This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

IN view of the interest attaching to the vaporisation and diffusion of solids, the following observations may be worthy of record.

The Diffusion of Solids

The mechanical resonant response of a solid depends on its shape, density, elastic moduli and dissipation. We describe here instrumentation and computational methods for acquiring and analyzing the resonant ultrasound spectrum of very small (0.001 cm3) samples as a function of temperature, and provide examples to demonstrate the power of the technique. The information acquired is in some cases comparable to that obtained from other more conventional ultrasonic measurement techniques, but one unique feature of resonant ultrasound spectroscopy (RUS) is that all moduli are determined simultaneously to very high accuracy. Thus in circumstances where high relative or absolute accuracy is required for very small crystalline or other anisotropic samples RUS can provide unique information. RUS is also sensitive to the fundamental symmetry of the object under test so that certain symmetry breaking effects are uniquely observable, and because transducers require neither couplant nor a flat surface, broken fragments of a material can be quickly screened for phase transitions and other temperature-dependent responses.

/pdf/resonant-ultrasound-spectroscopic-techniques-for-measurement-wt0fciyopd.pdf

Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids

Using the Landauer formulation of transport theory, we predict that dielectric quantum wires should exhibit quantized thermal conductance at low temperatures in a ballistic phonon regime. The quantum of thermal conductance is universal, independent of the characteristics of the material, and equal to ${\ensuremath{\pi}}^{2}{k}_{B}^{2}T/3h$ where ${k}_{B}$ is the Boltzmann constant, $h$ is Planck's constant, and $T$ is the temperature. Quantized thermal conductance should be experimentally observable in suspended nanostructures adiabatically coupled to reservoirs, devices that can be realized at the present time.

Quantized Thermal Conductance of Dielectric Quantum Wires

The remarkable optical properties of metal nanoparticles are governed by the excitation of localized surface plasmon resonances (LSPRs). The sensitivity of each LSPR mode, whose spatial distribution and resonant energy depend on the nanoparticle structure, composition and environment, has given rise to many potential photonic, optoelectronic, catalytic, photovoltaic, and gas- and bio-sensing applications. However, the precise interplay between the three-dimensional (3D) nanoparticle structure and the LSPRs is not always fully understood and a spectrally sensitive 3D imaging technique is needed to visualize the excitation on the nanometre scale. Here we show that 3D images related to LSPRs of an individual silver nanocube can be reconstructed through the application of electron energy-loss spectrum imaging, mapping the excitation across a range of orientations, with a novel combination of non-negative matrix factorization, compressed sensing and electron tomography. Our results extend the idea of substrate-mediated hybridization of dipolar and quadrupolar modes predicted by theory, simulations, and electron and optical spectroscopy, and provide experimental evidence of higher-energy mode hybridization. This work represents an advance both in the understanding of the optical response of noble-metal nanoparticles and in the probing, analysis and visualization of LSPRs.

http://www.emc2012.org.uk/documents/Abstracts/Abstracts/EMC2012_0766.pdf

Three-dimensional imaging of localized surface plasmon resonances of metal nanoparticles

The elastic constants and internal friction of induction hardened and unhardened SAE 1050 plain-carbon steel at ambient temperatures were determined by resonant ultrasonic spectroscopy. The hardened specimen contained only martensite and the unhardened specimen was ferrite-pearlite. Using an inverse Ritz algorithm with assumed orthorhombic symmetry, all nine independent elastic-stiffness coefficients were determined, and, from the resonance peak widths, all nine components of the internal-friction tensor were determined. Similar measurements and analysis on monocrystalline α-iron were performed. The steel has slight elastic anisotropy, and the isotropically approximated elastic moduli were lower in the martensite than in ferrite-pearlite: shear modulus by 3.6%, bulk modulus by 1.2%, Young modulus by 3.2%, and Poisson ratio by 1.5%. Isotropically approximated elastic moduli of α-iron were 0.6–1.3% higher than ferrite-pearlite. All components of the internal-friction in martensite were higher than those of ferrite-pearlite, but lower than those of α-iron.

/pdf/elastic-constants-and-internal-friction-of-martensitic-steel-2xg52mt6pj.pdf

Elastic constants and internal friction of martensitic steel, ferritic-pearlitic steel, and α-iron

We have used “trapped spin wave” or edge modes of magnetic precession to probe the magnetic environment near magnetic film edges magnetized perpendicular to the edge. Micromagnetic models of dynamics in stripes reveal that the edge mode frequency-field relationship depends on whether the edge surface is vertical or tapered, while the “bulk” modes are nearly unaffected. The models show the edge-mode frequency going to zero at the edge saturation field. This critical field becomes much less distinct for applied fields misaligned from the edge normal by as little as 1°. Ferromagnetic-resonance and Brillouin light-scattering measurements of the edge modes in an array of 480-nm-wide×12-nm-thick Ni80Fe20 stripes have a lower edge saturation field than the vertical edge models, but agree well with the model of 45°-tapered edges.

/pdf/characterization-of-magnetic-properties-at-edges-by-edge-2cc87g8p5n.pdf

Characterization of Magnetic Properties at Edges by Edge Mode Dynamics

Brillouin light scattering shows that shape distortions in ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ nanomagnets can have a dramatic effect on the measured collective linewidth of certain spin-wave modes. The intentional introduction of quantifiable asymmetric egglike shape distortion to an ideal elliptical structure lifts the degeneracy of end modes with concentrated amplitude at the nanomagnet edges. In contrast, modes with concentrated amplitude at the interior are significantly less affected by the distortion. The splitting of end modes by asymmetric distortions explains the large inhomogeneous linewidth broadening in end modes found in large ensembles of nanomagnets that contain a relatively small statistical variation in the degree of distortion.

Effects of shape distortions and imperfections on mode frequencies and collective linewidths in nanomagnets

Phonon spectra of polymeric linear nanostructures have been characterized using Brillouin light scattering. In addition to phonon modes similar to those present in uniform thin films, the phonon spectra of the nanolines reveal a new mode with a lower frequency that depends on the width of the nanolines. Classic wave theory and finite element analysis were combined to identify this new mode as a flexural vibration of the nanolines. Analysis of the phonon spectra gave estimates of elastic constants in the nanostructures and indicated that there is no significant deviation from bulk mechanical properties and no mechanical anisotropy in structures as small as 88nm.

/pdf/acoustic-modes-and-elastic-properties-of-polymeric-4q9d6bdxv6.pdf

Acoustic modes and elastic properties of polymeric nanostructures

http://absimage.aps.org/image/MAR06/MWS_MAR06-2005-001900.pdf

Ward L. Johnson

Papers

Elastic constants and internal friction of martensitic steel, ferritic-pearlitic steel, and α-iron

Characterization of Magnetic Properties at Edges by Edge Mode Dynamics

Effects of shape distortions and imperfections on mode frequencies and collective linewidths in nanomagnets

Acoustic modes and elastic properties of polymeric nanostructures

Acoustic modes and elastic properties of polymeric nanostructures