Natural preconditioners for saddle point systems

The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or discrete setting, so saddle point systems arising from discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as does, for example, the widely used sequential quadratic programming approach to nonlinear optimization.\ud This article concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee rapid convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness -- in terms of rapidity of convergence -- is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends

Computationally Efficient Steady--State Simulation Algorithms for Finite-Element Models of Electric Machines.

The finite element method is a powerful tool for analyzing the magnetic characteristics of electric machines, taking account of both complex geometry and nonlinear material properties. When efficiency is the main quantity of interest, loss calculations can be affected significantly due to the development of eddy currents as a result of Faraday’s law. These effects are captured by the periodic steady-state solution of the magnetic diffusion equation. A typical strategy for calculating this solution is to analyze an initial value problem over a time window of sufficient length so that the transient part of the solution becomes negligible. Unfortunately, because the time constants of electric machines are much smaller than their excitation period at peak power, the transient analysis strategy requires simulating the device over many periods to obtain an accurate steady-state solution. Two other categories of algorithms exist for directly calculating the steady-state solution of the magnetic diffusion equation; shooting methods and the harmonic balance method. Shooting methods search for the steady-state solution by solving a periodic boundary value problem. These methods have only been investigated using first order numerical integration techniques. The harmonic balance method is a Fourier spectral method applied in the time dimension. The standard iterative procedures used for the harmonic balance method do not work well for electric machine simulations due to the rotational motion of the rotor. This dissertation proposes several modifications of these steady-state algorithms which improve their overall performance. First, we demonstrate how shooting methods may be implemented efficiently using Runge-Kutta numerical integration methods with mild coefficient restrictions. Second, we develop a preconditioning strategy for the harmonic balance equations which is robust against large time constants, strong nonlinearities, and rotational motion. Third, we present an adaptive framework for refining the solutions based on a local error criterion which further reduces simulation time. Finally, we compare the performance of the algorithms on a practical model problem. This comparison demonstrates the superiority of the improved steady-state analysis methods, and the harmonic balance method in particular, over transient analysis.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113322/1/pries_1.pd

3D simulation of magneto-mechanical coupling in MRI scanners using high order FEM and POD

Magnetic Resonance Imaging (MRI) scanners have become an essential tool in the medi-cal industry due to their ability to produce high resolution images of the human body. To generate an image of the body, MRI scanners combine strong static magnetic fields with transient gradient magnetic fields. The interaction of these magnetic fields with the con-ducting components present in superconducting MRI scanners gives rise to an important problem in the design of new MRI scanners. The transient magnetic fields give rise to the appearance of eddy currents in conducting components. These eddy currents, in turn, result in electromagnetic stresses, which cause the conducting components to deform and vibrate. The vibrations are undesirable as they lead to a deterioration in image quality (with image artefacts) and to the generation of noise, which can cause patient discomfort. The eddy currents, in addition, lead to heat being dissipated and deposited into the cryo-stat, which is filled with helium in order to maintain the coils in a superconducting state. This deposition of heat can cause helium boil off and potentially result in a costly magnet quench. Understanding the mechanisms involved in the generation of these vibrations and the heat being deposited into the cryostat are, therefore, key for a successful MRI scanner design. This involves the solution of a coupled magneto-mechanical problem, which is the focus of this work.In this thesis, a new computational methodology for the solution of three-dimensional (3D) magneto-mechanical coupled problems with application to MRI scanner design is presented. To achieve this, first an accurate mathematical description of the magneto-mechanical coupling is presented, which is based on a Lagrangian formulation and the assumption of small displacements. Then, the problem is linearised using an AC-DC splitting of the fields, and a variational formulation for the solution of the linearised prob-lem in a time-harmonic setting is presented. The problem is then discretised using high order finite elements, where a combination of hierarchical H1 and H(curl) basis func-tions is used. An efficient staggered algorithm for the solution of the coupled system is proposed, which combines the DC and AC stages and makes use of preconditioned iter-ative solvers when appropriate. This finite element methodology is then applied to a set of challenging academic and industrially relevant problems in order to demonstrate its accuracy and efficiency.This finite element methodology results in the accurate and efficient solution of the magneto-mechanical problem of interest. However, in the design stage of a new MRI scanner, this coupled problem must be solved repeatedly for varying model parameters such as frequency or material properties. Thus, even if an efficient finite element solver is available for the solution of the coupled problem, the need for these repeated simulations result in a bottleneck in terms of computational cost, which leads to an increase in design time and its associated financial implications. Therefore, in order to optimise this process, the application of Reduced Order Modelling (ROM) techniques is considered. A ROM based on the Proper Orthogonal Decomposition (POD) method is presented and applied to a series of challenging MRI configurations. The accuracy and efficiency of this ROM is demonstrated by performing comparisons against the full order or high fidelity finite element software, showing great performance in terms of computational speed-up, which has major benefits in the optimisation of the design process of new MRI scanners

Improving Condition and Sensitivity of Linear Inverse Problems in Magnetic Applications

Die Identifikation nicht direkt zugänglicher Prozesse anhand gemessener Daten ist von großer Bedeutung in vielen Bereichen. Im Fokus dieser Arbeit liegen Applikationen in der Magnetostatik, Magnetokardiographie und Magnetinduktionstomographie. Ein Ansatz zur Identifikation besteht in der Lösung eines entsprechenden linear inversen Problems. Unglücklicherweise haben in den Daten enthaltene Fehler und Rauschen einen signifikanten Einfluss auf die inverse Lösung. Ziel dieser Arbeit ist die Reduktion der Einflüsse von Fehlern und Rauschen durch eine Verbesserung der Kondition des Problems, sowie eine Steigerung der Sensitivität der Messanordnungen. Zur Bestimmung der Kondition wird das Verhältnis des größten und mittleren Singulärwerts der Kernmatrix als neues Maß vorgeschlagen. Darüber hinaus werden Ansätze zur Analyse der Sensitivität hinsichtlich der Messung elektromagnetischer Quellen und der Erfassung elektrischer Leitfähigkeitsveränderungen präsentiert.Strategien zur Verbesserung von Kondition und Sensitivität werden in vier Simulationsstudien beschrieben. In der ersten Studie wird ein Tabu-Suche-Ansatz zur Optimierung der Anordnung magnetischer Sensoren vorgestellt. Anordnungen mit optimierte Sensorpositionen resultieren dabei in einer deutlich besseren Kondition als regelmäßige Anordnungen. In einer zweiten Studie werden Parameter adaptiert,welche den Quellenraum für die Bildgebung durch magnetische Nanopartikel definieren. Als eine Schlussfolgerung sollte der Quellenraum etwas größer als das Sensorareal definiert werden. Diese Arbeit zeigt ebenfalls, dass Variationen in den Sensorrichtungen für monoaxiale Sensorarrays zu einer Verbesserung der Kondition führen. Zudem wird die Sensitivität von Spulenanordnungen für die Magnetinduktionstomographie bewertet und verglichen. Durch Nutzung relativ großer Spulen, die das Messgebiet nahezu vollständig abdecken, können Kondition und Sensitivität wesentlich verbessert werden.Die präsentierten Methoden und Strategien ermöglichen eine substantielle Verbesserung der Kondition des linear inversen Problems bei der Analyse magnetischer Messungen. Insbesondere die Anordnung von Sensoren in Bezug auf das Messobjekt ist kritisch für die Kondition, sowie die Qualität inverser Lösungen. Die vorgestellten Methoden sind darüber hinaus für linear inverse Probleme in zahlreichen Bereichen einsetzbar.The identification and reconstruction of hidden, not directly accessible processes from measured data is important in many areas of research and engineering. This thesis focusses on applications in magnetostatics, magnetocardiography, and magneticinduction tomography. One approach to identify these processes is to solve a related linear inverse problem. Unfortunately, noise and errors in the data have a significant impact on inverse solutions.The aim of this work is to reduce the effects of noise and errors by improving the condition of the problem and to increase the sensitivity of measurement setups. To quantify the condition, we propose the ratio of the largest and the mean singular value of the kernel matrix. Moreover, we outline approaches to analyse quantitatively and qualitatively the sensitivity to electromagnetic sources and electrical conductivity changes.In four simulation studies, strategies to improve the condition and sensitivity inmagnetic applications are described. First, we present a tabu search algorithm to optimize arrangements of magnetic sensors. Optimized sensor arrays result in a considerably improved condition compared with regular arrangements. Second, we adapt parameters that define source space grids for magnetic nanoparticle imaging. One conclusion is that the source space should be defined slightly larger than the sensor area. Third, we demonstrate for mono-axial sensor arrays that variations in thesensor directions and small variations in the sensor positions lead to improvements of the condition, too. Finally, we evaluate and compare the sensitivities of six coil setups for magnetic induction tomography. Our investigations indicate a rapid decay of sensitivity by several orders of magnitude within a range of a few centimetres. By using relatively large coils that cover the measurement region almost completely, the condition and sensitivity can be improved clearly.The methods and strategies presented in this thesis facilitate substantial improvements of the condition for linear inverse problems in magnetic applications. In particular, the arrangement of sensors relative to the measurement object is critical to the condition and to the quality of inverse solutions. Moreover, the presented methods are applicable to linear inverse problems in various fields

Accelerating induction machine finite-element simulation with parallel processing

Finite element analysis used for detailed electromagnetic analysis and design of electric machines is computationally intensive. A means of accelerating two-dimensional transient finite element analysis, required for induction machine modeling, is explored using graphical processing units (GPUs) for parallel processing. The graphical processing units, widely used for image processing, can provide faster computation times than CPUs alone due to the thousands of small processors that comprise the GPUs. Computations that are suitable for parallel processing using GPUs are calculations that can be decomposed into subsections that are independent and can be computed in parallel and reassembled. The steps and components of the transient finite element simulation are analyzed to determine if using GPUs for calculations can speed up the simulation. The dominant steps of the finite element simulation are preconditioner formation, computation of the sparse iterative solution, and matrix-vector multiplication for magnetic flux density calculation. Due to the sparsity of the finite element problem, GPU-implementation of the sparse iterative solution did not result in faster computation times. The dominant speed-up achieved using the GPUs resulted from matrix-vector multiplication. Simulation results for a benchmark nonlinear magnetic material transient eddy current problem and linear magnetic material transient linear induction machine problem are presented. The finite element analysis program is implemented with MATLAB R2014a to compare sparse matrix format computations to readily available GPU matrix and vector formats and Compute Unified Device Architecture (CUDA) functions linked to MATLAB. Overall speed-up achieved for the simulations resulted in 1.2-3.5 times faster computation of the finite element solution using a hybrid CPU/GPU implementation over the CPU-only implementation. The variation in speed-up is dependent on the sparsity and number of unknowns of the problem