The Theory of Singularities. A Complement to Local Finite Element Models in Physics

The methods used to digitize and model physical problems in industrial applications are essentially local. The most widespread of these – finite element calculations – reduces the global matrix model to a series of local (elementary) matrices, first generated by a variational principle of the virtual power type (with local interpolations) and then numerically assembled. Particular techniques (finite differences, discrete elements, SPH) complement these by considering particular zones of influence, but these techniques are also ultimately concerned with the juxtaposition of local problems.

These methods have very strong operational qualities, which explain their success and their almost universal use. Their local character, which enables the global problem to be reduced to a series of isoparametric problems, is a powerful asset, but it is also the source of their limitations. Firstly, they require a complete local mesh of the entire domain in all its geometry and topology. Secondly, they are not optimal, since the assembled models – of large size – are structurally hollow, due to the local nature of the elements. In fact, special numerical techniques have been adapted to such systems. Finally, because of the standardization and simplicity of the local model, the elements require a certain number of restrictions on their nature and shape, leading to the risk of errors or local discrepancies in certain types of particular problem.

In fact, a first attempt to overcome these difficulties consisted in condensing substructures, which amounts to representing an assembly solely by its interfaces. Guyan's static condensation, followed by Craig-Bampton's dynamic modal synthesis, have led to major advances in this direction. But in all cases, the mesh used to generate the condensation remains local. The decisive step towards further progress therefore consists in going back upstream, to mathematically propose a boundary problem, intrinsically allowing only the limits of the problem to be meshed. This leads to boundary methods, the principle of which is presented in this article. The advantages are threefold: the size of the numerical problem is drastically reduced, its solid character is reinforced (which increases accuracy) and, finally, these techniques are well suited to certain irregular problems, making them both parallel and complementary to previous local methods.

In this article, integral methods are presented on the basis of simple problems that are representative of the effectiveness of these techniques. Indeed, many problems in physics are directly or indirectly linked to harmonic functions. Examples include regular or singular elastic problems, Lagrange plate bending, voltage in an electrostatic capacitor, etc.

Thus,...