Model Order Reduction Scheme Research Articles

A frequency-independent absorption function surrogate for perfectly matched layer in exterior acoustics

In many engineering applications, the solution of acoustic wave problems in the infinite domain is required over a broad frequency range with densely sampled increments. In order to achieve efficient numerical simulations via a spatial discretization, e.g. finite element method, additional artificial absorbing boundaries are necessary to truncate the computational domain into appropriate bounded sizes. One of the most commonly used non-reflecting techniques to attenuate propagating waves is known as the perfectly matched layer. However, the system matrices arising from the finite element treatment of the Helmholtz equation in the absorbing layers are frequency-dependent, implying that they must be formed and inverted at each frequency of interest. Such a procedure is rather troublesome for frequency sweeps. To address this, a surrogate of perfectly matched layers is proposed, which enables the corresponding system matrices to be independent of the frequency. Moreover, it avoids the use of a relatively large computational domain and relatively thick enclosed layers at low frequencies, thus improving the ability of perfectly matched layers across the entire frequency range. After that, an adaptive projection-based model order reduction scheme is further developed to reduce the computational complexity of exterior acoustic systems. A robust error indicator based on the relative error of two constructed reduced order models is accordingly introduced. The performance of the present solution framework is discussed and compared with other implementation strategies, in the context of multi-frequency solution of two-dimensional test models with single or multiple scatterers.

A stable model order reduction scheme for an acoustic finite element model with perfectly matched layers

Perfectly matched layers (PMLs) are often used in an exterior acoustic finite element model to simulate the Sommerfeld radiation condition. However, the time-domain model is often too large to efficiently handle the transient simulation due to the frequency-dependent stretching function and the number of elements needed per wavelength. Krylov subspace-based model order reduction can alleviate this problem by only matching the important moments of the original model. However, stability-preservation is not guaranteed, which is the key to perform time-domain simulation. This work proposes a stability-preserving model order reduction scheme for the time-domain PMLs. By assessing the stability conditions of time-domain PMLs, a two-step strategy is applied. Firstly, a one-sided split basis is used to preserve the form of Lyapunov function of the original model such that a stable intermediate reduced order model (ROM) results. Secondly, an unsplit basis is applied on the modal transformation of this intermediate ROM to reduce the size further. The proposed method is verified by numerical simulations.

Model Order Reduction Strategy for LTI Systems and Application to PID Controller Design

Model Order Reduction Strategy for LTI Systems and Application to PID Controller Design

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

A Robust Model Order Reduction Scheme for Lithium-Ion Batteries in Control-Oriented Vehicle Models

The role of batteries in electrification of vehicles is eminent; thus, a dynamic model that represents the physics-based phenomena of the battery system at a minimum computational cost is essential in the model-based design of electrified vehicle control systems. Furthermore, robustness of the reduced-order battery model when maintaining the dominant physics-based phenomena governing the dynamic behavior of the battery system is crucial. Characterization of the power signal applied to the lithium-ion battery in the energy management controller of a plug-in hybrid electric vehicle shows that there is a dominant frequency range in the input signal to the battery. This key feature can be considered as a basis to construct a reduced-order model in which the training input is different from the original power signal. The original idea in this paper is to generate the training input by applying a low-pass filter to the white-noise random signal to maintain the same dominant frequency range observed in the original power signal. Response of the reduced-order model, constructed using the proper orthogonal decomposition, compared to the high-fidelity battery model shows promising results; a maximum relative error of 1% was obtained for the battery state of charge while simulation time was reduced by 42.9%.

Reduced-order model for RBDO of multiple TMDs on eccentric L-shaped buildings subjected to seismic excitations

L-shaped buildings subjected to ground accelerations undergo coupled lateral and torsional vibrations that can result in critical stress concentrations at their re-entrant corners. Even so, a comprehensive literature review indicates that the Reliability-Based Design Optimization (RBDO) of multiple Tuned Mass Dampers (TMDs) in L-shaped buildings has not been studied yet, to the best of the authors’ knowledge. Indeed, the more refined structural model required to capture the wings motion, rather than assuming shear-framed behavior as performed on asymmetric rectangular buildings, becomes critical in an RBDO framework. Thus, integrating a non-modal Model Order Reduction (MOR) strategy allows for a massive reduction in the computational burden. Emerging from control engineering, these techniques do not form their orthogonal subspace based on modal information. Furthermore, they dispense previous snapshots of the system solution while being suitable for non-classically damped systems. Within this context, this paper presents a novel MOR-based RBDO study of multiple TMDs in L-shaped buildings subjected to seismic excitation. The MOR algorithm connects second-order Krylov subspaces to build the reduced-order model at a specific expansion frequency with a windowing strategy for sweeping the complete frequency domain. Then, it is integrated into the RBDO, aiming to determine bi-directional TMD configurations that minimize the building’s annual failure rate for the serviceability limit state. The application case consists of a 4-story L-shaped building with 2880 active degrees of freedom (DOFs). The results show that a reduction in failure probability of up to 65% can be achieved while reaching a speed-up factor (SUF) of more than 35.

Adaptive MOR techniques for non‐linear FE simulations of unit cells

AbstractThis work shows an adaptive Model Order Reduction (MOR) scheme for the computation of unit cells with non‐linear material behavior. A reduction of dimensionality and computational cost is achieved by projecting the high‐fidelity system of equations (SOE) from the full vector space to a lower dimensional subspace. An adaptive algorithm updates the projection operator, which is initially gained from a Proper Orthogonal Decomposition (POD), to ensure convergence and optimal size control of the reduced problem. Numerical examples show a comparison between the discretization of full displacements and fluctuations of a hexagonal unit cell, which show the independence of the MOR basis from the macroscopic displacements.

Investigation of proper orthogonal decomposition for echo state networks

Echo State Networks (ESN) are a type of Recurrent Neural Network that yields promising results in representing time series and nonlinear dynamic systems. Although they are equipped with a very efficient training procedure, Reservoir Computing strategies, such as the ESN, require high-order networks, i.e., many neurons, resulting in a large number of states that are magnitudes higher than the number of model inputs and outputs. A large number of states not only makes the time-step computation more costly but also may pose robustness issues, especially when applying ESNs to problems such as Model Predictive Control (MPC) and other optimal control problems. One way to circumvent this complexity issue is through Model Order Reduction strategies such as the Proper Orthogonal Decomposition (POD) and its variants (POD-DEIM), whereby we find an equivalent lower order representation to an already trained high dimension ESN. To this end, this work aims to investigate and analyze the performance of POD methods in Echo State Networks, evaluating their effectiveness through the Memory Capacity (MC) of the POD-reduced network compared to the original (full-order) ESN. We also perform experiments on two numerical case studies: a NARMA10 difference equation and an oil platform containing two wells and one riser. The results show that there is little loss of performance comparing the original ESN to a POD-reduced counterpart and that the performance of a POD-reduced ESN tends to be superior to a normal ESN of the same size. Also, the POD-reduced network achieves speedups of around 80% compared to the original ESN.

Model order reduction for seismic waveform modelling: inspiration from normal modes

SUMMARY The computational cost of full waveform simulation in seismological contexts is known to be expensive and generally requires large clusters of computers working in parallel. Although there have been many methods proposed over recent years to reduce this burden, in this work, we focus on a particular method called model order reduction (MOR) whereby a full waveform system of equations is projected onto a lower dimensional space to reduce computational and memory requirements at the cost of introducing approximation errors. In this paper, inspired by normal mode (NM) theory, we use the eigenmodes of the seismic wave equation to span this lower dimensional space. From this we argue that NM theory can be seen as an early form of MOR. Using this as inspiration, we demonstrate how free body oscillations and a form of Petrov–Galerkin projection can be applied in regional scale problems utilizing recent advanced eigensolvers to create a MOR scheme. We also demonstrate how this can be applied to inverse problems. We further conjecture that MOR will have an important role to play in future full waveform applications, particularly those of a time-critical nature such as seismic hazard monitoring.

A Fast Frequency Sweep approach for performance-based optimization of earthquake-resistant irregular large-scale buildings

Performance-Based Design (PBD) optimization of buildings subjected to seismic excitations has increasingly been applied to regular small-scale structural systems to minimize initial or life-cycle costs. Nonetheless, a comprehensive bibliographic survey reveals that studies dealing with irregular buildings are scarce, although they can present higher seismic vulnerability than regular buildings. Specifically, the PBD optimization of irregular buildings represented by large-scale finite element (FE) models, and considering dozens of discrete design variables, cannot be found to the best of the authors’ knowledge. Hence, this paper presents an original framework for the PBD optimization of irregular buildings subjected to seismic excitations, handling thousands of DOFs and dozens of discrete design variables. To relieve the computational burden, enabling the problem to be solved, a Model Order Reduction (MOR) strategy based on the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) approach is combined with a hybrid optimization algorithm, which integrates a local optimizer with a restart strategy. The former is employed to speed up the frequency-domain response, while the latter allows for tackling a non-convex and multimodal objective function with many discrete variables. The procedure is adaptatively devised to operate in a robust and fully black-box approach. The application case consists of a stiffness in-plane irregular H-shape building with 16,452 active DOFs and 34 discrete design variables. The proposed scheme largely reduces the computational burden, leading to speed-up factors (SUF) up to 19 times compared to direct frequency-domain solutions.

Adaptive methods for MOR in non‐linear FE simulations of unit cells

AbstractIn this contribution, an adaptive method for Model Order Reduction (MOR) is presented for simulating physically non‐linear material behavior of micro‐heterogeneous structures. The method is purely online and includes a Galerkin projection, which is performed using a projection operator initially gained by a Proper Orthogonal Decomposition (POD). By this projection, the number of degrees of freedom to solve for is decreased drastically and can be projected back to the vector space of the high fidelity problem after the solution process. The composition of the projection operator follows an adaptive algorithm, which evaluates an update of the operator if convergence is not ensured during the Newton Raphson scheme. Later on, during the simulation, the number of degrees of freedom to solve is controlled using a POD‐based control criterion for sorting out non‐relevant basis vectors from previous updates. This results in an efficient method and leads to a satisfying speedup of the simulation process. The performance and efficiency of the presented adaptive MOR scheme is investigated in numerical simulations of a hexagon‐shaped unit cell.

Analysis of an innovative sampling strategy based on k-means clustering algorithm for POD and POD-DEIM reduced order models of a 2-D reaction-diffusion system

In this work, a model-order reduction methodology based on proper orthogonal decomposition (POD) and Galërkin projection is presented and applied to the simulation of the self-ignition of a stockpile of solid fuel. Self-ignition is a phenomenon associated with steep changes in space and time, yielding high gradients of state variables which demand grid refinement and, thus, increase of the computational burden. To cope with this difficulty, first, a full order model (FOM), generated by finite-difference discretisation of the PDEs constituting the differential model, is employed to generate reference solutions. Two different POD-based formulations are proposed: the classical POD-Galërkin is employed to generate reduced order models (ROM), then discrete empirical interpolation method (DEIM) is employed to deal with nonlinearities in a more efficient manner. These reduction techniques are further supplemented with an innovative sampling approach based on k-means clustering. The resulting agile ROM is validated against the FOM. Both model-order reduction strategies, particularly the POD-DEIM model, reproduce the FOM solutions with high accuracy and much lower computational cost: The results of the application of a combination of the DEIM algorithm and k-means clustering show that the computational time for the calculation of one solution reduces up to 1020 times, while remaining able to reproduce all bifurcation points found with the FOM, thus demonstrating quantitative and qualitative agreement.

A reduced-order cutFEM approach to model complex moving sound sources

While traditional finite element based vibro-acoustic analysis of structures is done on stationary structures, in many occasions the structure under investigation is actually moving. To perform accurate finite element analysis on the vibro-acoustic performance of such structures requires the evaluation of a time-varying system, leading to several challenges. Of paramount importance are a strategy to update the mesh to the new configuration and an adequate time integration scheme that does not introduce large numerical artifacts. Additionally, the time-varying nature leads to an increase in computational complexity. In this paper a modeling strategy based on cut finite elements (cutFEM) is introduced that aims to tackle these challenges. The cutFEM approach can be seen as a fictitious domain approach in which the source is moving over a static mesh, cutting through elements where the actual source is located, using a level set description. The key advantage of this approach is that no remeshing and/or mesh morphing is required. It is shown how both implicit and explicit time integration schemes can be used to propagate the solution while the source is moving. Additionally, several model order reduction strategies are compared to speed up the online evaluation of the reduced order model.

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

In this paper, we explore the Riccati-based boundary feedback stabilization of the incompressible Navier–Stokes flow via the Krylov subspace techniques. Since the volume of data derived from the original models is gigantic, the feedback stabilization process through the Riccati equation is always infeasible. We apply aℋ2optimal model-order reduction scheme for reduced-order modeling, preserving the sparsity of the system. An extended form of the Krylov subspace-based two-sided iterative algorithm (TSIA) is implemented, where the computation of an equivalent Sylvester equation is included for minimizing the computation time and enhancing the stability of the reduced-order models with satisfying the Wilson conditions. Inverse projection approaches are applied to get the optimal feedback matrix from the reduced-order models. To validate the efficiency of the proposed techniques, transient behaviors of the target systems are observed incorporating the tabular and figurative comparisons with MATLAB simulations. Finally, to reveal the advancement of the proposed techniques, we compare our work with some existing works.

Model order reduction strategies for weakly dispersive waves

We focus on the numerical modeling of water waves by means of depth averaged models. We consider in particular PDE systems which consist in a nonlinear hyperbolic model plus a linear dispersive perturbation involving an elliptic operator. We propose two strategies to construct reduced order models for these problems, with the main focus being the control of the overhead related to the inversion of the elliptic operators, as well as the robustness with respect to variations of the flow parameters. In a first approach, only a linear reduction strategies is applied only to the elliptic component, while the computations of the nonlinear fluxes are still performed explicitly. This hybrid approach, referred to as pdROM, is compared to a hyper-reduction strategy based on the empirical interpolation method to reduce also the nonlinear fluxes. We evaluate the two approaches on a variety of benchmarks involving a generalized variant of the BBM–KdV model with a variable bottom, and a one-dimensional enhanced weakly dispersive shallow water system. The results show the potential of both approaches in terms of cost reduction, with a clear advantage for the pdROM in terms of robustness, and for the EIMROM in terms of cost reduction.

Uncertainty analysis based on reduced-order models for composite structures with interval field variables

Composite structures exhibit significant spatial variabilities in their response due to the variations in the individual lamina properties. Considering the difficulty of obtaining accurate probability distributions of laminae properties in practical engineering, interval fields are proposed as a beneficial supplement to random fields and a novel uncertainty analysis method has been developed for composite structures. This study focuses on the development of model order reduction strategies to reduce the dimensionality of the problem and achieve faster computations. First, the bounds and autocorrelation are quantified based on measurement data of interval fields, which are expressed as interval Karhunen-Loève expansion. Next, considering the multilayer structure of the laminate, an equivalent reduced-order model is developed to reduce the computational cost. The specific focus of this strategy is on the equivalence of the constitutive matrix on the fine-scale and coarse-scale models, while the reduced-order model is constructed directly based on the data of the constitutive matrix. Finally, the results of the numerical examples present that the proposed reduced-order model can accurately predict the uncertainty response and significantly improve the convergence speed of the interval Monte Carlo results in the uncertainty analysis of composite structures.

Parametric model order reduction for acoustic boundary element method systems through a multiparameter Krylov subspaces recycling strategy

AbstractThe recently introduced framework enabling model order reduction (MOR) for systems arising within the acoustic boundary element method (BEM), allows for fast frequency sweeps to obtain the detailed response of an acoustic system. Nevertheless, in many cases it is desirable to include additional parameters in the MOR scheme to facilitate iterations among different configurations of a specific design. In that context, this work proposes a multiparameter Krylov subspaces recycling scheme, based on the algorithms proposed in Reference 23 for a single parameter, aiming at the parametric MOR of acoustic BEM systems. The proposed scheme allows to include multiple parameters along with the non‐affine wavenumber kernel dependency and thus facilitates the fast solution of parametric acoustic systems. The article distinguishes among three different parametrizations of BEM systems: (i) a right‐hand side parametrization, (ii) an affine, and (iii) a non‐affine parametrization of the system matrix. The two former cases allow for the construction of a cheap residual error estimator and thus facilitate the deployment of a greedy sampling strategy. This work proposes the employment of such a greedy strategy in the context of Krylov subspaces recycling, which combined with the automatic Krylov subspaces recycling (AKR) algorithm, enables the construction of a global reduction basis incorporating such parametric variations. On the contrary, since in case of non‐affine parametrizations a cheap residual error estimator is not directly available, a multiparameter AKR algorithm is deployed to ensure the predefined residual level. The performance of the proposed methods is benchmarked against a reduced basis method‐type approach, which follows the same assumptions as the AKR for the case of non‐affine systems. The potential of the algorithms is demonstrated on academic BEM examples including source position, material, and shape parametrizations and significant reductions in the sampling required for the construction of the respective reduction bases are reported.

Gramian-based model reduction for unstable stochastic systems

This paper considers large-scale linear stochastic systems representing, e.g., spatially discretized stochastic partial differential equations. Since asymptotic stability can often not be ensured in such a stochastic setting (e.g., due to larger noise), the main focus is on establishing model order reduction (MOR) schemes applicable to unstable systems. MOR is vital to reduce the dimension of the problem in order to lower the enormous computational complexity of for instance sampling methods in high dimensions. In particular, a new type of Gramian-based MOR approach is proposed in this paper that can be used in very general settings. The considered Gramians are constructed to identify dominant subspaces of the stochastic system as pointed out in this work. Moreover, they can be computed via Lyapunov equations. However, covariance information of the underlying systems enters these equations which is not directly available. Therefore, efficient sampling-based methods relying on variance reduction techniques are established to derive the required covariances and hence the Gramians. Alternatively, an ansatz to compute the Gramians by deterministic approximations of covariance functions is investigated. An error bound for the studied MOR methods is proved yielding an a priori criterion for the choice of the reduced system dimension. This bound is new and beneficial even in the deterministic case. The paper is concluded by numerical experiments showing the efficiency of the proposed MOR schemes.

Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

<p style='text-indent:20px;'>We are interested in the numerical solution of coupled semilinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.</p>

Real-time Human Response Prediction Using a Non-intrusive Data-driven Model Reduction Scheme

Recent research in non-intrusive data-driven model order reduction (MOR) enabled accurate and efficient approximation of parameterized ordinary differential equations (ODEs). However, previous studies have focused on constant parameters, whereas time-dependent parameters have been neglected. The purpose of this paper is to introduce a novel two-step MOR scheme to tackle this issue. In a first step, classic MOR approaches are applied to calculate a low-dimensional representation of high-dimensional ODE solutions, i.e., to extract the most important features of simulation data. Based on this representation, a long short-term memory (LSTM) is trained to predict the reduced dynamics iteratively in a second step considering the parameters at the respective time step. The potential of this approach is demonstrated on an occupant model within a car driving scenario. The reduced model's response to time-varying accelerations matches the reference data with high accuracy for a limited amount of time. Furthermore, real-time capability is achieved. Accordingly, it is concluded that the presented method is well suited to approximate parameterized ODEs and can handle time-dependent parameters in contrast to common methods.