Natural Element Method for the Simulation of Structures and Processes

Francisco Chinesta, Serge Cescotto, Elias Cueto, Philippe Lorong

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848212206
• eBook ISBN: 9781118616925
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.

This book presents a recent state of the art on the foundations and applications of the meshless natural element method in computational mechanics, including structural mechanics and material forming processes involving solids and Newtonian and non-Newtonian fluids.

Chapter 1

Introduction

“Meshless” methods are alternative techniques to the finite element method in solving partial differential equations. While the finite element method derives an approximation based on the elements, using shape functions, the meshless methods allow us to derive an approximation at any point; thanks to the information provided by the surrounding nodes. In these approaches the concept of element is thus not used any more. Connectivity between the nodes is not defined any more by the mesh but only by the concepts of “vicinity” or “field of influence.” These methods were developed with the aim of avoiding the numerical problems involved in mesh construction. These problems have been discussed in many studies; it is, forexample, a question of simulation of manufacturing processes such as extrusion, injection, or setting forms by removal of matter where it is necessary to face extremely large distortions of the mesh. In other processes such as foundry, drilling, or laser welding, precisely knowing the position of the interface between the solid phase and the liquid phase is essential. In the simulation of processes such as cutting by adiabatic shearing which involves a localized deformation, possibly accompanied by the propagation of a fissure, it is necessary to carry out the simulation without the mesh being conceived influencing the direction of propagation of the shear band or the fissure. The appearance of a localized deformation requires a finer representation of the solution in certain areas of the domains, and it is thus necessary to be able to refine the mesh easily without the geometrical constraints known within the framework of finite elements (mainly in 3D) and the problems related to precise projection of the fields between the two meshes. The objective of the meshless methods is to eliminate the structure of the mesh and to build the approximation starting only from the nodes. Although structures with a geometrical character are necessary (to build node connectivity for the integration of the weak form associated with the equation to be solved and so on), these do not interfere, in general, with the quality of the solution and thus can be built independently. Even after being proposed at the end of 1970s, the “meshless” methods had to wait approximately 15 years before having a real development and an interest within the scientific community.

In the interval, little passion had been shown for them because of the numerous difficulties presented by the first techniques. The first “meshless” method seems to be the so-called smooth particle hydrodynamics (SPH) method (Lucy 1977), which was initially used to model astronomical phenomena in unbounded domains. This method, based on an approximation using the properties of the convolution product, has two disadvantages: low consistency and difficulty associated with the imposition of boundary conditions. In 1992, Nayrolles, Touzot and Villon proposed using a local approximation of least squares in a new method called the “diffuse elements method” (DEM). In 1994, Belytschko et al. proposed the “element-free Galerkin” (EFG) method based on the same principles as the preceding one but using “exact” derivatives of the shape functions. The method known as the “reproducing kernel particle method” (RKPM) introduced by Liu et al. in 1995 is an extension of the SPH method but with the reproduction of linear fields or polynomials of higher order being introduced, thanks to the correction function affecting the kernel function used in SPH method. Finally, the so-called partition of unity method introduced by Babuska in about 1996 is a general principle allowing us to enrich any function associated with a problem involving known physics, within the framework of finite elements and of meshless methods, by adding additional unknowns in the global system of equations. Thus, particular functions such as discontinuous functions and singular functions can be reproduced.

Lastly, more recently, the natural element method (NEM) rests on principles completely different from the previous ones. This method is halfway between meshless methods and the finite element method. The NEM proposes an interpolation based on the concepts of the Voronoi diagram and its natural neighbors. The Voronoi diagram associated with a cloud of nodes distributed over the domain to be studied is the Delaunay dual mesh. Thus, a mesh is being used for the construction of the interpolation. However, as the examples presented in this chapter show, the quality of interpolation produced does not depend on the form of the triangles (2D problems) or tetrahedrons (3D problems) present in the Delaunay mesh. The latter is built in a systematic way without requiring repositioning of nodes. With NEM the choice of support of shape functions is automatic and optimal in the sense that node vicinity is taken into account as much as possible to define the interpolation. Wit...

Table of contents

• Cover
• Title Page
• Copyright
• "Fondation Cetim" 2008 Award Foreword
• Acknowledgements
• Chapter 1: Introduction
• Chapter 2: Basics of the Natural Element Method
• Chapter 3: Numerical Aspects
• Chapter 4: Applications in the Mechanics of Structures and Processes
• Chapter 5: A Mixed Approach to the Natural Elements
• Chapter 6: Flow Models
• Chapter 7: Conclusion
• Bibliography
• Index