Collocation Methods

The collocation methods constitute a group of meshless method where the Partial Differential Equations (PDE) are solved at collocation points. The strong form of the PDE is used which makes this method “truly” meshless. The method has a great flexibility with regards to the placement of points as the nodes are not connected by elements. The insertion of new nodes in the domain does not require a full pre-processing of the model as only a part of the problem matrix is modified. The differential operators can be approximated using several methods such as the Generalized Finite Difference (GFD), the Discretization-Corrected Particle Strength Exchange (DC PSE), the Moving Least Square (MLS) or the Radial Basis-Finite Difference (RBF-FD).

The aim of this project is to be able to readily solve a given problem based on a CAD file and on a set of boundary conditions. The selection of the main parameters on which the method depends is critical in order to minimize the approximation error. The stability of the methods can be improved by selecting carefully the nodes on which the approximation is based and by using other stabilization methods such as the Finite Increment Calculus (FIC) or a Voronoi diagram. The derivatives approximation can also be enhanced using the gradient recovery method. This method can be used as a basis for a posteriori error estimation and model adaptability. The GFD method has primarily been considered in this project as it has been proven efficient and accurate compared to the other methods. This method, based on a Taylors’s series expansion of the unknow field, does not assume that the solution belongs to a certain solution space. It can then be used, in its default form, to solve a wide variety of problems.

Two-dimensional and three-dimensional problems in the field of continuum mechanics are considered. A C++ code, based on the MPI framework, has been developed for this project for distributed memory machines.

Gear with two tooths under pressure loading (left) and horseshoe under shear loading (right) – von Mises stress results.
Some References:

T. Liszka and J. Orkisz. The finite difference method at arbitrary irregular grids and its application in applied mechanics. Computers & Structures, 11(1-2):83–95, feb 1980.
J. Orkisz. Meshless finite difference method i. basic approach. In in: Idelshon, OÃśate, Duorkin (Eds.). Computational Mechanics, IACM, CINME, 1998.
J. Orkisz. Meshless finite difference method ii. adaptive approach. In in: Idelshon, OÃśate, Duorkin (Eds.). Computational Mechanics, IACM, CINME, 1998.
J.D. Eldredge, A. Leonard, and T. Colonius. A general deterministic treatment of derivatives in particle methods. Journal of Computational Physics, 180(2):686–709, aug 2002.
B. Schrader, S. Reboux, and I.F. Sbalzarini. Discretization correction of general integral PSE operators for particle methods. Journal of Computational Physics, 229(11):4159–4182, jun 2010.
E. Oñate. Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems. Computer Methods in Applied Mechanics and Engineering, 151(1-2):233–265, jan 1998.
M. Duflot. Application des méthodes sans maillage en mécanique de la rupture. PhD thesis, 2004.