Modeling of mechanical systems with uncertainties is extremely challenging and requires a careful analysis of a huge amount of data. Both, probabilistic modeling and nonprobabilistic modeling require either an extremely large ensemble of samples or the introduction of additional dimensions to the problem, thus, resulting also in an enormous computational cost growth. No matter whether the Monte‐Carlo sampling or Smolyak's sparse grids are used, which may theoretically overcome the curse of dimensionality, the system evaluation must be performed at least hundreds of times. This becomes possible only by using reduced order modeling and surrogate modeling. Moreover, special approximation techniques are needed to analyze the input data and to produce a parametric model of the system's uncertainties. In this paper, we describe the main challenges of approximation of uncertain data, order reduction, and surrogate modeling specifically for problems involving polymorphic uncertainty. Thereby some examples are presented to illustrate the challenges and solution methods.

We present a topology optimization procedure for flexible multibody systems, which is subjected to time-and space-dependent constraints. The flexible bodies are parameterized using the solid isotropic material with penalization approach. For the efficient solution of the optimization problem, the pointwise time-dependent constraints are discretized and aggregated into a single constraint using the Kreisselmeier-Steinhauser function. The procedure is tested by optimizing the mass of a flexible piston rod, while the deformation within the design domain is constrained.

The numerical simulation of acoustic problems is, for itself, a quite difficult task since the underlying systems are usually highly complex with a broad frequency range and high sensitivity. Due to this complexity and the corresponding computational burden, tasks like optimization and uncertainty quantification (UQ) are seldom performed in acoustics. Especially when dealing with polymorphic uncertainties where combined techniques of UQ might be required, a direct use of the model is not viable. To allow such engineering tasks, the construction of a cheap surrogate or reduced model is common practice in order to allow a large number of model evaluations at low costs. For acoustic systems, the construction of a reasonably accurate surrogate model can become a challenging task since many systems operate in the frequency domain where phenomena like resonance and interference can cause highly nonlinear responses. In this paper, a surrogate model based on the combination of a parametric model for capturing local nonlinearities and a random process regression for modelling the global trend is presented. The basic procedure based on previous works of the authors is extended to a predict the system response both for unobserved parameters and frequencies. The procedure is demonstrated for a representative example namely the acoustic simulation of a car interior, and further improvements in accuracy and efficiency by the usage of multilevel information are discussed.