Luc Knockaert

Matrix-Interpolation-Based Parametric Model Order Reduction for Multiconductor Transmission Lines With Delays

A novel parametric model order reduction technique based on matrix interpolation for multiconductor transmission lines (MTLs) with delays having design parameter variations is proposed in this brief. Matrix interpolation overcomes the oversize problem caused by input-output system-level interpolation-based parametric macromodels. The reduced state-space matrices are obtained using a higher-order Krylov subspace-based model order reduction technique, which is more efficient in comparison to the Gramian-based parametric modeling in which the projection matrix is computed using a Cholesky factorization. The design space is divided into cells, and then the Krylov subspaces computed for each cell are merged and then truncated using an adaptive truncation algorithm with respect to their singular values to obtain a compact common projection matrix. The resulting reduced-order state-space matrices and the delays are interpolated using positive interpolation schemes, making it computationally cheap and accurate for repeated system evaluations under different design parameter settings. The proposed technique is successfully applied to RLC (R-resistor, L-inductor, C-capacitance) and MTL circuits with delays.

Hybrid Nonlinear Modeling Using Adaptive Sampling

This paper proposes a direct method for the extraction of empirical-behavioral hybrid models using adaptive sampling. The empirical base is responsible for the functionality over a wide range of variables, especially in the extrapolation range. The behavioral part corrects the errors of the empirical part in the region of particular interest, thus, it improves the accuracy in the desired region. Employment of response surface methodology and adaptive sampling allows full automation of the hybrid model extraction and assures its compactness. We used this approach to build a hybrid model composed of a robust empirical model available in CAD tools and a Radial Basis Functions interpolation model with Gaussian basis function. We extracted the hybrid model from measurements of a 0.15 μm GaAs HEMT and compared it with the pure behavioral and pure empirical models. The hybrid model yields higher accuracy while maintaining extrapolation capabilities. Additionally, the extraction time of the hybrid model is relatively low. We also show that a good accuracy level can be achieved with a small number of measurements.

Polynomial Chaos-Based Macromodeling of General Linear Multiport Systems for Time-Domain Analysis

In this paper, we present a new effective methodology to build stochastic macromodels for the time-domain analysis of generic linear multiport systems. The proposed technique allows one to calculate a stable and passive polynomial chaos (PC)-based macromodel of a system under stochastic variations. The Galerkin projections method and a PC-based model of the system scattering parameters are used along with the Vector Fitting algorithm, leading to an accurate and efficient description of the system variability features. The proposed technique results to be very versatile and, thus, is well suited to be applied to many different and complex modern electrical systems (e.g., interconnections and filters). The accuracy and computational efficiency of the proposed technique are verified through comparison with the standard Monte Carlo analysis for two pertinent numerical examples, showing a maximum simulation speedup of 765 times.

A Kriging and stochastic collocation ensemble for uncertainty quantification in engineering applications

We propose a new surrogate modeling approach by combining two non-intrusive techniques: Kriging and Stochastic Collocation. The proposed method relies on building a sufficiently accurate Stochastic Collocation model which acts as a basis to construct a Kriging model on the residuals, to combine the accuracy and efficiency of Stochastic Collocation methods in describing stochastic quantities with the flexibility and modeling power of Kriging-based approaches. We investigate and compare performance of the proposed approach with state-of-art techniques over benchmark problems and practical engineering examples on various experimental designs.

Multipoint Model Order Reduction Using Reflective Exploration

Reduced order models obtained by model order reduction methods must be accurate over the whole frequency range of interest. Multipoint reduction algorithms allow to generate accurate reduced models. In this paper, we propose the use of a reflective exploration technique for obtaining the expansion points adaptively for the reduction algorithm. At each expansion point the corresponding projection matrix is computed. Then, the projection matrices are merged and truncated based on their singular values to obtain a compact reduced order model. Three conductor transmission line example is used to illustrate the technique.

RATIONAL MODELING OF MULTIVARIATE MULTI-FIDELITY DATA

Abstract. Accurate multi-fidelity modeling is of high importance in the present day engineering design process. It allows to model computationally expensive simulations at a reduced cost by combining simulations with variable fidelity levels. In this paper, a novel algorithm is proposed to build multivariate models from variable fidelity simulations using rational functions. The modeling is based on high-fidelity data and low-fidelity data that is sampled over a parameter space of interest. The former is assumed to be computationally expensive and sparse, whereas the latter is cheaper to obtain but comes at a lower accuracy. It is shown that accurate rational models can be built at a reduced cost by combining these types of data. The effectiveness of the algorithm is applied to several examples and confirmed by numerical results.

Stochastic macromodeling for hierarchical uncertainty quantification of nonlinear electronic systems

A hierarchical stochastic macromodeling approach is proposed for the efficient variability analysis of complex nonlinear electronic systems. A combination of the Transfer Function Trajectory and Polynomial Chaos methods is used to generate stochastic macromodels. In order to reduce the computational complexity of the model generation when the number of stochastic variables increases, a hierarchical system decomposition is used. Pertinent numerical results validate the proposed methodology.

Passive parametric macromodeling by using Sylvester state-space realizations

A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. The direct parameterization of poles and residues may be not appropriate, due to their possible non-smooth behavior with respect to design parameters. This is avoided in the proposed technique, by converting the pole-residue description to a Sylvester description which is computed for each root macromodel. This technique is used in combination with suitable parameterizing schemes for interpolating a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parametric macromodels. The key features of the present approach are first the choice of a proper pivot matrix and second, finding a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester technique for parametric macromodeling.

Multipoint model order reduction for systems with delays

An adaptive frequency sampling algorithm is proposed in this paper to automate the generation of reduced order models for systems with delays which can be represented as frequency dependent state-space matrices. Reflective exploration technique is used to obtain an optimum number of frequency samples for which the reduced state-space matrices per frequency is computed using a common projection matrix and is then interpolated to obtain the frequency response. The algorithm is illustrated using a numerical example.