In technical applications, uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of Wiener (Amer. J. Math. 60(4),897-936, 1938) was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from academics and from industry. For each of the applications, chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.

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In (Augustin et al. in European J. Appl. Math. 19:149-190, 2008) we considered the Polynomial Chaos Expansion for the treatment of uncertainties in industrial applications. For many applications the method has been proven to be a computationally superior alternative to Monte Carlo evaluations. In the current overview we compare the accuracy of Polynomial Chaos type methods for the propagation of uncertainties in nonlinear problems and verify them on two examples relevant for industry. For weakly nonlinear time-dependent models, the generalized Kalman filter equations define an efficient method, yielding good approximations if the quantities of interest are restricted to the first two moments of the solution. Secondly, stochastic collocation is discussed. The method is applied to delay differential equations and random ordinary differential equations. Finally, a generalized PC method is discussed which is based on a subdivision of the random space. This approach is even suitable for highly nonlinear models.

Modeling and simulation is an established scientific and industrial method to support engineers in their work in all lifecycle phases-from first concepts or tender to operation and service-of a technical system. Due to the fact of increasing complexity of such systems, e.g. plants, cyber-physical systems and infrastructures, system simulation is rapidly gaining impact. In this paper, a simulation architecture is presented and discussed on three different industrial applications, which offers a client-server concept to master the challenges of a lifecycle spanning simulation framework. Looking ahead, open software concepts for modeling, simulation and optimization will be required to cover new co-simulation techniques and to realize distributed, for example web-based simulation environments and tools.