Infimalfolgen nichtkonvexer Variationsprobleme haben aufgrund feiner Oszillationen häufig keinen starken Grenzwert in Sobolevräumen. Diese Oszillationen haben eine physikalische Bedeutung; Finite-Element-Approximationen können sie jedoch im Allgemeinen nicht auflösen. Relaxationsmethoden ersetzen die nichtkonvexe Energie durch ihre (semi)konvexe Hülle. Das entstehende makroskopische Modell ist degeneriert: es ist nicht strikt konvex und hat eventuell mehrere Minimalstellen. Die fehlende Kontrolle der primalen Variablen führt zu Schwierigkeiten bei der a priori und a posteriori Fehlerschätzung, wie der Zuverlässigkeits- Effizienz-Lücke und fehlender starker Konvergenz. Zur Überwindung dieser Schwierigkeiten erweitern Stabilisierungstechniken die relaxierte Energie um einen diskreten, positiv definiten Term. Bartels et al. (IFB, 2004) wenden Stabilisierung auf zweidimensionale Probleme an und beweisen dabei starke Konvergenz der Gradienten. Dieses Ergebnis ist auf glatte Lösungen und quasi-uniforme Netze beschränkt, was adaptive Netzverfeinerungen ausschließt. Die vorliegende Arbeit behandelt einen modifizierten Stabilisierungsterm und beweist auf unstrukturierten Netzen sowohl Konvergenz der Spannungstensoren, als auch starke Konvergenz der Gradienten für glatte Lösungen. Ferner wird der sogenannte Fluss-Fehlerschätzer hergeleitet und dessen Zuverlässigkeit und Effizienz gezeigt. Für Interface-Probleme mit stückweise glatter Lösung wird eine Verfeinerung des Fehlerschätzers entwickelt, die den Fehler der primalen Variablen und ihres Gradienten beschränkt und so starke Konvergenz der Gradienten sichert. Der verfeinerte Fehlerschätzer konvergiert schneller als der Fluss- Fehlerschätzer, und verringert so die Zuverlässigkeits-Effizienz-Lücke. Numerische Experimente mit fünf Benchmark-Tests der Mikrostruktursimulation und Topologieoptimierung ergänzen und bestätigen die theoretischen Ergebnisse. / Infimising sequences of nonconvex variational problems often do not converge strongly in Sobolev spaces due to fine oscillations. These oscillations are physically meaningful; finite element approximations, however, fail to resolve them in general. Relaxation methods replace the nonconvex energy with its (semi)convex hull. This leads to a macroscopic model which is degenerate in the sense that it is not strictly convex and possibly admits multiple minimisers. The lack of control on the primal variable leads to difficulties in the a priori and a posteriori finite element error analysis, such as the reliability-efficiency gap and no strong convergence. To overcome these difficulties, stabilisation techniques add a discrete positive definite term to the relaxed energy. Bartels et al. (IFB, 2004) apply stabilisation to two-dimensional problems and thereby prove strong convergence of gradients. This result is restricted to smooth solutions and quasi-uniform meshes, which prohibit adaptive mesh refinements. This thesis concerns a modified stabilisation term and proves convergence of the stress and, for smooth solutions, strong convergence of gradients, even on unstructured meshes. Furthermore, the thesis derives the so-called flux error estimator and proves its reliability and efficiency. For interface problems with piecewise smooth solutions, a refined version of this error estimator is developed, which provides control of the error of the primal variable and its gradient and thus yields strong convergence of gradients. The refined error estimator converges faster than the flux error estimator and therefore narrows the reliability-efficiency gap. Numerical experiments with five benchmark examples from computational microstructure and topology optimisation complement and confirm the theoretical results.

New automated vehicles have the chance of high improvements to road safety. Nevertheless, from today's perspective, accidents will always be a part of future mobility. Following the “Vision Zero”, this thesis proposes the quantification of the driving situation's criticality as the basis to intervene by newly integrated safety systems. In the example application of trajectory planning, a continuous, real-time, risk-based criticality measure is used to consider uncertainties by collision probabilities as well as technical accident severities. As result, a smooth transition between preventative driving, collision avoidance, and collision mitigation including impact point localization is enabled and shown in fleet data analyses, simulations, and real test drives. The feasibility in automated driving is shown with currently available test equipment on the testing ground. Systematic analyses show an improvement of 20-30 % technical accident severity with respect to the underlying scenarios. That means up to one-third less injury probability for the vehicle occupants. In conclusion, predicting the risk preventively has a high chance to increase the road safety and thus to take the “Vision Zero” one step further.:Abstract Acknowledgements Contents Nomenclature 1.1 Background 1.2 Problem statement and research question 1.3 Contribution 2 Fundamentals and relatedWork 2.1 Integral safety 2.1.1 Integral applications 2.1.2 Accident Severity 2.1.2.1 Severity measures 2.1.2.2 Severity data bases 2.1.2.3 Severity estimation 2.1.3 Risk assessment in the driving process 2.1.3.1 Uncertainty consideration 2.1.3.2 Risk as a measure 2.1.3.3 Criticality measures in automated driving functions 2.2 Operational motion planning 2.2.1 Performance of a driving function 2.2.1.1 Terms related to scenarios 2.2.1.2 Evaluation and approval of an automated driving function 2.2.2 Driving function architecture 2.2.2.1 Architecture 2.2.2.2 Planner 2.2.2.3 Reference planner 2.2.3 Ethical issues 3 Risk assessment 3.1 Environment model 3.2 Risk as expected value 3.3 Collision probability and most probable collision configuration 4 Accident severity prediction 4.1 Mathematical preliminaries 4.1.1 Methodical approach 4.1.2 Output definition for pedestrian collisions 4.1.3 Output definition for vehicle collisions 4.2 Prediction models 4.2.1 Eccentric impact model 4.2.2 Centric impact model 4.2.3 Multi-body system 4.2.4 Feedforward neural network 4.2.5 Random forest regression 4.3 Parameterisation 4.3.1 Reference database 4.3.2 Training strategy 4.3.3 Model evaluation 5 Risk based motion planning 5.1 Ego vehicle dynamic 5.2 Reward function 5.3 Tuning of the driving function 5.3.1 Tuning strategy 5.3.2 Tuning scenarios 5.3.3 Tuning results 6 Evaluation of the risk based driving function 6.1 Evaluation strategy 6.2 Evaluation scenarios 6.3 Test setup and simulation environment 6.4 Subsequent risk assessment of fleet data 6.4.1 GIDAS accident database 6.4.2 Fleet data Hamburg 6.5 Uncertainty-adaptive driving 6.6 Mitigation application 6.6.1 Real test drives on proving ground 6.6.2 Driving performance in simulation 7 Conclusion and Prospects References List of Tables List of Figures A Extension to the tuning process