Abstract. The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L 2 -and H −1 -gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches. Mathematics Subject Classification (2000). 74P15, 74P05, 74S03, 35K99.
In brake systems, some dynamic phenomena can worsen the performance (e.g., fading, hot banding), but a major part of the research concerns phenomena which reduce driving comfort (e.g., squeal, judder, or creep groan). These dynamic phenomena are caused by specific instabilities that lead to self-excited oscillations. In practice, these instabilities can be investigated using the Complex Eigenvalues Analysis (CEA), in which positive real parts of the eigenvalues are identified to characterize instable regions. Measurements on real brake test benches or tribometers show that the coefficient of friction (COF), μ , is not a constant, but dynamic, system variable. In order to consider this aspect, the Method of Augmented Dimensioning (MAD) has been introduced and implemented, which couples the mechanical degrees of freedom of the brake system with the degrees of freedom of the friction dynamics. In addition to this, instability prediction techniques can often determine whether a system is stable or instable, but cannot eliminate the instability phenomena on a real brake system. To address this, the current work deals with the quantification of the relevant polymorphic uncertainty of the friction dynamics, wherein the aleatory and epistemic uncertainties are described simultaneously. Aleatory uncertainty is concerned with the stochastic variability of the friction dynamics and incorporated with probabilistic methods (e.g., a Monte Carlo simulation), while the epistemic uncertainty resulting from model uncertainties is modeled via fuzzy methods. The existing measurement data are collected and processed through Data Driven Methods (DDM) for the identification of the dynamic friction models and corresponding parameters. Total Variation Regularization is used for the evaluation of derivatives within noisy data. Using an established minimal model for brake squealing, this paper addresses the question of probabilities for instabilities and the degree of certainty with which this conclusion can be made. The focus is on a comparison between the conventional Coulomb friction model and a dynamic friction model in combination with the MAD. This shows that the quality of the predictive accuracy improves dramatically with the more precise friction model.
In this contribution, several case studies with data uncertainties are presented which have been performed in individual projects as part of the DFG (German Research Foundation) Priority Programme SPP 1886 “Polymorphic uncertainty modelling for the numerical design of structures.” In all case studies numerical models with uncertainties are derived from engineering problems describing concepts for handling and incorporating measurement data, either of model input parameters or of the system response. The first case study deals with polymorphic uncertain data based on computer tomographic scans with respect to air voids which are acquired, simplified and integrated in numerical models of adhesive bonds. In the second case study, the variation sensitivity analysis is presented to provide suitable prior knowledge for numerical soil analyses, for example, in order to reduce required input data. The uncertainty in friction processes is treated in case study 3 whereby measurement data are used in data driven methods to improve the numerical predictions. In case study 4, the failure behavior of die‐cast window hinges, which is affected by an uncertain initial pore distribution, is investigated by means of a Markov chain approach. In the last two case studies, mathematical methods of statistical inference and updating algorithms for uncertainty models are shown. Due to the heterogeneous spectrum of problems, a generalized strategy for data modeling, acquisition, and assimilation is developed and applied on each case study.
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.
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