A new finite element model for reduced order electromagneticmodeling

This paper introduces a new formulation suitable for direct model order reduction of finite element approximations of electromagnetic systems using Krylov subspace methods. The proposed formulation utilizes a finite element model of Maxwell's curl equations to generate a state-space representation of the electromagnetic system most suitable for the implementation of model order reduction techniques based on Krylov subspaces. It is shown that, with a proper selection of the finite element interpolation functions for the fields, the proposed formulation is equivalent to the commonly used approximation of the vector wave equation with tangentially continuous vector finite elements. This equivalence is exploited to improve the computational efficiency of the model order reduction process     

Stable reduced order modeling of piezoelectric energy harvesting modules using implicit Schur complement

In this work, we present an improved technique for reduced order modeling of a micro-machined piezoelectric energy harvester, presented in Kudryavtsev et al. (2015), and a novel frequency-tunable piezoelectric energy harvester with segmented electrodes for improved power generation. A computationally efficient implicit Schur complement is developed for better conditioning of the numerical models. In combination with Krylov subspace-based model order reduction methods, it offers an efficient way to generate guaranteed stable reduced order models of multi-physical device models. We demonstrate an excellent match between the full-scale and the reduced order models of the harvester devices, and establish a new methodology for system-level simulation based on these reduced order numerical models.     

Computationally efficient and stable order reduction methods for a large-scale model of MEMS piezoelectric energy harvester

In this work, we present novel model order reduction approaches for a large-scale multiport model of a MEMS-based piezoelectric energy harvester. These techniques are computationally efficient and generate stable reduced order models. The first method proposed combines model reduction based on Krylov subspaces and a Schur complement transformation of the resulting system. The second method includes structure preserving Krylov subspaces based model order reduction. We demonstrate an excellent match between the full-scale and the reduced order models of the harvester device during harmonic simulation and the co-simulation of the reduced harvester model together with the power management circuitry.     

AWE implementation for electromagnetic FEM analysis

Although full wave electromagnetic systems are large and cumbersome to solve, typically only a few parameters, such as input impedance, S parameters, and far field pattern, are needed by the designer or analyst. A reduced order modelling of these parameters is therefore an important consideration in minimising the CPU requirements. The asymptotic waveform evaluation (AWE) method is one approach to construct a reduced order model of the input impedance or other useful electromagnetic parameters. The authors demonstrate its application and validity when used in conjunction with the finite element method to simulate full wave electromagnetic problems     

Multiple block structure-preserving reduced order modeling of interconnect circuits

In this paper, we propose a generalized multiple-block structure-preserving reduced order interconnect macromodeling method (BSPRIM). Our approach extends the structure-preserving model order reduction (MOR) method SPRIM [R.W. Freund, SPRIM: structure-preserving reduced-order interconnect macromodeling, in: Proceedings of International Conference on Computer Aided Design (ICCAD), 2004, pp. 80-87] into more general block forms. We first show how an SPRIM-like structure-preserving MOR method can be extended to deal with admittance RLC circuit matrices and show that the 2q moments are still matched and symmetry is preserved. Then we present the new BSPRIM method to deal with more circuit partitions for linear dynamic circuits formulated in impedance and admittance forms. The reduced models by BSPRIM will still match the 2q moments and preserve the circuit structure properties like symmetry as SPRIM does. We also show that BSPRIM can build the compact models with similar size and accuracy of that produced by traditional projection based methods but using less computation costs. Experimental results show that BSPRIM outperforms SPRIM in terms of accuracy with more partitions and outperforms PRIMA with less CPU times for generating the same accurate models.     

Least-squares model order reduction enhancements

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     

Multi-port dynamic compact thermal models of dual-chip package using model order reduction and metaheuristic optimization

Delphi-like boundary condition independent (BCI) compact thermal models (CTMs) are the standard for modelling single die packages. However their extraction, particularly in the transient case, will be time consuming due to complex numerical simulations for a large number of external conditions. Lately, new approaches to extract a BCI dynamical CTM (DCTM), based on model order reduction (MOR) were developed. Despite the numerous advantages of this recent method, the lack of numerical tools to integrate reduced-order models (ROM) makes it difficult to use at board level. In this study, a novel process flow for extracting Delphi-inspired BCI DCTMs is proposed. Thus a detailed three-dimensional model is replaced by a BCI-ROM model using FANTASTIC matrix reduction code to generate the data used in the creation of a Delphi-style BCI DCTM. That hybrid reduction method has been applied, at first on a single-chip package (QFN16) then on a dual-chip package (DFN12). Their derived CTM and DCTM have been compared in term of accuracy and creation time using, or not, MOR reduction technique. The results show that for a similar accuracy, the integration of MOR technique allows minimizing the time-consuming numerical simulations and consequently reduce the thermal network creation time by 80%.     

Model-order reduction by dominant subspace projection: error bound, subspace computation, and circuit applications

Balanced truncation is a well-known technique for model-order reduction with a known uniform reduction error bound. However, its practical application to large-scale problems is hampered by its cubic computational complexity. While model-order reduction by projection to approximate dominant subspaces without balancing has produced encouraging experimental results, the approximation error bound has not been fully analyzed. In this paper, a square-integral reduction error bound is derived for unbalanced dominant subspace projection by using a frequency-domain solution of the Lyapunov equation. Such an error bound is valid in both the frequency and time domains. Then, a dominant subspace computation scheme together with three Krylov subspace options is introduced. It is analytically justified that the Krylov subspace for moment matching at low frequencies is able to provide a better dominant subspace approximation than the Krylov subspace at high frequencies, while a rational Krylov subspace with a proper real shift parameter is capable of achieving superior approximation than the Krylov subspace at low frequency. A heuristic method of choosing a real shift parameter is also introduced based on its new connection to the discretization of a continuous-time model. The computation algorithm and theoretical analysis are then examined by several numerical examples to demonstrate the effectiveness. Finally, the dominant subspace computation scheme is applied to the model-order reduction of two large-scale interconnect circuit examples.     

基于负载牵引法的小型化功率模块设计

为了更准确地设计功率放大器,放弃了依靠调试的设计方法而采用负载牵引法进行大信号参数提取并以此为基础进行放大的设计.阐述了管芯大信号模型理论,介绍了利用负载牵引技术对管芯进行大信号参数的提取过程,运用ADS软件对功率放大器的微波性能如增益、驻波比和功率附加效率等进行了优化,并在此基础上设计了S波段小型化功率放大器模块.制作的单级功率放大器模块在500 MHz带宽内Gp＞9 dB,VSWR≤1.5:1,Pou≥34.6 dBm.     

On the design of NMR sensor for well-logging applications

The magnetic fields of antenna and magnet used in inside-out nuclear magnetic resonance (NMR) well-logging tool are computed using a finite-element method (FEM). A typical operating frequency of such tools is 2 MHz, at which the skin depth is about 47 μm for copper conductor. A direct application of FEM to evaluate power loss at such frequency, therefore, requires very fine discretization of the conductors, which, in turn, makes the problem numerically ill-conditioned. A perturbation technique along with FEM is used to evaluate the power loss in conductors that avoids the need for small discretization steps along the conductor thickness. The design of the magnet, on the other hand, is complicated by the fact that the model is nonlinear in nature because of the presence of ferrites and steel materials surrounding the magnet and because the size of the problem is usually quite large; quarter of a million unknowns is fairly common. A typical nonlinear FEM model requires about 35 h of central processing unit (CPU) time on a Sun Ultra 60 296 MHz workstation with one gigabyte of RAM. The magnet is built by stacking several magnet segments along the axial direction and the objective of the design is to magnetize these segments in such a way so as to produce a desired field profile in front of the magnet. It generally requires many executions of the nonlinear FEM model. An optimal control technique is used in conjunction with the FEM to speed up the design process. Very good agreement between the measured and computed antenna efficiency and magnetic field is obtained thus validating the numerical model     

Passive parameterized time-domain macromodels for high-speed transmission-line networks

There is a significant need for efficient and accurate macromodels of components during the design of microwave circuits. Increased integration levels in microwave devices and higher signal speeds have produced the need to include effects previously neglected during circuit simulations. Accurate prediction of these effects involve solution of large systems of equations, the direct simulation of which is prohibitively CPU expensive. In this paper, an algorithm is proposed to form passive parametrized macromodels of large linear networks that match the characteristics of the original network in time, as well as other design parameters of the circuit. A novel feature of the algorithm is the ability to incorporate a set of design parameters within the reduced model. The size of the reduced models obtained using the proposed algorithm were less than 5% when compared to the original circuit. A speedup of an order of magnitude was observed for typical high-speed transmission-line networks. The algorithm is general and can be applied to other disciplines such as thermal analysis.     

Hierarchical Krylov subspace based reduction of large interconnects

In this paper, we propose a new model order reduction approach for large interconnect circuits using hierarchical decomposition and the Krylov subspace projection-based model order reduction methods. The new approach, called hiePrimor, first partitions a large interconnect circuit into a number of smaller subcircuits and then performs the projection-based model order reduction on each of subcircuits in isolation and on the top-level circuit thereafter. The new approach is very amenable for exploiting the multi-core based parallel computing platforms to significantly speed up the reduction process. Theoretically we show that hiePrimor can deliver the same accuracy as the flat reduction method given the same reduction order and it can also preserve the passivity of the reduced models as well. We also show that partitioning has large impacts on the performance of hierarchical reduction and the minimum-span objective should be required to attain the best performance for hierarchical reduction. The proposed method is suitable for reducing large global interconnects like coupled bus, transmission lines, large clock nets in the post-layout stage. Experimental results demonstrate that hiePrimor can be significantly faster and more scalable than the flat projection methods like PRIMA and be order of magnitude faster than PRIMA with parallel computing without loss of accuracy. Interconnect circuits with up to 4 million nodes can be analyzed in a few minutes even in Matlab by the new method.     

Krylov space methods on state-space control models

We give an overview of various Lanczos/Krylov space methods and the way in which they are being used for solving certain problems in Control Systems Theory based on state-space models. The matrix methods used are based on Krylov sequences and are closely related to modern iterative methods for standard matrix problems such as sets of linear equations and eigenvalue calculations. We show how these methods can be applied to problems in Control Theory such as controllability, observability, and model reduction. All the methods are based on the use of state-space models, which may be very sparse and of high dimensionality. For example, we show how one may compute an approximate solution to a Lyapunov equation arising from a discrete-time linear dynamic system with a large sparse system matrix by the use of the Arnoldi algorithm, and so obtain an approximate Gramian matrix. This has applications in model reduction. The close relation between the matrix Lanczos algorithm and the algebraic structure of linear control systems is also explored.     

Multipoint Galerkin asymptotic waveform evaluation for model orderreduction of frequency domain FEM electromagnetic radiationproblems

In this paper, a model order reduction technique is presented. This technique, known as Galerkin asymptotic waveform evaluation (GAWE), or multipoint Galerkin asymptotic waveform evaluation (MGAWE) if multiple expansion points are considered simultaneously, can be used to reduce matrices describing electromagnetic (EM) phenomena generated through the finite element method (FEM) to a smaller space while still accurately approximating the characteristics of the original responses. The resulting solution procedure of using GAWE or MGAWE, to solve FEM equations allows for wideband frequency simulations with a reduction in total computation time. Numerical simulations using these methods are shown along with traditional methods such as using an LU decomposition at each frequency point of interest and asymptotic waveform evaluation (AWE). Comparisons in accuracy as well as computation time are also given     

Efficient IC hotspot thermal analysis via low-rank Model Order Reduction

Efficient full-chip thermal simulation is among the most challenging problems facing the EDA industry today, due to the need for solution of very large systems of equations that require unreasonably long computational times. However, in most cases, temperature is not required to be computed at every point of the IC but only at certain hotspots, in order to assess the circuit's compliance with thermal specifications. This makes the thermal analysis problem amenable to Model Order Reduction techniques. System-theoretic techniques like Balanced Truncation offer very reliable bounds for the approximation error, which can be used to control the order and accuracy of the reduced models during creation, at the expense of greater computational complexity to create them. In this paper, we propose a computationally efficient low-rank Balanced Truncation algorithm based on extended Krylov subspace method, which retains all the system-theoretic advantages in the reduction of model order for fast hotspot thermal simulation. Experimental results demonstrate around 97% order reduction and very tight accuracy bounds.     

Algebraic multigrid preconditioner for homogeneous scatterers in electromagnetics

Electromagnetic wave scattering from large and complex bodies is currently the most challenging problem in computational electromagnetics. There is an increasing need for more efficient algorithms with reduced computational complexity and memory requirements. In this work we solve the problem of electromagnetic wave scattering involving three-dimensional, homogeneous, arbitrarily shaped dielectric objects. The fast multipole method (FMM) is used along with the algebraic multigrid (AMG) method, that is employed as a preconditioner, in order to accelerate the convergence rate of the Krylov iterations. Our experimental results suggest much faster convergence compared to the non preconditioned FMM, and hence significant reduction to the overall computation time.     

Fast Simulation of steady-state temperature distributions in electronic components using multidimensional model reduction

In this paper, a model reduction technique is applied to the thermal modeling of electronic components and devices with complex geometries. The reduced-order model is capable of predicting a complete detailed three-dimensional temperature distribution in the original model. The small size and the simplicity of the reduced model allows for the very quick simulation of the device under a wide range of input parameters, such as different boundary conditions and power distributions. Use of the reduced-order model in a thermal design cycle can have a significant effect on both prediction accuracy and simulation efficiency. In the paper, the usefulness of this technique is demonstrated through examples from different electronic devices and packages. Accuracy of the reduced-order model is validated by comparison with the solution to a detailed numerical model.     

On the extraction of PiN diode design parameters for validation of integrated power converter design

Design of integrated power systems requires prototype-less approaches. Accurate simulations are necessary for analysis and verification purposes. Simulation relies on component models and associated parameters. The paper focuses on a step-by-step extraction procedure for the design parameters of a one-dimensional finite-element-method (FEM) model of the PiN diode. The design parameters are also available for diverse physics-based analytical models. The PiN diode remains a complex device to model particularly during switching transients. The paper demonstrates that a simple FEM model may be considered unknowingly of the device exact technology. Heterogeneous simulation is illustrated. The state-of-art of parameter extraction methods is briefly recalled. The proposed procedure is detailed. The diode model and extracted parameters are systematically validated from electro-thermal point-of-view. Validity domains are discussed.     

A New Method for Designing Causal Stable IIR Variable Fractional Delay Digital Filters

This paper studies the design of causal stable Farrow-based infinite-impulse response (IIR) variable fractional delay digital filters (VFDDFs), whose subfilters have a common denominator. This structure has the advantages of reduced implementation complexity and avoiding undesirable transient response when tuning the spectral parameter in the Farrow structure. The design of such IIR VFDDFs is based on a new model reduction technique which is able to incorporate prescribed flatness and peak error constraints to the IIR VFDDF under the second order cone programming framework. Design example is given to demonstrate the effectiveness of the proposed approach.