Abstract
In this expository article, a variety of computational methods, such as Collocation, Finite Volume, Finite Element, Boundary Element, MLPG (Mesh less Local Petrov Galerkin), Trefftz methods, and Method of Fundamental Solutions, etc., which are often used in isolated ways in contemporary literature are presented in a unified way, and are illustrated to solve a 4th order ordinary differential equation (beam on an elastic foundation). Both the primal formulation, which considers the 4th order ODE with displacement as the primitive variable, as well as two types of mixed formulations (one resulting in a set of 2 second-order ODEs, and the other resulting in a set of 4 first-order ODEs), which consider both displacement and its derivatives as mixed variables, are used as strong forms of the problem. Through integration by parts of the weighted residuals, different global and local, unsymmetric and symmetric weak-forms are derived. Both global (harmonics, polynomials, Radial Basis Functions, Trefftz and Fundamental solutions), and local interpolations (element-based interpolations, meshless Moving Least Squares) are used as trial functions of primal and mixed variables. By using Dirac Delta function, Heaviside function, Galerkin and Petrov Galerkin type of function, as well as fundamental solutions as test functions of various weak-forms, primal and mixed implementations of Collocation, Finite Volume, Finite Element, Boundary Element, Mesh less Local Petrov Galerkin (MLPG), Trefftz and Method of Fundamental Solutions are developed. Applications of these methods are illustrated for solving problems with well-posed boundary conditions (BCs), which are the physically-consistent BCs of a solid-body (beam on elastic foundation), as well as for ill-posed boundary conditions, where the Cauchy type of B.C. are over-prescribed on a part of the boundary. The advantages & disadvantages of various primal & mixed, symmetric & unsymmetric weak forms are discussed, on the admissible order of continuity for trial & test functions, the requirement of evaluating higher-order differentials, as well as the enforcement of well-posed & ill-posed BCs. The relationship between various trial & test functions and the resulting sparse or dense, symmetric or non-symmetric, well-conditioned or ill-conditioned coefficient matrices are also demonstrated and discussed. This paper thus presents a unification of a variety of concepts in developing numerical methods for problems of multidisciplinary engineering and sciences, which are often presented in an ad hoc manner, in contemporary literature. The MATLAB codes pertaining to all the methods presented here are presented for free download at the website: www.care.eng.uci.edu/pubs.htm. This expository article will be a part of soon to be published introductory textbook by Atluri and Dolt (2015).