It is well known that the finite difference time domain (FDTD) method is a powerful numerical analysis tool for solving electromagnetic problems. In a simulated area, in order to discretize an object which is much smaller than the others, a very small space increment is needed and hence the time step should be decreased too for stability consideration in traditional FDTD. The small space and time increments will respectively increase the memory requirement and calculation time. To overcome these problems, some numerical methods were developed, such as the subcell and nonuniform grid method, to handle the small feature size. This thesis describes an efficient method for generating FDTD subcell equations. We construct a second order macromodel system instead of the subcell region in conventional FDTD. The macromodel system can be reduced with model order reduction techniques (MOR) and then translated into new FDTD update equations. When the problem contains several objects of the same size and material properties, the MOR subcell has the advantage of reusability. This means that the reduce-order model of the object needs to be generated only once nonetheless can be applied to every position where the objects originally occupied.

This thesis concerns itself with a finite element model of nonproportional viscoelastic damping and its subsequent reduction. The Golla-Hughes-McTavish viscoelastic finite element has been shown to be an effective tool in modeling viscoelastic damping. Unlike previous models, it incorporates physical data into the model in the form of a curve fit of the complex modulus. This curve fit is expressed by minioscillators. The frequency dependence of the complex modulus is accounted for by the addition of internal, or dissipation, coordinates. The dissipation coordinates make the viscoelastic model several times larger than the original. The trade off for more accurate modeling of viscoelasticity is increased model size. Internally balanced model order reduction reduces the order of a state space model by considering the controllability/observability of each state. By definition, a model is internally balanced if its controllability and observability grammians are equal and diagonal. The grammians serve as a ranking of the controllability/observability of the states. The system can then be partitioned into most and least controllable/observable states; the latter can be statically reduced out of the system. The resulting model is smaller, but the transformed coordinates bear little resemblance to the original coordinates. A transformation matrix exists that transforms the reduced model back into original coordinates, and it is a subset of the transformation matrix leading to the balanced model. This whole procedure will be referred to as Yae's method within this thesis. By combining GHM and Yae's method, a finite element code results that models nonproportional viscoelastic damping of a clamped-free, homogeneous, Euler-Bernoulli beam, and is of a size comparable to the original elastic finite element model. The modal data before reduction compares well with published GHM results, and the modal data from the reduced model compares well with both. The error between the impulse response before and after reduction is negligible. The limitation of the code is that it cannot model sandwich beam behavior because it is based on Euler-Bernoulli beam theory; it can, however, model a purely viscoelastic beam. The same method, though, can be applied to more sophisticated beam models. Inaccurate results occur when modes with frequencies beyond the range covered by the curve fit appear in the model, or when poor data are used. For good data, and within the range modeled by the curve fit, the code gives accurate modal data and good impulse response predictions. / Master of Science

Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD. / Master of Science / The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

In this paper, a new methodology for predicting fluid free surface shape using Model Order Reduction (MOR) is presented. Proper Orthogonal Decomposition combined with a linear interpolation procedure for its coefficient is applied to a problem involving bubble dynamics near to a free surface. A model is developed to accurately and efficiently capture the variation of the free surface shape with different bubble parameters. In addition, a systematic approach is developed within the MOR framework to find the best initial locations and pressures for a set of bubbles beneath the quiescent free surface such that the resultant free surface attained is close to a desired shape. Predictions of the free surface in two-dimensions and three-dimensions are presented. / Singapore-MIT Alliance (SMA)

To ensure the robustness of an integrated circuit design, its power distribution network (PDN) must be validated beforehand against any voltage drop on VDD nets. However, due to the increasing size of PDNs, it is becoming difficult to verify them in a reasonable amount of time. Lately, much work has been done to develop Model Order Reduction (MOR) techniques to reduce the size of power grids but their focus is more on simulation. In verification, we are concerned about the safety of nodes, including the ones which have been eliminated in the reduction process. This work proposes a novel approach to systematically reduce the power grid and accurately compute an upper bound on the voltage drops at power grid nodes which are retained. Furthermore, a criterion for the safety of nodes which are removed is established based on the safety of other nearby nodes and a user specified margin.

To ensure the robustness of an integrated circuit design, its power distribution network (PDN) must be validated beforehand against any voltage drop on VDD nets. However, due to the increasing size of PDNs, it is becoming difficult to verify them in a reasonable amount of time. Lately, much work has been done to develop Model Order Reduction (MOR) techniques to reduce the size of power grids but their focus is more on simulation. In verification, we are concerned about the safety of nodes, including the ones which have been eliminated in the reduction process. This work proposes a novel approach to systematically reduce the power grid and accurately compute an upper bound on the voltage drops at power grid nodes which are retained. Furthermore, a criterion for the safety of nodes which are removed is established based on the safety of other nearby nodes and a user specified margin.

Nowadays the highest device integration affects the design process in several ways. The process variations (PV) significantly impact the circuit performance. As a consequence, a major consideration is determining the relation of the production yield to the technology based manufacturing variations. The traditional Monte Carlo based sampling analysis became computationally not effective due to employing complex device models with the large parameter set. The higher device integration requires dealing with numerous local and global parameters and can bottleneck the efforts of achieving fast design cycles.Statistical analysis can be facilitated by direct relation estimation a of circuit metrics to the set of PV parameters. The traditional transistor models use a large number of parameters and equations but various performance factors are possible to be related to small parameter set. A new macro model is proposed for CMOS complementary gates, where all static and dynamic characteristics are related to set of Finite Points of IV device curves. All timing and power related quantities can be predicted by evaluating the model equations. The dynamic characterization relies on charge distribution at each node. The affect of all PV is represented with characterizing the FP sensitivity. In overall the new gate model employ same computational structure for different devices in far more simple computational form.Large scale circuit analysis based on the FP models can be used for estimation of various global performance parameters. Timing performance (STA) is calculated from node to node, where at each step a new set of parameters (including PV) are introduced. Motivated by the limitations the traditional PCA, we simplify the overall computational cost with new efficient reduction technique. It turned out that the input output correlation of performance-parameters model is essential information for reduction. If the model is unknown, Sliced Inverse Regression (SIR) technique can be used to determine the Effective Reduction Space (e.d.r.). Optionally if the empiric performance analytic expression is known, the e.d.r. is found by Principle Hessian Method. In theoretical aspect the inverse reduction technique reduces parameters in the sense of their statistical significance.

<p>Dynamic modeling of Vapor Compression Cycles (VCC) is particularly important for designing and evaluating controls and fault detection and diagnosis (FDD) algorithms. As a result, transient modeling of VCCs has become an active area of research over the past two decades. Although a number of tools have been developed, the computational requirements for dynamic VCC simulations are still significant. A computationally efficient but accurate modeling approach is critically important to accelerate the design and assessment of control and FDD algorithms where a number of iterations with a variety of test input signals are required. Typically, the dynamics of compressors and expansion devices evolve on an order of magnitude faster than those of heat exchangers (HX) within VCC systems. As a result, most dynamic modeling efforts have focused on heat exchanger models. The switched moving boundary (SMB) method, which segments a heat exchanger depending on thermodynamic phase of the refrigerant, i.e. subcooled liquid, two-phase and superheated vapor, and moves control volumes as the length of each phase changes, can reduce the computation time compared with the finite volume (FV) method by solving fewer equations due to a smaller set of control volumes. Despite the computational benefit of the SMB, there is a well-known numerical issue associated with switching the model representations when a phase zone disappears or reappears inside of a heat exchanger. More importantly, the computational load is still challenging for many practical VCC systems. This thesis proposes an approach applying nonlinear model order reduction (MOR) methods to dynamic heat exchanger models to generate reduced order HX models, and then to couple them to quasi-static models of other VCC components to complete a reduced order VCC model. To enable the use of nonlinear model reduction techniques, a reformulated FV model is developed for matching the baseline MOR model structure, by using different pairs of thermodynamic states with some appropriate assumptions. Then a rigorous nonlinear model order reduction framework based on Proper Orthogonal Decomposition (POD) and the Discrete Empirical Interpolation Method (DEIM) is developed to generate reduced order HX models. </p><p> </p><p> The proposed reduced order modeling approach is implemented within a complete VCC model. Reduced order HX models are constructed for a centrifugal chiller test-stand at Herrick Labs, Purdue University, and are integrated with quasi-static models of compressor and expansion valve to form the complete cycle. The reduced cycle model is simulated in the Modelica-based platform to predict load-change transients, and is compared with measurements. The validation results indicate that the reduced order model executes 200 times faster than real time with negligible prediction errors.</p><br>

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Miniaturisation of electronic chips poses challenges at the design stage. The progressively decreasing circuit dimensions result in complex electrical behaviour that necessitates complex models. Simulation of complex circuit models involves extraordinarily large compu- tational complexity. Such complexity is better managed through Model Order Reduction. Model order reduction has been successful in large reductions in system order for most types of circuits, at high levels of accuracy. However, multiport circuits with large number of inputs/outputs, pose an additional computational challenge. A strategy based on exible clustering of interconnects results in more e cient reduction of multiport circuits. Clustering methods traditionally use Krylov-subspace methods such as PRIMA for the actual model reduction step. These clustering methods are unable to reduce the model order to the optimum extent. SVD-based methods like Truncated Balanced Realization have shown higher reduction potential than Krylov-subspace methods. In this thesis, the di erences in reduction potential and computational cost thereof between SVD-based methods and Krylov-subspace methods are identi ed, analyzed and quanti ed. A novel algorithm has been developed, utilizing a particular combination of both these methods to achieve better results. It enhances the clustering method for model reduction using Truncated Balanced Realization as a second-level reduction technique. The algorithm is tested and signi cant gains are illustrated. The proposed novel algorithm preserves the other advantages of the current clustering algorithm.