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Developed from the authors, combined total of 50 years undergraduate and graduate teaching experience, this book presents the finite element method formulated as a general-purpose numerical procedure for solving engineering problems governed by partial differential equations. 

Focusing on the formulation and application of the finite element method through the integration of finite element theory, code development, and software application, the book is both introductory and self-contained, as well as being a hands-on experience for any student.

A First Course in Finite Elements is the ideal practical introductory course for junior and senior undergraduate students from a variety of science and engineering disciplines.  The accompanying advanced topics at the end of each chapter also make it suitable for courses at graduate level, as well as for practitioners who need to attain or refresh their knowledge of finite elements through private study.

Practical Multiscaling covers fundamental modelling techniques aimed at bridging diverse temporal and spatial scales ranging from the atomic level to a full-scale product level. It focuses on practical multiscale methods that account for fine-scale (material) details but do not require their precise resolution. The text material evolved from over 20 years of teaching experience at Rensselaer and Columbia University, as well as from practical experience gained in the application of multiscale software.

This book comprehensively covers theory and implementation, providing a detailed exposition of the state-of-the-art multiscale theories and their insertion into conventional (single-scale) finite element code architecture. The robustness and design aspects of multiscale methods are also emphasised, which is accomplished via four building blocks: upscaling of information, systematic reduction of information, characterization of information utilizing experimental data, and material optimization. To ensure the reader gains hands-on experience, a companion website hosting a lite version of the multiscale design software (MDS-Lite) is available.

Key features:

  • Combines fundamental theory and practical methods of multiscale modelling

  • Covers the state-of-the-art multiscale theories and examines their practical usability in design

  • Covers applications of multiscale methods

  • Accompanied by a continuously updated website hosting the multiscale design software

  • Illustrated with colour images

Practical Multiscaling is an ideal textbook for graduate students studying multiscale science and engineering. It is also a must-have reference for government laboratories, researchers and practitioners in civil, aerospace, pharmaceutical, electronics, and automotive industries, and commercial software vendors.

Small scale features and processes occurring at a nanometer and femtoseconds scales have a profound impact on what happens at a larger scale and over extensive period of time. The primary objective of this volume is to reflect the-state-of-the art in multiscale mathematics, modeling and simulations and to address the following barriers: What is the information that needs to be transferred from one model or scale to another and what physical principles must be satisfied during the transfer of information? What are the optimal ways to achieve such transfer of information? How to quantify variability of physical parameters at multiple scales and how to account for it to ensure design robustness?

Various multiscale approaches in space and time presented in this Volume are grouped into two main categories: information-passing and concurrent. In the concurrent approaches, various scales are simultaneously resolved, whereas in the information-passing methods, the fine scale is modeled and its gross response is infused into the continuum scale. The issue of reliability of multiscale modeling and simulation tools is discussed in several, which focus on hierarchy of multiscale models and a posterior model error estimation including uncertainty quantification. Component software that can be effectively combined to address a wide range of multiscale simulations is described as well. Applications range from advanced materials, to nanoelectromechanical systems (NEMS), to biological systems, and nanoporous catalysts where physical phenomena operate across 12 orders of magnitude in time scales and 10 orders of magnitude in spatial scales. A valuable reference book for scientists, engineers and graduate students practicing in traditional engineering and science disciplines as well as in emerging fields of nanotechnology, biotechnology, microelectronics and energy.





ASSOCIATE EDITORS: Somnath Ghosh , Klauss Hackl , Thomas J. R. Hughes , Caglar Oskay , Karel Matous ,Tamar Schlick

The aim of the journal is to advance the research and practice in diverse areas of Multiscale Computational Science and Engineering. The journal will publish original papers and educational articles of general value to the field that will bridge the gap between modeling, simulation and design of products based on multiscale principles. The scope of the journal includes papers concerned with bridging of physical scales, ranging from the atomic level to full scale products and problems involving multiple physical processes interacting at multiple spatial and temporal scales. The emerging areas of computational nanotechnology and computational biotechnology and computational energy sciences are of particular interest to the journal. The journal is intended to be of interest and use to researchers and practitioners in academic, governmental and industrial communities.

EDITORS-IN-CHIEF: René de Borst and Charbel Farhat

EDITORS: Jacob Fish Isaac HarariAntonio HuertaKenjiro Terada

The International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems.

The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome.

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