Search Results (7)

As semiconductor technologies continue to scale aggressively, electromigration (EM) has become critical in modern VLSI design. Since traditional EM assessment methods fail to accurately capture the complex behavior of multi-segment interconnects, recent physics-based models have been developed to provide a more accurate representation of EM-induced stress evolution. However, numerical methods for these models result in large-scale systems, which are computationally expensive and impractical for complex interconnect structures. Model order reduction (MOR) has emerged as a key enabler for scalable EM analysis, with moment-matching (MM) techniques offering a favorable balance between efficiency and accuracy. However, conventional Krylov-based approaches often suffer from limited frequency resolution or high computational cost. Although the extended Krylov subspace (EKS) improves frequency coverage, its symmetric structure introduces significant overhead in large-scale scenarios. This work introduces a novel MOR technique based on the asymmetric extended Krylov subspace (AEKS), which improves upon the conventional EKS by incorporating a sparsity-aware and computationally efficient projection strategy. The proposed AEKS-based moment-matching framework dynamically adapts the Krylov subspace construction according to matrix sparsity, significantly reducing runtime without sacrificing accuracy. Experimental evaluation on IBM power grid benchmarks demonstrates the high accuracy of our method in both frequency-domain and transient EM simulations. The proposed approach delivers substantial runtime improvements of up to 15× over full-order simulations and 100× over COMSOL, while maintaining relative errors below 0.5%, even under time-varying current inputs. Full article

Model order reduction (MOR) is crucial for efficiently simulating large-scale RLCk models extracted from modern integrated circuits. Among MOR methods, balanced truncation offers strong theoretical error bounds but is computationally intensive and does not preserve passivity. In contrast, moment matching (MM) techniques are widely adopted in industrial tools due to their computational efficiency and ability to preserve passivity in RLCk models. Typically, MM approaches based on the rational Krylov subspace (RKS) are employed to produce reduced-order models (ROMs). However, the quality of the reduction is influenced by the selection of the number of moments and expansion points, which can be challenging to determine. This underlines the need for advanced strategies and reliable convergence criteria to adaptively control the reduction process and ensure accurate ROMs. This article introduces a frequency-aware multi-point MM (MPMM) method that adaptively constructs an RKS by closely monitoring the ROM transfer function. The proposed approach features automatic expansion point selection, local and global convergence criteria, and efficient implementation techniques. Compared to an established MM technique, MPMM achieves up to 16.3× smaller ROMs for the same accuracy, over 99.18% reduction in large-scale benchmarks, and up to 4× faster runtime. These advantages establish MPMM as a strong candidate for integration into industrial parasitic extraction tools. Full article

Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications. Full article

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. Full article

Modeling and design of on-chip interconnect, the interconnection between the components is becoming the fundamental roadblock in achieving high-speed integrated circuits. The scaling of interconnect in nanometer regime had shifted the paradime from device-dominated to interconnect-dominated design methodology. Driven by the expanding complexity of on-chip interconnects, a passivity preserving model order reduction (MOR) is essential for designing and estimating the performance for reliable operation of the integrated circuit. In this work, we developed a new frequency selective reduce norm spectral zero (RNSZ) projection method, which dynamically selects interpolation points using spectral zeros of the system. The proposed reduce-norm scheme can guarantee stability and passivity, while creating the reduced models, which are fairly accurate across selected narrow range of frequencies. The reduced order results indicate preservation of passivity and greater accuracy than the other model order reduction methods. Full article

This paper presents an analysis of the time complexity of algorithms prepared for solving heat transfer problems at nanoscale. The first algorithm uses the classic Dual-Phase-Lag model, whereas the second algorithm employs a reduced version of the model obtained using a Krylov subspace method. This manuscript includes a description of the finite difference method approximation prepared for analysis of the real microelectromechanical system (MEMS) structure manufactured by the Polish Institute of Electron Technology. In addition, an approximation scheme of the model, as well as the Krylov subspace-based model order reduction technique are also described. The paper considers simulation results obtained using both investigated versions of the Dual-Phase-Lag model. Moreover, the relative error generated by the reduced model, as well as the computational complexity of both algorithms, and a convergence of the proposed approach are analyzed. Finally, all analyses are discussed in detail. Full article

Microelectromechanical Systems (MEMS) is difficult to take transient analysis due to the tight coupling between the multiple energy domains, typically nonlinear. An effective increment-dimensional precise integration method (PIM) combined with the model order reduction (MOR) technique based on Krylov subspace is present to solve large-scale nonlinear finite element dynamics systems. The numerical example of V-beam electro-thermal actuator is shown to demonstrate that the MOR-PIM method can achieve high precision and fast speed when solving the nonlinear dynamic equation with large-scale freedom. Full article