Some Approximation Methods of Dynamics and Statics

Abstract

The deformed configuration of a body, in general, is determined by the field of displacement vectors u = u(x, y, z, t; X, Y, Z), ie a continuum possesses an infinite number of degrees of freedom. The basic partial differential equations of such distributed parameter systems, with associated boundary and initial conditions, even in the case of linear elastic solids, but with non-simple geometry, can hardly be solved in an exact manner. Two classes of approximation techniques are commonly used to overcome these difficulties: (1) The essential boundary conditions are built into the approximation that is not a solution of the basic differential equations. The Rayleigh-Ritz-Galerkin method based on such a set of admissible functions is discussed below, and examples of discretization by the finite-element method (FEM) are given. (2) A solution of the basic equations is known that takes on the prescribed boundary values at a number of discrete points only. Such an approximation is the output of the class of collocation methods. The boundary element method (BEM) should be mentioned here. Collocation methods are not discussed in this text any further.