Search

Problems of vibration in engineering applications, that are treated by the finite element method using consistent mass matrices, lead to a general eigenvalue problem $$ {\rm A}{\text{x = }}\lambda {\rm B}{\text{x}}, $$ (1) where A and B are the stiffness and mass matrices, respectively. The order n of the matrices is in general large, corresponding to the number of nodal variables. The matrices A and B are symmetric, and without loss of generality we can assume A and B to be positive definite. The matrices are sparse and have the same, in general irregular, sparsity structure. We look for the m smallest eigenvalues $$ 0 < {{\lambda }_{1}} < {{\lambda }_{2}} \leqslant {{\lambda }_{3}} \leqslant \ldots \leqslant {{\lambda }_{m}} $$ (2) and for the corresponding eigenvectors z1, z2,..., zm of (1) such that $$ {\text{A}}{{{\text{z}}}_{{\text{j}}}}{\text{ = }}{{\lambda }_{{\text{j}}}}{\text{B}}{{{\text{z}}}_{{\text{j}}}}{\text{,}}\quad {{{\text{z}}}_{{\text{j}}}}^{{\text{T}}}{\text{B}}{{{\text{z}}}_{{\text{j}}}}{\text{ = 1,}}\quad {\text{(j = 1,2,}} \ldots {\text{,m)}}{\text{.}} $$ (3) The number m of the desired eigenpairs (»j, zj) is small compared with the order n of the matrices.

The smallest eigenvalues and corresponding eigenvectors of Ax = λBx, where A and B are symmetric, positive definite and sparse matrices of high order, can be computed by minimizing the Rayleigh quotient by means of the method of conjugate gradients. An appropriate preconditioning of the eigenvalue problem results in an essential improvement of the convergence. It is interesting that an implicit realization of the preconditioning does not increase the required computational effort per iteration step. The algorithm is highly vectorizable and is indeed a very efficient method with respect to computational work and to storage requirements.

The general eigenvalue problems that arise from vibrational problems treated by the finite element method can be solved suitably by the method of coordinate overrelaxation. To speed up the convergence of the basic algorithm the symmetric coordinate overrelaxation with Chebyshev acceleration was developed. Theoretically the gain of convergence is promising, but this variant is not very satisfactory since a second parameter has to be chosen and since the modified method cannot help in case of adjacent eigenvalues. A surprisingly decisive improvement of the convergence property is yielded by the method of simultaneous group coordinate overrelaxation. Some representative examples illustrate the behaviour of the variants.

The method of conjugate gradients for the solution of very large symmetric and positive definite systems of linear equations represents an economical algorithm if certain techniques of preconditioning are applied. The procedure consists of an appropriate congruence transformation of the given system matrix A in order to reduce the condition number essentially, thus improving the convergence. Three known types of preconditioning with triangular matrices of the same structure as A are considered. Numerical experiments on the convergence behaviour of the resulting preconditioned cg-algorithm are reported for a series of representative examples originating from finite difference and finite element approximations for typical engineering problems.

The simple eigenvalue problems of the vibrating string and the vibrating beam are treated by the method of finite elements using the different types of quadratic up to quintic elements. The resulting equations are interpreted as difference equations and their local truncation errors are analyzed and discussed. To solve the general algebraic eigenvalue problems the method of coordinate overrelaxation is applied. It is found that the asymptotic convergence of the algorithm depends in a substantial way on the type of the element. Finally, the condition numbers of the stiffness matrices for quartic and quintic elements behave amazingly.

Theoretical results and practical improvements concerning the computation of the lowest eigenvalue and the corresponding eigen-vector of the general eigenvalue problem Ax = λBx by the method of coordinate overrelaxation are reported. There exists a class of pairs of matrices A and B for which the rate of convergence of the sequence of iterated vectors is completely independent on the spectrum. For the symmetric coordinate overrelaxation method with Chebyshev acceleration the optimal parameters can be determined in an adaptive way by using the information gained from some iteration steps.

Ein normal gebautes Gebis (normale Okklusion) ist bei unserer Bevolkerung durchaus nicht die Regel, sondern bei 50% unserer Kinder lassen sich Anomalien feststellen. Bei einer Untersuchung von 17’000 Basler Schulkindern durch O. Meyer hatten wir Muhe, hundert mit normaler Okklusion herauszufinden, und sogar bei diesen stellten sich spater zum Teil Anomalien ein. Zum Gluck handelt es sich meist um Stellungsanomalien einzelner Zahne, welche die Harmonie des Gesichtes weniger storen. Ich mache jedoch die Beobachtung, das heute die Mutter schon die kleinste Verschiebung der Zahne ihrer Kinder sehen und unsere Klinik zur Konsultation aufsuchen. Aber nicht nur den Kindern konnen wir helfen, sondern auch den erwachsenen Patienten, deren Zahl fortwahrend zunimmt. Neben den leichten Anomalien gibt es die recht schweren Kieferanomalien. Dass sind Storungen der Bislage, welche da Gesicht sehr verunstalten konnen und so die mit solchen Anomalien Behafteten in ihrem Fortkommen hindern und oft psychisch sehr bedrangen.