zone equivalent rocks (the so-called Flatreef) with enhanced contamination and mineralization traits that reflect a transition between Eastern and Western Lobe equivalent stratigraphy and Platreef-style complexity. Traditional magma mixing models have been re-examined in light of radiogenic isotopic evidence for crustal involvement early in the chromite precipitation or formation process, combined with evidence for associated heterogeneous fluid contents, cryptic layering profiles, and textural evidence. A wide variety of alternative ore-genesis models have been proposed as a consequence. The fundamental mechanics of magma chamber processes and the existence of the magma chamber as an entity have been called into question through various lines of evidence which have promoted the concept of progressive emplacement of the complex as a stack of not-necessarily-quite-sequentially intruded sills (with or without significant quantities of transported phenocrysts), emplaced into variably crystallized and compacted crystal-liquid mush mixtures, modified by compaction-driven late magmatic fluid (silicate and aqueous) activity. Alternatively, petrological and geochemical observations have been used to discount these interpretations in favour of more conventional cooling and gravity-driven accumulation of silicate and ore minerals in a large, liquid-dominated system.RÉSUMÉLe complexe de Bushveld a demeuré à la base d’études sur la nature fondamentale des processus pétrologiques de formation d’intrusions litées et des gîtes des éléments du groupe platine (ÉGP)-Cu-Ni hébergés dans les oxydes et les sulfures. Ces études ont comporté des découvertes sur l’étendue physique, à la fois latérale et interne, du magmatisme de Bushveld. Les variations latérales de la suite stratifiée et mafique à ultramafique Rustenburg du lobe nord du complexe ont également révélé des roches équivalentes pétrologiquement distinctes de la zone critique supérieure (le communément désigné Flatreef) avec des traits de contamination et de minéralisation accrus qui reflètent une transition entre la stratigraphie équivalente des lobes est et ouest et la complexité de type Platreef. Les modèles traditionnels de mélanges magmatiques ont été réexaminés à la lumière de preuves isotopiques radiogéniques indiquant une implication de la croûte au début du processus de précipitation ou de formation de la chromite, combinées à des preuves de contenu fluide hétérogène associé, de profils de litage cryptique et de preuves texturales. Ainsi, une grande variété de modèles alternatifs de genèse de minerai a été proposée. La mécanique fondamentale des processus de la chambre magmatique et l'existence de la chambre magmatique en tant qu'entité ont été remises en question au moyen de divers éléments de preuve qui ont mis en avant le concept de mise en place progressive du complexe sous forme d'un empilement non-nécessairement séquentiel de sills injectés (avec ou sans quantités significatives de phénocristaux transportés) mis en place dans des mélanges de bouillie cristaux/liquide à cristallisation et compaction variable, modifiés par une activité tardive de fluide magmatique (silicaté et aqueux) induite par la compaction. Alternativement, des observations pétrologiques et géochimiques ont été utilisées pour écarter ces interprétations en faveur d'un processus plus conventionnel de refroidissement et d’accumulation de minérais silicatés et minéralisés induite par la gravité dans un vaste système à dominance liquide.