George P.L. Automatic Mesh Generation and Finite Element Computation

Elsevier Science B.V., 1996. — 119 p.A large range of physical phenomena can be formalized in terms of partial differential equations and thus can be solved numerically using the finite element method.
This method was first conceived and used by engineers in the early fifties, in particular for problems with a structural mechanic nature. Due to the constant improvement of computers since this date, the finite element method became more and more popular for numerical simulation of more and more diverse physical problems, including elasticity, heat transfer analysis, electromagnetic, flow computation.
During this time, both theoretical foundations and practical aspects of the method were developed and, in this respect, various finite elements created, analyzed and then used in simulations. These latter became more and more sophisticated as applied in complicated problems and both their size (in terms of the number of nodes defined) and their complexity (in terms of the nature of the problem: time dependent problem, non linear problem, coupled problem) becomes greater and greater. On the other hand, their fields of application (in terms of the geometries dealt with) become more and more realistic and as a consequence more and more complex.
Mesh generation represents the first step of any finite element method that engineers have to implement practically after the necessary theoretical analysis of the problem they want to solve.
This task must be carefully done as the mesh is responsible for the accuracy of the solution computed with such a support. In particular, it is of great importance to capture the geometry of the domain where the problem is posed as well as possible, with special attention paid to the way the boundaries of the domain are approximated and to capture the physical behavior of the problem as accurately as possible. It means that both the number of nodes and elements and their nature must be adequate (in terms of shape, size, density, variation from region to region).
However, mesh construction can be very expensive in terms of time and thus is costly. This is the reason why the generation of meshes is a crucial point in the numerical implementation of the finite element method and one can therefore deplore that the techniques, algorithms and tricks related to the creation of meshes have probably not received adequate attention over the last years (at least not by people considering only the theoretical aspects of the finite element method).
The present article aims to make the different notions encountered when considering the mesh generation aspects of the finite element method as clear as possible and would pay special attention to the automatic mesh generation methods. The discussion consists of several chapters divided into sections. Chapter I introduces useful definitions such as conformal mesh, mesh structure, control structure and control space. Finally, a methodology for conceiving a mesh is proposed. Chapter II outlines the most popular mesh generation methods and algorithms to help the reader to be familiar with these. Manual method, product method, algebraic method and method using the solution of partial differential equations are shortly presented as solutions in the case where the domain enjoys some appropriate properties. The multiblock method is briefly
discussed as a semi-automatic solution a priori possible in general. Automatic methods, including quadtree-octree techniques, advancing-front and Voronoi type approach are introduced. The two latter are then fully detailed in Chapter III (advancing-front method) and Chapter IV (Voronoi type method) for the general situation (geometrical point of view) and in the case where properties to be satisfied are specified in advance (using a control space).