Numerical Initial Value Problems in Ordinary Differential Equations

In this paper, we adopt the so-called sparse polynomial chaos metamodel for the uncertainty quantification in the framework of high-dimensional problems. This metamodel is used to model a realistic electronic bus structure with a large number of uncertain input parameters such as those related to microstrip line geometries. It aims at estimating quantities of interest, such as statistical moments, probability density functions, and provides sensitivity analysis of a response. It drastically reduces the model computational cost with regard to brute force Monte Carlo (MC) simulation. The method presents a good performance and is validated in comparison with MC simulation.

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Variability Impact of Many Design Parameters: The Case of a Realistic Electronic Link

Microwave modeling and simulation are essential to designing microwave circuits and systems. Although fundamental concepts and approaches for modeling and simulation are mature, the drive to higher frequencies, tighter design margins, and more functionality/complexity of circuits continue to defy the capabilities of existing modeling and simulation methods. Newer algorithms are being developed to address the speed, accuracy and robustness of design algorithms. Coupled with the advent of more powerful computers and algorithms, microwave design automations are solving much more complex problems in much shorter time than previously imaginable. This paper describes the advances and state-of-the-art in automated modeling and simulation. Automated data-driven modeling approaches covering data sampling/generation, model structure adaptation, and model training/validation are described. Simulation of nonlinear microwave circuits is described covering formulations of simulation equations and advanced solution algorithms addressing problem size, convergence speed and solution accuracy. The descriptions highlight fundamental concepts, advanced methodologies, and future trends of development.

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Simulation and Automated Modeling of Microwave Circuits: State-of-the-Art and Emerging Trends

Sensitivity analysis is important for electromagnetic (EM)-based design. The existing adjoint EM sensitivity analysis methods have to solve large systems of EM equations repetitively for different frequencies. This article addresses this situation and proposes to speed up the EM sensitivity analysis over a frequency range by solving EM equations at only a single frequency. A new adjoint EM sensitivity analysis algorithm for the fast frequency sweep using the matrix Pade via Lanczos (MPVL) technique based on the finite-element method (FEM) is proposed in this article. MPVL is incorporated to relate the information of one frequency to the information of multiple frequencies. A large system of EM equations is then solved at a single frequency to predict the sensitivity information for the entire frequency band. Adjoint formulations are further derived to avoid the effect of the number of design variables. The adjoint EM sensitivity analysis using the proposed technique can obtain the same accuracy as the existing techniques while taking less time by avoiding repetitively solving large systems of EM equations for different frequencies and different design variables. The proposed technique is demonstrated by three EM examples of microwave components.

Adjoint EM Sensitivity Analysis for Fast Frequency Sweep Using Matrix Padé via Lanczos Technique Based on Finite-Element Method

A fast algorithm is presented for statistical analysis of large circuits with multiple stochastic parameters. The proposed method combines the merits of the parameterized model order-based techniques and numerical inversion of Laplace transform (NILT). The response of the reduced model at any given time point is expressed as a linear combination of the frequency-domain response at a relatively small number of predetermined complex frequency points. This eliminates the necessity for explicit representation of the dynamic model in the form of a set of differential equations. As a result, the moment vectors associated with frequency are excluded while forming the moments’ subspace, leading to much smaller reduced models. In addition, evaluation of the time-domain response of the reduced-order models using NILT is more efficient and highly parallelizable compared to time-stepping numerical integration techniques. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Variability Analysis via Parameterized Model Order Reduction and Numerical Inversion of Laplace Transform

In this paper, we propose a novel parameterized model order reduction technique for circuits described by the delayed partial element equivalent circuit method. The moment vectors associated with frequency are excluded while forming the moments’ subspace. This leads to obtaining very compact parameterized reduced order models. An implicit multiparameter moment matching algorithm is used for the moments related to the design parameters. Numerical results validate the accuracy and efficiency of the proposed technique in both frequency- and time-domain. The time-domain response of the reduced order models is obtained using the numerical inversion of Laplace transform, which avoids stability issues as well as the computationally expensive repeated interpolation steps associated with conventional time-stepping integration method for delayed differential equations.

Parameterized Model Order Reduction of Delayed PEEC Circuits

Full-wave time-domain computational electromagnetic (CEM) solvers, which are integral equation (IE)-based, may suffer from what is called “late-time instability” problems. This unstable behavior occurs for CEM solvers for very fast rise-time input signals. A multitude of techniques has been devised by researchers over the years to solve the problem. In this article, we pursue an approach for the stable solution of full-wave partial element equivalent circuit (PEEC) models for fast-rising input waveforms. In particular, step and impulse response will be considered that are the most challenging from a stability point of view. For the solver part, a conventional full-wave PEEC code is used that requires one to use retarded partial elements. Unfortunately, a PEEC, as well as impedance Z-(method of moment) solvers using suitable numerical time-stepping methods have stability problems, especially for fast rising impulse or step inputs. An important step forward is achieved in this work by providing a larger class of stable solutions well above the stability achieved for time-stepping methods in the last 50 years. The time-domain stability is achieved by replacing the stepping integration methods with a numerical inversion of Laplace transform (NILT) technique. The NILT transform starts out by applying it to a frequency-domain PEEC solution. The surprising result is that the NILT-based method has a variable time-dependent bandwidth that is advantageous for the full-wave IE solution stability. In this article, we give several examples that show that a PEEC-NILT solution provides accurate and stable results for impulse, step- and piece-wise linear input waveforms.

Time-Domain Analysis of Retarded Partial Element Equivalent Circuit Models Using Numerical Inversion of Laplace Transform

A new method is presented for high-dimensional variability analysis based on two main concepts, namely, node tearing for parameter partitioning and sparse grid interpolation. Node tearing is used to localize the parameters and, thus, reducing the number of stochastic parameters within the subcircuits and sparse grids reduce the required number of samples for a targeted accuracy. MC analysis of the overall circuit is carried out using interface equations of a much smaller dimension than the original circuit equations. Pertinent computational results are presented to validate the efficiency and accuracy of the proposed method.

Uncertainty Quantification Using Parameter Space Partitioning

This article presents a new algorithm for time-domain circuit simulation based on numerical inversion of Laplace transform (NILT). The proposed method, labeled NILT $n$ , shows that for the same computational cost of the conventional NILT, referred to as NILT0, the approximation error is reduced by a significant factor, permitting time-marching with much larger time steps and, consequently, saving significant computational cost per time step. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed approach.

Ye Tao

Papers

Variability Analysis via Parameterized Model Order Reduction and Numerical Inversion of Laplace Transform

Parameterized Model Order Reduction of Delayed PEEC Circuits

Time-Domain Analysis of Retarded Partial Element Equivalent Circuit Models Using Numerical Inversion of Laplace Transform

Uncertainty Quantification Using Parameter Space Partitioning

Fast and Stable Time-Domain Simulation Based on Modified Numerical Inversion of the Laplace Transform