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

The establishment of modern terrestrial life is indissociable from angiosperm evolution. While available molecular clock estimates of angiosperm age range from the Paleozoic to the Late Cretaceous, the fossil record is consistent with angiosperm diversification in the Early Cretaceous. The time-frame of angiosperm evolution is here estimated using a sample representing 87% of families and sequences of five plastid and nuclear markers, implementing penalized likelihood and Bayesian relaxed clocks. A literature-based review of the palaeontological record yielded calibrations for 137 phylogenetic nodes. The angiosperm crown age was bound within a confidence interval calculated with a method that considers the fossil record of the group. An Early Cretaceous crown angiosperm age was estimated with high confidence. Magnoliidae, Monocotyledoneae and Eudicotyledoneae diversified synchronously 135-130 million yr ago (Ma); Pentapetalae is 126-121 Ma; and Rosidae (123-115 Ma) preceded Asteridae (119-110 Ma). Family stem ages are continuously distributed between c. 140 and 20 Ma. This time-frame documents an early phylogenetic proliferation that led to the establishment of major angiosperm lineages, and the origin of over half of extant families, in the Cretaceous. While substantial amounts of angiosperm morphological and functional diversity have deep evolutionary roots, extant species richness was probably acquired later.

https://nph.onlinelibrary.wiley.com/doi/pdf/10.1111/nph.13264

A metacalibrated time-tree documents the early rise of flowering plant phylogenetic diversity

This paper presents a state-of-the-art review on a hot topic in the literature, i.e., vibration based energy harvesting techniques, including theory, modelling methods and the realizations of the piezoelectric, electromagnetic and electrostatic approaches. To minimize the requirement of external power source and maintenance for electric devices such as wireless sensor networks, the energy harvesting technique based on vibrations has been a dynamic field of studying interest over past years. One important limitation of existing energy harvesting techniques is that the power output performance is seriously subject to the resonant frequencies of ambient vibrations, which are often random and broadband. To solve this problem, researchers have concentrated on developing efficient energy harvesters by adopting new materials and optimising the harvesting devices. Particularly, among these approaches, different types of energy harvesters have been designed with consideration of nonlinear characteristics so that the frequency bandwidth for effective energy harvesting of energy harvesters can be broadened. This paper reviews three main and important vibration-to-electricity conversion mechanisms, their design theory or methods and potential applications in the literature. As one of important factors to estimate the power output performance, the energy conversion efficiency of different conversion mechanisms is also summarised. Finally, the challenging issues based on the existing methods and future requirement of energy harvesting are discussed.

A comprehensive review on vibration energy harvesting: Modelling and realization

This paper presents a framework to investigate the dynamics of overall vehicle-track systems with emphasis on theoretical modelling, numerical simulation and experimental validation. A three-dimensional vehicle-track coupled dynamics model is developed in which a typical railway passenger vehicle is modelled as a 35-degree-of-freedom multi-body system. A traditional ballasted track is modelled as two parallel continuous beams supported by a discrete-elastic foundation of three layers with sleepers and ballasts included. The non-ballasted slab track is modelled as two parallel continuous beams supported by a series of elastic rectangle plates on a viscoelastic foundation. The vehicle subsystem and the track subsystem are coupled through a wheel-rail spatial coupling model that considers rail vibrations in vertical, lateral and torsional directions. Random track irregularities expressed by track spectra are considered as system excitations by means of a time-frequency transformation technique. A fast explicit integration method is applied to solve the large nonlinear equations of motion of the system in the time domain. A computer program named TTISIM is developed to predict the vertical and lateral dynamic responses of the vehicle-track coupled system. The theoretical model is validated by full-scale field experiments, including the speed-up test on the Beijing-Qinhuangdao line and the high-speed running test on the Qinhuangdao-Shenyang line. Differences in the dynamic responses analysed by the vehicle-track coupled dynamics and by the classical vehicle dynamics are ascertained in the case of vehicles passing through curved tracks.

https://www.tandfonline.com/doi/pdf/10.1080/00423110802621561

Fundamentals of vehicle–track coupled dynamics

A review is presented of dynamic modelling of railway track and of the interaction of vehicle and track at frequencies which are sufficiently high for the track's dynamic behaviour to be significant. Since noise is one of the most important consequences of wheel/rail interaction at high frequencies, the maximum frequency of interest is about 5kHz: the limit of human hearing. The topic is reviewed both historically and in particular with reference to the application of modelling to the solution of practical problems. Good models of the rail, the sleeper and the wheelset are now available for the whole frequency range of interest. However, it is at present impossible to predict either the dynamic behaviour of the railpad and ballast or their long term behaviour. This is regarded as the most promising area for future research.

Modelling of Railway Track and Vehicle/Track Interaction at High Frequencies

Railways are an environmentally friendly means of transport well suited to modern society. However, noise and vibration are key obstacles to further development of the railway networks for high-speed intercity traffic, for freight and for suburban metros and light-rail. All too often noise problems are dealt with inefficiently due to lack of understanding of the problem. This book brings together coverage of the theory of railway noise and vibration with practical applications of noise control technology at source to solve noise and vibration problems from railways.

David Thompson

Papers

Railway Noise and Vibration: Mechanisms, Modelling and Means of Control

Prediction of ground vibration from trains using the wavenumber finite and boundary element methods

Experimental validation of the twins prediction program for rolling noise, part 1: description of the model and method

A theoretical model for ground vibration from trains generated by vertical track irregularities

A review of the modelling of wheel/rail noise generation