Particle Finite Element Method

This textbook teaches classical mechanics as one of the foundations of physics. It describes the mechanical stability and motion in physical systems ranging from the molecular to the galactic scale.
Aside from the standard topics of mechanics in the physics curriculum, this book includes an introduction to the theory of elasticity and its use in selected modern engineering applications, e.g. dynamic mechanical analysis of viscoelastic materials. The text also covers many aspects of numerical mechanics, ranging from the solution of ordinary differential equations, including molecular dynamics simulation of many particle systems, to the finite element method. Attendant Mathematica programs or parts thereof are provided in conjunction with selected examples. Numerous links allow the reader to connect to related subjects and research topics.

The numerical treatment of partial differential equations with particle methods and meshfree discretization techniques is a very active research field both in the mathematics and engineering community. Due to their independence of a mesh, particle schemes and meshfree methods can deal with large geometric changes of the domain more easily than classical discretization techniques. Furthermore, meshfree methods offer a promising approach for the coupling of particle models to continuous models.

Meshfree Particle Methods is a comprehensive and systematic exposition of particle methods, meshfree Galerkin and partitition of unity methods, molecular dynamics methods, and multiscale methods. Most theories, computational formulations, and simulation results presented are recent developments in meshfree methods. They were either just published recently or even have not been published yet, many of them resulting from the authors´ own research. The presentation of the technical content is heuristic and explanatory with a balance between mathematical rigor and engineering practice. It can be used as a graduate textbook or a comprehensive source for researchers, providing the state of the art on Meshfree Particle Methods.

The proceedings contain a selection of refereed contributions and are meant to serve as a source of reference for all those interested in the state of the art in computational fluid dynamics.

In the first contribution, Morbiducci and co-workers discuss the theoretical and methodological bases supporting the Lagrangian- and Euler-based methods, highlighting their application to cardiovascular flows. The second contribution, by the Ansón and van Lenthe groups, proposes an automated virtual bench test for evaluating the stability of custom shoulder implants without the necessity of mechanical testing. Urdeitx and Doweidar, in the third paper, also adopt the finite element method for developing a computational model aim to study cardiac cell behavior under mechano-electric stimulation. In the fourth contribution, Ayensa-Jiménez et al. develop a methodology to approximate the multidimensional probability density function of the parametric analysis obtained developing a mathematical model of the cancer evolution.