Model Order Reduction Techniques Research Articles

Fast electromigration stress analysis using Low-Rank Balanced Truncation for general interconnect and power grid structures

Electromigration (EM) has become one of the most significant challenges considering longterm reliability in integrated circuit design. The problem is caused by the large current density in circuit interconnections. However, in most cases, we are interested in the EM stress at specific points of the interconnect, such as vias, junctions and boundaries. As a result, Model Order Reduction (MOR) techniques can provide attractive methodologies to reduce the complexity of the original systems. System-theoretic techniques like Balanced Truncation (BT) offer very reliable bounds for the approximation error, compared to moment-matching methods. In this paper, we apply a computationally efficient Low-Rank BT procedure based on the extended Krylov subspace method, that can handle large-scale models and significantly accelerate the EM stress analysis for any general interconnect or power grid structure. For the experimental evaluation, we first consider a general structure interconnect tree to demonstrate the generalizability of the proposed method and then we employ our method on the industrial IBM power grid benchmarks where the its scalability is shown. For the latter, we demonstrate that our method can achieve a speedup up to 238× over a standard transient analysis method and a speedup up to 15× over COMSOL, while exhibiting negligible error.

Interaction of transverse pressure and in-plane internal stresses on bi-stability of electrostatically rectangular microplates

Bi-stability behavior is commonly observed in an electrostatically actuated pressurised or initially curved microplates. Additionally, presence of in-plane internal stresses considerably changes the mechanical behaviour of the electrostatically actuated microplate. In this paper, the effects of in-plane internal stresses on bi-stability criteria of pressurised rectangular microplates are investigated. For this, the governing differential equation of motion of a rectangular plate is derived using Kirchhoffs plate theory including von Karmans nonlinearity, transverse pressure, axial in-plane internal stresses, and non-linear electrostatic actuation force. Further, Galerkins based reduced-order modelling technique is applied to solve the equation. The problem has been solved for two different sets of plate boundary conditions: simply supported and clamped. The analysis reveals that the presence of tensile or compressive in-plane internal stresses significantly changes the transverse pressure range for the occurrence of bi-stability on the voltage-deflection equilibrium path. Further, we have investigated the variation of Required threshold gap between microplate and substrate for bi-stability in presence of in-plane internal stresses. Results of this investigation can be used to design more efficient microplate based bi-stable N/MEMS devices.

A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells

AbstractIn this contribution, we present a modelling and simulation framework for parametrised lithium-ion battery cells. We first derive a continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterised non-linear system of partial differential equations, the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modelling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modelling framework.

Non-intrusive system state reconstruction from indirect measurements: A novel approach based on Hybrid Data Assimilation methods

The problem of estimating in real-time the state of a system by combining experimental data and models has been extensively addressed in literature. In particular, there have been a lot of developments in reduced order modelling techniques integrated in a data assimilation framework. This coupling allows to reconstruct the variable of interest considering a priori knowledge (i.e. the mathematical model) and some measurements of it. However, in some engineering systems not all the variables of interest may be measurable and hence the typical framework cannot be applied. However, these methods can be extended to reconstruct the full state of a system, by means of partial observations only. In this work, a novel approach will be introduced, based on a two step method: first the measurements are used to determine the parameters describing the system; then, the full state is estimated by means of reduced order modelling techniques. This new approach will be compared with the state-of-the-art both on a simple thermal hydraulics case and on an innovative nuclear reactor design, the Molten Salt Fast Reactor.

Designing PID Controller Using improved Differential-Pole-clustering Approach of Model Order Reduction Techniques

The method of mathematical modelling that takes into account various physical systems, such as various tele-communications, High voltage transmission, and various chemical reactors, ultimately produced a variety of solutions for bigger order complex mathematical models with higher order. Order reduction is essential for such higher order systems (HOS) for reducing mathematical complexity for designing controllers. In order to improve upon several existing model order reduction approaches, the goal of this paper is to solve difficult higher order models using Differential Pole Cluster Mixed (DPM) Techniques. The accuracy of the proposed reduced model has been made via comparing step responses of original and reduced order models. In order to improve upon several existing model order reduction approaches. The higher order system PID Controller has been made using the proposed reduced-order approach.

Surrogate models for plasma displacement and current in 3D perturbed magnetohydrodynamic equilibria in tokamaks

A numerical database of over one thousand perturbed three-dimensional (3D) equilibria has been generated, constructed based on the MARS-F (Liu et al 2000 Phys. Plasmas 7 3681) computed plasma response to the externally applied 3D field sources in multiple tokamak devices. Perturbed 3D equilibria with the n = 1–4 (n is the toroidal mode number) toroidal periodicity are computed. Surrogate models are created for the computed perturbed 3D equilibrium utilizing model order reduction (MOR) techniques. In particular, retaining the first few eigenstates from the singular value decomposition (SVD) of the data is found to produce reasonably accurate MOR-representations for the key perturbed quantities, such as the perturbed parallel plasma current density and the plasma radial displacement. SVD also helps to reveal the core versus edge plasma response to the applied 3D field. For the database covering the conventional aspect ratio devices, about 95% of data can be represented by the truncated SVD-series with inclusion of only the first five eigenstates, achieving a relative error (RE) below 20%. The MOR-data is further utilized to train neural networks (NNs) to enable fast reconstruction of perturbed 3D equilibria, based on the two-dimensional equilibrium input and the 3D source field. The best NN-training is achieved for the MOR-data obtained with a global SVD approach, where the full set of samples used for NN training and testing are stretched and form a large matrix which is then subject to SVD. The fully connected multi-layer perceptron, with one or two hidden layers, can be trained to predict the MOR-data with less than 10% RE. As a key insight, a better strategy is to train separate NNs for the plasma response fields with different toroidal mode numbers. It is also better to apply MOR and to subsequently train NNs separately for conventional and low aspect ratio devices, due to enhanced toroidal coupling of Fourier spectra in the plasma response in the latter case.

A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls.

Nowadays, the shipbuilding industry is facing a radical change toward solutions with a smaller environmental impact. This can be achieved with low emissions engines, optimized shape designs with lower wave resistance and noise generation, and by reducing the metal raw materials used during the manufacturing. This work focuses on the last aspect by presenting a complete structural optimization pipeline for modern passenger ship hulls which exploits advanced model order reduction techniques to reduce the dimensionality of both input parameters and outputs of interest. We introduce a novel approach which incorporates parameter space reduction through active subspaces into the proper orthogonal decomposition with interpolation method. This is done in a multi-fidelity setting. We test the whole framework on a simplified model of a midship section and on the full model of a passenger ship, controlled by 20 and 16 parameters, respectively. We present a comprehensive error analysis and show the capabilities and usefulness of the methods especially during the preliminary design phase, finding new unconsidered designs while handling high dimensional parameterizations.

A Review of the Accuracy of Primal Assembly Model Order Reduction Techniques

Model order reduction (MOR) techniques have been used in structural dynamics for over sixty years as a quick and effective way to examine the dynamic behavior of complex finite element (FE) models. Various macroscopic phenomena arise from the physics and mechanics of the underlying microstructure, and MOR approaches can be employed in conjunction with the multiscale computational homogenization approach, which is deriving local macroscopic constitutive responses from underlying microstructure; to predict the dynamic behavior of heterogeneous macrostructures. The accuracy and efficiency of the most representative approaches from dynamic condensation, component mode synthesis (CMS), and general model reduction (GMR) methods are compared in this paper. The techniques considered are Guyan and the improved reduced system (IRS) from classical dynamic condensation, Craig-Bampton (CB) and enhanced Craig-Bampton (ECB) from CMS, IRS-based substructuring, GMR, and GMRPlus. All the substructuring methods employ primal assembly at the boundary interface of the substructures. The accuracy and computing efficiency of the techniques are compared in-depth using real-world engineering problems, and the effect of the number of substructures on the accuracy and computational cost is also examined.

Exhaustive modal analysis of large-scale power systems using model order reduction

This paper presents an efficient modal analysis methodology that computes all modes of any given large-scale power system in exhaustive manner using the model order reduction techniques. For this, a reduced order model is generated using the Balanced Truncation (BT) method for which the controllability and observability gramians are approximated using the low-rank Cholesky factors. This leads to a rapid identification of classes of coupled dynamic devices of the original system. Next, approximated oscillatory modes are computed for each class. Finally, the exact values of the oscillatory modes of the overall power system are determined by iterative computations (the Modified Arnoldi method) initialized to the approximated modes found at the first step. The proposed methodology is able to put into evidence all coupling modes of any given large-scale power system (containing power electronics or any other specific dynamic devices). No a priori knowledge about the pattern of oscillations is needed. The accuracy and efficiency of the proposed methodology are thoroughly validated on several power systems with different orders, including a large scale model of the interconnected European power system.

Numerical Calculation of Pick-Up Coils Frequency Response as a Useful Tool for Local Magnetic Field Sensors Design

The aim of this article is to present a useful numerical procedure suitable to accurately model pick-up probes for design-oriented purposes, reducing the required prototyping activity and allowing more accurate results with respect to analytic estimations, taking into account both inductive and capacitive effects in complex geometries. The proposed methodology is characterized by the following steps: 1) a partial element equivalent circuit (PEEC) formulation, speeded up by a parametric model order reduction (pMOR) technique, is used to efficiently model the probe in the frequency domain taking into account both inductive and capacitive effects; 2) a nonlinear least-squares solver is used to identify an equivalent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$RLC$ </tex-math></inline-formula> network describing the full behavior of the probe up to the first resonance frequency; and 3) a circuital model, considering the probe, the transmission line (TL), and the terminating impedance, is used to estimate the bandwidth of the complete system from the probe to the acquisition system. The experimental validation of the proposed method against the RFX-mod2 three-axis pick-up probe prototype is also presented, together with the application to the design of the magnetic sensors for the new Divertor tokamak test (DTT) facility.

Parametric reduced-order modeling enhancement for a geometrically imperfect component via hyper-reduction

In this paper, an improved nonlinear reduced-order modeling technique capable of describing the parameterized shape defect is presented. In the proposed framework, a set of defect-shapes are pre-determined based on a nominal configuration. Then, the reduced-order representation is created in a polynomial form comprising a set of reduced-tensor coefficients of defect and physical displacement fields. However, constructing reduced tensors using a large number of discretized elements usually requires enormous amounts of computational resources. Therefore, to reduce the computational expense, a quadratic-manifold-based energy-conserving sampling and weighting approach was employed to obtain the reduced tensors concerning only a few optimally selected elements. This approach can be used to conduct both time-transient and frequency response analyses on rotating mechanical components. It was found that the proposed approach can accurately estimate the broad defect-parametric variation. In particular, its computational efficiency demonstrated a significant improvement over that of existing approaches.

Adjoint-based phase reduction analysis of incompressible periodic flows

We establish the theoretical framework for adjoint-based phase reduction analysis for incompressible periodic flows. Through this adjoint-based method, we obtain spatiotemporal phase sensitivity fields through a single pair of forward and backward direct numerical simulations, as opposed to the impulse-based method that requires a very large number of simulations. Phase-based analysis involves perturbation analysis about a periodically varying base state and hence is tailored for the analysis of periodic flows. We formulate the phase description of periodic flows with respect to the potential and vortical perturbations in the flow field. The current phase-reduction analysis can also be implemented consistently in the immersed boundary projection method, which facilitates the analysis over arbitrarily-shaped bodies. We demonstrate the strength of the phase-based analysis for periodic flows over circular cylinder and symmetric airfoils at high incidence angles. The critical regions for phase modification in the cylinder flow are investigated and the locations of flow separation are shown to be the most sensitive regions. Further, the results reveal the influence of the angle of attack and airfoil thickness on the phase-sensitivity distribution of flows over various airfoils. The phase for such flows is defined based on the lift coefficient, and hence is influenced by the vortical structures responsible for lift production. The present framework sheds light on the connection between phase-sensitivity and vortex formation dynamics.

Dynamic mode decomposition analysis and fluid-mechanical aspects of viscoelastic fluid flows past a cylinder in laminar vortex shedding regime

This study presents an extensive numerical investigation to understand the effect of fluid viscoelasticity on the flow dynamics past a stationary cylinder in the laminar vortex shedding regime. The governing equations, namely, mass, momentum, and Oldroyd-B viscoelastic constitutive equations, have been solved at a fixed value of the Reynolds number of 100 and over a range of values of the Weissenberg number as 0≤Wi≤2 and polymer viscosity ratio as 0.5≤β≤0.85. In particular, for the first time, this study presents a detailed analysis of how the fluid viscoelasticity influences the coherent flow structures in this benchmark problem using the dynamic mode decomposition (DMD) technique, which is considered to be one of the widely used reduced order modeling techniques in the domain of fluid mechanics. We show that this technique can successfully identify the low-rank fluid structures in terms of the spatiotemporal modes from the time-resolved vorticity field snapshots and capture the essential flow features by very few modes. Furthermore, we observe a significant difference in the amplitude and frequency associated with these modes for Newtonian and viscoelastic fluids otherwise under the same conditions. This, in turn, explains the differences seen in the flow dynamics between the two types of fluids in an unambiguous way, such as why the fluid viscoelasticity suppresses the vortex shedding phenomenon and decreases the energy associated with the velocity fluctuations in viscoelastic fluids than that in Newtonian fluids. However, before performing the DMD analysis, we also present a detailed discussion on the various fluid-mechanical aspects of this flow system, such as streamline patterns, vorticity fields, drag and lift forces acting on the cylinder, etc. This will ultimately set a reference platform for delineating the importance of the DMD analysis to get further insight into flow physics.

A-EFIE With Model-Order Reduction Technique for Fast Analysis of Circuit Simulation

A fast frequency-sweep method for microwave circuit simulation based on a full-wave solver is presented in this letter. The complexity of the mathematical model of the augmented electric field integral equation (A-EFIE) system is reduced by employing the model-order reduction (MOR) technique, where the reduced-order subspace is constructed by singular value decomposition (SVD). Compared to the A-EFIE with the asymptotic waveform evaluation (AWE) technique, the A-EIFE with the SVD-based MOR technique can speed up the broadband circuit analysis more efficiently.

Generalized polynomial chaos based surrogate models for acoustics and vibrations

This work explores generalized polynomial chaos (GPC) surrogate models and their effectiveness for general acoustics and vibration applications. GPC is primarily known as an uncertainty quantification (UQ) technique and in that context the underlying polynomial-based model has been shown to be effective in mapping input probability distributions to the corresponding output probability distributions. In this study, GPC surrogate models – including those generated by both quadrature and regression methods – are evaluated for their effectiveness in non-UQ focused analyses. As points of comparison, Krylov subspace and other more traditional reduced order modeling techniques are demonstrated and compared to GPC models so that the differences may be better understood. Example problems with several different levels of complexity are used to show how the computational burden as well as the overall effectiveness of the method changes as the number of input variables that must be considered grows.

Learning coordinate systems for fast and accurate acoustic modeling

Many popular numerical methods, such as finite difference and spectral methods, rely on simple domain geometry and environmental smoothness. Unfortunately, these features are rarely found in real-world simulations. We propose learning coordinate transformations with deep neural networks to facilitate acoustic modeling in complex media. Using automatic differentiation, we obtain new coordinate systems by solving various optimization problems that map the computational domain to a rectangular grid. Different choices of the objective function are utilized to attain different goals, including (i) mapping complicated boundaries such as the seafloor to straight lines and (ii) reducing the rank of two-dimensional slices of the refractive index. The first choice allows for domain decomposition to accurately model sharp discontinuities in density and sound speed. The second drastically accelerates a non-intrusive reduced-order modeling technique referred to as the dynamical low-rank approximation. Using realistic ocean test cases, we compare the performance of our learned coordinate systems with domain decomposition to the classic approach of smoothing sharp discontinuities, and we compare full-rank, low-rank, and coordinate-system-accelerated low-rank solutions of the three-dimensional parabolic wave equation.

Parameter, input and state estimation for linear structural dynamics using parametric model order reduction and augmented Kalman filtering

Tracking the evolution in time of parameters, input and states of a structural dynamic system is often difficult, since their direct measurement can be problematic or even impossible. It is of great interest to estimate these quantities based on output-only data from a limited set of sensors. This work proposes an estimation technique for states, inputs and material parameters for structural dynamics models based on an Augmented Extended Kalman Filter. A parametric Model Order Reduction technique is proposed to construct a Reduced Order Model which maintains an explicit dependency on material parameters, enabling the parameter estimation thanks to a low computational cost and an efficient derivation of the linearized system. The choice of sensor configurations that ensure the observability of unknown quantities is discussed as well. The proposed methodology shows highly promising results and could be employed for model refinement or condition monitoring. The methodology is validated both numerically and experimentally, using data acquired on a scaled wind turbine blade, with errors on the estimated parameters lower than 3.5% with respect to experimentally identified parameter values.

Accelerating boundary element methods in wideband frequency sweep analysis by matrix-free model order reduction

In frequency-domain acoustic simulations, a wideband frequency sweep analysis is often required. This becomes particularly challenging for boundary element methods (BEM), as the system matrix has a complex dependency on frequency. Previous attempts have been made to accelerate the frequency sweep analysis for the conventional BEM i.e., without fast multipole or ℋ-matrices accelerations, which restrains the applications to limited scale problems due to the dense system matrix. This paper investigates and incorporates a rational Krylov projection based model order reduction (MOR) technology i.e., matrix-free method into various BEM formulations to address the critical challenge of accelerating frequency sweep analysis. Operating on the transfer functions of the BEM system’s input and output, the matrix-free MOR avoids the direct interference with the BEM kernels, which makes it feasible and attractive to work with existing fast BEM techniques such as ℋ-matrices and fast multipole method. The combined holistic BEM solution retains the merits of the acceleration from both MOR and fast BEM techniques. Significant speed-ups have been achieved by the matrix-free MOR for wideband BEM simulations. Numerical examples are presented to demonstrate the capability of the presented holistic approach in simulating various acoustic problems, including interior–exterior models, complex geometries and frequency dependent boundary conditions.

Periodic Coupled-Cluster Green's Function for Photoemission Spectra of Realistic Solids.

We present an efficient implementation of the coupled-cluster Green's function (CCGF) method for simulating photoemission spectra of periodic systems. We formulate the periodic CCGF approach with Brillouin zone sampling in the Gaussian basis at the coupled-cluster singles and doubles (CCSD) level. To enable CCGF calculations of realistic solids, we propose an active-space self-energy correction scheme by combining CCGF with the cheaper many-body perturbation theory (GW) and implement the model order reduction (MOR) frequency interpolation technique. We find that the active-space self-energy correction and MOR techniques significantly reduce the computational cost of CCGF while maintaining the high accuracy. We apply the developed CCGF approaches to compute spectral properties and band structure of silicon (Si) and zinc oxide (ZnO) crystals using triple-ζ Gaussian basis sets and medium-size k-point sampling and find good agreement with experimental measurements.

Fast solution of elasto-plastic problems by reduced order finite element method with manifold learning

Solution of elasto-plastic problems is often computationally intensive. The most effective way to accelerate the computation is to reduce the number of the degree of freedom of the model. Considering the fact that constitutive relations of solid materials are very diverse and usually quite complicated, the paper investigates a model order reduction technique which can be embedded into existing finite element frameworks and reuse their constitutive models. The semi-intrusive technique can thus be used to solve nontrivial problems. Two typical models, i.e., the perforated plate under tensile load and the pressure vessel head under cyclic load, are tested to evaluate the accuracy and the efficiency of the reduced order model. It is shown that a speedup of around two orders of magnitude can be achieved by the reduced order model while maintaining the same level of accuracy. Moreover, the requirements for processor and memory of the computer are also significantly reduced.