Zur systematik numerischer näherungsverfahren in der statik und dynamik der konstruktionen

Abstract

The availability of efficient computers has shifted considerable emphasis on the utilization of discontinuous methods in engineering. On the other hand, the availability of these facilities has also enlarged the applicability of analytical solutions of mechanical problems, especially the coupling of analytical and numerical methods. There are several reasons for not applying in every case only the prominent finite difference or finite element methods. In many cases other methods lead to solutions of sufficient accuracy under relatively small requirements for the size of computers and computer-time. It is worthwhile to classify the manifold methods and tchniques to realize the common roots and to show, how it is possible to develop further methods. The starting points of a classification are the different ways to describe mathematically a physical situation: differential equations (integro-differential equations) and variational principles. On the basis of these two different formulations it is possible to develop various methods to solve special, but also more general, problems in Structural Mechanics. The variational problem is the basis of the methods associated with the names of Ritz, Trefftz and Kantorowitsch, but also the most frequently used numerical method, the finite element method, is based thereupon. The differential equations are the starting point of various methods as finite differences, dynamic relaxation, and collocation (point matching), least squares and discrete least squares, applied to boundaries and domains. A special case is the application of linear programming to boundary value problems. In cases where there are restrictions in the values of displacements or forces some difficulties arise, and it is intelligent to apply a Simplex-algorithm. A general method to establish finite differences formulation to a whole range of differential equations is based on collocation. Some other methods (transfer matrices, Fourier transforms) can be included in the systematics. Some special methods for solving linear equations are applicable directly to the mechanical models (dynamic relaxation, Monte-Carlo method). Such way of application is very interesting and avoids the necessity to establish and to handle large systems of linear equations in the computer.