Peter Kaps

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

SummaryGeneralizedA(α)-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.

A study of Rosenbrock-type methods of high order

SummaryThis paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.

Rosenbrock methods for Stiff ODEs: A comparison of Richardson extrapolation and embedding technique

In [16], Rosenbrock methods of order four are investigated using an embedded method of order three for step size control. Here, we study such a method using Richardson extrapolation for step size control and compare the two techniques with each other. The usual belief that extrapolation is inferior to embedding is not true. Numerical results for the 25 examples of STIFF DETEST and for some more difficult problems show the following behaviour: For low tolerances (∼10−2) embedding is superior, for moderate tolerances (∼10−4) both techniques are comparable and for high tolerances (<10−5) extrapolation is superior. Under certain conditions the extrapolated value can be used for step continuation without stability problems.ZusammenfassungIn [16] werden Rosenbrockmethoden der Ordnung 4 untersucht, die zur Schrittweitensteuerung eine eingebettete Methode der Ordnung 3 verwenden. In dieser Arbeit werden neue solche Methoden hergeleitet, die auf Schrittweitensteuerung durch Richardsonextrapolation zugeschnitten sind, und beide Techniken miteinander verglichen. Es zeigt sich, daß im Gegensatz zur üblichen Meinung Extrapolation nicht schlechter ist als Einbettung. Die numerischen Ergebnisse für die 25 Beispiele aus STIFF DETEST und einige schwierigere Beispiele zeigen folgendes Verhalten: Für niedrige Genauigkeiten (∼10−2) sind eingebettete Methoden vorteilhaft, für mittlere Genauigkeiten (∼10−4) sind beide Techniken gleichwertig und für hohe Genauigkeiten (<10−5) ist Extrapolation überlegen. Unter bestimmten Voraussetzungen kann der extrapolierte Wert als Startwert für den nächsten Schritt verwendet werden.

Application of a variable-order semi-implicit Runge-Kutta method to chemical models

Abstract A semi-implicit Runge-Kutta method is proposed and application is made to models of chemical kinetics. The presented algorithm GRKV4 is based on an A-stable formula with variable order up to four and possesses an automatic stepsize control. The range of application of GRKV4 is discussed.

Calculation of the contact pressure between ski and snow during a carved turn in Alpine skiing

The macroscopic contact area between ski and snow and the contact pressure are crucial influencing factors for carved turns in Alpine skiing. In the present paper, a simulation model is developed to quantify these factors. The ski is modelled as an Euler–Bernoulli beam with variable cross section, camber, bending and torsional stiffness using measured data from skis. The reaction forces of the snow are decomposed in penetration and shear forces. For the penetration forces a hypoplastic constitutive law is applied incorporating elastic and plastic deformation of the snow at the contact area. For the shear forces metal cutting theory is used. Ski deformation, contact area and contact pressure are computed based on quasi‐static equilibrium between forces exerted by the skier and snow reaction forces. Parameter studies are performed to investigate the influence of edging and distributing the load between the inner and outer ski. Higher edging angles as well as loading both skis affected the contact pressure positively by increasing the resistance against shearing. The results of our study agree well with measurement data taken from literature. Based on the results, the importance of actions of the skier during carved turns is concluded.

Safety assessment of jumps in ski racing

The influence of important parameters on the flight trajectory for jumps in downhill World Cup races was investigated. To quantify the impact injury risk at landing, the parameter equivalent landing height (ELH) was introduced, which considered a variable slope inclination during the landing movement. Altogether, 145 runs at four different jumps in World Cup races and trainings were recorded and analyzed. A simulation model was developed to predict the flight phase of the skier. Drag and lift areas were selected by parameter identification to fit the simulation trajectory to the two‐dimensional data from the video analysis. The maximum values of the ELH which can be absorbed with muscle force was taken from the study of Minetti et al. for elite female and male ski racers. A sensitivity analysis based on the four jumps showed that ELH is mainly influenced by takeoff angle, takeoff speed, and the steepness of the landing surface. With the help of the developed simulation software, it should be possible to predict the ELH for jumps in advance. In case of an excessive ELH, improvements can be made by changing the takeoff inclination or the approach speed.

Modeling of the Ski-Snow Contact for a Carved Turn

Carved turns with alpine skis are investigated. During the movement of a ski, snow is loaded and unloaded. Compacted snow is not elastic, i.e. deformations remain. Such effects are modeled by a hypoplastic constitutive equation. During a turn the shovel digs into the snow and the tail maintains nearly the same penetration depth as the part under maximum load. This results in a higher resistance against shearing for the afterbody of the ski. In the present work we investigated the benefits of the hypoplastic against the elastic forcepenetration relationship. Simulation results for a sledge on two skis are compared to experimental track data.

Influence of Ski Bending Stiffness on the Turning Radius of Alpine Skis at Different Edging Angles and Velocities

Carved turns with Alpine Skis were investigated using a computer simulation model. Varied input data to the model were the bending stiffness of the skis, the edging angle, and the velocity. Results include the turn radius and the force distribution along the running surface of the skis.

Modeling the ski–snow contact in skiing turns using a hypoplastic vs an elastic force–penetration relation

A ski–snow interaction model is presented. The force between ski and snow is decomposed into a penetration force normal to the snow surface, a shear force parallel to it, and friction. The purpose of this study was to investigate the benefits of a hypoplastic vs an elastic contact for penetration in the simulation of skiing turns. To reduce the number of influencing factors, a sledge equipped with skis was considered. A forward dynamic simulation model for the sledge was implemented. For the evaluation of both contact models, the deviation between simulated trajectories and experimental track data was computed for turns of 67 and 42 m. Maximum deviations for these turns were 0.44 and 0.14 m for the hypoplastic contact, and 0.6 and 7.5 m for the elastic contact, respectively. In the hypoplastic contact, the penetration depth of the skis afterbody maintained nearly the same value as the part under maximum load, whereas it decreased in the elastic contact. Because the shear force is proportional to the penetration depth, the hypoplastic contact resulted in a higher shearing resistance. By replacing the sledge with a skier model, one may investigate more complex skier actions, skiing performance, or accident‐prone skiing maneuvers.

An approximate simulation model for initial luge track design.

Competitive and recreational sport on artificial ice tracks has grown in popularity. For track design one needs knowledge of the expected speed and acceleration of the luge on the ice track. The purpose of this study was to develop an approximate simulation model for luge in order to support the initial design of new ice tracks. Forces considered were weight, drag, friction, and surface reaction force. The trajectory of the luge on the ice track was estimated using a quasi-static force balance and a 1d equation of motion was solved along that trajectory. The drag area and the coefficient of friction for two runs were determined by parameter identification using split times of five sections of the Whistler Olympic ice track. The values obtained agreed with experimental data from ice friction and wind tunnel measurements. To validate the ability of the model to predict speed and accelerations normal to the track surface, a luge was equipped with an accelerometer to record the normal acceleration during the entire run. Simulated and measured normal accelerations agreed well. In a parameter study the vertical drop and the individual turn radii turned out to be the main variables that determine speed and acceleration. Thus the safety of a new ice track is mainly ensured in the planning phase, in which the use of a simulation model similar to this is essential.