<p> This thesis contributes to the development of adjoint variable methods (AVM) and space mapping (SM) technology for computer-aided electromagnetics (EM)-based modeling and design of microwave circuits and antennas.</p> <p> The AVM is known as an efficient approach to design sensitivity analysis for problems of high complexity. We propose a general self-adjoint approach to the sensitivity analysis of network parameters for an Method of Moments (MoM) solver. It requires neither an adjoint problem nor analytical system matrix derivatives. For the first time, we suggest practical and fast sensitivity solutions realized entirely outside the EM solver, which simplifies the implementation. We discuss: (1) features of commercial EM solvers which allow the user to compute network parameters and their sensitivities through a single full-wave simulation; (2) the accuracy of the computed derivatives; (3) the overhead of the sensitivity computation. Our approach is demonstrated by FEKO, which employs an MoM solver.</p> <p> One motivation for sensitivity analysis is gradient-based optimization. The sensitivity evaluation providing the Jacobian is a bottleneck of optimization with full-wave simulators. We propose an approach, which employs the self-adjoint sensitivity analysis of network parameters and Broyden's update for practical EM design optimization. The Broyden's update is carried out at the system matrix level, so that the computational overhead of the Jacobian is negligible while the accuracy is acceptable for optimization. To improve the robustness of the Broyden update in the sensitivity analysis, we propose a switching criterion between the Broyden and the finite-difference estimation of the system matrix derivatives.</p> <p> In the second part, we apply for the first time a space mapping technique to antenna design. We exploit a coarse mesh MoM solver as the coarse model and align it with the fine mesh MoM solution through space mapping. Two SM plans are employed: I. implicit SM and output SM, and II. input SM and output SM. A novel local meshing method is proposed to avoid inconsistencies in the coarse model. The proposed techniques are implemented through the new user-friendly SMF system. In a double annular ring antenna example, the S-parameter is optimized. The finite ground size effect for the MoM is efficiently solved by SM Plan I and the design specification is satisfied after only three iterations. In a patch antenna example, we optimize the impedance through both plans. Comparisons are made. Coarseness in the coarse model and its effect on the SM performance is also discussed.</p> / Thesis / Master of Applied Science (MASc)

<p> The thesis presents an efficient self-adjoint approach to the S-parameter sensitivity analysis based on full-wave electromagnetic (EM) time-domain simulations with two commonly used numerical techniques: the finite-difference time-domain (FDTD) method and the transmission-line matrix (TLM) method. Without any additional simulations, we extract the response gradient with respect to all the design variables making use of the full-wave solution already generated by the system analysis. It allows the computation of the S-parameter derivatives as an independent post-process with negligible overhead. The sole requirement is the ability of the solver to export the field solution at user-defined points. Most in-house and commercial solvers have this ability, which makes our approach readily applicable to practical design problems.</p> <p> In the TLM-based self-adjoint techniques, we propose an algorithm to convert the electrical and magnetic field solutions into TLM voltages. The TLM-based discrete adjoint variable method (AVM) is originally developed to use incident and reflected voltages as the state variables. Our conversion algorithm makes the TLM-AVM method applicable to all time-domain commercial solvers, FDTD simulators included, with comparable accuracy and less memory overhead. Our approach is illustrated through waveguide examples using a TLM-based commercial simulator.</p> <p> Currently, our TLM-based self-adjoint approach is limited to loss-free homogeneous problems. However, our FDTD-based self-adjoint approach is valid for lossy inhomogeneous cases as well. The FDTD-based self-adjoint technique needs only the E-field values as the state variables. In order to make it also applicable to a TLM-based solver, whose mesh grid is displaced from the FDTD grid, we interpolate the E-field solution from the TLM mesh to that on the FDTD mesh. Our FDTD-based approach is validated through the response derivatives computation with respect to both shape and constitutive parameters in waveguide and antenna structures. The response derivatives can be used not only to guide a gradient-based optimizer, but also to provide a sufficient good initial guess for the solution of nonlinear inverse problems.</p> <p> Suggestions for further research are provided.</p> / Thesis / Master of Applied Science (MASc)

<p> This thesis proposes a sensitivity solver for frequency-domain electromagnetic (EM) simulators based on volume methods such as the finite-element method (FEM). The proposed sensitivity solver computes S-parameter Jacobians directly from the field solutions available from the EM simulation. It exploits the computational efficiency of the self-adjoint sensitivity analysis (SASA) approach where only one EM simulation suffices to obtain both the responses and their gradients in the designable parameter space. The proposed sensitivity solver adopts the system equations of the finite-difference frequency-domain (FDFD) method.</p> <p> There are three major advantages to this development: (1) the Jacobian computation is completely independent of the simulation engine, its grid and its system equations; (2) the implementation is straightforward and in the form of a post-processing algorithm operating on the exported field solutions; and (3) it is computationally very efficient-time requirements are negligible in comparison with conventional field-based optimization procedures utilizing Jacobians computed via response-level finite differences or parameter sweeps.</p> <p> The accuracy and the efficiency of the proposed sensitivity solver are verified in the sensitivity analysis and the gradient-based optimization of filters and antennas. Compared to the finite-difference approximation, drastic reduction of the time required by the overall optimization process is achieved. All examples use a commercial finite-element simulator.</p> <p> Suggestions for future research are provided.</p> / Thesis / Master of Applied Science (MASc)

This study has two parts. In the first part we develop a computational approach to the solution of an inverse modelling problem concerning the material properties of electrolytes used in Lithium-ion batteries. The dependence of the diffusion coefficient and the transference number on the concentration of Lithium ions is reconstructed based on the concentration data obtained from an in-situ NMR imaging experiment. This experiment is modelled by a system of 1D time-dependent Partial Differential Equations (PDE) describing the evolution of the concentration of Lithium ions with prescribed initial concentration and fluxes at the boundary. The material properties that appear in this model are reconstructed by solving a variational optimization problem in which the least-square error between the experimental and simulated concentration values is minimized. The uncertainty of the reconstruction is characterized by assuming that the material properties are random variables and their probability distribution estimated using a novel combination of Monte-Carlo approach and Bayesian statistics. In the second part of this study, we carefully analyze a number of secondary effects such as ion pairing and dendrite growth that may influence the estimation of the material properties and develop mathematical models to include these effects. We then use reconstructions of material properties based on inverse modelling along with their uncertainty estimates as a framework to validate or invalidate the models. The significance of certain secondary effects is assessed based on the influence they have on the reconstructed material properties. / Thesis / Doctor of Philosophy (PhD)

High-accuracy electromagnetic design and analysis of electric machines is enhanced by the use of various numerical methods. These methods solve Maxwell’s equations to determine the distribution of the electric and magnetic fields throughout the considered machine structure. Due to the complicated architectures of the machines and the nonlinearity of the utilized magnetic materials, it is a very challenging task to obtain an analytical solution and, in most cases, only a numerical solution is possible. The finite element method (FEM) is one of the standard numerical methods for electromagnetic field analysis. The considered machine domain is divided into finite elements to which the field equations are applied. FEM solvers are utilized to develop optimization procedures to assist in achieving a design that meets the required specifications without violating the design constraints. The design process of electric machines involves adjusting the machine parameters. This is usually done through experience, intuition, and heuristic approaches using FEM software which gives results for various parameter changes. There is no guarantee that the achieved design is the optimal one. An alternative approach to the design of electric machines exploits robust gradient-based optimization algorithms that are guaranteed to converge to a locally-optimal model. The gradient-based approaches utilize the sensitivities of the performance characteristics with respect to the design parameters. These sensitivities are classically calculated using finite difference approximations which require repeated simulations with perturbed parameter values. The cost of evaluating these sensitivities can be significant for a slow finite element simulation or when the number of parameters is large. The adjoint variable method (AVM) offers an alternative approach for efficiently estimating response sensitivities. Using at most one extra not-iterative simulation, the sensitivities of the response to all parameters are estimated. Here, a MATLAB tool has been developed to automate the design process of switched reluctance motors (SRMs). The tool extracts the mesh data of an initial motor model from a commercial FEM software, JMAG. It then solves for magnetic vector potential throughout the considered SRM domain using FEM taking into consideration the nonlinearity of the magnetic material and the motor dynamic performance. The tool calculates various electromagnetic quantities such as electromagnetic torque, torque ripple, phase flux linkage, x and y components of flux density, air-region stored magnetic energy, phase voltage, and phase dynamic currents. The tool uses a structural mapping technique to parametrize various design parameters of SRMs. These parameters are back iron thickness, teeth height, pole arc angle, and pole taper angle of both stator and rotor. Moreover, it calculates the sensitivities of various electromagnetic quantities with respect to all these geometric design parameters in addition to the number of turn per phase using the AVM method. The tool applies interior point optimization algorithm to simultaneously optimize the motor geometry, number of turns per phase, and the drive-circuit control parameters (reference current, and turn-on and turn-off angles) to increase the motor average dynamic torque. It also applies the ON/OFF topology optimization algorithm to optimize the geometries of the stator teeth for proper distribution of the magnetic material to reduce the RMS torque ripple. A 6/14 SRM has been automatically designed using the developed MATLAB tool to achieve the same performance specifications as 6110E Evergreen surface-mounted PM brushless DC motor which is commercially available for an HVAC system. / Thesis / Doctor of Philosophy (PhD)

This thesis deals with non-linear non-quadratic optimal control problems in an autonomous system and a related iterative numerical method, the Kleinman-Newton method, for solving the problem. The thesis proves the local convergence of Kleinman-Newton method using the contraction mapping theorem and then describes how this Kleinman-Newton method may be used to numerically solve for the optimal control and the corresponding solution. In order to show the proof and the related numerical work, it is necessary to review some of earlier work in the beginning of Chapter 1 [Zhang], and to introduce the Kleinman-Newton method at the end of the chapter. In Chapter 2 we will demonstrate the proof. In Chapter 3 we will show the related numerical work and results. / Ph. D.

A time spectral and coupled adjoint based sensitivity analysis of rotor blade is carried out in this study. The time spectral method is an efficient technique to solve unsteady periodic problems by transforming unsteady equation of motion to a steady state one. Due to the availability of the governing equations in the steady form, the steady form of the adjoint equations can be applied for the sensitivity analysis of the coupled fluid-structure system. An expensive computational time and memory requirement for the unsteady adjoint sensitivity analysis is thus avoided. A coupled analysis of fluid, structural, and flight dynamics is carried out through a CFD/CSD/CA coupling procedure that combines FSI analysis with enforced trim condition. Coupled sensitivity analysis results and their validations are presented and compared with aerodynamics only sensitivity analysis results. The fluid-structure coupled adjoint based sensitivity analysis will be applied to the shape optimization of a rotor blade in the future work. Minimization of required power is the objective of the optimization problem with constraints on thrust and drag of the rotor. The bump functions are considered as the design variables. Rotor blade shape changes are obtained by using the bump function on the surface of the airfoil sections along the span. / Doctor of Philosophy / The work in this dissertation is motivated by the reducing the computational cost at the early design stage with guaranteed accuracy. In the research, the author proposes that the goal can be achieve through coupled adjoint based sensitivity analysis using a fluid structure interaction in time spectral form. Adjoint based sensitivity analysis is very efficient for solving design problems with a large number of design variables. The time spectral approach is used to overcome inefficient calculation of rotor flows by expressing flow and structural state variables as Fourier series with small number of harmonics. The accuracy and the efficiency of flow solver are examined by simulating UH-60A forward flight condition. A significant reduction in the computational cost is achieved by its Fourier series form of the periodic time response and the assumption of periodic steady state. A good agreement between time accurate and time spectral analysis is noted for the high speed forward flight condition of UH-60A configuration. Prediction from both methods also agree quite well with the experimental data. The adjoint based sensitivity analysis results are compared with the finite difference sensitivity analysis results. Even with presence of small discrepancies, these two results show a good agreement to each other. Coupled sensitivity analysis includes not only the effect of fluid state changes but also the contribution of structural deformation.

Numerical solutions obtained through the use of Computational Fluid Dynamics (CFD) are subject to discretization error, which is locally generated by truncation error. The discretization error is extremely difficult to properly estimate and this in turn leads to uncertainty over the quality of the numerical solutions obtained via CFD methods and the engineering functionals computed using these solutions. Adjoint error estimation techniques specifically seek to estimate the error in functionals, but are dependent upon accurate truncation error estimates. This work examines the application of new, single-grid, truncation error estimation procedures to the problem of adjoint error estimation for both the quasi-1D and 2D Euler equations. The new truncation error estimation techniques are based on local reconstructions of the computed solutions and comparisons are made for the quasi-1D study in order to determine the most appropriate solution variables to reconstruct as well as the most appropriate reconstruction method. In addition, comparisons are made between the single-grid truncation error estimates and methods based on uniformally refining or coarsening the underlying numerical mesh on which the computed solutions are obtained. A method based on an refined grid error estimate is shown to work well for a non-isentropic flow for the quasi-1D Euler equations, but all truncation error estimations methods ultimately result in over prediction of functional discretization error in the presence of a shock in 2D. Alternatives to adjoint methods, which can only estimate the error in a single functional for each adjoint solution obtained, are examined for the 2D Euler equations. The defection correction method and error transport equations are capable of locally improving the entire computed solution, allowing for error estimates in multiple functionals. It is found that all three functional discretization error estimates perform similarly for the same truncation error estimate, although the defect correction method is the most costly from a computational viewpoint. Comparisons are made between truncation error and adjoint weighted truncation error based adaptive indicators. For the quasi-1D Euler equations it is found that both methods are competitive, however the truncation error based method is cheaper as a separate adjoint solve is avoided. For the 2D Euler equations, the truncation error estimates on the adapted meshes suffer due to a lack of smooth grid transformations which are used in reconstructing the computed solutions. In order to complete this work, a new CFD code incorporating a variety of best practices from the field of Computer Science is developed as well as a new method of performing code verification using the method of manufactured solutions which is significantly easier to implement than traditional manufactured solution techniques. / Ph. D.

Computational fluid dynamics (CFD) simulations can provide tremendous insight into complex physical processes and are often faster and more cost-effective to execute than experiments. However, each CFD result inherently contains numerical errors that can significantly degrade the accuracy of a simulation. Discretization error is typically the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since the generation, transport, and diffusion of these errors is a highly nonlinear function of the computational grid and discretization scheme. As CFD is increasingly used in engineering design and analysis, it is imperative that CFD practitioners be able to accurately quantify discretization errors to minimize risk and improve the performance of engineering systems. In this work, improvements are made to the accuracy and efficiency of existing error estimation techniques. Discretization error is estimated by deriving and solving an error transport equation (ETE) for the local discretization error everywhere in the computational domain. Truncation error is shown to act as the local source for discretization error in numerical solutions. An equivalence between adjoint methods and ETE methods for functional error estimation is presented. This adjoint/ETE equivalence is exploited to efficiently obtain error estimates for multiple output functionals and to extend the higher-order properties of adjoint methods to ETE methods. Higher-order discretization error estimates are obtained when truncation error estimates are sufficiently accurate. Truncation error estimates are demonstrated to deteriorate on grids with a non-smooth variation in grid metrics (e.g., unstructured grids) regardless of how smooth the underlying exact solution may be. The loss of accuracy is shown to stem from noise in the discrete solution on the order of discretization error. When using conventional least-squares reconstruction techniques, this noise is exactly captured and introduces a lower-order error into the truncation error estimate. A novel reconstruction method based on polyharmonic smoothing splines is developed to smoothly reconstruct the discrete solution and improve the accuracy of error estimates. Furthermore, a method for iteratively improving discretization error estimates is devised. Efficiency and robustness considerations are discussed. Results are presented for several inviscid and viscous flow problems. To facilitate the study of discretization error estimation, a new, higher-order finite-volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design. / Ph. D. / Computational fluid dynamics (CFD) is a branch of computational physics at the intersection of fluid mechanics and scientific computing in which the governing equations of fluid motion, such as the Euler and Navier-Stokes equations, are solved numerically on a computer. CFD is utilized in numerous fields including biomedical engineering, meteorology, oceanography, and aerospace engineering. CFD simulations can provide tremendous insight into physical processes and are often preferred over experiments because they can be performed more quickly, are typically more cost-effective, and can provide data in regions where it may be difficult to measure. While CFD can be an extremely powerful tool, CFD simulations are inherently subject to numerical errors. These errors, which are generated when the governing equations of fluid motion are solved on a computer, can have a significant impact on the accuracy of a CFD solution. If numerical errors are not accurately quantified, ill-informed decision-making can lead to poor system performance, increased risk of injury, or even system failure. In this work, research efforts are focused on numerical error estimation for the finite -volume method, arguably the most widely used numerical algorithm for solving CFD problems. The error estimation techniques provided herein target discretization error, the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since these errors are generated, convected, and diffused by the same physical processes embedded in the governing equations. In this work, improvements are made to the accuracy and efficiency of existing discretization error estimation techniques. Results are presented for several inviscid and viscous flow problems. To facilitate the study of these error estimators, a new, higher-order finite -volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design.

Actuellement, le principal obstacle à la mise en œuvre de la FWI élastique en trois dimensions sur des cas d'étude réalistes réside dans le coût de calcul associé aux taches de modélisation sismique. Pour surmonter cette difficulté, je propose deux contributions. Tout d'abord, je propose de calculer le gradient de la fonctionnelle avec la méthode de l'état adjoint à partir d'une forme symétrisée des équations de l'élastodynamique formulées sous forme d'un système du premier ordre en vitesse-contrainte. Cette formulation auto-adjointe des équations de l'élastodynamique permet de calculer les champs incidents et adjoints intervenant dans l'expression du gradient avec un seul opérateur de modélisation numérique. Le gradient ainsi calculé facilite également l'interfaçage de plusieurs outils de modélisation avec l'algorithme d'inversion. Deuxièmement, j'explore dans cette thèse dans quelle mesure les encodages des sources avec des algorithmes d'optimisation du second-ordre de quasi-Newton et de Newton tronqué permettait de réduire encore le coût de la FWI. Finalement, le problème d'optimisation associé à la FWI est mal posé, nécessitant ainsi d'ajouter des contraintes de régularisation à la fonctionnelle à minimiser. Je montre ici comment une régularisation fondée sur la variation totale du modèle fournissait une représentation adéquate des modèles du sous-sol en préservant le caractère discontinu des interfaces lithologiques. Pour améliorer les images du sous-sol, je propose un algorithme de débruitage fondé sur une variation totale locale au sein duquel j'incorpore l'information structurale fournie par une image migrée pour préserver les structures de faible dimension. / Currently, the main limitation to perform 3D elastic full waveform inversion on a production level is the computational cost it represents. With this in mind, we provide two contributions. First, we develop a self adjoint formulation of the isotropic first order velocity-stress elastic equations that allow to implement only one forward modeling operator in the gradient computation. Second, we combine Newton and quasi-Newton optimization methods with source encoding techniques to see to what extent the computational cost could be further reduced. Finally, the optimization process associated to FWI is ill posed and requires regularization constraints. I show that the total variation of the model as a regularization term provides and adequate description of earth models, preserving the discontinuous character of the lithological layers. To improve the quality of the images, we propose a local total variation denoising algorithm based on the incorporation of the information provided by a migrated image.