Model Order Reduction Scheme Research Articles

Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to Bayesian optimization.

In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element. We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model can be built.

Reduced-order modeling of linear time invariant systems using big bang big crunch optimization and time moment matching method

In this paper, a new approach is proposed to approximate the high-order linear time invariant (LTI) system into its low-order model. The proposed approach is a mixed method of model order reduction scheme consisting of recently developed big bang big crunch optimization algorithm and the time-moment matching method. This proposed method is applicable to single-input single-output, multi-input multi-output system, and time delayed LTI systems. The proposed approach is substantiated with various numerical examples of low and high-order systems. The results show that the reduced-order models preserve both transient and steady state conditions of original systems. Further, the results are also compared with the existing approaches of reduced order modeling which show exceptional improvement in integral square error (ISE) and other time domain specifications.

Reduced order modelling of liquid-filled pipe systems

In the design of vibration-sensitive structures, large three-dimensional numerical models are frequently used, those occasionally being too large to be simulated in a practical manner. More specifically, the analyses of vibrating complex water-pipe systems require efficient model order reduction strategies. In the paper, a numerical procedure for model order reduction of a water-pipe system housed in a vibration-sensitive research facility is presented. The finite element model employed includes a water-pipe system where fluid–structure interaction effects are accounted for. The reduced model is developed using component mode synthesis and a reduction of the interface degrees of freedom, enabling time-efficient dynamic analyses. The model was proven to be time-efficient and to generate output of high accuracy.

A Fully Adaptive Scheme for Model Order Reduction Based on Moment Matching

A fully adaptive model order reduction scheme based on moment matching is proposed to derive the reduced-order models of linear time-invariant (LTI) systems. According to the given error tolerance, the order of the reduced-order model as well as the expansion points for the transfer function is automatically determined on the fly during the process of model reduction. In this sense, the reduced-order model is automatically obtained without assigning the number of moments and expansion points in a priori , which is a prerequisite for the standard implementation of model reduction based on moment matching. The proposed adaptive scheme is found to be efficient when it is tested on various LTI systems.

Reduced order modelling for static and dynamic aeroelastic predictions with multidisciplinary approach

We implement reduced order modelling techniques for aeroelastic predictions of the HIRENASD and S4T wings in order to represent CFD based high-fidelity solutions efficiently. Model reduction techniques such as non-intrusive Polynomial Chaos Expansion and Proper Orthogonal Decomposition are applied to both static and dynamic aeroelastic cases. The high-fidelity solutions are obtained by fluid structure interaction analysis using a 3D Euler unsteady aerodynamic solver and structural modal solution from a finite element solver. The model order reduction strategy is based on a multidisciplinary approach since both structural and aerodynamic input parameters are employed. The model order reduction is performed not only to represent the high-fidelity computational analyses when small variations of input parameters are considered but also to characterize the flutter responses of the S4T wing in a broad range of input values over the entire flight regime for Mach numbers between 0.60 and 1.20. The efficient aeroelastic analyses performed using the developed reduced order models agreed well with the high-fidelity computational analyses.

Stochastic model order reduction in uncertainty quantification of composite structures

Multilayer fibre reinforced composites exhibit significant spatial variabilities in their response due to the variations in the individual lamina properties. Uncertainty quantification of these structures can be carried out using the polynomial chaos based stochastic finite element method (SFEM). The variations in the individual laminae properties are modelled as independent random fields. The SFEM method involves two stages of discretisation. First, the displacement fields are discretised into variables along the spatial dimension using traditional FEM. Next, the random fields are discretised into a vector of correlated random variables using polynomial chaos expansions. As the random field discretisation is carried out for each individual laminae, the number of random variables entering the formulation becomes large, making uncertainty quantification computationally prohibitive. This study focusses on the development of model order reduction strategies that enable reducing the stochastic dimensionality of the problem and enable faster computations. This is achieved by developing a probabilistically equivalent structure scale model using two approaches – one based on probabilistic equivalence of a nodal response and the other seeking equivalence on the probabilistic characteristics of the constitutive matrix. A set of numerical examples are presented to highlight the salient features of the proposed developments.

Model Order Reduction and domain decomposition strategies for the solution of the dynamic elastic–plastic structural problem

Abstract A new strategy for the efficient solution of highly nonlinear structural problems is proposed, based on the combined use of Domain Decomposition (DD) and snapshots version of the Proper Orthogonal Decomposition (POD) techniques. The formulation here presented is tailored for applications in elastic–plastic structural dynamics. In this context the POD is applied to domains that remain elastic and a double strategy to update the reduced basis is adopted. First, the Singular Value Decomposition (SVD) proposed allows to update the reduced basis as soon as a new snapshot is stored; secondly, an online adaptation technique of the reduced space is performed, through a plastic check during the reduced analysis. The applications show that the computation time necessary for solving elastic–plastic problems can be reduced of approximately 50%, while keeping accuracy comparable to that obtained for the full model with a classical monolithic method.

Automated Reduced Model Order Selection

This letter proposes to automate generation of reduced-order models used for accelerated S-parameter computation by applying a posteriori model error estimators. So far, a posteriori error estimators were used in Reduced Basis Method (RBM) and Proper Orthogonal Decomposition (POD) to select frequency points at which basis vectors are generated. This letter shows how a posteriori error estimators can be applied to automatically select the order of the reduced model in second-order Model Order Reduction (MOR) methods. Three different error estimators are investigated and compared in order to arrive at a new MOR scheme that is fast, reliable, and fully automated. The effectiveness of the proposed approach is verified by very high accuracy of the computed scattering parameters ( S-parameters) for an example of a waveguide filter over a prescribed frequency band.

Thermal Noise Compliant Synthesis of Linear Lumped Macromodels

This paper addresses the synthesis of equivalent circuits from black box state-space macromodels, as produced by model order reduction or rational curve fitting schemes. The emphasis is here on thermal noise compliance, intended as the guarantee that the produced netlists can be safely used in standard circuit solvers to perform thermal noise analysis, in addition to usual DC, AC, and transient simulations. Due to the fact that SNR is a key figure of merit in nearly all signal processing analog circuits, noise analysis is mandatory in design and verification of most analog and RF/millimeter-wave electronic applications. However, common macromodel synthesis approaches rely on components that do not (and cannot) have an associated thermal noise model, such as controlled sources or negative circuit elements. Therefore, macromodel-based noise analyses are generally not possible with currently available approaches. We propose a circuit realization derived from the classical resistance extraction synthesis, with suitable modifications for enhancing macromodel sparsity and efficiency. The resulting equivalent netlist, which is compatible with any standard circuit solver, is shown to produce exact noise characteristics, even if its elements are derived through a mathematical procedure, totally unrelated to the actual topology of the physical system under modeling. The procedure is validated by several examples.

Transient quasi-static Ritz vector (TQSRV) method by Krylov subspaces and eigenvectors for efficient contact dynamic finite element simulation

This paper presents a novel model order reduction (MOR) method called the transient quasi-static Ritz vector (TQSRV) method for efficient transient finite element (FE) analysis. Comparing with frequency response analysis, linear transient FE analysis with a fixed time step takes less computation time as an effective dynamic stiffness matrix assembled before time marching procedure is factorized once. Nevertheless as the number of degrees of freedom of a FE model has been dramatically increased for accurate engineering simulation, even the state-of-the-art computer and software often face their limitations. For fast but accurate transient FE analysis, we present a new MOR scheme called the TQSRV method with Krylov bases spanned at multiple angular velocities and several lowest eigenvectors. By calculating transient responses of reduced FE models and comparing it with the responses of full FE models, the effectiveness and accuracy of the TQSRV method are demonstrated.

Transient response analysis of randomly parametrized finite element systems based on approximate balanced reduction

A model order reduction scheme of the transient response of large-scale randomly parametrized linear finite element system in state space form has been proposed. The reduced order model realization is aimed at preserving the invariant properties of the dynamic system model based on the dominant coupling characteristics of the specified system inputs and outputs. An a-priori model reduction strategy based on the balanced truncation method has been proposed in conjunction with the stochastic spectral Galerkin finite element method. Approximation of the dominant modes of the controllability Gramian has been performed with iterative Arnoldi scheme applied to Lyapunov equations. The reduced order representation of the randomly parametrized dynamical system has been obtained with Arnoldi–Lyapunov vector basis using an implicit time stepping algorithm. The performance and the computational efficacy of the proposed scheme has been illustrated with examples of randomly parametrized advection–diffusion–reaction problem under the action of transient external forcing functions. The convergence of the proposed reduced order scheme has been shown with a-posteriori error estimates.

Efficient Variability Analysis of Electromagnetic Systems Via Polynomial Chaos and Model Order Reduction

We present a novel technique to perform the model-order reduction (MOR) of multiport systems under the effect of statistical variability of geometrical or electrical parameters. The proposed approach combines a deterministic MOR phase with the use of the Polynomial Chaos (PC) expansion to perform the variability analysis of the system under study very efficiently. The combination of MOR and PC techniques generates a final reduced-order model able to accurately perform stochastic computations and variability analysis. The novel proposed method guarantees a high-degree of flexibility, since different MOR schemes can be used and different types of modern electrical systems (e.g., filters and connectors) can be modeled. The accuracy and efficiency of the proposed approach is verified by means of two numerical examples and compared with other existing variability analysis techniques.

Weighted Reduced Basis Method for Stochastic Optimal Control Problems with Elliptic PDE Constraint

In this paper we develop and analyze an efficient computational method for solving stochastic optimal control problems constrained by an elliptic partial differential equation (PDE) with random input data. We first prove both existence and uniqueness of the optimal solution. Regularity of the optimal solution in the stochastic space is studied in view of the analysis of stochastic approximation error. For numerical approximation, we employ a finite element method for the discretization of physical variables, and a stochastic collocation method for the discretization of random variables. In order to alleviate the computational effort, we develop a model order reduction strategy based on a weighted reduced basis method. A global error analysis of the numerical approximation is carried out, and several numerical tests are performed to verify our analysis.

An on-line time dependent parametric model order reduction scheme with focus on dynamic stress recovery

In view of the tendency towards ever lighter and more powerful machines and even shorter design cycles, it becomes essential to have virtual prototyping tools that allow for fast and reliable numerical simulations. Current state-of-the-art structural dynamics and flexible multibody simulation techniques usually involve the solution of matrix systems with thousands to millions of variables. Model order reduction schemes are used to keep computational effort affordable at the expense of a minimal loss of accuracy. These techniques typically face difficulties with systems in which flexible bodies can be loaded in many degrees of freedom and rarely allow for accurate local stress and strain evaluation. The present work proposes to address both issues for the particular but frequent case of moving loads or boundary conditions. This behavior is found in most of the contact problems and systems that include sliding components for which many loading or boundary locations are possible but only a few of them are active at a certain moment in time. The proposed scheme exploits a reduction vector space that continuously varies in time by means of a parametric definition of the external load position. Contrary to the majority of the parametric model order reduction schemes that allow mainly for quasi-static parametric variations, the proposed approach can be used efficiently for time simulation of dynamically varying parametric models. This is achieved by considering the implicit time dependency of the reduction vector space using Galerkin projections or alternatively by direct substitution of the reduced kinetic energy, potential energy and generalized forces in the Lagrange equations. It is shown, by developing a consistent mathematical framework, that the price to pay for the very compact reduction space obtained is the evaluation of some extra terms in the equations of motion. Numerical examples are used to assess the accuracy of the proposed method. Results show the potential of this strategy with particular focus on displacement and stress fields and furthermore highlight its real-time potential. Moreover the developed framework together with the numerical results allow for a deeper physical understanding of the complex phenomena related to this category of time varying multiple-input/multiple output systems.

Static modes switching in gear contact simulation

Contact modeling is an active research area in the field of multibody dynamics. Despite the important research effort, two main challenging issues, namely accuracy and speed, are not yet jointly solved. One main issue remains which is the lack of model order reduction schemes capable to efficiently treat systems where multiple, a priori unknown, input–output locations are present. This work first analyzes the importance of including the necessary residual attachment modes by numerical simulation of two gears meshing in an ad-hoc flexible multibody model. Given the large number of residual attachment modes needed, the methodology named static modes switching is extended and successfully applied to improve efficiency. The method proposes an on-line selection of residual attachment modes for accurate local deformation prediction. The applicability to impact problems is discussed through numerical experiments and the automatic selection strategy is based purely on geometrical information. Results show that the method can be applied to gear meshing simulation, obtaining a high level of accuracy while preserving computational efficiency. Comparisons are made between modally reduced models, full non-linear finite element and the proposed strategy.

On the applicability of static modes switching in gear contact applications

Contact modeling is an active research area in the field of multibody dynamics. Despite the important research effort of the last decades, two main challenging issues, namely accuracy and speed, are far from being jointly solved. One main issue remains the current lack of model order reduction schemes capable of efficiently treat systems where multiple, a priori unknown, input-output locations are present. This work discusses how a methodology named static modes switching can be extended for its use in gear contact simulation. The method proposes an on-line strategy for the selection of residual attachment modes for accurate local deformation prediction. The applicability of the method is discussed by means of several numerical experiments. The result is an indication that the static modes switching underlying idea can be applied also in case of impulsive phenomena like gear impacts. Moreover, a numerical study on penalty factor values and number of eigenmodes and residual attachment modes shows an interesting relation between these three quantities, the dynamic behavior of the system, and the solution accuracy. Finally, some simulations are performed to compare the adopted model order reduction strategy with a reference nonlinear finite element simulation.

Modelling and simulation strategy for parametric transient electromagnetic simulations

This paper presents an efficient two-step reduction strategy for parametric transient electromagnetic simulations. The first step involves the conversion of a large system of parametrised differential algebraic equations (PDAEs), obtained after finite element (FE) discretisation into a significantly smaller system of ordinary differential equations, while symbolically preserving the parameters. The parametric ODE (PODE) model so obtained, having an affine parametric dependence can be further reduced using system theoretic parametric model order reduction strategies. In this work, two techniques of achieving such reductions are presented: aggregate projection matrix approach and multi-dimensional moment matching. Both these methods result in symbolic retention of parameters in the reduced model. The transient electromagnetic simulations corresponding to different values of the parameters can then be carried out in the reduced space at a drastically reduced computational cost. However, it was observed that ...

Gramian‐based model order reduction of parameterized time‐delay systems

ABSTRACTTime‐delay systems (TDSs) frequently arise in circuit simulation especially in high‐frequency applications. Model order reduction (MOR) techniques can be used to facilitate the simulation of TDSs. On the other hand, many kinds of variations, such as temperature and geometric uncertainties, can have significant impact on the transient responses of TDSs. Therefore, it is important to preserve parametric dependence during the MOR procedure. This paper presents a new parameterized MOR scheme for TDSs with parameter variations. We derive parameterized reduced‐order models (ROMs) for TDSs using balanced truncation by approximating the Gramians in the multi‐dimensional space of parameters. The resulting ROMs can preserve the parametric dependence, making it efficient for repeated simulations under different parameter settings. Numerical examples are presented to verify the accuracy and efficiency of our proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.

Improved component mode synthesis and variants

This survey focuses on the two known model order reduction schemes being widely integrated in various commercial finite element packages, namely, the static and dynamic condensation methods. The advantages as well as the corresponding drawbacks have been extensively analyzed in several papers throughout the last decades. Based on combining the beneficial properties of the aforementioned methods, several alternative reduction methodologies are outlined in this paper, i.e., the generalized improved reduction system method, the generalized component mode synthesis and the improved component mode synthesis with its generalized version, which incorporate in a more efficient way the system’s inertia terms. Therefore, the associated error regarding higher frequency ranges of interest is better controlled. Basis of these methodologies is the so-called master and slave degrees of freedom partitioning, the right selection of which highly influences the reduced order model’s dynamics. The methods are tested and verified on a rather small three-dimensional bar structure and on the lever part of a turbocharger’s variable turbine geometry. Several reduced order models are generated by varying both the number of Craig–Bampton modes and the selection of the required master degrees of freedom. A comparison is conducted based on the modal criterion of the corresponding eigenvectors and the associated computation time required.

Decentralized and Passive Model Order Reduction of Linear Networks With Massive Ports

It is well known that model order reduction for circuits with many terminals remains a challenging problem. One reason is that existing approaches are based on a centralized framework, in which each input-output pair is implicitly assumed to be equally interacted and the matrix-valued transfer function is assumed to be fully populated. In this paper, we attempt to address this long-standing problem using a decentralized model order reduction scheme, in which a multi-input multi-output system is decoupled into a number of subsystems and each subsystem corresponds to one output and several dominant inputs. The decoupling process is based on the relative gain array, which measures the degree of interaction of each input-output pair. For each decoupled subsystem, passive reduction can be easily achieved using existing reduction techniques. The proposed method is suitable for resistance-dominant interconnects such as on-chip power grids, substrate planes where extremely compact models can be obtained. Simulation results demonstrate the advantage of the proposed method compared to the existing approaches.