Finite difference method

Abstract: Picard-Vessiot rings.- Algorithms for difference equations.- The inverse problem for difference equations.- The ring S of sequences.- An excursion in positive characteristic.- Difference modules over .- Classification and canonical forms.- Semi-regular difference equations.- Mild difference equations.- Examples of equations and galois groups.- Wild difference equations.- q-difference equations.

Abstract: WHEN solving elliptic equations by the method of fmite differences we have to deal with systems of linear algebraic equations, often of a very high order. Given a sufficiently high order of the system, the familiar iterative methods of solution of such systems are very slowly convergent. Numerous works have been devoted to methods of speeding up the convergence of the iterations. These speedingup methods can be split provisionally into two groups. The first group includes methods which use the spectrum of the iterative operators; they are described in detail in text-books [l] and [2]. The second, rather indefinite group includes the so-called “relaxation” methods, that are based essentially on “intuition”, the “computer’s experience”; they are regarded as applicable for nonmechanical computation by a sufficiently experienced group of workers, but as little suited to being carried out on digital computers; it is usually suggested that the relaxation method can be extremely effective [3]. The present article describes a method that we have developed for iterative solution of elliptic difference equations, using an idea not unlike the relaxation method. The present method was put into practice on a digital computer and gave good results. Le us take Poisson’s equation in a rectangular domain:

Abstract: N the aerospace industry, the stress analysis of complex axisymmetric structures of arbitrary shape subjected to thermal and mechanical loads is of considerable interest. Rocket nozzles and cases, solid-propellant grains, and spacecraft heat shields are practical examples of such structures. Although the governing differential equations for solids of revolution have been known for many years, closed form solutions have been obtained for only a limited number of structures; thus, the stress analyst must rely on experimental or numerical techniques. The finite difference method has been the most popular of the numerical techniques; however, for structures of composite materials and of arbitrary geometry, this procedure is difficult to apply. In the present investigation, the finite element idealization is used as the basic numerical procedure. This technique has been applied successfully in the stress analysis of many complex structures.1"3 In Ref. 4 impressive results were obtained in the analysis of axisymmetric shells approximated by a series of truncated cone elements. The approach, which is presented here, is similar in many respects to existing methods used in the analysis of two-dimensional stress problems.5"7 Recently, the finite element method was applied to the structural analysis of axisymmetric solids subjected to axisymmetric loads.8 In the present paper, the finite element method is used in the determination of stresses and displacements developed within elastic solids of revolution which are subjected to axisymmetric or nonaxisymmetr ic loads. Emphasis is placed on the application of the technique to complex aerospace structures.