
Preface 
In 1990 I terminated my work as scientific employee at the chair of mechanics Prof. Heinz as well as the educational and research institute for mechanics Prof. Ballmann at the RWTH Aachen. My thesis had the topic 'A characteristics algorithm for the calculation of the transient transverse oscillations of helicopter rotor blades or other quasilinear flexible structures'. Afterwards I took over an engineer's office for structural engineering and formed from this the company for 'Civil engineering Mueller & Partner ltd'.
At the chair of mechanics as well as at the educational and research institute for mechanics at the RWTH Aachen questions about the numeric solutions of the principal equations of continuum mechanics were among other things of special interest. Apart from the handling of the equations of motion for fluid mechanics also stress waves in flexible solids were treated. The wave phenomena are simulated with numeric solution procedures, which are based on the characteristics theory.
In my thesis I worked out a numericel computationel method for the simulation of the propagation of deformations and stresses in quasilinear flexible structures. The regarded structures may by simple beams, wings of large stretching or other essentially one dimensional linear technical combined structures with variable cross section (Timoshenko beam). The deformation of the beam is described by a system of 12 partial, hyperbolic differential equations in time direction and one direction in space. The solution of these equations was carried out by means of characteristics differential scheme in matrix form, which enables it to handle with a comparatively reasonable effort the implementation of an appropriate computational program. The calculation is done thereby for all divergent wave velocities on the characteristics grids attached in each case. In this way unwanted numeric dissipations was avoidet and dispersion effects were almost completely eliminated, because the CFL NUMBER for all characteristics becomes alike unity. Spatial interpolations, which pretend false phase velocities, are avoided thereby. As technical area of application among other things a magneto elastic problem from the magnetic suspension railway was regarded. In the following years, starting from 1990, the computational program was successfully used for the numeric simulation and calculation of the suspension pins of the German Transrapit magnetic transport system .
Besides some of my colleagues at the chair of mechanics as well as the educational and research institute for mechanics on suggestion of Prof. Ballmann treated also stress waves in planes, plates and rotationall symmetric volumes. The treatment of the deformations of these quasitwodimensional structures are described by systems of partial, hyperbolic differential equations in time direction and two directions in space. With the solutions of these systems by means of the characteristics theory, in contrast to systems with one direction in space, socalled transverse derivatives appears in the characteristics differential scheme, for whose handling special numeric techniques are necessary (Method of Bicharacteristics). Hereby in the solution point the unknown transverse derivatives in the system of equations with explicit scheme of second order in spacetime direction are apriori eliminated. This succeeds by building of linear combinations of the appearing difference equations suitable by for the respectively regarded system. The transverse derivatives and the function values of the initial value surface are approximated by an second degree polynom of regression. The coefficients of the polynom in the two directions in space are obtained for each unknown field size in each grid point with the procedure of the least square method. Another possibility also used for the handling of the transverse derivatives consists of dividing the time step on the basis of the initial values up to the solution in the solution points according to the number of appearing directions in space and forming for each supstep time only appropriate linear characteristics, whereby the transverse derivatives for this appearing are to be determined in first order in spacetime direction with a scheme only from the initial value surface of the regarded supstep time ('Dimensional Splitting', 'Method of Fractional Step'). To construct from this a scheme second order in time direction furthermore investigations are necessary.
With the so won solution scheme for hyperbolic systems, which are based on the characteristics theory, excellend resultsare obtained compared with other solution methods, since the characteristics method is best adapted to the wave character of the physical problem. By the problems described above, which result by handling the transverse derivatives with multidimensional systems, the implementation of an appropriate solution scheme in a computational program is relatively complex and timeconsuming. In my thesis for quasilinear hyperbolic systems a solution scheme was developed, which is consistently formulated in matrix form and is particularly well suitable thereby for an implementation in a computational program. Therefore the question, whether a solution algorithm is possible for multidimensional systems in matrix form witch benefits of the linear characteristic scheme, like e.g.. CFL NUMBERS of unity, for the avoidance of unwanted numeric dissipations and dispersion effects. This question can be answered positively.
By the application of a general elimination procedure for the unknown transverse derivatives in the solution point an bicharacteristic algorithm (Bison: Bicharacteristic Solid Numeric) was developed, witch benefits of the one dimensional scheme also for multidimensional systems. The advantages are among other: A closed explicit algorithm in matrix form for any hyperbolic systems; Scheme second order in spacetime direction; Highest CFL NUMBERS of equal unity or near unity for all kinds of waves by using different bicharacteristic grids belonging to the corresponding wave velocities; Stable and convergent procedure for large number of time steps; Renouncement of an polynom of regression on the initial value surface; No 'Dimensional Splitting'; Relative simple for a general implementation of the scheme into a computational program; Special grid points with special schemes at the edge, at a corner or at a crack tips are developed partially and also relative simple to implement in a computer program; Relative short computing times; Specially qualified for implementations on parallel computers without problems.