Flachaire, France Rémi (2023) New numerical integration methods applied to peridynamics PRE - Research Project, ENSTA.
Full text not available from this repository.
Abstract
The finite element method is classically used to numerically solve partial differential equations. For example, we use it in continuum mechanics to predict the behavior of a structure. However, the formulation of classical equations involving differentials is no longer valid at the level of discontinuities. The control and prediction of cracks becomes complicated. It is then necessary to use mathematical artifices (weak formulation, for example) which depend on the case to be treated. In this context, alternative non-local models have been developed. The peridynamic theory [5], that can be considered as a generalization of the classical mechanics of continuums, provides for example such class of nonlocal models. Peridynamics has a non-local character, as it has an internal length scale called the horizon. It is based on a simple principle : to reformulate the equations of continuum mechanics such as not to involve differentials of the displacement field. By using an integral formulation, it becomes interesting for the modeling and simulation of fracture. The peridynamic theory has been validated experimentally, through the use of data from several wave propagation applications, as well as crack initiation and propagation experiments [1]. Two fields of research in peridynamics can be identified. First, the surface effect. It’s the border effect in a nonlocal model such as ours. Different ways of looking at borders can be treated. Serge Prudhomme and Patrick Diehl, in the article [4] present two methods : the variable horizon method and the extended domain method. 1D Neuman and Dirichlet conditions are used at the boundaries. As this report was submitted a few weeks before the end of the internship, this problem of borders has not yet been fully processed. We will concentrate on a second research problem : how to numerically approach the integrals in the peridynamic problem ? We will propose new integration methods. A typical discretization proposed by Silling in 2000 [6] is the nodal discretization called EMU, which is a collocation approach [7]. The idea is to choose a finite-dimensional space of candidate solutions and a number of points in the domain (called collocation points), and select the solution that satisfies the given equation at the collocation points. However, other discretization approaches are available : continuous and discontinuous finite elements, Gaussian quadrature [8] and spatial discretization [7] [2]. This paper focuses on the collocation approach which is widely [3] adopted in current peridynamic simulations. We will present the problem in the section I. By focusing on the collocation method, we restrict ourselves to using the collocation points as integration nodes for the approximation of the integral. The classical method consists in applying the trapezium method to a circular domain. The problem is that many approximations are made because of the shape of the domain. We will develop this method in the section II. The following sections will be dedicated to the development of new integration methods, which we have called ring methods. They offer better results than the classical method for a large spectrum of functions (sections VII and VIII). Then, the integration methods will be applied to the peridynamic problem.
Item Type: | Thesis (PRE - Research Project) |
---|---|
Uncontrolled Keywords: | peridynamics, numerical calculation of integral, nonlocal models, cracks |
Subjects: | Mathematics and Applications |
ID Code: | 9738 |
Deposited By: | Rémi FLACHAIRE |
Deposited On: | 28 nov. 2024 15:04 |
Dernière modification: | 28 nov. 2024 15:04 |
Repository Staff Only: item control page