A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

In this article, two model order reduction (MOR) methods of Krylov subspace and Laguerre orthogonal polynomials were employed to the numerical simulation of heat transfer characteristics of ground heat exchangers and surrounding ground. The results show that the relative errors between the direct solution and two MOR methods are less than 0.1% under more than certain orders. Considering both the relative errors and time consumptions, for two MOR methods, it is workable to take about 1% of the original system order as the reduced system order. The larger the nodes of space and time, the more obvious the efficiency of two MOR methods.

Fast Simulation Methods for Dynamic Heat Transfer through Building Envelope Based on Model-order-Reduction

In this work, the influence of model order reduction (MOR) methods on optical aberrations is analyzed within a dynamical–optical simulation of a high precision optomechanical system. Therefore, an integrated modeling process and new methods have to be introduced for the computation and investigation of the overall dynamical–optical behavior. For instance, this optical system can be a telescope optic or a lithographic objective. In order to derive a simplified mechanical model for transient time simulations with low computational cost, the method of elastic multibody systems in combination with MOR methods can be used. For this, software tools and interfaces are defined and created. Furthermore, mechanical and optical simulation models are derived and implemented. With these, on the one hand, the mechanical sensitivity can be investigated for arbitrary external excitations and on the other hand, the related optical behavior can be predicted. In order to clarify these methods, academic examples are chosen and the influences of the MOR methods and simulation strategies are analyzed. Finally, the systems are investigated with respect to the mechanical–optical frequency responses, and in conclusion, some recommendations for the application of reduction methods are given.

Machine-Learning based model order reduction of a biomechanical model of the human tongue

This thesis presents the result of the study on model order reduction (MOR) methods, that can be applied to coupled systems. The goal of the research was to develop reduction techniques, that preserve special properties of coupled or interconnected system, e.g. block-structure of the underlying matrices. On the other hand, the new techniques should also be able to benefit from the knowledge, that the system they are applied to, consists of two (or more) sub-systems or describes some phenomena in different physical domains. As a result of this study, two main approaches are proposed. Their general description is given in the following paragraphs. First, the Separate Bases Reduction (SBR) algorithm is developed, which is a projection based MOR technique that uses Krylov subspaces as reduction bases. The novelty of this method is that SBR algorithm, unlike standard reduction methods designed for coupled problems, uses an uncoupled formulation of the system. In other words, an appropriate Krylov subspace is built for each of the sub-system constituting the interconnected system. As a result, the computational costs of application of the SBR algorithm, with respect to time and memory storage needed for calculations, is lower than in case of MOR methods that use the coupled formulation of the system. Moreover, the blockdiagonal form of the reduction matrices allows for preservation of the block-structure of the system matrices and keeps the sub-systems (or different physical domains) still recognizable in the reduced-order model. The SBR algorithm was successfully applied to a few test cases, resulting in the reduced systems that approximate the original ones with accuracy comparable to the accuracy of systems reduced by means of other blockstructure preserving MOR methods. The second topic of the research focuses on the couplings between the sub-systems. Here, the off-diagonal blocks of the system matrices that correspond to the couplings, are approximated by matrices of lower rank. As a main tool, generalized singular value decomposition (GSVD) is used, which allows to find the most important components of a coupling block with respect to one of the sub-systems. Although this method does not reduce the dimension of the considered problem, it gives benefits if used before application of a MOR technique. First of all, the use of low-rank approximations of the coupling blocks can decrease the computational costs of the the Krylov subspaces construction needed for reduction. If the couplings can be approximated by sufficiently low-rank blocks, the necessary matrix inverse calculation can be performed cheaper, by application of the Sherman-Morrison formula. Moreover, the undesired growth of the reduction bases, in case of use of the SBR algorithm to sub-systems with many inputs (outputs), can be lowered by use of only dominant components of the input (output) space. The conducted experiments showed, that for some cases, the number of the components used to define the couplings can be significantly reduced.

In electrodynamic field computations the continuous Maxwell equations are typically discretized in the space variables, i. e the continuous space is mapped onto a finite set of discrete elements leading to a system of differential equations constituting the Maxwell grid equations. These dynamical systems can be very large. Due to limited computational, accuracy and storage capabilities, simplified models, obtained by means of model order reduction (MOR) methods, which capture the main features of the original model are then successfully used instead of the original models. Most commonly MOR via projection is used. Variation of model parameters like geometrical or material parameters give rise to multivariate dynamical systems. It is aimed that also the simplified models keep this parameter dependence. In this work, MOR methods are presented for multivariate systems based on the finite integration technique (FIT). The methods are applied to numerical examples with both geometrical and material variations.

Summary form only given. Phased array characterization, and electromagnetic (EM) and circuit (CKT) co-simulation of packaging, components and PCBs both involve large number of ports. As such, obtaining their transfer (system) function matrix requires multiple simulations, each with a different port excitation. Even fast methodologies such as the model order reduction (MOR) and other fast CEM methods, experience severe slow-down attributed to the large number of ports. This is due to the fact that the transfer (system) function matrix, i.e. s-matrix or Y or Z-matrix is dense (P × P, P is number of ports), and traditionally each row or column is obtained through a full-wave solution of the model at hand. Traditionally such problems are best tractable with direct factorization matrix methods such as the multifrontal LU decomposition, where after the factorization stage, the transfer matrix is rapidly obtained through P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> forward/backward substitutions. Alternatively block Krylov iterative methods (Y. Saad, Iterative Methods for Sparse Linear Systems, 2003) or multiple right-hand-side Krylov methods (C. Smith et al., IEEE Trans Antenna Propag, Nov 1989) could be used. This work will exploit the inherent rank-deficiency of such transfer matrices, to reduce the computational cost. Randomized singular value decomposition (SVD) algorithms (N. Halko et al., SIAM Review, pp.217-288, 2011) will be used to compute rank-deficient representations of such transfer matrices or sub-blocks. These methods use random projections along with low complexity, often out-of-core, linear algebra operations to find the dominant subspace of a matrix. As a result, the randomized SVD of an m × n dense matrix requires O(mnlog(k)) (k min(m, n)) floating-point operations instead of the O(mn2) of the standard full SVD. To extract the frequency dependence of such transfer function matrices, these randomized SVD-based methods will be combined with deterministic, robust and error controllable model order reduction methods as well as direct factorization schemes appropriately tailored to expedite the computation of selective inverse matrix entries. The talk will first outline the importance of fast many-port device CEM solvers, and highlight their computational challenges. Then, it will present the proposed computational framework that combines the randomized SVD, model order reduction method and fast selective inverse direct solvers. To verify the accuracy and efficiency of the proposed framework, problems ranging from antenna array, signal integrity and the computation of the Dirichlet-toNeumann map will be considered.

Three-dimensional electromagnetic methods are fundamental tools for the analysis and design of high-speed systems. These methods often generate large systems of equations, and model order reduction (MOR) methods are used to reduce such a high complexity. When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. Design space optimization and exploration are usually performed during a typical design process that consequently requires repeated simulations for different design parameter values. Efficient performing of these design activities calls for parameterized model order reduction (PMOR) methods, which are able to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as layout or substrate features. We propose a novel PMOR method for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, a procedure to find scaling and frequency shifting coefficients and positive interpolation schemes. The proposed scaling and frequency shifting coefficients enhance and improve the modeling capability of standard positive interpolation schemes and allow accurate modeling of highly dynamic systems with a limited amount of initial univariate models in the design space. The proposed method is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach.

In the transient analysis of an engineering power electronics device, the order of its equivalent circuit model is excessive large. To eliminate this issue, some model order reduction (MOR) methods are proposed in the literature. Compared to other MOR methods, the structure-preserving reduced-order interconnect macromodeling (SPRIM) based on Krylov subspaces will achieve a higher reduction radio and precision for large multi-port Resistor-Capacitor-Inductor (RCL) circuits. However, for very wide band frequency transients, the performance of a Krylov subspace-based MOR method is not satisfactory. Moreover, the selection of the expansion point in this method has not been comprehensively studied in the literature. From this point of view, a broadband enhanced structure-preserving reduced-order interconnect macromodeling (SPRIM) method is proposed to reduce the order of equation sets of a transient interconnect circuit model. In addition, a method is introduced to determine the optimal expansion point at each frequency in the proposed method. The proposed method is validated by the numerical results on a transient problem of an insulated-gate bipolar transistor (IGBT)-based inverter busbar under different exciting conditions.

In this contribution, investigations on model order reduction for coupled systems composed from components of a passenger car are shown. In today's development processes, the simulation of mechanical components is indispensable and large Finite Element models are often used for this purpose. For the calculation of time‐domain or frequency‐domain analyses, for example, a lot of computing power is required. However, with the application of model order reduction methods, this effort can be reduced, but this results in a trade‐off between the reduction error and the computational time. Since the computation of reduction bases for complete systems can be computationally expensive, it is of interest to be able to reduce components individually and then assemble them into a reduced overall model. This can result in both, a saving of computational effort when creating the bases, as well as a saving of the required memory space. Furthermore, there are many possible combinations of components in the modular systems of today's automotive industry, which emphasizes the model order reduction by parts and not by assemblies. In this work, methods of model order reduction for coupled systems are presented and will be tested on components in the chassis of a sports car. Therefore, an assembly consisting of a brake disc and wheel rim together with the wheel hub are investigated. For this purpose, the eigenmodes and transfer functions of the overall model, the reduced overall model and the assembly built from individual reduced bodies are compared.

Nonlinear model order reduction based on tensor Kronecker product expansion with Arnoldi process

Two enhancements to the least-squares (LS) discrete-time model order reduction (MOR) method are presented: scaling and frequency response matching. Scaling generally improves the low-frequency fit between the reduced-order model (ROM) and the original model. For exact gains at specific frequencies, optional frequency response constraints can easily be added to the LS MOR method. An example is presented that illustrates these enhancements. The example model is reduced with the Hankel norm, weighted impulse response gramian, and LS MOR methods. Plots of error versus frequency are given for each of the three MOR methods. >

Continual technology innovations make it possible to fabricate electronic devices on the order of 10nm. In this nanoscale regime, quantum physics becomes critically important, like energy quantization effects of the narrow channel and the leakage currents due to tunneling. It has also been utilized to build novel devices, such as the band-to-band tunneling field-effect transistors (FETs). Therefore, it presages accurate quantum transport simulations, which not only allow quantitative understanding of the device performances but also provide physical insight and guidelines for device optimizations.&#13;\n&#13;\nHowever, quantum transport simulations usually require solving repeatedly the Green’s function or the wave function of the whole device region with open boundary treatment, which are computationally cumbersome. Moreover, to overcome the short-channel effects, modern devices usually employ multi-gate structures that are three-dimensional, making the computation very challenging. It is the major target of this thesis to enhance the simulation efficiency by proposing several fast numerical algorithms. The other target is to apply these algorithms to study the physics and performances of some emerging electronic devices.&#13;\n&#13;\nFirst, an efficient method is implemented for real space simulations with the effective mass approximation. Based on the wave function approach, asymptotic waveform evaluation combined with a complex frequency hopping algorithm is successfully adopted to characterize electron conduction over a wide energy range. Good accuracy and efficiency are demonstrated by simulating several n-type multi-gate silicon FETs. This technique is valid for arbitrary potential distribution and device geometry, making it a powerful tool for studying n-type silicon nanowire (SiNW) FETs in the presence of charged impurity and surface roughness scattering.&#13;\n&#13;\nSecond, a model order reduction (MOR) method is proposed for multiband simulation of nanowire structures. Employing three- or six-band k.p Hamiltonian, the non-equilibrium Green’s function (NEGF) equations are projected into a much smaller subspace constructed by sampling the Bloch modes of each cross-section layer. Together with special sampling schemes and Krylov subspace methods for solving the eigenmodes, large cross-section p-type SiNW FETs can be simulated. A novel device, junctionless FET, is then investigated. It is found that its doping density, channel orientation, and channel size need to be carefully optimized in order to outperform the classical inversion-mode FET.&#13;\n&#13;\nWith a spurious band elimination process, the MOR method is subsequently extended to the eight-band k.p model, allowing simulation of band-to-band tunneling devices. In particular, tunneling FETs with indium arsenide (InAs) nanowire channel are studied, considering different channel orientations and configurations with source pockets. Results suggest that source pocket has no significant impact on the performances of the nanowire device due to its good electrostatic integrity.&#13;\n&#13;\nAt last, improvements are made for open boundary treatment in atomistic simulations. The trick is to condense the Hamiltonian matrix of the periodic leads before calculating the surface Green’s function. It is very useful for treating leads with long unit cells.

Michiels et al. (SIAM J. Matrix Anal. Appl. 32(4):1399–1421, 2011) proposed a Krylov-based model order reduction (MOR) method for time-delay systems. In this paper, we present an efficient process, which requires less memory consumption, to accomplish the model reduction. Memory efficiency is achieved by replacing the classical Arnoldi process in the MOR method with a two-level orthogonalization Arnoldi (TOAR) process. The resulting memory requirement is reduced from quadratic dependency of the reduced order to linear dependency. Besides, this TOAR process can also be applied to reduce the original delay system into a reduced-order delay system. Numerical experiments are given to illustrate the feasibility and effectiveness of our method.

Many new promising model order reduction (MOR) methods and algorithms were developed during the last decade. Industry and academic research institutions intend to test, validate, compare, and use these new promising MOR techniques with their own models. Therefore, an MOR toolbox bridging the gap between theoretical, algorithmic, and numerical developments to an end-user-oriented program, usable by non-experts, was developed called ‘Model Order Reduction of Elastic Multibody Systems’ (Morembs). A C++ implementation as well as a Matlab implementation including an intuitive graphical user interface is available. Import from various FE programs is possible, and the reduced elastic bodies can be exported to a variety of programs to simulate the compact models. In the course of the various projects, many improvements on the algorithmic side were added. As we learned over the years, there is not one ‘optimal’ MOR method. ‘Optimal’ MOR depends on circumstances, like boundary conditions, excitation spectra, further model usage. The toolbox is now used, e.g., in solid mechanics, biomechanics, vehicle dynamics, control of flexible structures, or crash simulations. In all these use cases, the toolbox allows the user to facilitate their well-known modeling and simulation environment. Only the critical MOR process during preprocessing is performed with Morembs, which helps to compare the various MOR techniques to find the most suited one.