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A Free Primer on the Kinematics of Discrete Elastic Rods

oliver oreilly's picture

A new book, A Primer on the Kinematics of Discrete Elastic Rods, has just been published by Springer and is freely accessible to faculty and students at

https://link.springer.com/book/10.1007/978-3-319-76965-3

The book is a result of efforts by Khalid Jawed (now an Assistant Professor at UCLA) and a U.C. Berkeley graduate student, Alyssa Novelia, and myself to understand the novel formulation of Kirchhoff's celebrated rod theory was published by Bergou et al. We hope that our book provides an accessible introduction to this remarkable formulation. In Bergou et al.’s formulation, an elastic rod is discretized into a series of segments (or edges) connecting vertices (or nodes). The edges are free to stretch and rotate relative to their adjacent neighbors. The relative rotations of the cross-sections of the rod are modeled with the help of a pair of material vectors that are associated with each edge. The original formulation has been extended in a variety of directions including an extension to viscous threads and sound generation. The discrete elastic rod (DER) formulation is computationally cheap and, as a result, is used in computer graphics to render images of hairs and trees and is the technical underpinning behind the  Bristle Brush feature in Adobe Illustrator and Adobe Photoshop.

Bergou et al.'s DER formulation uses ideas from the nascent field of discrete differential geometry and concepts such as holonomy from  classic differential geometry. As a result, understanding the DER formulation (even for students who have exceptional backgrounds in continuum mechanics) can be challenging. Indeed, initially we were unable to rederive many of the key results in the papers by Bergou et al. (Bergou2010,Bergou2008) and the related works by Audoly et al. (Audoly2012) and Kaldor et al. (Kaldor2010). The remarkable simulations in these four papers provided sufficient motivation for us to eventually prove the main results contained in the aforementioned papers.  Again, we hope that our book provides an accessible introduction to this remarkable formulation.

 

 

 

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