Ulrich Römer

Quantification of Uncertainty in the Field Quality of Magnets Originating from Material Measurements

A challenge in accelerator magnet design is the strong nonlinear behavior due to magnetic saturation. In practice, the underlying nonlinear saturation curve is modeled according to measurement data that typically contain uncertainties. The electromagnetic fields and in particular the multipole coefficients that heavily affect the particle beam dynamics inherit this uncertainty. In this paper, we propose a stochastic model to describe the uncertainties and we demonstrate the use of generalized polynomial chaos for the uncertainty quantification of the multipole coefficients. In contrast to previous works we propose to start the stochastic analysis from uncertain measurement data instead of uncertain material properties and we propose to determine the sensitivities by a Sobol decomposition.

Approximation of Moments for the Nonlinear Magnetoquasistatic Problem With Material Uncertainties

In this paper, we study the magnetoquasistatic problem with uncertainties in the nonlinear magnetic material characteristic. In the case of small input uncertainties, adjoint techniques can be used to efficiently approximate the statistics of the quantities of interest. We carry out the corresponding sensitivity analysis and investigate the methods approximation properties. Numerical results discussing the approximation error are given for an electrical transformer.

High Dimensional Uncertainty Quantification for an Electrothermal Field Problem using Stochastic Collocation on Sparse Grids and Tensor Train Decomposition

Summary The temperature developed in bondwires of integrated circuits (ICs) is a possible source of malfunction and has to be taken into account during the design phase of an IC. Because of manufacturing tolerances, a bondwires geometrical characteristics are uncertain parameters, and as such, their impact has to be examined with the use of uncertainty quantification methods. Considering a stochastic electrothermal problem featuring 12 bondwire-related uncertainties, we want to quantify the impact of the uncertain inputs onto the temperature developed during the duty cycle of an IC. For this reason, we apply the stochastic collocation method on sparse grids, which is considered the current state-of-the-art. We also implement an approach based on the recently introduced low-rank tensor decompositions, in particular the tensor train decomposition, which in theory promises to break the curse of dimensionality. A comparison of both methods is presented, with respect to accuracy and computational effort.

Multilevel Monte Carlo simulation of the eddy current problem with random parameters

The multilevel Monte Carlo method is applied to an academic example in the field of electromagnetism. The method exhibits a reduced variance by assigning the samples to multiple models with a varying spatial resolution. For the given example it is found that the main costs of the method are spent on the coarsest level.

Low-Dimensional Stochastic Modeling of the Electrical Properties of Biological Tissues

Uncertainty quantification plays an important role in biomedical engineering as measurement data is often unavailable and literature data shows a wide variability. Using state of the art methods one encounters difficulties when the number of random inputs is large. This is the case, e.g., when using composite Cole-Cole equations to model random electrical properties. It is shown how the number of parameters can be significantly reduced by the Karhunen-Loève expansion.

Determination of bond wire failure probabilities in microelectronic packages

This work deals with the computation of industry-relevant bond wire failure probabilities in microelectronic packages. Under operating conditions, a package is subject to Joule heating that can lead to electrothermally induced failures. Manufacturing tolerances result, e.g., in uncertain bond wire geometries that often induce very small failure probabilities requiring a high number of Monte Carlo (MC) samples to be computed. Therefore, a hybrid MC sampling scheme that combines the use of an expensive computer model with a cheap surrogate is used. The fraction of surrogate evaluations is maximized using an iterative procedure, yielding accurate results at reduced cost. Moreover, the scheme is non-intrusive, i.e., existing code can be reused. The algorithm is used to compute the failure probability for an example package and the computational savings are assessed by performing a surrogate efficiency study.

Balancing modeling and discretization errors in the numerical approximation of magnetostatic fields with uncertainties

This work is concerned with error control in the numerical approximation of magnetic fields. It is shown how linearization and discretization error can be balanced with respect to a desired numerical accuracy, reflecting an uncertainty in the solution. The uncertainty itself is roughly estimated using a gradient based worst-case approach. The different error measures are illustrated using a numerical example.

Magnetoquasistatic Model and its Numerical Approximation

We proceed with a more mathematical and detailed investigation of the magnetoquasistatic model as presented in Chap. 2. After establishing a weak formulation, we consider approximation in space, linearization of the nonlinear model and discretization with respect to time, respectively.

Parametric Model, Continuity and First Order Sensitivity Analysis

The subject of this chapter is a detailed description of a parametric magnetoquasistatic model, generalizing the deterministic setting of Chap. 3. To this end we choose a continuous setting, i.e., parametrization is discussed on the differential equation level. Moreover, in a first step we allow for a general, possibly infinite dimensional parametrization, before discussing its finite dimensional approximation later on. Continuity and differentiability results will be established for different kind of inputs. In particular the sensitivity analysis presented in Sects. 4.4, 4.5 will be a key tool for propagating uncertainties in Chaps. 5 and 6.

Magnetoquasistatic Approximation of Maxwell’s Equations, Uncertainty Quantification Principles

Starting from the classical form of Maxwell’s equations the magnetoquasistatic approximation will be derived and justified. Additionally, some key notions from the area of uncertainty quantification, verification and validation will be established.