Element Free Galerkin Method: Theory Programming and Applications

Abstract

With the advent of high speed digital computers and innovative numerical techniques, there is volcanic proliferation in the domain of handling complex structural problem using umpteen number of approximate methods viz. FEM, Finite Difference Method (FDM), Finite Volume Method (FVM), Finite Strip Method, etc. In today's world of challenge, a buzz word in the corner is the development of an approach which is not banking on the "Meshing Concept" of the geometry or continuum but working on the contemporary "Mesh Free" approach. This can handle a large number of non- conventional problems including those of fracture mechanics and non-linear domains

and thermal analysis field. Researchers are still working hard in the maze of this new emerging research area and are still struggling and working very hard in the area of the Mesh free methods. An humble attempt has been made in this piece of work to first understand the subject which is almost at its infancy in the world, with only handful of researchers are working on this topic. The detailed literature survey has been carried out to explore and understand various methods developed in mid 1990's and their possible application areas. Thrust has been given on the understanding of wide gamut of approaches, with prime focus on Element Free Galerkin (EFG) method only. In the present work, because of availability of limited literature in this domain; efforts have been made in getting an insight into the modus-operandi involved. EFG method uses, least square interpolants to construct the trial and test functions for variational principle (weak form); the dependant variable and its gradient are continuous in the entire domain. Procedure for construction of shape function using moving least square (MLS) approximation is presented. Various terms related to EFG method such as support domain, weak forms, choice of weight function and EFG formulations are presented. Software for analysis of 1D and 2D problems by EFG method are developed and tested. First a step by step procedure to solve a 1D bar using EFG with interim vi computations is presented. Further a problem of 1D bar problem of temperature domain is analyzed and results are plotted. In both the types of problems, results are in good agreement with available results. A problem of 1D cantilever beam is analyzed, here number of gauss point are taken as twice the number of nodes. Results converge very fast as number of nodes increases. To check the versatility of the method as well as of programs, various other boundary conditions such as simply supported, propped cantilever and fixed beams are also considered for analysis. Finally two dimensional plane stress problems are studied. Detailed flowchart is presented. The program in C language is developed for structural analysis. The Timoshenko beam problem analysis is carried out by two methods i.e. Lagrange multiplier method and Penalty method. The results are discussed and difficulties of both the methods are enumerated. In chapter 7, important conclusions, summary and future scope are described.