Partial differential equations are a versatile tool to model and eventually simulate physical phenomena. An important aspect in view of the reliability and relevance of such simulations are uncertainties arising from unknown parameters and measurement errors. In particular, the modelling and discretization of uncertainties of the computational domain requires special care. Such uncertainties emerge in a natural fashion when considering products fabricated by line production which are subject to manufacturing tolerances or shapes which are obtained by remote sensing techniques, like e.g. ultrasound or magnetic resonance imaging. This minisymposium is dedicated to recent developments in the numerical treatment of shape uncertainties in partial differential equations and welcomes contributions addressing analytical aspects, forward modelling, assimilation of measurement data, optimization, and applications.
16:30
Numerical aspects of the domain mapping method for elliptic PDEs on random domains
Michael Multerer | Università della Svizzera italiana | Switzerland
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Michael Multerer | Università della Svizzera italiana | Switzerland
Data uncertainties arise in a natural fashion in applications where model parameters are obtained from measurements and estimates. Being able to quantify such uncertainties can greatly improve the relevance and reliability of model predictions and moreover provide valuable insights into statistical properties of quantities of interest. In this talk, we address different numerical aspects of uncertainty quantification for elliptic partial differential equations on random domains. Starting from the modeling of random domains via random deformation fields, we discuss how the corresponding Karhunen-Loève expansion can efficiently be computed. Afterwards, we recapitulate available regularity results, which facilitate the use of higher order quadrature methods for the computation of quantities of interest. Finally, we present some numerical results to
illustrate the presented approach.
17:00
Adaptive Stochastic Galerkin FEM for randomly perturbed domains
Martin Eigel | WIAS Berlin | Germany
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Martin Eigel | WIAS Berlin | Germany
A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field affinely depending on a countable number of random variables. The examined high-dimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, we describe an a posteriori adaptive algorithm in efficient tensor formats which is based on a recently developed residual error estimator for the lognormal elliptic problem.
17:30
Yield Optimization with Adaptive Newton-MC
Mona Fuhrländer | TU Darmstadt | Germany
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Mona Fuhrländer | TU Darmstadt | Germany
Sebastian Schöps | TU Darmstadt | Germany
Uncertainties in the manufacturing process of electromagnetic components may lead to deviations in the geometry or material parameters. This may lead to rejections due to malfunctioning of the device. We quantify the impact of uncertainty and optimize the design to minimize rejections. Simulations (finite element method) are used to build the model, based on partial differential equations (PDE). Its uncertain geometry and material parameters can be modeled as random variables. Then, the probability that one realization in a manufacturing process fulfills all predefined performance requirements is called the yield. For yield estimation, we use a hybrid approach combining the efficiency of stochastic collocation and the accuracy of classical Monte Carlo (MC) analysis. To maximize the yield, which is equivalent to minimizing the failure probability, we present an adaptive Newton-MC approach. This balances the accuracy and the computational effort by adjusting the size of the MC sampling adaptively. As performance requirements, we consider restrictions involving the PDEs describing the electromagnetic field, i.e., Maxwell’s equations in
frequency domain.
This work is supported by the ’Excellence Initiative’ of the German Federal and State Governments
and the Graduate School of Computational Engineering at TU Darmstadt.
18:00
Risk Averse Design of Tall Buildings Under Uncertain Wind Loading
Brendan Keith | TU München | Germany
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Brendan Keith | TU München | Germany
Anoop Kodakkal | TU München | Germany
Andreas Apostolatos | TU München | Germany
Simon Urbainczyk | TU München | Germany
Roland Wüchner | TU München | Germany
Barbara Wohlmuth | TU München | Germany
Kai-Uwe Bletzinger | TU München | Germany
Design decisions for tall buildings require a careful focus on wind loading, largely due to the inherently uncertain nature of the local atmospheric boundary layer. In industry, extreme values of top floor acceleration, top floor displacement, and base reactions are the principal Quantities of Interest (QoI) for designers and other decision makers. In this talk, demands for tall building designs are taken from common industrial practice and put into a mathematical setting using recently proposed principles from risk averse stochastic PDE-constrained optimization. In particular, the Buffered Failure Probability is used to construct chance constraints on various representative design spaces, thereby implicitly characterizing the largest feasible set in which to seek a cost-effective design. Such a mathematical formulation is found to be more promising to deliver the requirements of human comfort in tall buildings (i.e., serviceability criteria) when compared to more conventional approaches which involve only optimization with respect to the mean and/or RMS value of the predicted acceleration. Shape modification of several common global design features, as well as two specific local features - namely, rounded corners and recessed corners - are explored. The QoI mentioned above are also used as performance measures to compare amongst the various optimal designs, depending on the particular use case and engineering specifications.