Meshless Methods for solving large-scale problems - Metsis, Panagiotis (Σχολή Πολιτικών Μηχανικών, 2014)
The present work deals with meshless methods and especially the Element Free Galerkin Method (EFGM) which is more often used by the computational mechanics community. EFGM requires only nodal data and no element connectivity is needed to construct the shape functions. However a global background cell structure is necessary for the numerical integration. Meshless methods have a number of virtues due to the absence of mesh related problems. Effortless modeling of the problem geometry is accompanied by the natural development of adaptive schemes where nodes may be added anywhere in the domain irrespectively of the initial arrangement. Furthermore, increased smoothness of the solution and its derivatives is accomplished since the shape function field has higher order properties. Especially in stress analysis results are more accurate and improved stress fields are achieved when nodes are added to stress concentration areas only. Crack propagation analysis is also improved, since the node arrangement is able to move together with the face of the crack. The aforementioned benefits come with great computational cost in comparison to other - well established- methods such as the Finite Element Method (FEM). The aim of this work is to improve the overall computational efficiency of meshless methods and gain a deeper understanding of their inherent properties. The main focus is on EFGM but the majority of the contributions may be applied on other similar methods also. The first attempt to reduce the computational cost is by employing domain decomposition techniques on the physical as well as on the algebraic domains. A novel balancing technique is proposed, by which the overlapping stiffness coefficients of each subdomain are replaced by the mean value of the subdomain contribution in the boundary dof. The proposed technique vastly decreases the number of solution iterations. Also a novel non-overlapping decomposition is proposed. The subdomain matrices are no longer ill conditioned, resulting in much faster convergence of the iterative procedure. The assembly of the stiffness matrix of EFGM is accountable for a significant percentage of the total computing time. In order to resolve this issue, a novel Node-Gauss point correlation is proposed. The search for interaction between Gauss points and nodes is performed only on specific regions, reducing time. Furthermore, a novel Node pair-wise formulation of stiffness matrix is proposed. The proposed, alternative way, to perform the aforementioned computation is the creation of global stiffness coefficients directly at the interacting nodes. Another issue of concern in EFGM is the calculation of the shape functions and shape functions derivatives that are used to construct the stiffness matrix. A novel formulation of the shape functions by parts (initial and added) is proposed. The contribution of added nodes is thoroughly investigated and separated from the initial part of the shape function field by using the binomial inverse theorem. A hierarchical refinement of the stiffness matrix is proposed. Then, a second, purely hierarchical, refinement of the stiffness matrix is proposed. In that way the computational effort for the assembly of the stiffness matrix is reduced, where specially tailored solution procedures can be applied that avoid refactorization of the initial stiffness matrix.