1 - Fundamentals of the finite element method

Abstract

Various phenomena treated in science and engineering are often described in terms of differential equations formulated by using their continuum. Solving differential equations under various conditions such as boundary or initial conditions leads to the understanding of the phenomena and can predict the future of these phenomena (determinism). Exact solutions for differential equations, however, are generally difficult to obtain. Numerical methods are adopted to obtain approximate solutions for differential equations. Among those numerical methods, those that approximate continua with infinite degree of freedom by discrete body with finite degree of freedom are called ‘discrete analysis’. Popular discrete analyses are the finite difference method, the method of weighted residuals, and the Rayleigh–Ritz method. Via these methods of discrete analysis, differential equations are reduced to simultaneous linear algebraic equations and thus can be solved numerically.